Local Lyapunov exponents: Sublimiting growth rates of linear random differential equations

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 1963 Wolfgang Siege...

Author: Wolfgang Siegert

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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1963

Wolfgang Siegert

Local Lyapunov Exponents Sublimiting Growth Rates of Linear Random Differential Equations

123

Wolfgang Siegert Allianz Lebensversicherungs - AG Reinsburgstrasse 19 70178 Stuttgart Germany [email protected]

ISBN 978-3-540-85963-5 e-ISBN 978-3-540-85964-2 DOI: 10.1007/978-3-540-85964-2 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008934460 Mathematics Subject Classiﬁcation (2000): 60F10, 60H10, 37H15, 34F04, 34C11, 58J35, 91B28, 37N10, 92D15, 92D25 c 2009 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publishing Services Printed on acid-free paper 987654321 springer.com

Preface

Establishing a new concept of local Lyapunov exponents, two separate theories are brought together, namely Lyapunov exponents and the theory of large deviations. Speciﬁcally, for the stochastic diﬀerential system dZtε = A (Xtε ) Ztε dt √ dXtε = b (Xtε ) dt + ε σ (Xtε ) dWt the new concept is introduced. Due to stationarity, the Lyapunov exponents of Ztε (which by Oseledets’ Multiplicative Ergodic Theorem describe the exponential growth rates of Ztε ) do not depend on the initial position x of X ε . Now the goal of this work is to provide a Lyapunov-type number for each regime of the drift b. As this characteristic number shall depend on the domain in which X ε , a dynamical system perturbed by additive white noise, is starting, it yields a concept of locality for the Lyapunov exponents of Ztε . Furthermore, the locality of such local Lyapunov exponents is to be understood as reﬂecting the quasi-deterministic behavior of X ε which asserts that in the limit of small noise, ε → 0, the process X ε has metastable states depending on its initial value as well as on the time scale chosen (Freidlin-Wentzell theory). Up to now local Lyapunov exponents have been deﬁned as ﬁnite time versions of Lyapunov exponents by several authors, but here we target at investigating the large time asymptotics t → ∞. So the goal is to connect the large parameters t and ε−1 in the customary deﬁnition of the Lyapunov exponents in order to approach the sublimiting distributions (Freidlin) which are supported by the metastable states of X ε . The local Lyapunov exponent is then understood to be the exponential growth rate of Z ε on the time scale chosen, subject to convergence in probability as ε → 0. Notably, the system itself changes in the sense that the noise intensity converges to zero with the time horizon depending on the noise intensity parameter. In contrast to this new concept the Lyapunov exponents as obtained by the Multiplicative Ergodic Theorem reﬂect the information of limit distributions, i.e. of invariant

v

vi

Preface

measures, as time increases to inﬁnity for the system parameter ε > 0 being ﬁxed. As a result we prove that the local Lyapunov exponent is bounded from above by the largest real part of the spectrum of the matrix A evaluated at the metastable state corresponding to the time scale; the respective bound from below holds true with the smallest real part of an eigenvalue of A at the corresponding metastable state. Assuming that A takes its values in the diagonal matrices, it is shown that its eigenvalues at the respective metastable state are precisely the possible local Lyapunov exponents. Moreover, in a “strongly” hypoelliptic situation it can be proved that only the largest eigenvalue is observed under convergence in probability. The latter result is regarded as sublimiting Furstenberg-Khasminskii formula, since the resulting limit is obtained as a (trivial) integral which produces the top eigenvalue. For the above tasks the prerequisites which fundamentally consist of knowing the exit probabilities of all the stochastic systems involved will be provided in detail: For this purpose, an integrated account of the theory for nondegenerate stochastic diﬀerential systems (Freidlin and Wentzell) and of the exit probabilities for degenerate stochastic systems (Hern´andez-Lerma) is given in chapters 2 and 3. The subsequent ﬁnal chapter is the heart of the book. Here, all the results are proven and discussed. Acknowledgements Foremost, I would like to thank Prof. Peter Imkeller for posing the problem considered in this book as well as for his support during my doctoral studies. Furthermore, I would like to thank Prof. Ludwig Arnold and Prof. Peter Kloeden for their interest in my work. I greatly acknowledge many fruitful discussions with Dr. Ilya Pavlyukevich and his precious comments on the draft. Moreover, I am grateful to Prof. Onesimo Hern´ andez-Lerma and Prof. Wolfgang Kliemann for their valuable comments and for sharing my enthusiasm. The ﬁnancial support by the German Research Foundation (Deutsche Forschungsgemeinschaft) through the Research Training Group “Stochastic Processes and Probabilistic Analysis” as well as through the Collaborative Research Centre “SFB 649 Economic Risk” is gratefully acknowledged. Last but not least, I thank my family, notably my wife Barbara and my parents, for their permanent backing.

Berlin/Stuttgart July 2008

Wolfgang Siegert

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Linear diﬀerential systems with parameter excitation . . . . . 1.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Spherical coordinates for linear systems . . . . . . . . . . . . . . . . . . . 1.3 The Multiplicative Ergodic Theorem: Lyapunov exponents . . . 1.4 The deterministic case: Lyapunov exponents for asymptotically constant linear systems . . . . . . . . . . . . . . . . . 1.5 Sample systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 10 12 20

1

2

Locality and time scales of the underlying non-degenerate stochastic system: Freidlin-Wentzell theory . . . . . . . . . . . . . . . 2.1 Preliminaries and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The limiting distribution (stationary measure) . . . . . . . . . . . . . 2.3 The large deviations principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Exit probabilities for non-degenerate systems . . . . . . . . . . . . . . 2.5 Sublimiting distributions: Metastability and quasi-deterministic behavior . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Sample systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28 44 53 55 60 68 72 91 108

3

Exit probabilities for degenerate systems . . . . . . . . . . . . . . . . . 125 3.1 Exit probabilities for degenerate systems depending on a small parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.2 Uniform consequence for the exit probability . . . . . . . . . . . . . . . 140

4

Local Lyapunov exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Local Lyapunov exponents: upper and lower bound . . . . . . . . . 4.2 The local growth rate of the determinant . . . . . . . . . . . . . . . . . . 4.3 Local Lyapunov exponents in the diagonal case . . . . . . . . . . . . .

143 144 156 157

vii

viii

Contents

4.4 Local Lyapunov exponents in the two-dimensional, general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Qualitative theory of nonlinear real noise systems on time scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 The local Lyapunov exponent . . . . . . . . . . . . . . . . . . . . . . 4.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177 178 186 226

Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

List of Figures

1

A prototypical potential function U : Rd → R with two wells . .

1.1 1.2

Geometrical interpretation of the quantity vt := (cos αt )2 . . . . . 42 The switching curve and tendencies of ¯ α) = − x + 1 (λ2 − λ1 ) sin 2α . . . . . . . . . . . . . . . . . . . . . . . . . 48 h(x, 2

2.1 2.2 2.3 2.4 2.5 2.6 2.7

Sketch of the potential function U1 . . . . . . . . . . . . . . . . . . . . . . . . Sketch of the density pε ∼ e−2U1 /ε . . . . . . . . . . . . . . . . . . . . . . . . . Sketch of ϕ and ϕδ within the domains D and D∗ . . . . . . . . . . . ε,x ε,x Sketch of the stopping times τm and θm+1 . ................ Visualization of the 1-graphs on {1, 2, 3} . . . . . . . . . . . . . . . . . . . A two well potential function U : Rd → R (cf. ﬁgure 1) . . . . . . . Schematic depiction of the thermohaline circulation . . . . . . . . . .

64 65 74 86 93 110 116

4.1 4.2

The tendencies of the Verhulst drift vector ﬁeld . . . . . . . . . . . . . The tendencies of the comparison drift vector ﬁeld given the sublimiting statistics of X ε,x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Drift vector ﬁeld F with one attracting switching curve and the sublimiting statistics of X ε,x . . . . . . . . . . . . . . . . . . . . . . . . . . The tendencies and switching curves of the sample drift vector ﬁeld F (x, y) = 12 − y + y(1 − y)x on R2 . . . . . . . . . . . . . . Strictly separated switching surfaces on Rd . . . . . . . . . . . . . . . . . ¯ α) = − x + 1 (λ2 − λ1 ) sin 2α and The drift vector ﬁeld h(x, 2 the sublimiting statistics of X ε,x0 . . . . . . . . . . . . . . . . . . . . . . . . . .

164

4.3 4.4 4.5 4.6

3

168 180 184 185 224

ix

Introduction

During the last decades intense research has been devoted to dynamical systems subject to random perturbations: Considerable eﬀort has been dedicated to investigate exit times and exit locations from given domains and how they relate to the respective deterministic dynamical system. Building upon considerations in physics and chemistry (see e.g. the classical paper by Kramers [Kr 40]) the theory of large deviations by Freidlin and Wentzell ([Fr-We 98], [Fr 00]) provides the correct mathematical framework for tackling these problems in case of Gaussian perturbations. This theory sets up the precise time scales for transitions of non-degenerate stochastic systems between certain regimes. The behavior of such systems is called metastable. The theory of random dynamical systems, on the other hand, considers stochastic processes which satisfy a certain ﬂow property, the cocycleproperty. The main cornerstone here is the Multiplicative Ergodic Theorem by Oseledets [Os 68]; also see Arnold [Ar 98]: This theorem assigns Lyapunov exponents to linear random dynamical systems. These are the exponential growth rates as time grows large for ﬁxed intensities of the underlying noise. The following work now attempts to close the gap between these two stochastic disciplines: It does not study the exponential growth rate for a ﬁxed noise intensity and large time (resulting in the Lyapunov exponents), but considers the exponential growth, if the time horizon depends on the noise intensity. Thus one considers the Lyapunov characteristics on time scales. Since these time scales correspond to metastable points, the Lyapunov exponents are localized by connecting the large time asymptotics to the limit of vanishing noise intensity. What one usually does when dealing with “local Lyapunov exponents” is to replace the inﬁnite time limit (characterizing the Lyapunov exponents) by a large, but ﬁnite time horizon; see Abarbanel et al. [Ab-Brw-Ke 91] and [Ab-Brw-Ke 92], Wolﬀ [Wo 92], Pikovsky [Pk 93], Pikovsky and Feudel [Pk-Fe 95] and Bailey et al. [Ba-El-Ny 97]. A similar discussion in the same spirit is undertaken by Monahan [Mo 02]: In the case of the Maas model he describes the concept of a “local Lyapunov exponent” for which the inﬁnite W. Siegert, Local Lyapunov Exponents. Lecture Notes in Mathematics 1963. c Springer-Verlag Berlin Heidelberg 2009

1

2

Introduction

time limit is replaced by a large, but ﬁnite time horizon; this time then needs to be large enough for the system to sample the local attractor, but smaller than the average escape time of the regime. This rationale is also applied when calculating the respective exponents numerically. Now generally speaking, the problem in the case of an elliptic stochastic diﬀerential system is that switches to other regimes occur with strictly positive probability—no matter how small the time horizon is chosen. On the other hand, e.g. as computers can necessarily work with ﬁnite calculation horizons only, our concept of localizing Lyapunov exponents by means of time scales justiﬁes the above ﬁnite-time procedure, if time scales are chosen appropriately. Let us further describe this rationale: The base systems under consideration are dynamical systems with additive white noise perturbations; for simplicity let the process X ε be deﬁned by the stochastic diﬀerential equation (SDE) of gradient type √ dXtε = − ∇U (Xtε ) dt + ε dWt for the moment. It describes the motion of a particle in a potential landscape which is derived from the real-valued, diﬀerentiable function U deﬁned on Rd . The linearization of X ε is then given as the solution of the linear, real noise driven diﬀerential equation dZtε = − HU (Xtε ) Ztε dt , the so called variational equation, in which HU (x) denotes the Hesse matrix of second derivatives of U at x. The variational equation governs the evolution of “inﬁnitesimal disturbances” of X0ε . The Lyapunov exponents of the system are now deﬁned as the exponential growth rates lim

t→∞

1 log | Ztε ( . , x, z) | . t

Their existence is assured by Oseledets’ [Os 68] Multiplicative Ergodic Theorem and their number does not exceed the dimension d of the state space of X ε . Due to the stationarity of the ﬂow X ε in Oseledets’ theorem, the Lyapunov exponents do not depend on the initial condition x = X0ε,x . Suppose that the potential function U has the qualitative shape as depicted in ﬁgure 1. Moreover, let Λ01 (x) ≥ . . . ≥ Λ0d (x) denote the (decreasingly indexed) eigenvalues of − HU (x). The goal of this contribution is to provide a Lyapunov-type number for each regime of the potential function U . As this characteristic number shall depend on the initial point (more precisely, on the well in which the stochastic solution X ε is starting), it shall yield a concept of locality for Lyapunov exponents. The

Introduction

3 U

v V

K1

K3

K2

Fig. 1 A prototypical potential function U : Rd → R with two wells

main motivation for the concept thus deﬁned is that in the small-noise-limit the particle X ε stays in the initial shallow well near K1 for an exponentially long time (Kramers’ law, Freidlin-Wentzell theory) during which we can “see” the shallow well; afterwards the particle ﬁnally overcomes the barrier at K3 and the deeper well around K2 dominates the picture; mathematically this is made precise by observing that K1 and K2 are the supports of the sublimiting distribution on the corresponding time scales (Freidlin [Fr 77] and [Fr 00]). Furthermore, K1 and K2 are the metastable points of the potential. In order to capture this metastable behavior we connect the parameters t and ε by ε(t) :=

ζ log t

for a scaling parameter ζ > 0 for approaching the sublimit distributions. Hence, we consider ε(t) 1 log Zt ( . , x, z) t and we then conjecture that this random variable converges in probability to Λ0j (Kk ) as t → ∞, where k ∈ {1, 2} depends on the initial position x and on the time scale parameter ζ of Xtε ; more precisely, if x is in the K1 -regime and if also ζ < 2v, then k = 1; otherwise, k = 2. This conjecture is proven in section 4.3 under the additional assumption that HU ( . ) only takes its values in the diagonal matrices; the index j ∈ {1, . . . , d} then, of course, depends on the initial direction z of Z. We call these limit numbers the local Lyapunov exponents of Z. In the general case, i.e. abstaining from the diagonality condition, section 4.4 gives conditions under which the above

4

Introduction

exponential growth rate converges in probability to the top eigenvalue of −HU (Kk ), Λ01 (Kk ) , where Kk again denotes the metastable state for the time scale chosen; this limit then is the local Lyapunov exponent of Z. In other words, deﬁning T (ε) := eζ/ε , the previous discussion concerns the convergence in probability of 1 log ZTε (ε) ( . , x, z) T (ε) as ε → 0. In comparison with the previously mentioned concept by Abarbanel et al. [Ab-Brw-Ke 91] and [Ab-Brw-Ke 92], Wolﬀ [Wo 92], Pikovsky [Pk 93], Pikovsky and Feudel [Pk-Fe 95] and Bailey et al. [Ba-El-Ny 97] who would take the ﬁnite time growth rate 1 log ZTε ( . , x, z) T for ﬁxed T and ε as “local Lyapunov exponent”, one therefore obtains a rigorous explanation for how to correctly choose the time horizon depending on the underlying noise intensity ε. Furthermore, we would like to comment on a second type of “local Lyapunov exponents” which can be found in the literature: Let d = 1, then the drift of Z ε in the variational equation is − HU (x) ≡ −U (x) , the negative curvature of U at x. Several authors then call this number −U (x) the “local” or “local in phase space” or “instantaneous” Lyapunov exponent; see Fujisaka [Fu 83], van den Broeck and Nicolis [vB-Ni 93], Witt et al. [Wt-Ne-Kt 97] and Pikovsky and Feudel [Pk-Fe 95]. However, this is not in accordance with our understanding, since the corresponding stochastic system X ε does not stay near an arbitrary initial point x, but is conﬁned to one of the metastable points K1 , K2 by the drift −U . The system Z ε mainly samples −U (Kk ) for k ∈ {1, 2} , but neglects contributions of some other −U (x). Therefore our result for the one-dimensional situation (see remark 4.1.4) only has the values −U (K1 )

and

− U (K2 )

as local Lyapunov exponents in the sense of our deﬁnition in contrast to Fujisaka [Fu 83], van den Broeck and Nicolis [vB-Ni 93], Witt et al. [Wt-Ne-Kt 97] and Pikovsky and Feudel [Pk-Fe 95]. In other words, this expresses the fact that the instantaneous rates −U (x) do not have equal

Introduction

5

rights, but the Dirac measures δK1 and δK2 are the sublimiting distributions reﬂecting the preferences of X ε on the time scales. A third, deterministic concept of locality of Lyapunov exponents diﬀerent from ours has been introduced by Eden, Foias and Temam [Ed-Fo-Tm 91]. It has already been indicated above that Kramers’ law, made precise by Freidlin and Wentzell, plays a dominant role in the following considerations. It is interesting that already the classical Eyring-Kramers formula for the exit times of X ε contains the eigenvalues of the Hesse matrix of U at K1 , K2 ε ε and K3 . Namely let τ12 (and τ21 ) denote the time at which X ε enters the K2 -well when started in K1 (and vice versa). Then in the limit as ε → 0, under non-degeneracy assumptions, the following asymptotic expressions are known to hold for the mean exit times, d 2π i=1 Λ0i (K3 ) 2v/ε ε E τ12 ≈ 0 e d Λ1 (K3 ) Λ0 (K1 ) i=1

and ε E τ21

i

d 2π i=1 Λ0i (K3 ) 2V /ε ≈ 0 e d 0 Λ1 (K3 ) i=1 Λi (K2 )

which are cited from Bovier at al. [Bv-Ec-Gd-Kn 04] in the above notation. Abstracting the previous considerations one detects that the underlying diﬀusion does not have to stem from an SDE of gradient type; the Freidlin-Wentzell theory of large deviations and metastability also admits more general drift functions b and also state dependent noise coeﬃcients σ under certain assumptions. Overall, the two characteristic features of the above stochastic diﬀerential system (X ε , Z ε ) are the following: Firstly, it is degenerate in the sense that in the equation for Z ε there is no noise component but only a drift coeﬃcient depending on X ε ; in other words, the diﬀerential equation of Z ε is random, driven by the real noise process X ε . Secondly, the diﬀerential system is linear with respect to Z ε . Further abstracting the system matrix A is by no means restricted to be the negative of the Hesse matrix of some potential function; it can be any matrix valued function deﬁned on the state space of X ε . Moreover, we will also admit that the state spaces of X ε and Z ε can have diﬀerent dimensions. Hence, the general object of the subsequent considerations is the real-noise driven, linear stochastic diﬀerential system dZtε = A (Xtε ) Ztε dt √ dXtε = b (Xtε ) dt + ε σ (Xtε ) dWt ,

(1)

where ε ≥ 0, W is a Wiener process on Rd , A ∈ C(Rd , Rn×n ) (or A ∈ C(Rd , Cn×n )), the drift b : Rd → Rd does not necessarily stem from a

6

Introduction

potential, but exhibits several regimes and the values of σ : Rd → Rd×d are close to idRd . All in all, X ε is a diﬀusion in Rd with small noise intensity and Z ε is a linear cocycle in Rn . Generalizing the previously sketched idea the local Lyapunov exponents are the possible limits of ε(t) 1 log Zt ( . , x, z) t as t → ∞, where ε(t) ≡

ζ log t

for a scaling parameter ζ > 0, depending on the initial condition x of X ε and the initial direction z of Z ε . The goal of proving convergence in probability for this exponential growth rate on a time scale is organized as follows: In the ﬁrst chapter we collect known results on linear, real noise driven diﬀerential systems such as the Multiplicative Ergodic Theorem, their decomposition in spherical coordinates and the deterministic Hartman-WintnerPerron theorem. The latter two subjects yield coordinate systems which will decisively come into play in chapter 4. Beforehand, chapter 2 gives an account of the Freidlin-Wentzell theory as needed for describing the locality, metastability and sublimiting distributions. Since this theory is based on the fundamental exit time law for non-degenerate stochastic systems, this result is recalled in detail. The system Z ε of (1) is a degenerate stochastic diﬀerential system by deﬁnition, since there is no stochastic diﬀerential in the Z-component. This is in particular also true for the angle of Z ε . As the behavior of the latter process needs to be investigated in chapter 4 chapter 3 recalls known results on its exit probabilities; more precisely, we give an account of the theorems by Hern´ andez-Lerma concerning exit probabilities of degenerate systems which are not covered by Freidlin-Wentzell theory. Chapter 4 ﬁnally investigates the exponential growth rates on the time scales: Firstly, it is proven that the top real part of an eigenvalue at the metastable state is an upper bound for the local Lyapunov exponent; likewise, the smallest real part of an eigenvalue at the metastable state is a lower bound for the local Lyapunov exponent; see section 4.1. Since det Z ε is a process in R and the situation is quite tractable for a one-dimensional state space, a consequence for the exponential growth rate of det Z ε can be drawn which is subject to section 4.2. The coordinates from the Hartman-Wintner-Perron theorem allow to explicitly calculate the exponential growth rate under the additional assumption that A solely takes diagonal values; see section 4.3. Finally, section 4.4 gives criteria under which one can obtain convergence in the general, two-dimensional case: Here the result by Hern´andez-Lerma comes into play as the decomposition of |Ztε | by means of spherical coordinates demands an assertion on the angle process of Ztε . More precisely,

Introduction

7

a statement concerning the Lebesgue measure of the times which the angle process spends at the switching curves of its drift can be deduced; this result which is analogous to Freidlin’s [Fr 00] metastability theorem then allows to calculate a sublimiting version of the Furstenberg-Khasminskii formula. A ﬁnal remark is necessary concerning the use of the notion of “time scales”: There is now a very elaborate theory of “dynamic equations on time scales”; this theory has been invented by Hilger [Hi 88]; also see Bohner and Peterson [Boh-Pet 01]. This concept understands “time scales” (also called “measure chains”) as certain time sets which are underlying to the systems under consideration. Hence, the investigation of an ordinary diﬀerential equation (ODE) means to work with the “time scale” T = R+ or R; an ordinary diﬀerence equation is understood as dynamic equation on the “time scale” T = N0 or Z. However, in this paper here the physical time set is always R+ and a time scale in our context is understood as a time horizon T ≡ T (ε) which depends on the parameter ε > 0 of the base SDE of X ε . The symbol 2 will be used to mark the end of a proof. In order to avoid latent ambiguities, it will also be employed sporadically to ﬁnish remarks and examples where necessary.

Chapter 1

Linear diﬀerential systems with parameter excitation

This chapter intends to introduce to the general setting of linear stochastic systems Z˙ t = A(t) Zt , d Zt denotes where (A(t))t≥0 is a matrix valued stochastic process and Z˙ t ≡ dt the derivative with respect to the “time” variable t. Such diﬀerential systems are called parametrically excited (perturbed) or real noise linear systems; in the engineering literature the terminology rheo linear system is also used. The system matrix is assumed to be a continuous mapping deﬁned on the state space of a Markov process which serves as stochastic input for the system. Hence, the above matrix process is of the form

A(t) = A(Xt ), where (Xt )t≥0 is the input process. Note for preciseness that some authors further diﬀerentiate the nature of the noise by distinguishing “real noise” on the one hand which is deﬁned on the two-sided time set R and Markovian noise on the other hand which is a Markov process with time set R+ ; see Arnold and Kliemann [Ar-Kl 83, p.4]. In this book both terms will be used interchangeably in the latter sense. Linear real noise systems with statedependent coeﬃcient matrix A(Xt ) and Markovian input noise Xt as above have for example been investigated by Frisch [Fs 66]. In this reference it is argued heuristically how a Fokker-Planck equation might be obtained for (Xt , Zt )t , if Xt is stationary. Real noise systems with Markovian input noise are also subject to the considerations of Kats and Krasovskii [Ka-Kv 60]; however, these authors consider the case that (Xt )t≥0 is a Markov chain with ﬁnite state space. Now the setting of our work also assumes that the input process has ﬁnitely many “states of preference”, but is a continuous process deﬁned by a stochastic diﬀerential equation with respect to Brownian motion. These preferential (“metastable”) states correspond to certain time scales which will be made precise in the next chapter.

W. Siegert, Local Lyapunov Exponents. Lecture Notes in Mathematics 1963. c Springer-Verlag Berlin Heidelberg 2009

9

10

1 Linear diﬀerential systems with parameter excitation

Since the real noise process (Xt )t≥0 will be deﬁned by some SDE, the coupled system (X, Z) is also given by the resulting SDE. Note that this diﬀerential equation is degenerate by deﬁnition in the sense that no noise term, i.e. no summand of the type g(Xt , Zt ) dBt containing the diﬀerential of a Brownian motion (or some more general stochastic process) B, is visible in the Z-direction, but solely in the X-variable; see e.g. Bunke [Bu 72, Ch.6] for standard results on such equations. A diﬀerent type of randomly perturbed linear SDE is the so called white noise case; see e.g. Khasminskii [Kh 67], [Kh 80] and [Kh 60]. Here, there is no real noise input but stochastic integration with respect to Brownian motion in the SDE for Z itself. The latter type of systems is not subject to our investigations, but also will be commented on in this chapter. From an applications’ point of view real noise stochastic systems are often considered as more realistic for describing “real”-world problems. The reason is that most processes to be modeled are concerned with variables of a speciﬁed magnitude or even restricted to a bounded interval such as concentrations (e.g. in chemistry), population ﬂuctuations (in life sciences) or strictly positive parameters in technical systems; see e.g. Kliemann [Kl 80, App.IV: p.3] and [Kl 83b], Arnold et al. [Ar-Hh-Lf 78], Ahmadi and Morshedi [Ah-Mr 78] and Griesbaum [Gb 99] for further examples and discussions. In contrast using white noise instead is mostly considered as too drastic an idealization; see Kliemann [Kl 80, App.IV: p.3] and Wihstutz [Wh 75, p.3].

1.1 The model The system which lies at the heart of these investigations is the real-noise driven system dZtε = A (Xtε ) Ztε dt √ dXtε = b (Xtε ) dt + ε σ (Xtε ) dWt

(1)

where A ∈ C(Rd , Kn×n ) is a continuous matrix function (K = R or C); d, n ∈ N denote the dimensions of the state spaces of X ε and Z ε , respectively; ε ≥ 0 parametrizes the intensity of (Wt )t≥0 which denotes a Brownian motion in Rd , deﬁned on a complete probability space (Ω, F , P), and X ε is a diﬀusion, deﬁned by the above SDE with coeﬃcient functions b ∈ C ∞ (Rd , Rd ) and σ ∈ C ∞ (Rd , Rd×d ). A detailed study of the SDE for X ε will follow in the next chapter specifying the assumptions; see (2.1) and the corresponding set of assumptions 2.1.1. For the moment, just assume that there exists a unique non-exploding solution1 X ε,x , where the superscript x ∈ Rd denotes its initial value. 1 For standard terminology concerning SDEs, such as solution, uniqueness and nonexplosiveness, we refer to Hackenbroch and Thalmaier [Hb-Th 94, Ch.6], Khasminskii [Kh 80, Ch.I-III] and Freidlin and Wentzell [Fr-We 98, Ch.1] among many others.

1.1 The model

11

Since the stochastic process X ε,x is non-explosive and has (P-almost surely) continuous paths, the SDE (1) deﬁnes a continuous stochastic process (X ε , Z ε ) in Rd × Kn ; more precisely, the diﬀerential equation for Z ε , dZtε = A Xtε,x (ω) Ztε dt , Z0ε = z ∈ Kn , (1.1) is a random diﬀerential equation (RDE) which ω-wise, i.e. for any realization Xtε,x (ω), is solved as an ODE; hence, the continuity of t → A(Xtε,x (ω)) guarantees existence and uniqueness of the solution path Ztε (ω, x, z) in Kn (see e.g. Coppel [Cp 65, p.42]) which together with X ε,x forms the solution (X ε , Z ε ) of (1) in Rd × Kn , starting in (x, z). Collecting these solution paths (possibly deﬁned as 0 on an exceptional subset of Ω of zero P-measure) yields a well-deﬁned mapping Z ε : R+ × Ω × Rd × Kn −→ Kn , (t, ω, x, z) → Z ε (t, ω, x, z) := Z ε (t, ω, x) z := Ztε (ω, x) z := Ztε (ω, x, z) and these notations will be used equivalently. Being linear in the z-variable the mappings Z ε (t, ω, x) form a matrix process which (as the corresponding fundamental matrix) solves the random matrix diﬀerential equation dZtε = A Xtε,x (ω) Ztε dt , Z0ε = idKn . Furthermore, as the fundamental solution of a linear equation, Z ε (t, ω, x) takes its values in the set of invertible matrices, its inverse being governed by the matrix diﬀerential equation d(Z −1 )t = − (Z −1 )t A(Xtε,x ) dt . In particular, the Wronski-determinant det(Ztε (ω, x)) does not vanish and diﬀerentiating pathwise with respect to t yields the random diﬀerential equation d det(Ztε (ω, x)) = trace A Xtε,x (ω) det(Ztε (ω, x)) dt, det(Z0ε (ω, x)) = 1, and therefore one obtains the Jacobi equation (Liouville equation) t

det(Ztε (ω, x)) = det(Z0ε (ω, x)) exp trace A Xuε,x (ω) du (1.2) 0

t trace A Xuε,x (ω) du ; = exp 0

see e.g. Coppel [Cp 65, p.44] for the pathwise calculations. Note that for ε = 0 all objects remain well-deﬁned as the solutions of the resulting ODE.

12

1 Linear diﬀerential systems with parameter excitation

Under appropriate conditions it can be proved that X ε and hence also Z ε depend continuously on the parameter ε > 0; see Blagovescenskii and Freidlin [Bla-Fr 61]. However, this will not be used in the sequel. In this work we are interested in Lyapunov exponents, i.e. exponential growth rates 1 log | Ztε (ω, x, z) | , t which are realized for large times t = T (ε) ; such growth rates will then be called local Lyapunov exponents. In the following section spherical coordinates will be introduced in order to get an integral decomposition for the growth rate mentioned previously. This representation which is also an ingredient of the Furstenberg-Khasminskii formula (see p.27) provides quantitative information concerning |Z ε | and will play a fundamental role in subsection 4.4.2.

1.2 Spherical coordinates for linear systems In the following the system Z ε will be decomposed by means of spherical coordinates; for this section we take K = R, i.e. Ztε (ω, x, z) is an element of Rn where n ∈ N. In the literature on stochastic systems the use of such coordinates (also called the projection method ) is accredited to Khasminskii [Kh 67] and Infante [In 68]; see e.g. Kliemann [Kl 79, p.465] and [Kl 80, p.144] and Crauel [Cra 84, p.13]. It is, however, interesting to note that this use of spherical coordinates is by no means restricted to the stochastic case. In fact, this method had been a well established tool in investigating deterministic linear diﬀerential systems before; see e.g. Levi-Civita [LC 11] and Wintner [Wi 50] and [Wi 57]. Since the matrix Z ε (ω, x) is invertible, the process Ztε (ω, x, z) is nonzero for z = 0 and can hence be characterized by its radial and spherical components via εt (ω, x, z) := | Ztε (ω, x, z) | ∈ (0, ∞) and ψtε (ω, x, z) :=

Ztε (ω, x, z) ∈ S n−1 , εt (ω, x, z)

respectively, if z = 0 ; here, S n−1 := {y ∈ Rn : |y| = 1} denotes the unit sphere in Rn . For the system Z ε as given by (1), the deﬁning pathwise diﬀerential equation (1.1) is equivalent to the system of the two RDEs d εt = Q(Xtε , ψtε ) εt dt ,

(1.3)

dψtε = h(Xtε , ψtε ) dt ,

(1.4)

1.2 Spherical coordinates for linear systems

13

where

Q(x, ψ) := Q A(x), ψ := A(x)ψ , ψ

and

h(x, ψ) := h A(x), ψ := A(x)ψ − A(x)ψ , ψ ψ ;

here, . , . denotes the standard scalar product of Rn . This follows in a d straightforward manner by calculating the pathwise derivative dt for ﬁxed (ω, x, z) where z = 0, since d d ε ≡ Z ε , Z ε 1/2 dt t dt t t 1 1 d Z ε , Z ε = 2 εt dt t t

1 1 d ε ε Z ,Z = 2 2 εt dt t t 1 = ε A(Xtε ) Ztε , Ztε ≡ A(Xtε ) ψtε , ψtε εt t and hence d d ε εt dt Ztε − Ztε dt t d Ztε d ε ψt ≡ = dt dt εt ( εt )2 εt A(Xtε ) Ztε − Ztε A(Xtε ) ψtε , ψtε εt = ( εt )2 ε ε ≡ A(Xt ) ψt − A(Xtε ) ψtε , ψtε ψtε

due to (1.1); conversely, these diﬀerential equations for ( ε , ψ ε ) imply that d ε ε d ε Z ≡ ( ψ ) dt t dt t t d ε d ε t + εt ψ = ψtε dt dt t = ψtε A(Xtε ) ψtε , ψtε εt + εt A(Xtε ) ψtε − εt A(Xtε ) ψtε , ψtε ψtε ≡ A(Xtε ) Ztε . Note that h(A, ψ) ≡ Aψ − Aψ, ψ ψ vanishes at an element ψ ∈ S n−1 , if and only if ψ is an eigenvector of the matrix A. The above vector ﬁeld h(A, . ) is considered as the projection of the linear vector ﬁeld ψ → Aψ onto S n−1 for a ﬁxed A ∈ Rn×n . Due to the multiplicative structure of the RDE for ε , it can be integrated pathwise by separation of variables, thus giving t

| Ztε (ω, x, z) | ≡ εt (ω, x, z) = | z | exp Q Xuε,x (ω), ψuε (ω, x, z) du 0

14

1 Linear diﬀerential systems with parameter excitation

and therefore log |z| 1 1 log |Ztε (ω, x, z)| = + t t t

t

Q Xuε,x (ω), ψuε (ω, x, z) du.

(1.5)

0

Now since h(x, −ψ) = −h(x, ψ) for all x and ψ, the SDE for ψtε is symmetric with respect to the origin; more precisely, h(x, . ) can be viewed as a vector ﬁeld on the projective space P n−1 , i.e. on S n−1 where opposite points are identiﬁed; therefore, ψ ε is also considered as a P n−1 -valued process. The SDE for ε and formula (1.5) remain unaﬀected, since Q(x, −ψ) = Q(x, ψ) for all x and ψ. Furthermore, the projective process ψ ε is decoupled from the radial process ε ; therefore, (X ε , ψ ε ) is a Markov process in Rd × S n−1 or Rd × P n−1 , respectively, whose generating partial diﬀerential operator is given by ∂ Lε := G ε + h , ∂ψ where G ε :=

d

bi

i=1

d ε ∂ ∂2 + aij ∂xi 2 i,j=1 ∂xi ∂xj

ε

is the generator of X to be discussed later; see section 2.2 . Remark 1.2.1 (n = 2). Consider the two-dimensional case, i.e. Kn = R2 . Here, we can canonically identify ψ ∈ S 2 with its angle α ∈ [0, 2π) via

cos α = ψ , sin α for which we will use the abbreviation . ψ = α in the following. In particular, the spherical process ψtε on S 1 deﬁnes the angle process αεt ∈ [0, 2π) by

cos αεt := ψtε . sin αεt The previous SDE (1.4), dψtε = h(Xtε , ψtε ) dt , implies the following equation for αεt : ¯ ε , αε ) dt , dαεt = h(X t t ¯ being deﬁned by for the drift h ¯ ¯ A(x), α , h(x, α) := h

(1.6)

1.2 Spherical coordinates for linear systems

15

where ¯ A, α := − a12 sin2 α + a21 cos2 α + a22 − a11 sin α cos α h

for matrices A =

a11 a12 a21 a22

∈ R2×2 ;

. see Arnold and Kliemann [Ar-Kl 83, p.59]. More precisely, let α = also cos α sin α =: ψ(α) and deﬁne

d − sin α ψ(α) = ; ψ (α) := dα cos α then one directly calculates that ¯ h(A, α) =

Aψ(α) , ψ (α)

for any A ∈ R2×2 ; furthermore, (ψ(α), ψ (α)) is an orthonormal basis of R2 for any α and one obtains altogether for the pathwise derivatives that d d ε ψ ≡ ψ(αεt ) dt t dt d ε α = ψ (αεt ) dt t ε ε ¯ = ψ (αεt ) h(A(X t ), αt ) ε ε = A(Xt )ψ(αt ) , ψ (αεt ) ψ (αεt ) = A(Xtε )ψ(αεt ) − A(Xtε )ψ(αεt ) , ψ(αεt ) ψ(αεt ) ≡ h(A(Xtε ), ψtε ) which proves that the RDEs for ψ ε and αε are equivalent, since this calculation can be read in both directions. ¯ that h(A ¯ Note that it follows from the above deﬁnition of h + c I2 , . ) = 2×2 ¯ h(A, . ), for all A ∈ R and c ∈ R, where I2 denotes the two-dimensional unit matrix; an example for such a system is given by Ahmadi and Morshedi [Ah-Mr 78, Sec.IV]; also see Kliemann and R¨ umelin [Kl-Rm 81, p.17]. Also note that if A ∈ R2×2 is symmetric, then ¯h reads ¯ h(A, α) ≡ − a12 sin2 α + a21 cos2 α + (a22 − a11 ) sin α cos α 1 = a12 cos 2α + (a22 − a11 ) sin 2α . 2 In any case the RDE (1.3) for ε now reads ¯ tε , αεt ) εt dt , d εt = Q(X

16

1 Linear diﬀerential systems with parameter excitation

¯ being deﬁned by for Q ¯ α) := Q(A(x), ¯ Q(x, α) , where ¯ Q(A, α) := a11 cos2 α + a22 sin2 α + (a12 + a21 ) sin α cos α

(1.7)

is just designed such that Q(A, ψ) ≡ Aψ, ψ

cos α cos α a11 a12 , ≡ a21 a22 sin α sin α = a11 cos2 α + a22 sin2 α + (a12 + a21 ) sin α cos α ¯ ≡ Q(A, α) , . if α = ψ. Formula (1.5) hence becomes log |z| 1 1 log |Ztε (ω, x, z)| = + t t t

t

¯ X ε,x (ω), αε (ω, x, z) du. Q u u

(1.8)

0

Furthermore, the drift function ¯ h(x, . ) is π-periodic which corresponds to the fact that h(x, −ψ) = −h(x, ψ) for ψ ∈ S 1 ; αε is the solution of the above real noise SDE modulo π. Thus (Xtε , αεt ) is a Markov process in R2 × [0, π) with generator ¯ ∂ . L¯ε := G ε + h ∂α Another way of describing the motion of the angle is by means of ξtε := tan αεt whenever αεt ∈ / {(2k + 1)π/2 : k ∈ Z}; deﬁning F (A, ξ) := a21 − a12 ξ 2 + [ a22 − a11 ] ξ for A ∈ R2×2 and ξ ∈ R, the dynamics of ξ ε is described by the diﬀerential equation dξtε = F (A(Xtε ), ξtε ) ≡ a21 (Xtε ) − a12 (Xtε ) (ξtε )2 + a22 (Xtε ) − a11 (Xtε ) ξtε dt ,

(1.9)

1.2 Spherical coordinates for linear systems

17

since 1 d ε ¯ ξt = h(Xtε , αεt ) dt cos2 αεt 1 − a12 (Xtε ) sin2 αεt + a21 (Xtε ) cos2 αεt ≡ cos2 αεt + a22 (Xtε ) − a11 (Xtε ) sin αεt cos αεt = a21 (Xtε ) − a12 (Xtε ) (ξtε )2 + a22 (Xtε ) − a11 (Xtε ) ξtε . Remark 1.2.2. The above pathwise considerations can be rewritten for any linear diﬀerential equation with time dependent system matrix, Z˙ t = A(t) Zt

(t ≥ 0)

in Rn , as ˙ t = Q(t, ψt ) t , ψ˙ t = h(t, ψt ) , where t := | Zt | ∈ (0, ∞) and ψt := Ztt ∈ S n−1 whenever Z0 = 0. Here, the notation is adapted as Q(t, ψ) := Q A(t), ψ ≡ A(t)ψ , ψ and

h(t, ψ) := h A(t), ψ := A(t)ψ − A(t)ψ , ψ ψ .

In particular, 1 1 log | Z0 | log | Zt | = + t t t

t

Q(u, ψu ) du . 0

For the two-dimensional case n = 2,

a11 (t) a12 (t) A(t) := , a21 (t) a22 (t) . and the then well deﬁned angle process αt = ψt , i.e.

cos αt := ψt , sin αt it follows that ¯ αt ) , α˙ t = h(t,

(1.10)

18

1 Linear diﬀerential systems with parameter excitation

where ¯ ¯ h(t, α) := h(A(t), α) ≡ − a12 (t) sin2 α + a21 (t) cos2 α + (a22 (t) − a11 (t)) sin α cos α ≡ A(t)ψ(α) , ψ (α) . The formula (1.10) becomes 1 1 log | Z0 | log | Zt | = + t t t

t

¯ αu ) du , Q(u,

(1.11)

0

¯ α) := Q(A(t), ¯ where Q(t, α). Furthermore, ξt := tan αt is driven by the diﬀerential equation ξ˙t = F (A(t), ξt ) ≡ a21 (t) − a12 (t) ξt2 + [ a22 (t) − a11 (t) ]ξt . The following deﬁnition is taken from Arnold and Kliemann [Ar-Kl 83, p.13] and from Kliemann [Kl 79, p.464]. Deﬁnition 1.2.3 (Switching surfaces of a drift function). Let F : Rd × R → R be a continuous function. The points (x, y) ∈ Rd × R with F (x, y) = 0 deﬁne connected surfaces in Rd ×R. These are called the switching surfaces (of F ). If d = 1, these sets will also be called the switching curves (of F ). This terminology will now be used for investigating the behavior of ξ ε and αε , respectively. Here, we follow Arnold and Kliemann [Ar-Kl 83, p.59f.]; also see Kliemann [Kl 80, p.148f.] and [Kl 79, p.465f.]. Deﬁnition-Remark 1.2.4 (Switching surfaces corresponding to (1.1)). Consider the RDE (1.1) which is the pathwise diﬀerential system deﬁned by the SDE (1) and suppose that Kn = R2 , i.e. the system matrix is given by a mapping A ∈ C(Rd , R2×2 ). By (1.9) A( . ) induces the drift vector ﬁeld for ξ ε ≡ tan αε , F (x, ξ) := F (A(x), ξ) ≡ a21 (x) − a12 (x) ξ 2 + [ a22 (x) − a11 (x) ] ξ , where x ∈ Rd and ξ ∈ R. According to the previous deﬁnition 1.2.3 we look for the zeros, F (x, ξ) = F (A(x), ξ) = 0 ;

1.2 Spherical coordinates for linear systems

19

from the solution formula for quadratic equations it follows that one gets no switching curve, if a12 = 0 and (a22 − a11 )2 < −4a12 a21 ; one switching curve ξ1 (A(x)) =

a22 − a11 , if a12 = 0 and (a22 − a11 )2 = −4a12 a21 , 2a12

and ξ1 (A(x)) = ∞ , if a12 = 0 and a11 = a22 and a21 = 0 , respectively; two switching curves ξ1,2 (A(x)) =

(a22 − a11 )2 + 4a12 a21 , 2a12 0 and (a22 − a11 )2 > −4a12 a21 , =

(a22 − a11 ) ± if a12

and ξ1 (A(x)) = ∞ , ξ2 (A(x)) =

a21 , if a12 = 0 and a11 = a22 , a11 − a22

respectively; inﬁnitely many switching curves ξ ∈ R ∪ {∞} , if a12 = 0 and a11 = a22 and a21 = 0 . The above switching curves for F , i.e. for the motion of ξ ε ≡ tan αε , directly translate into switching curves of the drift ¯ h(A(x), α) ≡ − a12 (x) sin2 α + a21 (x) cos2 α + (a22 (x) − a11 (x)) sin α cos α ¯ of the angle motion; since h(A(x), . ) is π-periodic, we can restrict our investigations on an interval of that length, e.g. (− π2 , π2 ] ; then we get the following ¯ switching curves of h: Ai (x) := Ai (A(x)) := arctan ξi (A(x)) , if there are i ∈ {1, 2} many zeros ξi (A(x)) of F (A(x), ξ). In the last case of inﬁnitely many switching curves any α is a zero of ¯ h(A(x), . ). It follows from the above that a12 (x) = 0 implies that π2 is a switching curve; furthermore, if a21 (x) = 0, then 0 is a switching curve.

20

1 Linear diﬀerential systems with parameter excitation

In case that there are exactly two switching curves A1,2 (A(x)), the previous notation is redeﬁned such that A1 (A(x)) is attracting and A2 (A(x)) is repelling, in the sense that ¯ α → h(A(x), α) changes its sign from + to − at A1 (A(x)) and changes from − to + at A2 (A(x)). Remark 1.2.5. The above considerations have been accomplished in the real noise situation of the diﬀerential equation (1). Actually, this technique of projecting the linear system onto the sphere using spherical and angular coordinates can also be transferred to the white noise case; the Itˆo-formula then provides the SDEs for the radial, spherical and angular component, respectively. We do not give details here, since this goes beyond the framework of this book, but refer to the literature instead: See Khasminskii [Kh 67] and [Kh 80, Sec.VI.7-VI.9], Nishioka [Nk 76], B¨ ohme [Bm 80], Auslender and Milshtein [Al-Mi 82], Arnold et al. [Ar-Oe-Pd 86], Arnold and Kliemann [Ar-Kl 87a], Pinsky and Wihstutz [Pi-Wh 88], Pardoux and Wihstutz [Pd-Wh 88] and [Pd-Wh 92] and Imkeller and Lederer [Im-Ld 99] and [Im-Ld 01].

1.3 The Multiplicative Ergodic Theorem: Lyapunov exponents In this section the “classical” Lyapunov exponents are to be discussed as the result of Oseledets’ [Os 68] Multiplicative Ergodic Theorem. Since this result and its arguments are not to be used in the sequel, we refer to Arnold [Ar 98] for details. Here, the solution of the RDE (1.1) is modeled as random dynamical system (RDS) over the canonical metric dynamical system. More precisely, let Ω := C(R+ , Rd ) denote the path space of X ε , dXtε = b (Xtε ) dt +

√ ε σ (Xtε ) dWt

(t ≥ 0) ,

where ε > 0. Ω is ﬁxed as canonical probability space in this section. If endowed with the topology of uniform convergence on compacts, Ω is a Polish space which can then be equipped with its Borel-σ-algebra; the latter is the trace σ-algebra of B(Rd )R+ in Ω, i.e. F := B(Ω) = Ω ∩ B(Rd )R+ ;

1.3 The Multiplicative Ergodic Theorem: Lyapunov exponents

21

see Arnold [Ar 98, p.544f.] and Hackenbroch and Thalmaier [Hb-Th 94, 1.24]. For all t ∈ R+ , there is the canonical shift transformation on Ω deﬁned by θ(t) : Ω → Ω ,

ω → θ(t)ω := ω(t + • ) ≡ s → ω(t + s) .

Here, the mapping (t, ω) → θ(t)ω is continuous, hence measurable with respect to the underlying Borel-σ-algebras. Let (Ptε )t denote the Markov transition probabilities of the Markov process X ε . Furthermore, suppose that there exists a unique stationary probability distribution for X ε , i.e. a unique probability measure2 ρε on Rd which is invariant with respect to the Markov transition probabilities (Ptε )t in the sense that ρε ( . ) = Ptε (x, . ) ρε (dx) (t ≥ 0) ; Rd

conditions assuring existence and uniqueness of such a measure ρε will be given in section 2.2 where the SDE of X ε is investigated in detail; see p.62f. Then there is a unique probability Pρε on (Rd )R+ , B(Rd )R+ such that the coordinate process is a (time-homogeneous) Markov process with transition semigroup (Ptε )t and initial distribution ρε ; see e.g. Arnold [Ar 98, p.548] and Hackenbroch and Thalmaier [Hb-Th 94, 2.5]. Since the outer measure of Ω is full, Pρ∗ε (Ω) = 1, due to the continuity of the paths of X ε , Pρε also induces a probability distribution on F ; see e.g. Hackenbroch and Thalmaier ε [Hb-Th 94, p.43f.]. It will be denoted by the same symbol Pρε . Let X ε,ρ denote the process with distribution Pρε on the path space Ω, i.e. the system X ε with initial distribution ρε . Note that the above stationarity of the measure ρε (stationarity of the canonical Markov process), i.e. the (Ptε )t -invariance of ρε , is equivalent to Pρε being (θ(t))t -invariant in the sense that Pρε ◦ θ(t)−1 = Pρε

for all t ∈ R+ ;

this fact is also expressed by stating that all shifts θ(t) preserve the measure Pρε ; see Arnold [Ar 98, p.545,549]. Altogether, the above system (Ω, F , Pρε , (θ(t)t∈R+ ) satisﬁes 1) (ω, t) → θ(t)ω is F ⊗ B(R+ )/F -measurable, 2) θ(0) =idΩ and θ(s + t) = θ(s) ◦ θ(t) for all s, t ∈ R+ , 3) Pρε ◦ θ(t)−1 = Pρε for all t ∈ R+ ,

which are the characterizing properties for calling Ω, F , Pρε , (θ(t)t∈R+ a metric (measure preserving) dynamical system; see Arnold [Ar 98, p.536f.]. 2

This measure ρε is not to be confused with the radial component process εt of the previous section.

22

1 Linear diﬀerential systems with parameter excitation

Furthermore, this metric dynamical system is ergodic meaning that Pρε (A) ∈ {0, 1} on all sets A ∈ F which are (θ(t))t∈R+ -invariant, i.e. θ(t)−1 A = A for all t ∈ R+ . This is due to the fact that the measures which are ergodic in this sense are the extreme points of the convex set of (θ(t))t -invariant measures; however, due to the postulated uniqueness of ρε , there is only one such measure which is Pρε ; see Arnold [Ar 98, p.539]. ε Let X ε,ρ denote the stationary Markov process which realizes Pρε as deﬁned above. This process is now speciﬁed as stochastic input for the diﬀerential system (1); more precisely, consider the RDE ε dZtε = A Xtε,ρ (ω) Ztε dt ,

Z0ε = z ∈ Kn

and let Φε : R+ × Ω × Kn −→ Kn , (t, ω, z) −→ Φε (t, ω, z) ≡ Φε (t, ω)z t ε =z + A Xuε,ρ (ω) Φε (u, ω, z) du 0

denote its unique (up to indistinguishability) solution. Then the following holds true: 1) Φε is B(R+ ) ⊗ F ⊗ B(Kn )/B(Kn )-measurable, 2) Φε is a cocycle over (θ(t))t∈R+ in the sense that for all ω ∈ Ω and s, t ∈ R+ : Φε (0, ω) = idKn

and

Φε (t + s, ω) = Φε (t, θ(s)ω) ◦ Φε (s, ω) ,

3) Φε ( . , ω, . ) is continuous for any ω ∈ Ω, 4) Φε (t, ω) ≡ Φε (t, ω, . ) is linear for any t ∈ R+ and ω ∈ Ω ; see Arnold [Ar 98, 2.2.12]. These properties are summarized by calling Φε a linearrandom dynamical system (RDS) on Kn over the metric dynamical system Ω, F , Pρε , (θ(t)t∈R+ with time R+ ; see Arnold [Ar 98, p.5f.]. In particular, since Φε is deﬁned by the characterizing RDE (1), it will be called the RDS generated by (1). Being the solution of a linear diﬀerential equation the linear operator Φε (t, ω) is even invertible, i.e. Φε (t, ω) ∈ GL(n, K) for all t ∈ R+ and ω ∈ Ω. The pathwise Jacobi (Liouville) equation (1.2) now reads t

ε det(Φε (t, ω)) = exp trace A Xuε,ρ (ω) du . (1.12) 0

1.3 The Multiplicative Ergodic Theorem: Lyapunov exponents

23

The adjoint matrix of Φε (t, ω) will be denoted by Φε (t, ω)∗ in the sequel. Then the Multiplicative Ergodic Theorem states the following in the above setting: Theorem 1.3.1 (Multiplicative Ergodic Theorem). Suppose that X ε has a unique stationary distribution ρε and let Φε be the linear, invertible RDS generated by (1), where ε > 0. Assume that β + ∈ L1 (Pρε ) and β − ∈ L1 (Pρε ) which are deﬁned by β ± (ω) := sup log+ Φε (t, ω)±1 , 0≤t≤1

where log+ a := max(0, log a) and . denotes the operator norm. Then ε ∈ F of full measure, Pρε (Ω ε ) = 1, there exists a (θ(t))t∈R+ -invariant set Ω such that the following holds: 1/2t lim Φε (t, ω)∗ Φε (t, ω) =: Ψε (ω)

t→∞

ε and is non-negative deﬁnite, its eigenvalues being given exists for any ω ∈ Ω by n values ε ε ε eΛ1 ≥ eΛ2 ≥ · · · ≥ eΛn > 0 , ε ; writing the where Λε1 ≥ Λε2 ≥ · · · ≥ Λεn > −∞ do not depend on ω ∈ Ω distinct numbers in this list of eigenvalues of Ψε (ω) as ε

ε

ε

eλ1 > eλ2 > · · · > eλpε > 0 and deﬁning dεi as the counting multiplicity of λεi in the list {Λε1 , . . . , Λεn }, it follows that dεi = dim Uiε (ω)

ε and i ∈ {1, . . . , pε } , for all ω ∈ Ω ε

where Uiε (ω) denotes the eigenspace of Ψε (ω) corresponding to eλi . Further deﬁne Upε (ω) ⊕ · · · ⊕ Uiε (ω) , i ∈ {1, . . . , pε } ε Vi (ω) := {0} , i = pε + 1 ε , entailing the ﬂag for any ω ∈ Ω ε {0} ≡ Vp+1 (ω) ⊂ Vpε (ω) ⊂ · · · ⊂ V1ε (ω) = Kn .

Then the Lyapunov exponent λε (ω, z) := lim

t→∞

1 log | Φε (t, ω)z | t

24

1 Linear diﬀerential systems with parameter excitation

ε and z ∈ Kn \ {0} and it holds that exists for all ω ∈ Ω ε z ∈ Viε (ω) \ Vi+1 (ω) ⇐⇒ λε (ω, z) = λεi .

The latter fact is equivalent to the following characterization, Viε (ω) = z ∈ Kn : λε (ω, z) ≤ λεi . ˜ ε and t ∈ R+ it holds that For all ω ∈ Ω λε (θ(t)ω, Φε (t, ω)z) = λε (ω, z)

(z ∈ Kn \ {0})

and hence Φε (t, ω) Viε (ω) = Viε (θ(t)ω)

(i ∈ {1, . . . , pε }) .

The Multiplicative Ergodic Theorem has initially been proved by Oseledets [Os 68]. The one-sided version considered above is taken from Arnold [Ar 98, 3.4.1]. The fact that the proof there is presented for the case K = R is no restriction: According to Arnold [Ar 98, 3.4.10.(ii)], all arguments hold true for the complex case, K = C, as can be also read oﬀ from Ruelle’s [Ru 82] generalization to real or complex Hilbert spaces. The numbers Λε1 , Λε2 , . . . , Λεn are the possible exponential growth rates of ε Φ due to the above theorem and are called the Lyapunov exponents of Φε (under ρε ). Furthermore the proof of the Multiplicative Ergodic Theorem (via the Furstenberg-Kesten Theorem) and the above version (1.12) of the pathwise Jacobi (Liouville) equation yield together with the ergodic theorem that n i=1

ε

Λεi

≡

p

dεi λεi

i=1

1 log | det Φε (t, . ) | Pρε -a.s. t t

ε 1 trace A Xuε,ρ ( . ) du = lim t→∞ t 0 ≡ trace A(x) ρε (dx) Pρε -a.s. , = lim

t→∞

(1.13)

Rd

provided that A ∈ L1 (ρε ). This formula is called the trace formula for the sum of the Lyapunov exponents; see Arnold [Ar 98, 3.4.15, 3.3.11 & 3.3.4] and Oseledets [Os 68, p.203]. Further note that the previous condition A ∈ L1 (ρε )

(1.14)

1.3 The Multiplicative Ergodic Theorem: Lyapunov exponents

25

is suﬃcient for the integrability assumption β ± ∈ L1 (Pρε ) of the Multiplicative Ergodic Theorem, since by the deﬁning RDEs for Φε and (Φε )−1 , ε Φ (t, ω)±1 ≤ 1 +

t

ε A Xuε,ρ (ω) Φε (u, ω)±1 du

0

and hence by the general version of the Gronwall lemma (see e.g. Arnold [Ar 98, p.557]), t

ε ε,ρε Φ (t, ω)±1 ≤ exp A Xu (ω) du 0

which implies that ± β (ω) Pρε (dω) ≡

sup log+ Φε (t, ω)±1 Pρε (dω)

Ω 0≤t≤1

Ω

t

ε,ρε sup log exp A Xu (ω) du Pρε (dω)

≤

+

Ω 0≤t≤1 1

ε,ρε A Xu (ω) du Pρε (dω)

= Ω 1

=

0

0

1

= 0 = Rd

0

ε,ρε A Xu (ω) Pρε (dω) du Ω ε −1 (dx) du A(x) Pρε ◦ Xuε,ρ Rd

A(x) ρε (dx)

<∞ ; also see Arnold [Ar 98, 3.4.15 & 4.2.10]. At the end of this section we would like to comment on further research concerning Lyapunov exponents, notably the so-called FurstenbergKhasminskii formula and the associated law of large numbers which are due to Arnold et al. [Ar-Kl-Oe 86, Th.4.1]. This remark is in particular intended to underline the basic diﬀerence between their setting and the framework of our work: While Arnold et al. [Ar-Kl-Oe 86] make sure that the generator ¯ ∂ of (X ε , αε ) is hypoelliptic (where the notation is as in the L¯ε ≡ G ε + h ∂α ∂ is hypoelprevious section), it will be proposed here that the operator Lε + ∂t ¯ liptic: By assuming that h is “strongly hypoelliptic” on the sets of interest as will be made precise in deﬁnition 4.4.4, it will be made sure that (X ε , αε ) ˆ ε, α can “essentially” be replaced by a process called (X ˆ ε ) such that its cor∂ ε ˆ responding operator L + ∂t is hypoelliptic according to 3.1.3 (c); see the proofs of theorems 4.4.6 and 4.4.7. However, note that the hypoellipticity of ∂ is stronger than hypoellipticity of Lε ; see Arnold et al. [Ar-Kl-Oe 86, Lε + ∂t p.104]. The reason is basically that the hypoellipticity of Lε is related to the

26

1 Linear diﬀerential systems with parameter excitation

existence of a C ∞ density of the invariant measure (limiting distribution), ∂ , meaning “ellipticity” in the sense of Ichiwhile the hypoellipticity of Lε + ∂t hara and Kunita [Ic-Ku 74], assured that already the ﬁnite time transition probabilities have C ∞ densities; also see Arnold [Ar 84, p.794]. Remark 1.3.2. Again assume that (1.14) is satisﬁed and suppose that K = R, A is analytic and that σ( . ) = idRd for simplicity in addition to the previous assumptions. Using the spherical and angular coordinates as deﬁned in the previous section for Φε , one gets the processes ψ ε (t, ω, ψ) ≡

Φε (t, ω, z) | Φε (t, ω, z) |

starting in

ψ ε (0, ω, ψ) = ψ :=

z |z|

and in dimension n = 2, . αε (t, ω, α) = ψ ε (t, ω, ψ)

starting in

. α = ψ,

respectively. These systems can be regarded as random dynamical systems over (θ(t))t∈R+ in their own right; see Arnold [Ar 98, 6.2.1]. Now suppose that the operator Lε is hypoelliptic, i.e. that for any distribution v ≡ v(x) on Rd , it follows that v is a C ∞ function in every open subset ormander [H¨ o 67, p.149ﬀ.] has shown of Rd , where Lε v is a C ∞ function. H¨ that this is equivalent to proposing that

√ e1 √ ed b(x) dim LA , ε , ... , ε (x, ψ) = d + n − 1 h(x, ψ) 0 0 for all (x, ψ) ∈ Rd × P n−1 , where LA{V} denotes the Lie-algebra generated by the set {V} of vector ﬁelds, ei denotes the i-th canonical basis vector (column vector) of Rd and where the simplifying assumption that σ = idRd such that d ε ∂ ∂ + ∆ Lε = bi (x) + h(x, ψ) ∂x ∂ψ 2 i i=1 has also been used; further see Arnold [Ar 84, p.794]. Due to Arnold et al. [Ar-Kl-Oe 86, Prp.2.2 & Rem.2.2], this is equivalent to demanding that dim LA h(x, . ) : x ∈ Rd (ψ) = n − 1 for all ψ ∈ P n−1 . (1.15) If n = 2, this hypoellipticity condition (1.15) is equivalent to assuming that for any ψ ∈ P 1 , there is x ∈ Rd such that h(x, ψ) does not vanish; see Arnold et al. [Ar-Kl-Oe 86, Prp.2.1] and Pardoux and Wihstutz [Pd-Wh 88, p.444] and [Pd-Wh 92, p.291]. As Arnold et al. [Ar-Kl-Oe 86, Cor.3.1] further show under the above hypoellipticity assumption (1.15), the Markov process (X ε , ψ ε ) has a unique stationary (invariant with respect to Markov transition probabilities) probability distribution µε on Rd × P n−1 whose Rd -marginal is ρε ; µε has

1.3 The Multiplicative Ergodic Theorem: Lyapunov exponents

27

support Rd × C, where C is an “invariant control set” of a certain associated control problem; we do not go further into details but refer to Arnold et al. [Ar-Kl-Oe 86, p.88-95] instead. The stationary measure µε has a density mε which satisﬁes the forward equation (Fokker-Planck equation) (Lε )∗ mε = 0, where (Lε )∗ denotes the formal adjoint operator of Lε . Using the formula (1.5) and the ergodic theorem one gets the following Furstenberg-Khasminskii formula ε λ := Q(x, ψ) µε (dx, dψ) Rd ×P n−1

for the Lyapunov exponent (exponential growth rate) of the unique stationary process (X ε , ψ ε ). Furthermore, λε coincides with the top Lyapunov exponent from the Multiplicative Ergodic Theorem, λε = λε1 ≡ Λε1 , and the following law of large numbers holds true: For all z ∈ Rn \ {0}, λε (ω, z) ≡ lim

t→∞

1 log | Φε (t, ω)z | = λε t

for Pρε -almost all ω .

(1.16)

For the proofs of these results see Arnold et al. [Ar-Kl-Oe 86, Th.4.1]. For n = 2, the above assertions can also be stated in terms of the angle process αε (t, ω, α): The hypoellipticity condition (1.15) is met if and only if ¯ α) does not vanish. In this case the for any α ∈ [0, π) at least one vector h(x, ε ε ¯ε above reasoning implies that (X , α ) has a unique stationary measure µ ∞ ε ε ∗ ε ¯ with a C density m ¯ satisfying the Fokker-Planck-equation (L ) m ¯ = 0, as well as the periodicity constraint m ¯ ε (x, 0) = m ¯ ε (x, π)

for all x ∈ Rd .

Unfortunately no explicit formula for m ¯ ε is known to us. Its Rd -marginal is ε ρ and the Furstenberg-Khasminskii formula for the almost surely observed top Lyapunov exponent now reads 1 t ¯ ε ε ε ¯ α) m Q(Xu , αu )du = Q(x, ¯ ε (x, α) dxdα , (1.17) λ = lim t→∞ t Rd ×[0,π) 0 ¯ had been deﬁned in the previous section. where Q

Remark 1.3.3. Our goal in this paper is to obtain local Lyapunov characteristic numbers, i.e. growth numbers for Ztε for certain time scales t ≤ T (ε). For this aim neither the MET nor the Furstenberg-Khasminskii formula are applicable, since these theorems cover the case that the time t tends to ∞ for ﬁxed ε and hence the respective invariant measures ρε and µε arise which precisely describe this asymptotic behavior. However, our rationale here is not to observe the asymptotic behavior as t → ∞, but “before”, i.e. on shorter time scales T (ε). Therefore we need to use an argument which reproduces the

28

1 Linear diﬀerential systems with parameter excitation

“limit-matrices” A(Ki ) on the time scales on which the “metastable” state Ki is observed; see deﬁnition 2.5.4 for the precise explanation of a metastable state corresponding to a time scale. For that purpose we next recall, what happens in the deterministic case (ε = 0) dzt = A(t) zt , dt t→∞

if A(t) has a limit matrix A, i.e. A(t) −−−→ A. The easiest way would be to obtain a closed form for the dynamical system zt and to manipulate this explicit expression; e.g. zt = eAt z0 for the constant case A(t) ≡ A. However, this does not work, since even in the general deterministic case, if A( . ) is nonconstant, the formula for the propagator, involving the time-order-operator, seems too bulky for further manipulations. Hence we investigate the dynamics of the absolute value t ≡ |zt | and relate it to the Jordan decomposition of the limit matrix.

1.4 The deterministic case: Lyapunov exponents for asymptotically constant linear systems This section is dedicated to discussing the deterministic case ε = 0 in equation (1), dZt0 = A Xt0 Zt0 dt dXt0 = b Xt0 dt . In the situations we are interested in, the drift vector ﬁeld b conﬁnes the deterministic system X 0,x to converge to a certain attracting point Ki , t→∞

→ Ki , Xt0,x −−−− where i ∈ {1, . . . , l} denotes an index determined by the initial value x; see assumption 2.1.1(K). Since A( . ) is assumed to be continuous in (1), it follows that t→∞

A(t) := A(Xt0,x ) −−−− → A(Ki ) ; in other words, the (deterministic) coeﬃcient matrix A(t) is asymptotically constant. In the sequel, we will consider general asymptotically constant, linear, deterministic diﬀerential systems z˙t = A(t) zt which we write as dzt = A(t) zt dt

(1.18)

1.4 The deterministic case: Asymptotically constant linear systems

29

again, where t → A(t) is a continuous function taking its values in Kn×n , the n × n matrices with real or complex entries and zt ∈ Kn . Assuming that the system (1.18) is asymptotically constant as deﬁned above, i.e. that there is a ﬁxed matrix A such that t→∞

A(t) −−−−→ A (where Kn×n is equipped with the operator norm . ), one can rewrite (1.18) as (1.19) dzt = [ A + G(t) ] zt dt , where of course

t→∞

G(t) := A(t) − A −−−−→ 0 . In this deterministic case (1.19), the statements to be discussed in the sequel are due to Hartman and Wintner [Ha-Wi 55] and Perron [Pe 29]; the reference underlying the exposition here is Coppel [Cp 65, Ch.IV]. As one expects from the well-known case of linear deterministic systems with constant coeﬃcients, z˙t = A zt , the Lyapunov exponents will turn out to be the real parts of the eigenvalues of A and the proof uses the Jordan decomposition of A. However, a closed form of eAt only exists in the case of constant coeﬃcients, G(t) ≡ 0. The theory of asymptotic integration by Hartman and Wintner then treats the general situation. This will then motivate our general rationale in the stochastic case for the system (1): The “local” Lyapunov exponents (if existing as stochastic limits) should be obtained as the real parts of the eigenvalues of a certain “sublimit” matrix Aµ(x,ζ) , where the deviation G( . ) from Aµ(x,ζ) gets small in a stochastic sense; see proposition 4.1.1. The deterministic considerations here shall be started by recalling some facts concerning the simplest possible case of a linear system with constant coeﬃcients, z˙t = A zt . Remark 1.4.1 (Decomposition of the state space). For the matrix A ∈ Rn×n (or Cn×n ) and a given λ ∈ R, the state space Kn (Rn or Cn ), on which the linear transformation A acts, can be uniquely decomposed into a direct sum Kn = E<λ ⊕ Eλ ⊕ E>λ such that the three subspaces E<λ , Eλ and E>λ are invariant under A and the eigenvalues (i.e. the complex roots of the respective characteristic polynomial) of A restricted to these spaces have real parts less than, equal to or greater than λ, respectively. Namely, let D<λ , Dλ and D>λ denote open domains in C including precisely the respective eigenvalues; if the respective domain is nonempty, then we assume that its boundary is rectiﬁable; the subspaces

30

1 Linear diﬀerential systems with parameter excitation

E<λ , Eλ and E>λ are then given as the images of the corresponding Riesz projections P<λ , Pλ and P>λ deﬁned by 1 −1 ( χ In − A ) dχ , ν ∈ <λ ; λ ; >λ , Pν := 2πi ∂Dν where In denotes the n×n unit matrix; Pν is deﬁned to be the zero projection, if Dν is empty. In particular, for the linear ordinary diﬀerential equation with constant coeﬃcients z˙t = A zt , or dzt = A zt dt , one recovers the (Oseledets-)splitting Kn = E1 ⊕ · · · ⊕ Ep and the Lyapunov spectrum λ1 > λ2 > · · · > λp consisting of the distinct real parts of eigenvalues of A, where Ei is the sum of the generalized eigenspaces to eigenvalues with real part equal to λi and is dynamically characterized as lim

t→±∞

1 log | zt | = λi ⇐⇒ z0 ∈ Ei \ {0} . t

In this terminology, the above spaces E<λ , Eλ and E>λ are given as the direct sums of the corresponding spaces E1 , . . . , Ep . The solution ﬂow can also be described in the Riesz calculus as 1 ( χ In − A )−1 etχ dχ , etA = 2πi ∂D where the interior of the domain D now contains all eigenvalues of A and ∂D is supposed to be rectiﬁable. For the above statements see e.g. Coppel [Cp 65, Sec. II.1 & III.2], Riesz and Sz¨ okefalvi-Nagy [Ri-Na 55, Ch.XI], Kato [Kt 80, §I.5] and Arnold [Ar 98, 3.2.3]. Now we discuss the result by Hartman and Wintner [Ha-Wi 55] and Perron [Pe 29]. As the above cited Oseledets-splitting and the corresponding Lyat punov spectrum for the constant coeﬃcient diﬀerential equation dz dt = A zt is derived from the Jordan canonical form of the matrix A, it does not come as a surprise that the same strategy is used for the perturbed equation (1.19). The following lemma will turn out to be crucial in this argumentation. In the sequel let t0 be an arbitrary initial time of the system and let . denote the operator-norm. Furthermore, d/dt and the dot “ · ” will be used interchangeably for the derivative with respect to the “time” variable t.

1.4 The deterministic case: Asymptotically constant linear systems

31

Lemma 1.4.2 (Riccati-type diﬀerential inequality). Let v C 1 ([t0 , ∞), R) satisfy the diﬀerential inequality v(t) ˙ ≥ β v(t) − γ(t) (t > t0 ) ,

∈

where γ ∈ C 0 ([t0 , ∞), R+ ) fulﬁlls

t+1

t→∞

γ(u) du −−−−→ 0 t

and β is a real-valued function; let v ∗ ∈ R be a subsequential limit point of v and assume that β is continuous at v ∗ . Then β(v ∗ ) ≤ 0 . In particular, if v satisﬁes the Riccati-type diﬀerential inequality v(t) ˙ ≥ b v(t) 1 − v(t) − γ(t) (t > t0 ) , where b > 0 is constant and γ is as above, then either lim sup v(t) ≤ 0

or

t→∞

lim inf v(t) ≥ 1 . t→∞

Proof. 1) In order to obtain a contradiction we assume that β(v ∗ ) > 0. As v ∗ is a continuity point of β, this implies that one can choose constants η1 , η2 > 0 such that β(v) ≥ η1 > 0

v ∈ Bη2 (v ∗ ) ,

for all

where the latter set Bη2 (v ∗ ) denotes the closed ball with center v ∗ and radius η2 . t+1

Due to the convergence assumption on t γ(u)du , one can ﬁx a positive constant η3 < min(η1 , η2 ) and a time t1 ≥ t0 such that

t+1

γ(u) du ≤ η3

(t ≥ t1 ) ;

t

in particular, this implies that

t+T

γ(u) du ≤ (T + 1) η3

(t ≥ t1 , T > 0) .

t

Let [τ, τ + T ] denote some time interval such that τ ≥ t1 , T > 0 and v(t) ∈ Bη2 (v ∗ )

for all

t ∈ [τ, τ + T ] .

32

1 Linear diﬀerential systems with parameter excitation

Applying the proposed diﬀerential inequality v(t) ˙ ≥ β(v(t)) − γ(t) over this τ +• time interval [τ, τ + T ] and using the previous estimate of τ γ(u) du now yields that τ +s τ +s v(τ + s) − v(τ ) = v(u) ˙ du ≥ β v(u) − γ(u) du τ

τ

≥ η1 s − (s + 1) η3 > η3 s − (s + 1) η3

(1.20)

= − η3 for all s ∈ [0, T ]. Next we use this inequality (1.20) to prove that t1 can be enlarged such that also for all t ≥ t1 ; |v(t) − v ∗ | < η2 for this purpose let [ τ, τ + T ] denote some interval such that τ ≥ t1 , T > 0 and ( τ ≤ t ≤ τ + T) ; |v(t) − v ∗ | < η2 − η3 as v ∗ is a subsequential limit of t → v(t), such time intervals exist for any τ) ∈ previous choice of t1 ; due to the above reasoning (1.20) and since v( [v ∗ − (η2 − η3 ), v ∗ + (η2 − η3 )], v ( τ + s )

(1.20)

>

v( τ ) − η3 ≥ v ∗ − η2

(0 ≤ s ≤ T)

so that t → v(t) cannot exit the interval [v ∗ − η2 , v ∗ + η2 ] via the lower boundary; on the other hand, the same reasoning, since also v( τ + T) ∈ ∗ ∗ [v − (η2 − η3 ), v + (η2 − η3 )], yields that v ( τ)

(1.20)

<

v τ + T + η3 ≤ v ∗ + η2

so that t → v(t) cannot enter the interval [v ∗ − η2 , v ∗ + η2 ] via the upper boundary; this proves altogether that t1 can be chosen large enough such that for all t ≥ t1 , |v(t) − v ∗ | < η2 as had been claimed above. In particular, (1.20) can be applied for any τ ≥ t1 and s := T > 0, hence implying that T →∞

v(τ + T ) ≥ [v(τ ) − η3 ] + (η1 − η3 ) T −−−−→ ∞ in contradiction to the previously proven boundedness |v(t) − v ∗ | < η2 for large t.

1.4 The deterministic case: Asymptotically constant linear systems

33

2) Now we consider the Riccati-part of the statement, i.e. the case that β(v) := b v (1 − v) . Since b > 0, β is a downward sloped quadratic function which is strictly positive on (0, 1), strictly negative on R \ [0, 1] and vanishes at 0 and 1. Now let v ∗ be some subsequential limit point of v. Then the ﬁrst part of the lemma, shown above, yields that β(v ∗ ) ≤ 0 which — according to the special choice of β — is only possible for v ∗ ∈ / (0, 1). But this is just a reformulation of the above claim. Theorem 1.4.3 (Hartman-Wintner-Perron). Let K denote the real or complex number ﬁeld, K = R or C, and consider the linear ODE dzt = [ A + G(t) ] zt dt

(1.19)

in Kn , where A ∈ Kn×n is constant and the continuous map G : [t0 , ∞) → Kn×n satisﬁes t+1 t→∞ G(u) du −−−−→ 0 . (1.21) t

Then either zt = 0 for all large t or the Lyapunov exponent of zt exists and is equal to the real part of one of the eigenvalues Λj of A, i.e.3 : λ(zt ) := lim

t→∞

1 log | zt | ∈ { −∞ , Re(Λ1 ) , . . . , Re(Λn ) } . t

Proof. We ﬁrst choose an enumeration of the eigenvalues such that Re(Λ1 ) ≥ Re(Λ2 ) ≥ · · · ≥ Re(Λn ) . After a constant, invertible coordinate transformation we can assume that A is given in (complex) Jordan canonical form ⎛ ⎞ Λ1 0 0 ⎜ a Λ2 0 ⎟ ⎜ ⎟ ⎜ 0 a Λ3 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ Λ 4 A = ⎜ ⎟ ⎜ ⎟ .. ⎜ ⎟ . ⎜ ⎟ ⎜ ⎟ ⎝ Λn−1 0 ⎠ a Λn 3 Here we adopt to the common convention that the Lyapunov exponent of the trivial solution zt = 0 is deﬁned as λ(0) := −∞. Furthermore, Re( . ) denotes the real part operation as usual.

34

1 Linear diﬀerential systems with parameter excitation

where in this example Λ1 = Λ2 = Λ3 and Λn−1 = Λn , so that the boxes depict lower Jordan blocks of the following qualitative shape: ⎞ ⎛ Λ 0 ⎟ ⎜ a Λ 0 ⎟ ⎜ a Λ 0 ⎟ ⎜ ⎟ ⎜ a . . . ⎟ ; ⎜ .. .. .. J := ⎜ ⎟ ⎟ ⎜ a Λ 0 ⎟ ⎜ ⎝ a Λ 0 ⎠ a Λ here, a > 0 is some positive parameter and all matrix entries not mentioned are zero. Note that we have taken the “ordinary” lower Jordan blocks J 1 and carried out another transformation z˜t := Czt , where C := diag(1, a, . . . , an−1 ) ; then d d z˜t = C zt = CAzt + CG(t)zt ≡ CAC −1 z˜t + CG(t)C −1 z˜t ; dt dt hence, we can redeﬁne A as CAC −1 and G(t) as CG(t)C −1 ; after this transformation the Jordan-blocks of A are of the form Jja and G(t) still enjoys the prescribed assumption. The possible limits of 1t log |zt | are invariant under such invertible transformation of variables (or more generally under so-called Lyapunov-transformations). More precisely, the norm |C . | on Kn is equivalent to the “old” norm | . | as Kn is ﬁnite dimensional; since t → ∞, factors from converting these norms into each other, end up as vanishing summands; therefore, we can redeﬁne zt as z˜t for the following considerations without loss of generality. d zt = [ A + G(t) ] zt , is of the form Hence, the underlying ODE (1.19), dt d 1 z = Λ1 z 1 + (Gz)1 dt d i z = ai−1 z i−1 + Λi z i + (Gz)i dt

(i = 2, . . . , n) ,

where the superscript i indicates the i-th coordinate and ai := Ai+1,i ∈ {0, a}

(i = 1, . . . , n − 1)

denotes the n − 1 entries of A below the diagonal. Putting a0 := 0 as well as z 0 := 0 for deﬁniteness and deﬁning

1.4 The deterministic case: Asymptotically constant linear systems

35

it := zti , this implies that

d i i d i 2 d i i d i i z¯ z = z z + z¯ z = dt dt dt dt

d i z = 2 Re z¯i dt i = 2 Re z¯ ai−1 z i−1 + Λi z i + (Gz)i = 2ai−1 Re(¯ z i z i−1 ) + 2 Re(Λi ) ( i )2 + 2 Re(¯ z i (Gz)i )

(1.22)

for all i = 1, . . . , n , where “ ¯ ” denotes complex conjugation; hence, we can estimate d i 2 i 2 z i z i−1 ) + 2 Re(¯ z i (Gz)i ) dt − 2 Re(Λi ) = 2ai−1 Re(¯ (1.23) i i−1 i i ≤ 2ai−1 + 2 |(Gz) | . Now deﬁne λ1 > λ2 > · · · > λp as the distinct numbers in the list Re(Λ1 ) ≥ Re(Λ2 ) ≥ · · · ≥ Re(Λn ) . Note in particular that by deﬁnition, for all k = 1, . . . , p i := min{ i : Re(Λi ) = λk }

implies that

ai−1 = 0 .

(1.24)

We now deﬁne auxiliary processes describing the norms of the projection of zt onto the subspaces belonging to the diﬀerent real parts of eigenvalues; more precisely: Lkt := ( it )2 , k = 1, . . . , p {i : Re(Λi )=λk }

Mtk :=

Ntk

⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

( it )2

,k = 1 , k = 2, . . . , p

{i : Re(Λi )>λk }

⎪ n ⎪ ⎪ ⎪ ⎪ ( it )2 ≡ | zt |2 ⎩ ⎧ ⎪ ⎨

:=

i=1

( it )2

,k = p + 1 , k = 1, . . . , p

{i : Re(Λi )≤λk }

⎪ ⎩0

,k = p + 1 .

36

1 Linear diﬀerential systems with parameter excitation

Then | zt |2 = Mtk + Ntk = L1t + · · · + Lpt . Now it follows that k dL k dt − 2λk L + , i 2 d( ) = − 2λk ( i )2 dt {i : Re(Λi )=λk } ≤2 ai−1 i i−1 + 2 {i : Re(Λi )=λk }

⎛

⎞1/2 ⎛

≤2⎝

( i )2 ⎠

{i : Re(Λi )=λk }

⎛

i |(Gz)i |

{i : Re(Λi )=λk }

⎝

⎞1/2

a2i−1 ( i−1 )2 ⎠

{i : Re(Λi )=λk }

⎞1/2 .1/2 n i 2⎠ i 2 ( ) |(Gz) |

{i : Re(Λi )=λk }

i=1

+ 2⎝ ⎛

⎞ 1/2

= 2 (Lk )1/2 ⎝

a2i−1 ( i−1 )2 ⎠ + 2 (Lk )1/2 |(Gz)|

{i : Re(Λi )=λk , i−1 ≥ i }

⎛

≤ 2 a (Lk )1/2 ⎝

(1.25)

⎞1/2

( i )2 ⎠

+ 2 (Lk )1/2 G |z|

{i : Re(Λi )=λk }

1/2 = 2 a Lk + 2 G (Lk )1/2 M k + N k . / Mk + Nk k 2 a + 2 G =L Lk

(1.26)

for all k = 1, . . . , p , where we used in consecutive order the deﬁnition of Lkt , the calculation (1.23), the Cauchy-Schwarz inequality, (1.24), the fact that ai ∈ {0, a} and (1.25). Similarly, it follows from the deﬁnition of Mtk , (1.22), the estimate that Re(a b) ≥ −|a| |b| for complex numbers a and b, (1.24) and (1.25) that d Mk = dt =

{i : Re(Λi )>λk }

{i:Re(Λi )>λk }

0

d i 2 ( ) dt 1 2ai−1 Re(¯ z i z i−1 ) + 2 Re(Λi ) ( i )2 + 2 Re z¯i (Gz)i

1.4 The deterministic case: Asymptotically constant linear systems

≥ −2

ai−1 i i−1 + 2λk−1

{i : Re(Λi )>λk }

( i )2

{i : Re(Λi )>λk }

− 2

37

| (Gz) | i

i

{i : Re(Λi )>λk }

≥ − 2 a M k + 2 λk−1 M k − 2 G (M k )1/2 |z| √ = 2 (λk−1 − a) M k − 2 G M k M k + N k

(1.27)

for all k = 2, . . . , p + 1. In the same manner it follows from the deﬁnition of Ntk , (1.22), the estimate that Re(a b) ≤ +|a| |b| for complex numbers a and b, (1.24) and (1.25) that d Nk = dt

{i : Re(Λi )≤λk }

=

d i 2 ( ) dt

2ai−1 Re(¯ z i z i−1 ) + 2 Re(Λi ) ( i )2 + 2 Re(¯ z i (Gz)i )

{i:Re(Λi )≤λk }

≤2

{i : Re(Λi )≤λk }

+ 2

ai−1 i i−1 + 2λk

( i )2

{i : Re(Λi )≤λk }

| (Gz) | i

i

{i : Re(Λi )≤λk }

≤ 2 a N k + 2 λk N k + 2 G (N k )1/2 |z| √ = 2 (λk + a) N k + 2 G N k M k + N k .

(1.28)

for all k = 1, . . . , p. From (1.27) and (1.28) one gets for |z|2 ≡ M p+1 ≡ N 1 that d |zt |2 ≤ 2 λ1 + a + G(t) |zt |2 , 2 λp − a − G(t) |zt |2 ≤ dt so that integration over t ≥ t0 yields that t G(s) ds |zt0 | ≤ |zt | exp (λp − a)(t − t0 ) − t0

t ≤ exp (λ1 + a)(t − t0 ) + G(s) ds |zt0 | . t0

This proves again (besides the invertibility of the fundamental matrix solution) that if zt1 = 0 for some time t1 , then also zt = 0 for all times t ≥ t1 . Furthermore, due to the standing assumption (1.21) it follows that 1 t

t

t→∞

G(s) ds −−−−→ 0 t0

38

1 Linear diﬀerential systems with parameter excitation

and hence the previous string of inequalities implies the statement about the Lyapunov exponent for a non-vanishing solution in case p = 1: 1 t→∞ log | zt | −−−−→ λ1 t since a can be chosen arbitrarily small. Thus it is left to show the claim concerning the Lyapunov exponents in case p ≥ 2; again a > 0 serves as the small parameter and since p ≥ 2, we can choose it such that 2a <

min

k=1,...,p−1

λk − λk+1 .

(1.29)

One needs to ﬁnd for each non-vanishing solution zt an index J ∈ {1, . . . , p} such that lim

t→∞

1 log | zt | = λJ . t

(1.30)

Having excluded the case that z vanishes after some time, we can consider the auxiliary processes vtk :=

Mtk Mtk ≡ ∈ [0, 1] k 2 | zt | Mt + Ntk

for k = 1, . . . , p + 1; in particular v 1 ≡ 0 and v p+1 ≡ 1. These processes satisfy N k M˙ k − M k N˙ k v˙ k = , (M k + N k )2 where · denotes the derivative with respect to time, vk ( 1 − vk ) =

d dt ,

as usual; furthermore,

Mk Nk ; (M k + N k )2

since for the positive quantities M k , N k also the trivial manipulation √ √ (N k M k + M k N k )2 ≤ 2(N k )2 M k + 2(M k )2 N k ≤ 3(N k )2 M k + 3(M k )2 N k + (M k )3 + (N k )3 = (M k + N k )3 holds true, we get together with (1.27) and (1.28) that v˙ k =

N k M˙ k − M k N˙ k (M k + N k )2

1.4 The deterministic case: Asymptotically constant linear systems

39

0 1 √ 1 k k k k + Nk N 2(λ − a)M − 2 G M M k−1 (M k + N k )2 0 1 √ − M k 2 ( λk + a ) N k + 2 G N k M k + N k 1 2 M k N k (λk−1 − λk − 2a) = (M k + N k )2 √ √ − 2 G M k + N k N k M k + M k N k ≥

≥

1 2 M k N k (λk−1 − λk − 2a) − 2 G (M k + N k )2 (M k + N k )2

= 2 (λk−1 − λk − 2a) v k ( 1 − v k ) − 2 G . Hence, lemma 1.4.2 applies to the Riccati-type diﬀerential inequality v˙ ≥ b v 1 − v − γ , with v := v k , b := 2 (λk−1 − λk − 2a) > 0 due to (1.29) and γ := 2 G due to (1.21). Therefore, as v ≡ v k takes its values in [0, 1], lim vtk ∈ {0, 1}

t→∞

(k = 1, . . . , p + 1) .

Since v 1 ≡ 0 and v p+1 ≡ 1, the following quantity is thus well-deﬁned: J := max k = 1, . . . , p : lim vtk = 0 . t→∞

This implies, as M k+1 = L1 + · · · + Lk , that for k = 1, . . . , p Mtk+1 − Mtk Lkt 0 , k = J ; k+1 k t→∞ = = vt − vt −−−−→ 1 , k = J . |zt |2 |zt |2

(1.31)

In particular one also gets Lkt | zt |2 t→∞ Lkt = −−−−→ 0 J | zt |2 LJ Lt t

(k = J ).

The importance about (1.31) is that it reduces the study of |zt | to large t, namely 2 log

LJ t − log |zt | =

(1.32) 2 LJ t for

LJ t→∞ 1 log t 2 −−−−→ 0 2 |zt |

and hence the claim (1.30) concerning the Lyapunov exponent λ(zt ) reduces to proving that lim

t→∞

1 log LJ t = 2 λJ . t

40

1 Linear diﬀerential systems with parameter excitation

Considering (1.26) for k = J after dividing by LJ t (= 0) now yields: t

1 1 d J J log Lu − 2 λJ du lim sup log Lt − 2λJ = lim sup t t du t→∞ t→∞ t0 t 1 d log LJ ≤ lim sup − 2λ J du u t→∞ t t0 du 3 (1.26) 1 t MuJ + NuJ ≤ 2a + 2 lim sup G(u) du LJ t→∞ t t0 u t (1.31) 1 ≤ 2 a + 4 lim sup G(u) du t t0 t→∞ (1.21)

= 2a.

Since a can be chosen arbitrarily small, this shows that the Lyapunov exponent λ(zt ) exists and is equal to λJ . Remark 1.4.4 (Asymptotic growth of the projections). Due to the structure of the diﬀerential equation in Jordan canonical form in the proof of theorem 1.4.3, it follows in particular from remark 1.4.1 that for any k = 1, . . . , p 2 Lkt = Pλk zt . Thus (1.32) implies that by choosing λ := λJ ≡ λ(zt ), the Lyapunov exponent of one particular solution zt , one gets that P≶λ zt t→∞ −−−−→ 0 . | Pλ zt | Remark 1.4.5. The proof of the above theorem 1.4.3 shows that the linearity of the perturbation G(t)z is not crucial. The argument also goes through unchanged for the nonlinearly perturbed equation dzt = [ A zt + g(t, zt ) ] dt , where the continuous map g : [t0 , ∞) × Rn → Rn satisﬁes | g(t, z) | ≤ γ(t) |z| together with

t+1

t→∞

γ(u) du −−−−→ 0 . t

1.4 The deterministic case: Asymptotically constant linear systems

41

Remark 1.4.6 (Kn = R2 ). As in theorem 1.4.3 again consider the linear ODE (1.19) dzt = [ A + G(t) ] zt dt under the same assumptions as imposed there; let Kn = R2 for simplicity. The exponential growth rate limt→∞ 1t log | zt | has been calculated as Re(Λ1 ) or Re(Λ2 ) in the non-trivial cases zt ≡ 0. Despite these numerically exact results, the formulas (1.10) or (1.11), 1 log | z0 | 1 log | zt | = + t t t

t

¯ αu ) du , Q(u, 0

which evaluate the growth rate of zt by a functional of its angle αt , have not been used in the course of the proof. It is the purpose of this remark to bridge this gap in the illustrative two-dimensional situation. If p = 1, i.e. if Re(Λ1 ) = Re(Λ2 ), we get in the notion of the previous proof (see p.35) that the norms of the relevant projections are L1t = Mt2 = Nt1 = |zt |2 and Mt1 = Nt2 = 0; the proof then estimates |zt |2 and does not contain a conclusion concerning the angle. If p = 2, i.e. if Re(Λ1 ) > Re(Λ2 ), the situation changes. Both eigenvalues are real, λ1 := Λ1 > Λ2 =: λ2 , each of which has an eigenvector. After the transformation mentioned at the beginning of the previous proof,

λ1 0 A = 0 λ2 and the canonical unit vector ei is the eigenvector corresponding to λi , i = 1, 2. Then it follows from the notion of p.35 that Lkt = ( kt )2 ⎧ ⎪ ⎨0 k Mt = ( 1t )2 ⎪ ⎩4 2

, k ∈ {1, 2} ,

i 2 2 i=1 ( t ) ≡ | zt | ⎧4 2 i 2 2 ⎪ ⎨ i=1 ( t ) ≡ | zt | Ntk = ( 2t )2 ⎪ ⎩ 0

,k = 1 ,k = 2 ,k = 3 , ,k = 1 ,k = 2 ,k = 3 ,

where as before it ≡ | zti |. The auxiliary processes v 1 , v 2 and v 3 as deﬁned M1 M3 on p.38 read in this case as vt1 = | ztt|2 ≡ 0 , vt3 = | ztt|2 ≡ 1 and vt := vt2 :=

Mt2 ( 1t )2 ≡ = (cos αt )2 , 1 | zt |2 ( t )2 + ( 2t )2

42

1 Linear diﬀerential systems with parameter excitation

.................. .. ......... ........ . t ............. .. . . . . . . . . . . . ......... ...... . . . . . . . . . . . ....... t .. ..... . . . . . . . . . .... .... ..... ... ........ ...... ..... ... ................ ........ ..... . .... .. . ... . ............. . . . . . ... .. . ... . . . . . . ... .. .. . . . . . ... .. .. . . . . . ... .. ... . . . . ... .. ... ... ... t .. ... ... .. ..... .. ... .. ....... .. . . .... ... ... ... .. . . .. ... . . ... ... . .. .. ... . . .. . . t . ... ... . .. ... .. ... .. 1 . . ... . . . t ... ... ... ... ... ... ... t ... .. . ... ... ... ... ... ... ... ... ... . . . ... .... .... ... ..... ..... ...... ..... ...... ...... . ....... . . . . ... ........ ........ .......... ................ .......... ...................................

e2 ............. .................r

z

s |z |

sin α

α

| cos α | √ =: v

r

e

Fig. 1.1 Geometrical interpretation of the quantity vt := (cos αt )2

where αt denotes the angle of zt as measured canonically with respect to the e1 -coordinate axis. As ﬁgure 1.1 illustrates, vt = cos2 αt quantiﬁes the distance between |zztt | and e2 and hence induces a metric on the projective space S 1 . Another popular choice to obtain a metric on S 1 is to work with | sin αt | which measures the distance between the projective lines |zztt | and e1 ; the latter metric is commonly used in the proof of the Multiplicative Ergodic Theorem; see Arnold [Ar 98, Prop. 3.2.8 & Lem. 3.4.6]. The system vt then satisﬁes the ODE d 1 2 d 2 2 ( 2t )2 dt ( t ) − ( 1t )2 dt ( t ) v˙ t = 4 |zt | from which the following Riccati-type diﬀerential inequality follows (see p.38f.), v˙ t ≥ 2 (λ1 − λ2 ) vt ( 1 − vt ) − 2 G(t) . Lemma 1.4.2 applies with b := 2 (λ1 − λ2 ) > 0 and γ := 2 G due to (1.21). Therefore, lim vt ∈ {0, 1} t→∞

and the following quantity is thus well-deﬁned, 1 , if limt→∞ vt = 1 , J := 2 , if limt→∞ vt = 0 .

1.4 The deterministic case: Asymptotically constant linear systems

43

. It displays the direction eJ to which the projective line ψt := |zztt | = αt converges; see ﬁgure 1.1; due to continuity the angle also converges, α∞ := limt→∞ αt ∈ {0, π2 , π, 3π 2 }. Therefore, it follows altogether that t→∞ ¯ ¯ A + G(t), αt ≡ Q A + G(t), ψt − α∞ ) −−− → Q A, eJ ≡ Q(A, Q and hence the formulas (1.10) and (1.11) imply that 1 log |z0 | 1 log |zt | = + t t t

t

t→∞ ¯ ¯ A + G(u), αu du − −−→ Q(A, α∞ ) ∈ {λ1 , λ2 }, Q

0

¯ (see (1.7)) as well as the diagonal where in the last step the deﬁnition of Q form of A have been used. Thus the gap between theorem 1.4.3 and the formulas (1.10) and (1.11) is closed. The following theorem shows that every real part of an eigenvalue can indeed be seen as a Lyapunov exponent. Since it will not be used in the sequel, the arguments will not be provided here. Instead, see Coppel [Cp 65, p.100–102] for the proof which consists of using remark 1.4.1, deﬁning an appropriate integral transformation and applying the ﬁxed point theorem by Schauder and Tychonoﬀ then. Theorem 1.4.7. Suppose that the assumptions of the Hartman-WintnerPerron theorem 1.4.3 hold, i.e. that we consider dzt = [ A + G(t) ] zt dt ,

(1.19)

where A is a constant matrix and G is a continuous matrix function again satisfying t+1 t→∞ G(u) du −−−−→ 0 . (1.21) t

Then for any real part λ of an eigenvalue Λi of A, λ ∈ { Re(Λ1 ) , . . . , Re(Λn ) } , there exists an η ≡ η(A) > 0 and a time T ≡ T (A, G) > t0 such that for all t1 ≥ T and for all χ<λ ∈ E<λ and χλ ∈ Eλ satisfying | χ<λ | < η | χλ | , the equation (1.19) has a unique solution (zt )t≥t1 such that P<λ zt1 = χ<λ , and λ(zt ) ≡ lim

t→∞

Pλ zt1 = χλ

1 log | zt | = λ . t

44

1 Linear diﬀerential systems with parameter excitation

Remark 1.4.8. As has been already mentioned, the results of this section are due to Hartman and Wintner [Ha-Wi 55]. There are many more papers on asymptotically constant linear ODEs of which we only mention Levinson [Lv 48] and Harris and Lutz [Harr-Lutz 77]; also see Eastham [Ea 89] and the references therein for an overview. There also exists a version of the Hartman-Wintner theorem for functional diﬀerential equations which is due to Pituk [Pu 99]. A vast amount of the literature is furthermore dedicated to the non-linear case, that is to considering ODEs x˙ t = f (x, t) , which are asymptotically autonomous; this means that the time-dependent vector ﬁeld f ( . , t) approaches an autonomous (time-independent) vector ﬁeld f ( . ) as t → ∞ in a certain sense; see e.g. Markus [Mar 56] and Strauss and Yorke [Stra-Yor 67].

1.5 Sample systems In this section several examples are presented which shall illustrate where linear, real noise driven stochastic systems (1) appear; these sample systems can be regarded as toy models for the diﬀerent situations described by the deﬁnitions and assumptions in chapter 4. Example 1.5.1 (Linearized SDEs with constant noise coeﬃcient σ). Let X ε be the diﬀusion given by the SDE √ dXtε = b (Xtε ) dt + ε σ dWt , where b ∈ C ∞ (Rd , Rd ), ε ≥ 0, and the Brownian motion W are as before and where σ ∈ GL(d, R) is now supposed to be a constant (invertible) matrix. Deﬁning the matrix-valued mapping A : Rd → Rd×d as the Jacobian of b,

∂ bi (x) A(x) := Db(x) := , ∂xj i,j=1,...,d yields the linearized (variational) equation dZtε = A (Xtε ) Ztε dt √ dXtε = b (Xtε ) dt + ε σ dWt

(1.33)

in the sense that the coeﬃcient functions of the SDE for X ε are linearized, i.e. √ diﬀerentiated with respect to the space variable x ; as the noise coeﬃcient ε σ does not depend on x by assumption, its derivative vanishes and thus there is no noise component of Z ε meaning that the resulting stochastic

1.5 Sample systems

45

system is indeed real noise driven; also see Arnold [Ar 88, p.194f.]. By taking derivatives it follows, in particular, that the dimensions of the state spaces of Z ε and X ε are equal, n = d. For further assertions (and the corresponding assumptions), concerning how Z ε can then be considered as the linearization of X ε itself, we would like to refer to Blagovescenskii and Freidlin [Bla-Fr 61], Khasminskii [Kh 80, Sec.V.6], Bismut [Bis 81] and Arnold [Ar 98, Sec.2.3] among others. By considering the linearized equation as above, it follows in other words that any system X ε to be investigated later (satisfying the general assumptions 2.1.1, more precisely) for which σ is constant serves as an example of (1) by taking the Jacobian A := Db. Since in most cases σ ≡ idRd for simplicity anyway, the sample models as will be discussed in section 2.6 already provide a whole class of sample systems for (1). To be speciﬁc two cases of linearized SDEs will be discussed in the following two examples. Example 1.5.2 (Multi-dimensional Ornstein-Uhlenbeck process). Consider the SDE √ dXtε = AXtε dt + ε σ dWt , where A and σ are constant elements of Rd×d , the latter being invertible in addition. This SDE is understood as random perturbation of the linear, deterministic dynamical system X˙ t0 = AXt0 . The linearization (1.33) of this SDE is dZtε = AZtε dt ,

Z0ε = z ∈ Rd ;

it does not depend on X ε and is therefore independent of ε, Ztε (ω, x, z) = Ztε (z) = Xt0,z ; also see Arnold [Ar 88, p.194]. In this case the Lyapunov exponents of the system (1.33) from the Multiplicative Ergodic Theorem 1.3.1 coincide with the Lyapunov exponents from 1.4.1 and are given by the real part of the eigenvalues of A. Since the stochasticity is not present after the linearization, it also amounts to call the real parts of the eigenvalues of A the local Lyapunov exponents of the system X ε . This system X ε will be further discussed in 2.6.1 under the assumption that A be normal and with the speciﬁcation σ = idRd ; these two conditions then allow to calculate a certain “cost” function (quasipotential). All in all this system X ε together with its linearization serves as a bridge between the stochastic case (i.e. the Multiplicative Ergodic Theorem 1.3.1) and the familiar deterministic fact 1.4.1, since the resulting Lyapunov exponents coincide and since it additionally incorporates a simple example for the FreidlinWentzell theory (with only one metastable state) which will come into play later; see 2.6.1. Example 1.5.3 (Linearized gradient SDE with constant σ. Potential U 1 ). Now consider the linearized SDE (1.33) in case that the drift is derived

46

1 Linear diﬀerential systems with parameter excitation

as the gradient of a function which is then called the potential (function) of the drift. More precisely, consider √ dXtε = − ∇U (Xtε ) dt + ε σ dWt , where the drift b := −∇U is given by U ∈ C ∞ (Rd , R), as for example in ﬁgure 1, and where again σ ∈ GL(d, R). An important special case is given by σ = idRd for which this SDE will be considered later as equation (2.2). By diﬀerentiation it follows that the linear component of the variational system (1.33) is given by dZtε = − HU Xtε Ztε dt , where the coeﬃcient matrix A(x) := −HU (x) is deﬁned via the Hesse matrix 2

∂ U (x) , HU (x) := ∂xi ∂xj i,j=1,...,d the symmetric matrix of second derivatives of U at x. Again note that n = d due to the linearization. For n = d = 2 the formulas as obtained in remark 1.2.1 read as follows: ¯ of the nonlinear real noise SDE (1.6), dαε = h(X ¯ ε , αε ) dt , for The drift h t t t ε ε the angle α of Z is ¯ α) ≡ h ¯ −H (x), α h(x, U

2 ∂2U ∂2U ∂ U 2 2 = sin α cos α ( sin α − cos α ) + − ∂x1 ∂x2 ∂x21 ∂x22

1 ∂ 2U ∂2U ∂2U sin 2α cos 2α + − =− ∂x1 ∂x2 2 ∂x21 ∂x22 for the drift of the angle process. According to (1.7) the depiction (1.8) of the growth rate, 1 1 t ¯ ε,x log | z | ε log | Zt (ω, x, z) | = + Q Xu (ω), αεu (ω, x, z) du , t t t 0 is determined by the kernel function ¯ α) ≡ Q ¯ −H (x), α Q(x, U =−

∂2U ∂2U ∂2U cos2 α − sin2 α − 2 sin α cos α . 2 2 ∂x1 ∂x2 ∂x1 ∂x2

The symmetry of the Hesse matrix implies that there are either two or inﬁnitely many switching curves of the angle drift ¯h; see 1.2.4. Note that the latter assertion is also a consequence of the spectral decomposition theorem for symmetric matrices.

1.5 Sample systems

47

In order to examine a speciﬁc numerical example consider the potential function U1 (x) :=

3 4 2 3 x − x31 − 3 x21 + c x1 x2 + x42 , 2 1 3 2

(1.34)

where c ∈ R is a constant. This function U1 is plotted in ﬁgure 2.1 for c = 1. Here, the coeﬃcient matrix of the linearized system is given by

−c − 18x21 + 4 x1 + 6 A1 (x) := − HU1 (x) = ; −c − 18 x22 furthermore, one gets ¯h(x, α) = c ( sin2 α − cos2 α ) + ( 18x2 − 18 x2 − 4 x1 − 6 ) sin α cos α 1 2 = − c cos 2α + ( 9x21 − 9 x22 − 2 x1 − 3 ) sin 2α . ¯ In particular, k π2 : k ∈ Z are zeros of h(x, . ) for any x, if c = 0. If c = 0, ¯ α) = 0 ; in particular, h ¯ is one can ﬁnd for any α an x ∈ R2 such that h(x, hypoelliptic in the sense of (1.15) then; also see deﬁnition 4.4.4. However, ¯h is not “strongly hypoelliptic” in the sense of deﬁnition 4.4.4 as will be discussed in remark 4.4.11. The integral kernel for the exponential growth rate in (1.8) is calculated as ¯ α) = ( − 18x2 + 4 x1 + 6 ) cos2 α − 18 x2 sin2 α − 2 c sin α cos α . Q(x, 1 2 Example 1.5.4 (Diagonal matrix plus small skew-symmetric perturbation). Consider the coeﬃcient matrix . . . 0 1 λ1 x λ1 0 2×2 +x = A : R→R , A(x) := 0 λ2 −x λ2 −1 0 for the system (1), where λ1 > λ2 . In particular, the dimensions are n = 2 and d = 1 in (1), the latter fact meaning that X ε enters as one-dimensional diﬀusion √ dXtε = − U (Xtε ) dt + ε dWt , where we used the fact that in dimension d = 1 any drift can be written as the gradient b = −U of a potential function by direct integration U (x) := x − 0 b(y)dy and where for simplicity σ := idR ≡ 1. For the system Z ε as deﬁned by (1) with the above coeﬃcient matrix function A it follows from remark 1.2.1 that its angle αεt follows the RDE (1.6), dαεt = ¯h(Xtε , αεt ) dt, with velocity

48

1 Linear diﬀerential systems with parameter excitation

¯ α) ≡ ¯ h(x, h A(x), α

= − x sin2 α − x cos2 α + λ2 − λ1 sin α cos α λ2 − λ1 sin 2α ; = −x + 2

hence, the switching curve can be written in the simple form x=

λ2 − λ1 sin 2α ; 2

¯ is sketched in this switching curve together with the tendencies of the drift h ﬁgure 1.2. The formula for the exponential growth rate (1.8) is determined by the integrand function (1.7) which reads ¯ α) ≡ Q(A(x), ¯ Q(x, α) 2 = λ1 cos α + λ2 sin2 α + (x − x) sin α cos α = λ1 + (λ2 − λ1 ) sin2 α here. For these facts also see Arnold [Ar 79, p.136f.].

α

6

3π/2

6

π π/2

?

π/4

6

1 (λ2 2

− λ1 )

6

?

1 (λ1 2

− λ2 ) x

−π/4 −π/2 −π

? ?

−3π/2

Fig. 1.2 The switching curve and tendencies of ¯ h(x, α) = − x +

1 (λ2 2

− λ1 ) sin 2α

1.5 Sample systems

49

The above matrix function . . . 0 1 λ1 x λ1 0 +x = A(x) ≡ 0 λ2 −x λ2 −1 0 has been investigated by Arnold [Ar 79] for Ornstein-Uhlenbeck noise as an example of an unstable (if λ1 > 0) system being stabilized by random parameter noise; also see Arnold and Kliemann [Ar-Kl 83, p.67f.]. However, here we are not interested in stabilization (as t → ∞), but in the behavior on time scales; more precisely, the parameter noise X ε will be assumed as small on the time scales under consideration which embodies that 0 is a metastable point of X ε ; in other words, 0 will be supposed to be a local of minimum 0 1 λ 0 U entailing that the skew-symmetric “perturbation” −1 0 of 01 λ is 2 small on the time scales on which X ε sojourns near 0. This terminology of metastability will be made precise in the subsequent chapter. The matrix function A of this example provides a toy model for the following investigations; more precisely, our ﬁndings of subsection 4.4.2 will turn out to be applicable to this model; see example 4.4.14. λ1 0 0 1 Furthermore, note that “superposing” the matrices and −1 0 0 λ2 is also a popular sample model for investigating the parameter dependence of Lyapunov exponents in the case of white noise; see Pardoux and Wihstutz [Pd-Wh 88, p.455f.] and [Pd-Wh 92, p.293]. After having discussed our main examples in 1.5.3 and 1.5.4 we would like to close this section by giving references for further sample systems without going into details any further. Example 1.5.5 (nonlinear, stochastic systems driven by real noise). Here, one considers nonlinear systems d yt = F (Xt , yt ) , dt where the motion of the vector yt ∈ Rn describes the evolution of the system under consideration and the other inﬂuences are modeled as perturbing real noise process X ≡ X ε , its state space being Rd for example; F is a drift function on the joint state space which is assumed to be diﬀerentiable. If O is an equilibrium of F (x, . ), i.e. if F (x, O) = 0 for all x, then linearization at O leads to the real noise system d Zt = A(Xt ) Zt , dt where A(x) := Dy F (x, y)|y=O , where Dy denotes the diﬀerentiation operator with respect to the y-variable; see e.g. Arnold and Kliemann [Ar-Kl 83,

50

1 Linear diﬀerential systems with parameter excitation

p.68]; such a system with dichotomous (“telegraphic”) noise, i.e. Xt being a Markov process with two states, has been considered by Arnold and Kloeden [Ar-Kd 89, p.1242f.]; the latter authors also point towards a real-world system in electrohydrodynamics: See Behn et al. [Bh-Lg-Jh 98] and M¨ uller and Behn [M-Bh 87] for details. Another physical example for such a linearized system with telegraphic noise is the LRC circuit described by Kats and Martynyuk [Ka-My 02, p.44 & p.20f.]; note that their linearized system only takes its values in the diagonal matrices. Such a situation will constitute the subject of our investigations in section 4.3. For applications e.g. to economics, we refer to Ruelle [Ru 88] and Weintraub [Wt 70] among others. Example 1.5.6 (Harmonic oscillator with real-noise input). Consider the random diﬀerential equation z¨ + 2 β + a2 Xt2 z˙ + 2 + a1 Xt1 z = 0 , where , β ∈ R and a1 , a2 ≥ 0 are constants and Xt1 , Xt2 are the components of a stochastic process X ≡ X ε ∈ R2 . Deﬁning - . - . Z1 z := , z˙ Z2 this equation can be rewritten as usual as - . 5. .6 - . . d Z1 0 1 0 0 0 0 Z1 1 2 = + a2 X t . + a1 X t dt Z2 −2 −2 β −1 0 0 −1 Z2 In this example a1 X 1 is considered as random restoring force, whereas a2 X 2 is thought of as random perturbation of the damping constant β. In case that X is a ﬁxed stochastic process, the related control theoretic analysis is undertaken for diﬀerent speciﬁcations of the parameters by Benderskii and Pastur [Be-Pt 75], Kliemann and R¨ umelin [Kl-Rm 81, p.18-24], Arnold and Kliemann [Ar-Kl 83, p.12 & p.63ﬀ.], Kliemann [Kl 88, p.91ﬀ.], Kliemann [Kl 80, p.120,158], Kliemann [Kl 79, p.468], Wihstutz [Wh 75], Arnold and Wihstutz [Ar-Wh 78] and R¨ umelin [Rm 78] and [Rm 79]; also see Pinsky and Wihstutz [Pi-Wh 91], Arnold et al. [Ar-Pp-Wh 86], Pinsky [Pi 86], Arnold [Ar 98, p.160] and Hern´ andez-Lerma [HL 79, p.39f.]. Applications to engineering systems are given by Griesbaum [Gb 99, p.22]; for the related modeling of the ship roll motion in particular of the capsizing of vessels see Colonius and Kliemann [Cu-Kl 00, p.497] and [Cu-Kl 97, p.137f.]. Benderskii [Be 69] considers the following oscillator with telegraphic noise, z¨ + pz˙ + (q − γ rt ) z = 0 ,

1.5 Sample systems

51

where p, q, γ > 0 are constants and (rt )t is a random process assuming the values ±1. The times at which (rt )t changes its sign form a Poisson process with a certain intensity. The case that X is a ﬁxed telegraphic noise process (i.e. a stationary, ergodic two-state Markov process) is treated by Arnold and Kloeden [Ar-Kd 89, p.1269f.]; more precisely, the latter authors investigate asymptotic formulas for a1 → 0 and a1 → ∞ for the case that β = a2 = 0; this case has also been considered by Leizarowitz [Lz 89] who assumes the real noise to be a ﬁnite-state Markov process. For white noise perturbations of the harmonic oscillator see Pardoux and Wihstutz [Pd-Wh 88, p.450] and Pinsky and Wihstutz [Pi-Wh 88] among others. Triangular matrices have been of interest, too: For explicit calculations on upper triangular matrices see Arnold [Ar 98, 3.3.13 & 3.4.16] and the references therein. Lower triangular matrices are e.g. considered by Kliemann [Kl 80, p.157] and Kliemann and R¨ umelin [Kl-Rm 81, p.15]. Brockett and Willems [Bro-Wil 72, p.253] propose a model for a closed loop dynamics with feedback interaction which leads to a linear real noise diﬀerential system.

Chapter 2

Locality and time scales of the underlying non-degenerate stochastic system: Freidlin-Wentzell theory

This chapter is devoted to substantiate the concept of locality (metastability, quasi-deterministic approximation) to be used. The framework here is set up by the SDE √ (t ≥ 0) , dXtε,x = b (Xtε,x ) dt + ε σ (Xtε,x ) dWt (2.1) ε,x d X0 = x ∈ R in Rd . Here, the coeﬃcients are functions b ∈ C ∞ (Rd , Rd ) and σ ∈ C ∞ (Rd , Rd×d ) and ε ≥ 0 parametrizes the intensity of (Wt )t≥0 which denotes a Brownian motion in Rd on1 a standard ﬁltered probability space (Ω, F , P, (Ft )t≥0 ). Hence, for any ε > 0 and x ∈ Rd , the stochastic process X ε,x solving (2.1) is a diﬀusion with drift b and covariance ε a, i.e. the generator of X ε is given by the diﬀerential operator G ε f :=

d i=1

bi

d ε ∂f ∂2f + aij ∂xi 2 i,j=1 ∂xi ∂xj

(f ∈ C 2 (Rd , R)) ,

where the positive semi-deﬁnite matrix a is deﬁned by a := σ σ ∗ . An important special case of equation (2.1) is given by the gradient SDE √ X0ε,x = x ∈ Rd , (2.2) dXtε,x = − ∇U (Xtε,x ) dt + ε dWt , where the drift b := −∇U is given by a potential U ∈ C ∞ (Rd , R), as for example in ﬁgure 1, and where for simplicity σ ≡ idRd ; also cf. example 1.5.3. The above equations are stochastic diﬀerential equations in the Itˆo sense. If σ is constant, as for example in (2.2), then (2.1) coincides with its Stratonovich version. However, this is not true in general; more precisely, 1

More precisely, W is adapted to the ﬁltration (Ft )t≥0 which is supposed to be standard, i.e. to satisfy “the usual conditions”, see e.g. Hackenbroch and Thalmaier [Hb-Th 94, 3.3.].

W. Siegert, Local Lyapunov Exponents. Lecture Notes in Mathematics 1963. c Springer-Verlag Berlin Heidelberg 2009

53

54

2 Locality and time scales of the underlying non-degenerate system

replacing the Itˆ o diﬀerential dWt in (2.1) by the Stratonovich integral ◦dWt , correction terms would need to be taken into account; this issue will be further commented on in remark 2.4.13. The behavior of the above SDEs (2.2) and (2.1) has been the subject of a tremendous amount of ongoing research. Its foundations have been laid out by Smoluchowski, Einstein, Langevin, Andronov et al. [And-Pon-Vi 33] and Kramers [Kr 40]; see e.g. also Chandrasekhar et al. [Cha-Kac-Smo 86]. The corresponding mathematical theory of large deviations is due to Freidlin and Wentzell; see e.g. [We-Fr 70] and [Fr-We 98]. In their work (2.1) describes the deterministic diﬀerential system dXt0 = b Xt0 dt

(t ≥ 0) ,

√ subject to the small (as ε → 0) random perturbations ε σ dWt ; furthermore, the exponential rate for large deviations of X ε from the deterministic trajectory X 0 is calculated. Freidlin and Wentzell also deduce that the long-time behavior of the stochastic system X ε,x has a deterministic component which can be described by a hierarchy of cycles in case that the drift b has ﬁnitely many stable attractors; for example, one can think of a potential function U for SDE (2.2) with ﬁnitely many wells as in ﬁgure 1. By virtue of this hierarchy of cycles one can assign to (Lebesgue almost-) every initial value x and each typical time scale eζ/ε a certain subset Kµ(x,ζ) of the state space Rd such that X ε,x spends most of the time until eζ/ε in a neighborhood of Kµ(x,ζ) . This set Kµ(x,ζ) will be called metastable with respect to the point x and the time scale eζ/ε . Another feature of this phenomenon is that the transition probability Px Xeεζ/ε ∈ . has diﬀerent limits for diﬀerent choices of the time scale and these limits do depend on the initial value x. This fact is expressed by saying that X ε,x has sublimiting distributions supported by the sets Kµ(x,ζ) . As already mentioned, these results are due to Freidlin and Wentzell (see [Fr 77], [Fr 00] and [Fr-We 98, § 6.6]). The existence of “metastable states” corresponding to initial conditions and time scales as described above is abbreviated by speaking of metastability or locality. Furthermore, the asymptotic behavior of X ε on the times scales will be described by basins of attraction, a hierarchy of cycles, metastable states, exit rates, rotation rates and so on. Although X ε is a stochastic process all the previous objects are non random. This technique of reducing the stochastic dynamics of the SDE (2.1) to a deterministic description is called quasi-deterministic approximation for the long-time behavior of the dynamical system perturbed by small noise (Freidlin [Fr 00]). Note that the terminology of “sublimiting distributions” emphasizes the asymptotic behavior on time scales eζ/ε . In contrast the limiting distribution,

2.1 Preliminaries and assumptions

55

also called the invariant or stationary measure, is the limit of the transition probabilities and hence depicts the asymptotics of large times for a ﬁxed noise intensity ε. In order to highlight this difference between the sublimiting and the limiting distribution also the latter object will be discussed in detail; see section 2.2. In this chapter an outline of the theory of large deviations, exit probabilities for non-degenerate systems and metastability shall be given. However, in order not to overburden the scope of this book, we allow ourselves only to sketch or even omit some of the proofs; when doing so, references to the underlying work by Freidlin and Wentzell as well as to Dembo and Zeitouni [De-Zt 98] are given. The terminus “metastable state” has been coined in statistical physics: The empirical description of “metastable thermodynamic phases” is characterized by the following properties, listed by Penrose and Lebowitz [Per-Leb 71]: 1) “Only one thermodynamic phase is present”, 2) “a system that starts in this state is likely to take a long time to get out” and 3) “once the system has gotten out, it is unlikely to return”. Physically speaking, it is the goal of the current chapter to detect points Kµ in Rd featuring such heuristic properties in the framework set up by the SDE (2.1). The importance of such SDEs (2.1) and (2.2) for applications shall then be illustrated in the concluding section of this chapter by discussing some sample systems.

2.1 Preliminaries and assumptions Let the matrix a ≡ σ σ ∗ be invertible. The action functional for (2.1) is x x deﬁned by 1ε I0T , where the functional I0T is given by x ¯+ , : C([0, T ], Rd ) −→ R I0T ⎧ T 0 1 2 ⎪ 1 ⎪ ⎪ a(fs )−1/2 f˙s − b(fs ) ds , f absolutely continuous ⎨ 2 0 x (f ) := I0T and f0 = x , ⎪ ⎪ ⎪ ⎩∞ , otherwise .

Here, a(x)−1/2 denotes the (unique) d × d-matrix whose square is the positive deﬁnite matrix a(x)−1 ≡ [σ(x) σ(x)∗ ]−1 . Also a(x)−1/2 is a symmetric matrix; hence, one can rewrite the integrand as 0 1 ; f˙s − b(fs ) , a(fs )−1 f˙s − b(fs )

56

2 Locality and time scales of the underlying non-degenerate system

see e.g. Freidlin [Fr 68] on factorization theorems and how diﬀerentiability properties are preserved. x depends on b, σ, x and a ﬁxed time horizon The above functional I0T T > 0, but not on ε ; it is the rate function for the large deviation principle xy x as the restriction of I0T to for X ε,x as we will later see. We also deﬁne I0T all paths f such that f (T ) = y. The quasipotential for the SDE (2.1) is then deﬁned as V : Rd × Rd −→ R+ , xy (.) V (x, y) := inf I0T T >0 x ≡ inf I0T (f ) : f ∈ C([0, T ], Rd) , f0 = x , fT = y , T > 0 ;

it describes the “minimal cost” of forcing X ε to connect x and y eventually. Now the assumptions are listed under which (2.1) is investigated: Assumption 2.1.1 (on the coeﬃcients of SDE (2.1)). The drift b ∈ C ∞ (Rd , Rd ) and the diﬀusion term σ ∈ C ∞ (Rd , Rd×d ) are supposed to further satisfy: (S) For all ε > 0 and x ∈ Rd , there exists a unique strong, non-exploding solution X ε,x to the SDE (2.1). (E) σ takes its values in the invertible matrices, σ −1 ∈ C ∞ (Rd , Rd×d ), and a ≡ σσ ∗ is strictly positive deﬁnite, i.e. there is a constant c > 0 such that c |x2 |2 ≤ a(x1 )x2 , x2 ≤ c−1 |x2 |2

(x1 , x2 ∈ Rd ) .

(K) There exist ﬁnitely many points K1 , . . . , Kl ∈ Rd such that (K1) any trajectory Xt0,x (except those in a ﬁnite union of lower dimensional submanifolds of Rd ) of the deterministic system dXt0 = b Xt0 dt is attracted to one of the points Ki as t → ∞ ; (K2) each Ki is stable, i.e. there exist open sets Ui ⊃ Ki such that Ui is attracted to Ki under the deterministic motion X 0 . (V)

lim V (0, y) = ∞ .

|y|→∞

(G) The system is generic in the sense that there are no symmetries in the function V . More precisely, it is assumed that the minima and maxima in equations (2.11)–(2.17) below are attained at only one point; this means (in the terminology coined below) that the main state, the rotation rate and the exit rate of cycles of any order shall be well deﬁned.

2.1 Preliminaries and assumptions

57

Remark 2.1.2 (on the set of assumptions 2.1.1). on (S): If σ(x) = idRd for all x ∈ Rd , a suﬃcient condition for (S) is e.g. that for some constant c > 0 and all x ∈ Rd , x, b(x) ≤ c 1 + |x|2 ; see Stroock and Varadhan [Str-Vdh 97, Th. 10.2.2]. Since (K) forces all trajectories to converge to one of the ﬁnitely many attractors Ki , one might think of a drift b for which x, b(x) < 0 outside some suﬃciently large ball. on (E): Condition (E) (“ellipticity” of (2.1) ) makes sure that the generator G ε is an elliptic diﬀerential operator and also bounds the covariance a from above. (E) makes sure that the Markov transition probabilities Ptε (x, . ) have Lebesgue densities; together with assumptions (B) deﬁned below, (E) guarantees the existence of an invariant probability measure ρε with Lebesgue density pε (x) > 0 to which the Markov transition probabilities converge; see the next section. The crucial point here is that we will use time scales T (ε) “below which” this limiting distribution ρε is observed, so this invariant measure does not play a decisive role in the subsequent analysis; however, we will use it to emphasize the diﬀerent behaviors of the transition probabilities in the limiting and the sublimiting cases, respectively; see sections 2.2 and 2.5. From an applications’ point of view the existence of an invariant probability is a reasonable property; see section 2.6. on (E): Due to assumption (E), V is indeed a well deﬁned function in R+ . on (K): This assumption, taken from Freidlin [Fr 77], allows to approximate the behavior of X ε by the one of a certain Markov chain whose ﬁnite state space is given by the union of small spheres around the points Ki . It is this construction which backs the exit time law 2.5.3 for cycles. For those initial conditions x for which X 0,x does not converge to one of the points K1 , . . . , Kl , there exist unstable equilibria Kl+1 , . . . , Kl to which the respective trajectories converge; these trajectories separate the domains of attraction D1 , . . . , Dl of the respective points K1 , . . . , Kl and therefore form separatrices of the deterministic dynamical system X 0 ; hence, these trajectories (and their initial values) constitute a ﬁnite union of lower dimensional submanifolds of Rd separating the domains D 1 , . . . , Dl . Conversely, one could start with all equilibria K1 , . . . , Kl of the deterministic dynamical system X 0 and then single out the stable ones K1 , . . . , Kl , where l ≤ l ; see Freidlin [Fr 00, p.337].

58

2 Locality and time scales of the underlying non-degenerate system

on (K): The sets Ki need not be one point sets: Instead one can also allow K 1 , . . . , K l ⊂ Rd to be just compact sets. However, since the concepts of the current chapter will be applied to the case where the Ki ’s are one point sets, we formulate this assumption as it will be used. In this more general situation, where the Ki ’s are compact sets, the statements of this chapter would hold true, if in (K) we add two more subassumptions: (K3) if x, y ∈ Ki , then x ∼ y (1 ≤ i ≤ l); (K4) if x ∈ Ki and y ∈ Kj , then x y (i = j) ; thereby we made use of the following equivalence relation which is induced by the quasipotential V : x ∼ y :⇐⇒ V (x, y) = V (y, x) = 0 . In this general situation these sets Ki are for example the limit cycles of the ODE x˙ = b(x) ; consider e.g. the drift

−x2 − x1 (x21 + x22 )2 − 3(x21 + x22 ) + 2 , b(x) := x1 − x2 (x21 + x22 )2 − 3(x21 + x22 ) + 2 which is taken from Jetschke [Je 89, p.55]; note that this drift decomposes orthogonally as b = −∇U + L (see 2.4.5 on the implications for the quasipotential), where

1 3 −x2 U (x) := |x|6 − |x|4 + |x|2 and L(x) := x1 6 4

;

in polar coordinates, := |x| and α := arctan xx21 , this ODE x˙ = b(x) can be rewritten as ˙ = − 5 + 3 3 − 2 ≡ − U ( ) = − ( 2 − 1) ( 2 − 2) α˙ = 1 ; since U (x) = U ( ) only depends on the absolute value of the position, this implies that b satisﬁes (K1)–(K4) with the stable sets K1 := {0} and K2 := { x ∈ R2 : |x| = 2 }. The invariant set { x ∈ R2 : |x| = 1 } violates the stability criterion (K2) and hence constitutes the saddle for the radial motion. An even simpler example for this phenomenon of a limit cycle is the twodimensional family of drifts

2.1 Preliminaries and assumptions

59

−x2 + x1 η − (x21 + x22 ) bη (x) := x1 + x2 η − (x21 + x22 ) which represents the normal form of the Hopf-bifurcation, where η ∈ R denotes the bifurcation parameter; cf. Guckenheimer and Holmes [Gu-Hl 83, p.146f.], Andronov et al. [And-Pon-Vi 33, §5] and Jetschke [Je 89, p.158;52,226]; a slight generalization of this system is considered by Leng et al. [Lg-SN-Tw 92, Ex.3]. Also in this case the drift decomposes orthogonally as bη = −∇Uη + L , where 1 η Uη (x) := |x|4 − |x|2 4 2

and

−x2 ; L(x) := x1

in polar coordinates this ODE can be rewritten as ˙ = − 3 + η ≡ − Uη ( ) = (η − 2 ) α˙ = 1

;

again Uη (x) = Uη ( ) only depends on the absolute value of the position, and bη satisﬁes (K) with the only stable set {0} , η≤0 K(η) := √ 2 { x ∈ R : |x| = η } , η > 0 . If η ≤ 0, the qualitative behavior of xt resembles the motion in example 2.6.1; if η > 0, then the origin is a repelling ﬁxed point. on (K): A sample drift which violates the stability assumption (K) is the following “mock” Van der Pol-ODE given in polar coordinates by ˙ = (1 − ) α˙ = 2 − cos α ; cf. Zeeman [Ze 88, p.126]. on (V): This assumption assures that the sets {y : V (x, y) ≤ c} which exhaust the state space Rd are all compact for any x and c ≥ 0. on (E) and (V): Instead of considering a diﬀusion on Rd , we can also take a diﬀusion X ε on a d-dimensional compact manifold M . In this case (E) and (V) are redundant except from assuming a to be positive deﬁnite on M . The (unique) stationary measure is already ﬁnite in this case; see example 2.6.11.

60

2 Locality and time scales of the underlying non-degenerate system

The compact setting is for example assumed by Wentzell and Freidlin [We-Fr 69] and Freidlin and Wentzell [Fr-We 98, §§ 6.4, 6.6]. One could equally well consider the “mixed” case with state space M × Rd for a compact manifold M under suitably adopted assumptions. on (G): As already mentioned, this assumption will be made precise below. In order to illustrate what might go wrong here, note that (G) is violated for example, if the potential function U in SDE (2.2) has wells of equal depth.

2.2 The limiting distribution (stationary measure) In this section we collect some facts on the connection between SDEs and partial diﬀerential equations (PDEs) to illustrate the behavior of the transition probabilities and of invariant measures. Since the contents of this section is well documented in textbooks, we refrain from giving proofs here and refer instead to standard monographs as e.g. Friedman [Fri 75, Ch. 6], Friedman [Fri 76, Sec. 14.4], Khasminskii [Kh 80, Ch. III & IV], Stroock and Varadhan [Str-Vdh 97], Wentzell [We 79, §11.2] and Fleming and Rishel [Fl-Ris 75, Ch.V]. Also see the expositions by Freidlin and Wentzell [Fr-We 98, Ch.1 & 4], Khasminskii [Kh 60], Ichihara and Kunita [Ic-Ku 74], Andronov et al. [And-Pon-Vi 33], Bernstein [Bs 33], Horsthemke and Lefever [Hh-Lf 85, Sec. 4.4] and Hackenbroch and Thalmaier [Hb-Th 94, Sec. 6.6]; recent results are due to Pardoux and Veretennikov [Pd-Ver 01] for example. In order to emphasize the diﬀerence between the “forward variables” and the “backward variables” in the “forward equation” and the “backward equation” to come, we allow for time-dependent coeﬃcients in this section, i.e. we consider the following SDE in Rd , √ dXtε,x = b (t, Xtε,x ) dt + ε σ (t, Xtε,x ) dWt (t ≥ s ≥ 0) , (2.3) ε,x d Xs = x ∈ R , where the coeﬃcients are functions b ∈ C ∞ (R+ × Rd , Rd ) and σ ∈ C ∞ (R+ × Rd , Rd×d ) and where s ≥ 0 denotes some initial time; assume that any σ(t, . ) satisﬁes the ellipticity condition (E) for some universal constant c > 0. For simplicity it is furthermore assumed that all coeﬃcient functions b and a ≡ (σσ ∗ ) as well as their ﬁrst derivatives are bounded (with respect to the respective norms). The latter assumption can be relaxed (see Friedman [Fri 75, p.147]); however, since the main focus of this paper lies on the behavior on bounded domains and during time scales, this assumption does not restrict the scope of this section for illustrative purposes.

2.2 The limiting distribution (stationary measure)

61

The partial diﬀerential operator associated with (2.3) is G ε (s, x) :=

d

bi (s, x)

i=1

d ∂ ε ∂2 + aij (s, x) . ∂xi 2 i,j=1 ∂xi ∂xj

In the sequel let Ps,x denote the probability measure P conditioned on events concerning the process X ε starting in x at time s ≥ 0 ; Es,x then denotes the corresponding expected value operator. ε {Xtε ∈ . } then possess densities with The transition probabilities Ps,x respect to the Lebesgue measure, ε {Xtε ∈ B} = pε (s, x; t, y) dy (s < t , B ∈ B(Rd ) ) . Ps,x B

These densities solve the backward (parabolic) equation −

d d ∂ pε (s, x; t, y) ∂ pε (s, x; t, y) ε ∂ 2 pε (s, x; t, y) = bi (s, x) + aij (s, x) ∂s ∂xi 2 i,j=1 ∂xi ∂xj i=1

≡ G ε (s, x) pε (s, x; t, y) ,

a PDE with respect to the backward (i.e. past) variables (s, x). Equivalently, the densities solve the forward (parabolic) equation d ∂ pε (s, x; t, y) ∂ = − bi (t, y) pε (s, x; t, y) ∂t ∂yi i=1 d ε ∂2 aij (t, y) pε (s, x; t, y) 2 i,j=1 ∂yi ∂yj ∗ =: G ε (t, y) pε (s, x; t, y) ,

+

a PDE with respect to the forward (i.e. future) variables (t, y); (G ε )∗ is the formal adjoint operator of G ε . Moreover, the densities are the fundamental solutions of the backward and the forward equation. In the time-homogeneous case, i.e. if b and σ are independent of t as in (2.1), one then gets for the transition densities pε (s, x; t, y) = pε (0, x; t − s, y) =: pεt−s (x, y)

(s < t) ,

from the above PDEs the Kolmogorov backward equation d d ∂ pεt (x, y) ε ∂ 2 pεt (x, y) ∂ pεt (x, y) = bi (x) + aij (x) ∂t ∂xi 2 i,j=1 ∂xi ∂xj i=1

≡ G ε (x) pεt (x, y)

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2 Locality and time scales of the underlying non-degenerate system

and the equivalent Kolmogorov forward equation (Fokker-Planck equation) d d ε ∂ pεt (x, y) ∂ ∂2 = − bi (y) pεt (x, y) + aij (y) pεt (x, y) ∂t ∂y 2 ∂y ∂y i i j i=1 i,j=1 ε ∗ ε ≡ G (y) pt (x, y) .

Further on, consider the time-homogeneous case (2.1). A stationary probability distribution for X ε is a probability measure ρε on Rd which is invariant with respect to the Markov transition probabilities (Ptε )t , Ptε (x, . ) := Px { Xtε ∈ . } ≡ P{ Xtε,x ∈ . } in the sense that

ρε ( . ) = Rd

Ptε (x, . ) ρε (dx)

or equivalently f (x) ρε (dx) = Rd

(t ≥ 0, x ∈ Rd ) ,

Rd

Ttε f (x) ρε (dx)

(t ≥ 0) ;

(f ∈ C c (Rd , R) ) ,

where Ttε f (x)

:= Rd

f (y) Ptε (x, dy) .

The stationary measure ρε is the “equilibrium (limiting) distribution as t → ∞”, in the sense that for any x ∈ Rd t→∞ Ttε f (x) −−−−→ f (y) ρε (dy) (f ∈ C b (Rd , R) ) Rd

and t→∞

Ptε (x, B) −−−−→ ρε (B)

(2.4)

for all B ∈ B(Rd ) such that ρε (∂B) = 0. If a stationary probability distribution exists, assumption (E) implies its uniqueness and the existence of a Lebesgue density pε (x) > 0 of ρε , the reason being that (E) entails strong ellipticity in the Lie-algebraic sense of Ichihara and Kunita [Ic-Ku 74, p.250]. The densities pεt then converge to the “equilibrium distribution” in the sense that for all x, y ∈ Rd t→∞

pεt (x, y) −−−−→ pε (y) ; pε , being the Lebesgue density of ρε , uniquely solves the respective autonomous diﬀerential equations

2.2 The limiting distribution (stationary measure)

63

0 = G ε (x) pε (x) (Kolmogorov backward equation) and 0 = (G ε (x))∗ pε (x) (Kolmogorov forward equation, Fokker-Planck equation) pε (x) dx = 1. For any f ∈ C b (Rd , R) and ε ∈ (0, ε0 ), the function

such

that

uεf (t, x) := Ex f (Xtε ) ≡ E [ f (Xtε,x ) ] is the unique solution of the Cauchy problem for (2.1): d d ∂uε ε ∂uε ∂ 2 uε = G ε uε ≡ bi + aij ∂t ∂xi 2 i,j=1 ∂xi ∂xj i=1

(t > 0),

(2.5)

uε (0, . ) = f. This function uεf converges to the “equilibrium distribution as t → ∞” as well, in the sense that for any x ∈ Rd t→∞ uεf (t, x) −−−−→ f (y) ρε (dy) . (2.6) Rd

Concluding these remarks, we note that a general, suﬃcient condition for the existence of a stationary probability distribution ρε is (E) together with (B) There exists a bounded domain S ⊂ Rd with smooth boundary such that for all compact sets K ⊂ Rd , sup Ex τSε,x < ∞ ,

x∈K

where τSε,x denotes the hitting time of S for the process X ε,x given by (2.1). (for a proof see Khasminskii [Kh 80, Sec.IV.4] and [Kh 60]). Remark 2.2.1 (Explicit formula for the stationary density in the gradient case). Suppose that the SDE under consideration is (2.2), i.e.: σ = idRd and b = −∇U for some suitable potential function U ∈ C ∞ (Rd , R) (U being in C 2 (Rd , R) would suﬃce here). Note that in dimension d = 1 any drift is the gradient x of a potential function, U (x) := − 0 b(s)ds. Then the above Fokker-Planck equation takes the form

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2 Locality and time scales of the underlying non-degenerate system

d ε ∂ ∂U ε + pε = 0 , (G ) p ≡ p ∂xi ∂xi 2 i=1 ε ∗

ε

where denotes the Laplace operator. A direct calculation of the respective derivatives shows that x → e−2 U(x)/ε fulﬁlls the above PDE ; hence, a stationary probability measure for (2.2) exists if and only if Nε := e− 2U(x)/ε dx < ∞ , Rd

in which case pε (x) = Nε−1 e− 2U(x)/ε

(x ∈ Rd ) .

Jacquot [Ja 92, p.347] shows that a suﬃcient condition for the ﬁniteness of the normalization constant Nε is the following: (U)

U → ∞ and |∇U | → ∞ as |x| → ∞ .

Note, however, that nevertheless (U) does not imply that U has positive curvature for all suﬃciently large x, as can be seen for example from U (x) := x2 2 − 2 cos x, where x ∈ R. Example 2.2.2 (Density pε for the potential function U 1 ). Consider for example (2.2) with the potential function U1 , 3 4 2 3 x1 − x31 − 3 x21 + x1 x2 + x42 , 2 3 2

U1 (x) :=

as in (1.34) of example 1.5.3, where c := 1. The potential is sketched in the ﬁgure 2.1, where the diﬀerent depths of the wells are emphasized by drawing level sets, too. The density pε of the stationary measure has the following qualitative shape as depicted in the contour plot of ﬁgure 2.2: The peaks are located at 10 8 6 4 2 0

−2

−1 −0.5 x_2

0 0.5 1

−2

−1.5

−1

−0.5

Fig. 2.1 The potential function U1 (x1 , x2 ) =

0.5

0

1

1.5

2

x_1

3 2

x41 −

2 3

x31 − 3 x21 + x1 x2 +

3 2

x42

2.2 The limiting distribution (stationary measure)

65

2.5 2 1.5 1 0.5 0

x_2

−1 −0.5

0 0.5 1

−2

−1.5

−1

−0.5

0 x_1

0.5

1

1.5

2

Fig. 2.2 Sketch of the density pε ∼ e−2U1 /ε

the (local) minima of the potential and the mass concentrates at the global minimum in the small noise limit ε → 0. This feature is characteristic for all potential SDEs (2.2): The local minima of U correspond to the local maxima of the density of the invariant distribution; these are the preferential states of the process X ε as t → ∞. Similarly, in the non-gradient case the system X ε from (2.1) accumulates near the points K1 , . . . , Kl ; a corresponding statement for X ε on time scales T (ε) will appear later (see theorems 2.5.5 and 2.5.6). For the Ornstein-Uhlenbeck process even the time-dependent transition densities can be obtained explicitly: Example 2.2.3 (Ornstein-Uhlenbeck process, one-well potential function). Consider the special case of a linear drift b(x) = −βx for a constant β > 0 in dimension d = 1 and choose σ := 1. Then the SDE (2.2) (or (2.1) respectively) becomes √ dXtε = −U (Xtε ) dt + ε dWt , X0ε = x0 , x

where U (x) := − 0 b(y)dy = β2 x2 denotes a quadratic one-well potential. Then the transition probability densities pεt (x, . ) are given by

1 (x − x0 αt )2 , exp − pεt (x, x0 ) = 2 σt2 2π σt2 where

αt := e−β t

and

σt2 :=

ε (1 − e−2βt ) . 2β

This follows from a direct veriﬁcation of the Kolmogorov forward equation (Fokker-Planck equation). In other words, Xtε is normally distributed where the mean is given by the trajectory of the deterministic system, Xt0 = x0 e−βt , and the variance is σt2 .

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2 Locality and time scales of the underlying non-degenerate system

Remark 2.2.4 (SDE of gradient type whose potential exhibits singularities). Consider the above SDE of gradient type (2.2) √ dXtε,x = −∇U (Xtε,x ) dt + ε dWt ; in the previous situation the drift b ≡ −∇U was supposed to be an element of C ∞ (Rd , Rd ) (or at least C 1 (Rd , Rd )); in particular, U is deﬁned on the whole of Rd and the stationary density is calculated as in 2.2.1. For some applications it is however necessary to diminish the state space Rd to an open set D. In the upcoming course of this exposition we will be ε concerned with the exit times τD of X ε from open sets D ⊂ Rd . However, ε one could also force the system X not to leave D at all. For this purpose ﬁx a potential function U ∈ C(Rd , R ∪ {+∞}) such that for D :=

x ∈ Rd : U (x) < ∞

the restriction of U to D is C 1 , U D ∈ C 1 (D, R) . Further assume that

|∇U (x)|2 e−4 U (x)/ε dx < ∞

D

and Nε := where the convention is used. Then

Rd

e− 2U(x)/ε dx < ∞ ,

e− 2U( . )/ε := 0 on D pε (x) = Nε−1 e− 2U(x)/ε

is the Lebesgue density of a probability measure ρε and there exists a process (Xtε )t≥0 with initial distribution P ◦ (X0ε )−1 = ρε which is a weak2 solution of the gradient SDE (2.2), √ dXtε = −∇U (Xtε ) dt + ε dWt , up to a terminal time T∞ for which Pρε {T∞ < ∞} = 0 ; here Pρε denotes the law of (Xtε )t≥0 starting with the stationary distribution ρε and T∞ being terminal means that it is the limit of an increasing sequence of stopping times.3 2 3

See e.g. Hackenbroch and Thalmaier [Hb-Th 94, 6.42]. See e.g. Hackenbroch and Thalmaier [Hb-Th 94, 6.16].

2.2 The limiting distribution (stationary measure)

67

This result is due to Meyer and Zheng [My-Zh 85] and has been cited in its above form by Kunz [Kz 02, p.16f.,27]. We ﬁnish this section by quoting a general result which does not contain an assertion on the stationary measure. However, it suitably concludes the above considerations on PDEs corresponding to uniformly non-degenerate SDEs (2.3) and will be used later in sketching the proof of theorem 3.1.6. A reference for this theorem is Friedman [Fri 75, p.146f.] among others. Theorem 2.2.5. Consider the SDE dXtx = b (t, Xtx ) dt + σ (t, Xtx ) dWt Xsx

(t ≥ s ≥ 0) ,

=x ∈R , d

where the coeﬃcients are functions b ∈ C ∞ (R+ × Rd, Rd ) and σ ∈ C ∞ (R+ × Rd , Rd×d ); assume that any σ(t, . ) satisﬁes the ellipticity condition (E) for some universal constant c > 0. The partial diﬀerential operator corresponding to this SDE is G(s, x) :=

d i=1

bi (s, x)

d ∂ 1 ∂2 + aij (s, x) . ∂xi 2 i,j=1 ∂xi ∂xj

Fix some bounded, open domain D ⊂ Rd with smooth boundary ∂D, some time horizon T > 0 and let θD := min τD (s, x) , T , where τD (s, x) := inf{t ≥ 0 : Xtx ∈ / D} denotes the ﬁrst exit time4 of X•x from D. Furthermore, let ψ ∈ C(∂Q, R) be some continuous function5 deﬁned on the boundary of Q := (0, T ) × D. Then the boundary value problem

∂ + G(s, x) v = 0 on Q ∂s v = ψ on ({T } × D) ∪ ([0, T ] × ∂D) is uniquely solved by v(s, x) := Es,x ψ θD , XθD . 4

Since D is open, τ ε,x is a stopping time with respect to the underlying standard ﬁltration (Ft )t≥0 ; see e.g. Hackenbroch and Thalmaier [Hb-Th 94, 3.12.(ii)]. 5 This function ψ(s, x) is not to be confused with the stochastic process ψtε (ω) as deﬁned in equation (1.4). Since these two objects will not be considered simultaneously, there is no ambiguity.

68

2 Locality and time scales of the underlying non-degenerate system

2.3 The large deviations principle This section is intended to sketch the fundamental principles for the SDE (2.1) from which the exit time law shall be deduced in the next section. Standard references underlying the following exposition are the monographs by Freidlin and Wentzell [Fr-We 98] and Dembo and Zeitouni [De-Zt 98]. At the beginning of this chapter the action functional and the quasipotential have already been mentioned (see p.55f.) in order to set up the assumptions 2.1.1. This section clariﬁes that the action functional indeed provides the rate function of a large deviation principle; in the next section the importance of the quasipotential in exit time considerations will be accounted for, thus illustrating its interpretation as cost function. The general setup for a large deviation principle is that of a family {µε }ε>0 of probability measures on a space E which we assume for simplicity to be a separable metric space equipped with its completed Borel-σ-Algebra B(E); the goal is to characterize the behavior of {µε }ε>0 on B(E) as ε → 0 in terms of a rate function I on an exponential scale. Deﬁnition 2.3.1 (Large Deviation Principle (LDP)). Let E be a separable metric space with its (completed) Borel-σ-algebra B(E). 1) A function I : E → [0, ∞] is a rate function, if it is lower semicontinuous, i.e. if the level sets {x ∈ E : I(x) ≤ α} are closed; since E is metric, an equivalent condition is that for all x ∈ E, lim inf I(xn ) ≥ I(x) . xn →x

A rate function I is good, if the level sets {x ∈ E : I(x) ≤ α} are compact. 2) A family {µε }ε>0 of probability measures on B(E) satisﬁes the large deviation principle (LDP) with rate function I, if − inf◦ I ≤ lim inf ε log µε (Ψ) ≤ lim sup ε log µε (Ψ) ≤ − inf I Ψ

ε→0

ε→0

(2.7)

Ψ

for all Ψ ∈ B(E) , where Ψ◦ and Ψ denote the topological interior and closure of the set Ψ, respectively. A large deviation principle is “pushed forward” by continuous mappings; this is the content of the following proposition. For the straightforward veriﬁcation we refer to Freidlin and Wentzell [Fr-We 98, p.81] and Dembo and Zeitouni [De-Zt 98, p.127]. The latter reference also contains more general versions of the contraction principle such as the “almost continuous case”, where the function F is measurable and a suitable limit of continuous functions; see [De-Zt 98, p.133].

2.3 The large deviations principle

69

Proposition 2.3.2 (Contraction principle - continuous case). Let F : E 1 → E 2 be a continuous function between separable metric spaces E 1 , E 2 and let I (1) : E 1 → [0, ∞] be a good rate function. Then I (2) : E 2 → [0, ∞] ,

I (2) (x) :=

inf

F−1 ({x})

I (1) ( . )

(where inf ∅ ≡ ∞) is a good rate function on E 2 . If, in addition, I (1) governs a LDP for a family of probability measures {νε }ε>0 on E 1 , then I (2) governs a LDP for {µε }ε>0 := { νε ◦ F−1 }ε>0 on E 2 . Proposition 2.3.2 is used in proving the following theorem 2.3.4. Beforehand, the corresponding notation is ﬁxed: Notation 2.3.3. Let T > 0 be an arbitrary time horizon. Then the following function spaces over the time interval [0, T ] are deﬁned: Cx := Cx ([0, T ], Rd) := {f : [0, T ] → Rd continuous, f0 = x}

(x ∈ Rd )

and H1 := H1 ([0, T ], R ) := d

·

gs ds : g ∈ L ([0, T ], R ) 2

d

,

0

the absolutely continuous functions starting in 0 with square integrable derivative. Furthermore, let pr[0,T ] denote the restriction map pr[0,T ] : (Rd )[0,∞) → (Rd )[0,T ] ,

pr[0,T ] (f ) := f [0,T ] .

Theorem 2.3.4 (LDP for strong solutions of SDE). Let X ε,x be the solution of (2.1), √ dXtε,x = b(Xtε,x ) dt + ε σ(Xtε,x ) dWt , X0ε,x = x , where b ∈ C ∞ (Rd , Rd ) and σ ∈ C ∞ (Rd , Rd×d ) are assumed to be bounded together with their ﬁrst derivatives. Let µε denote the law of X ε,x on Cx ([0, T ], Rd ) for a ﬁxed time T > 0 , µε := P ◦

pr[0,T ] ◦ X•ε,x

−1

.

Then {µε }ε>0 satisﬁes the LDP with good rate function I(f ) ≡ I[0,T ],x (f ) :=

{ g∈H1 : ft = x +

t 0

inf b(fs ) ds +

t 0

σ(fs ) g˙ s ds }

1 2

T 2

| g˙ t | dt . 0

70

2 Locality and time scales of the underlying non-degenerate system

If a ≡ σσ ∗ is strictly positive deﬁnite, as in assumption (E), then this x as deﬁned on p.55, i.e. rate function is the action functional I0T ⎧ T 2 ⎨1 a(fs )−1/2 [f˙s − b(fs )] ds I(f ) = 2 0 ⎩ ∞

, f − x ∈ H1 , f −x∈ / H1 .

Sketch of Proof. 1) √ Consider the case that b( . ) = 0, σ( . ) = idRd and Schilder’s x = 0 ﬁrst, i.e. Xtε,x = εWt . In this case the assertion is due to √ theorem (Dembo and Zeitouni [De-Zt 98, p.185f.]): The law of εWt on C0 ([0, T ], Rd ), √ νε := P ◦ (pr[0,T ] ◦ ( ε W• ))−1 , satisﬁes the LDP with good rate function T 1 |g˙ s |2 ds BM I[0,T ],0 (g) := 2 0 ∞

, g ∈ H1 ,g ∈ / H1 .

2) Now consider a general drift b and initial condition x, but again let σ( . ) = idRd , √ dXtε,x = b(Xtε,x ) dt + ε dWt , X0ε,x = x . For any g ∈ C0 , the integral equation t b(fs )ds + gt ft = x +

(t ∈ [0, T ])

0

admits a unique continuous solution f ; this gives rise to a well-deﬁned mapping F : C0 → Cx , F(g) := f . The map F is itself continuous by the Lipschitz continuity of b and Gronwall’s Lemma. Thus the contraction principle 2.3.2 applies and yields an LDP, as √ pr[0,T ] ◦ X•ε,x = F pr[0,T ] ◦ ε W• by the SDE for X ε,x which implies that µε := νε ◦ F−1 is the law of X ε,x on Cx . Due to the contraction principle 2.3.2 and Schilder’s theorem the corresponding rate function is I(f ) = =

inf

{g∈C0 : f =F(g)}

BM I[0,T ],0 (g)

inf

{ g∈H1 : f (t) = x +

t 0

b(fs ) ds + gt }

1 2

T 2

| g˙ t | dt ; 0

2.3 The large deviations principle

71

in the last equation it has been used that g ∈ H1 if and only if f = F(g) ∈ H1 + x ; in this case one obtains that g˙ t = f˙t − b(ft ) which yields together with the last equation that 2 1 T ˙ I(f ) = ft − b(ft ) dt ; 2 0 otherwise, if f = F(g) ∈ / H1 + x, then g ∈ / H1 and I(f ) = ∞ due to Schilder’s theorem. 3) Finally consider a general diﬀusion coeﬃcient σ; in this case the map F deﬁned analogously as above as F(g) := f via t t b(fs ) ds + σ(fs )g˙ s ds (t ∈ [0, T ]) ft = x + 0

0

is not necessarily continuous. Therefore, one approximates X ε,x by X ε,m,x ,

√ ε,m,x ε,m,x ε,m,x dt + ε σ X mt dWt , = b X mt X0ε,m,x = x dXt m

m

and uses the “almost continuous version” of the contraction principle which had already mentioned before; see Dembo and Zeitouni [De-Zt 98, p.214f., 133]. Further details concerning the above results are contained in the theorems 4.1.1 and 5.3.2. by Freidlin and Wentzell [Fr-We 98], in Wentzell and Freidlin [We-Fr 70, §1] as well as in Dembo and Zeitouni [De-Zt 98, Sec.5.6]. Remark 2.3.5. Consider the situation of the above theorem and assume that a is strictly positive deﬁnite. Then the rate function (action functional)

x I(f ) = I0T (f ) ≡

⎧ 1 ⎪ ⎪ ⎨

2

⎪ ⎪ ⎩

∞

T 0

0 12 −1/2 ˙ fs − b(fs ) ds , f absolutely continuous a(fs )

and f0 = x, , otherwise

vanishes if and only if f is a solution path of the ODE x˙ = b(x) on the time interval [0, T ] ; the action functional hence weighs the deviation of a path f from being a deterministic solution in the L2 -norm. The following continuity consequence of the LDP 2.3.4 will be used in the proof of the exit time law 2.4.6 (more precisely, in the lemmas 2.4.8, 2.4.9 and 2.4.10); it states that the above LDP also holds true uniformly with respect to the initial condition. For a proof see Dembo and Zeitouni [De-Zt 98, Cor.5.6.15]. A similar assertion will appear later in the case of degenerate SDEs; see corollary 3.2.1.

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2 Locality and time scales of the underlying non-degenerate system

Theorem 2.3.6 (Uniform asymptotics). Consider the situation of the above theorem 2.3.4 and ﬁx some compact set K ⊂ Rd . Then it follows for all open sets G ⊂ C([0, T ], Rd ), lim inf ε log inf P {X•ε,y ∈ G} ≥ − sup inf I[0,T ],y (f ) , ε→0

y∈K

y∈K f ∈G

and for all closed sets F ⊂ C([0, T ], Rd ), lim sup ε log sup P {X•ε,y ∈ F } ≤ − ε→0

y∈K

inf

y∈K, f ∈F

I[0,T ],y (f ) .

2.4 Exit probabilities for non-degenerate systems In this section we are interested in the noise-induced exit (exit time, exit location) from a neighborhood of an equilibrium point of the corresponding deterministic system. Throughout this section we consider the SDE (2.1) √ X0ε,x = x dXtε,x = b(Xtε,x ) dt + ε σ(Xtε,x ) dWt , under the assumptions 2.1.1. Again, the law of (Xtε,x )t≥0 is denoted by Px and Ex is the corresponding expected value. Notation 2.4.1 (First exit time). Let D be a bounded, open domain in Rd with smooth boundary ∂D. Then the random variable ε,x τ ε ≡ τ ε,x ≡ τD := inf{t ≥ 0 : Xtε,x ∈ / D}

denotes the ﬁrst exit time of X•ε,x from D. Since D is open, τ ε,x is a stopping time6 with respect to the underlying standard ﬁltration (Ft )t≥0 . Remark 2.4.2. The ﬁrst exit time τ ε and the ﬁrst exit location can be characterized for any ε in terms of solutions of PDEs involving the generator G ε of X ε : 1) f1 (t, x) := Px {τ ε ≤ t} is the unique solution of ε ∂f (t, x) , t > 0 , x ∈ D G f (t, x) = ∂t ⎧ ⎨ 0 , t = 0, x ∈ D f (t, x) = ⎩ 1 , t > 0 , x ∈ ∂D ; let Q := (0, ∞) × D; then f1 is continuous at all points of Q \ {(0, x) : x ∈ ∂D}. 6

See e.g.: Hackenbroch and Thalmaier [Hb-Th 94, 3.12.(ii)].

2.4 Exit probabilities for non-degenerate systems

73

2) f2 (x) := Ex τ ε is the unique solution of G ε f = −1 , on D f = 0 , on ∂D . 3) For any g ∈ C(∂D, R), f3 (x) := Ex (g(Xτεε )) is the unique solution of G ε f = 0 , on D f = g , on ∂D ; f2 and f3 are continuous on D. These diﬀerential equations are similar in spirit to the ones investigated in section 2.2 ; they are well known: 1) can be found e.g. in Friedman [Fri 76, p.347] and Freidlin and Wentzell [Fr-We 98, p.107], 2) and 3) are cited from Dembo and Zeitouni [De-Zt 98, p.222]; also see Hackenbroch and Thalmaier [Hb-Th 94, Sec.6.6]. The above PDEs are diﬃcult to solve, especially in higher dimensions, and will not be used in the sequel. Instead, the asymptotic behavior of τ ε as ε → 0 is investigated by means of the LDP for X ε . Theorem 2.4.3 (Consequence of the LDP 2.3.4 for the ﬁrst exit time). Let D be a bounded, open domain in Rd with smooth boundary ∂D and ﬁrst-exit time / D} , τ ε,x ≡ inf{t ≥ 0 : Xtε,x ∈ where X ε,x is the solution of the SDE (2.1) under assumption (E), starting in x ∈ D. Furthermore, let I be the rate function (action functional) for X ε,x on Cx ([0, T ], Rd); see 2.3.4. Then it follows for t ∈ [0, T ] that ⎧ ⎫ ⎨ ⎬ /D lim ε log Px {τ ε ≤ t} = − inf I[0,T ],x (f ) : f ∈ Cx ([0, T ], Rd), ∃ f (s) ∈ ε→0 ⎩ ⎭ s∈[0,t] ≡ − inf V (x, y; s) : s ∈ [0, t], y ∈ /D , where V (x, y; s) := inf { I[0,s],x (f ) : f ∈ Cx ([0, s], Rd ) , f (s) = y} . Proof. For simplicity, the proof shall be given only for the case that σ is a constant (invertible) matrix; in doing so we follow Freidlin and Wentzell (theorem 4.1.2 and example 3.3.5 in [Fr-We 98]) who consider σ = idRd . For the general case the reader is referred to Wentzell and Freidlin [We-Fr 70, Th.2.1] and Friedman [Fri 76, Th.14.4.1].

74

2 Locality and time scales of the underlying non-degenerate system

By theorem 2.3.4 the LDP holds for the law µε of X ε,x on Cx with rate function I ≡ I[0,T ],x , so (2.7) applies and it is thus left to verify: inf◦ I[0,T ],x ≤ inf I[0,T ],x , Ψ

where

Ψ :=

Ψ

f ∈ Cx ([0, T ], Rd) : ∃

f (s) ∈ /D

.

s∈[0,t]

As Ψ is open in Cx , Ψ is closed, Ψ = Ψ . Furthermore, inf I[0,T ],x < ∞ , Ψ

because for some (even for any) ﬁxed y ∈ ∂D, f ∈ Ψ, where f (u) := x + u t (y − x), and I[0,T ],x (f ) < ∞ due to the assumption (E). Suppose that the above claim were false, i.e. that inf◦ I[0,T ],x > inf I[0,T ],x . Ψ

Ψ

Hence, there exists ϕ ∈ Ψ \ Ψ◦ such that inf Ψ◦ I > I(ϕ). In particular, ϕ is absolutely continuous, since I(ϕ) < ∞. As ϕ ∈ Ψ, there exists s ≤ t such that ϕ(s) ∈ /D.

..................... ...... ......... ..... ...... .... ..... ... .... . . ... .. . . ... ... ... . ... .. . ... .. . ... .. . ... .. . ... ... .... ... ... ... δ ... .. ... .. ... ... .... ... ... .... .. .... ... .... .. .. . .... ... .... ... .... .... .... ... .. ... .... .. .... .... .. ... .. .... .... . ... ..... .... .. .... ... ... .... ..... ... .... .. ... ... ... .... ..... ... ... ... .. .. ... .. ... ... .. .. ... .. .... ... .. .. .. .. .. .. ... .. . . .. .. .. ... . . . .... ... .... ... ... .. ... ..... . . .. . ... . . . ... ... ... .... ... ...... ..... ... ....... ........ ... ................................... . . . . . . . . ... . . .......................................................... .... ... . ... ... ... ... ... ... ... ... .... . . . . ..... ..... ...... ..... ........ ........................

D

ϕ(t)

q

ϕ (t)

q

ϕ(s) q

q xδ

q

x

Fig. 2.3 Sketch of ϕ and ϕδ within the domains D and D∗

D∗

2.4 Exit probabilities for non-degenerate systems

75

Now for any δ > 0 ﬁx some xδ ∈ D∩B(ϕ(s), δ) , where B(ϕ(s), δ) denotes the open ball with center ϕ(s) and radius δ . The function ϕδ (r) := ϕ(r) +

r δ x − ϕ(s) s

(r ∈ [0, T ])

is also absolutely continuous and belongs to Ψ◦ . Hence, the proof is comδ→0 pleted, if it is shown that I(ϕδ ) −−−→ I(ϕ) , in contradiction to the choice of ϕ as inf Ψ◦ I > I(ϕ) . For this purpose deﬁne the auxiliary functions 1 Aδr := a− 2 ϕ˙ δ (r) − b(ϕδ (r)) and

1 Br := a− 2 ϕ˙ (r) − b( ϕ(r))

(r ∈ [0, T ]) (r ∈ [0, T ]) ,

where a ≡ σσ ∗ , in order to get I(ϕδ ) − I(ϕ) 2 2 1 1 T − 1 δ = a 2 ϕ˙ (r) − b(ϕδ (r)) − a− 2 ϕ˙ (r) − b( ϕ(r)) dr 2 0 T 2 1 δ 2 Ar dr − Br ≡ 2 0 T 1 δ2 δ 2 2 δ δ |Ar | + |Br | − Ar , Br − Br , Ar dr Ar , Br − |Br | + = 2 0 T 1 δ (Ar − Br ), (Aδr − Br ) (Aδr − Br ), Br + dr = 2 0 T 2 δ 1 T δ Ar − Br dr Ar − Br , Br dr + = 2 0 0 2 δ 1 δ A − B L2 ([0,T ],Rd ) . ≡ (A − B), B L2 ([0,T ],Rd ) + 2 Therefore the claim has been reduced to showing that δ→0

Aδ − B −−−→ 0 in L2 ([0, T ], Rd ) ; a suﬃcient condition for this assertion is that δ→0

Aδr − Br −−−→ 0 uniformly with respect to r ∈ [0, T ] . For this purpose ﬁx an open domain D∗ ⊂ Rd such that D∗ ⊃ D ∪ {ϕ(r) : r ∈ [0, T ]} ∪ {ϕδ (r) : r ∈ [0, T ], δ ∈ (0, 1]} ; δ→0

since ϕδ −−−→ ϕ uniformly on [0, T ], this domain D∗ can be chosen to be bounded. Since b has been assumed to be C ∞ in (2.1), there is a (local)

76

2 Locality and time scales of the underlying non-degenerate system

Lipschitz constant C < ∞ such that | b(w1 ) − b(w2 ) | ≤ C |w1 − w2 |

(w1 , w2 ∈ D∗ ) .

Therefore, using that ϕ˙ δ (r) = ϕ(r) ˙ + 1s (xδ − ϕ(s)) , it follows altogether that 0 1 δ Ar − Br ≡ a− 12 ϕ˙ δ (r) − b(ϕδ (r)) − a− 12 ϕ˙ (r) − b( ϕ(r)) 1 1 1 1 = a− 2 (xδ − ϕ(s)) + a− 2 ϕ(r) ˙ − a− 2 ϕ(r) ˙ s − 12 δ − 12 − a b(ϕ (r)) + a b( ϕ(r)) 1 1 − 12 δ ≤ a x − ϕ(s) + a− 2 b(ϕδ (r)) − b(ϕ(r)) s

11 δ −2 δ ≤ a x − ϕ(s) + C ϕ (r) − ϕ(r) s δ→0

−−−→ 0 , uniformly with respect to r ∈ [0, T ] .

The above theorem yields information about the distribution of exit times of non-degenerate stochastic systems. The corresponding assertion concerning degenerate systems by Hern´andez-Lerma will appear later in 3.1.7. Theorem 2.4.3 furthermore motivates the following deﬁnition which had been anticipated at the beginning of this chapter (see p.56) for formulating assumption (V) in 2.1.1; the corresponding cost function in the context of degenerate systems will then appear in 3.1.2. Deﬁnition 2.4.4 (Quasipotential). Let I be the good rate function for the solution X ε of (2.1) as provided by theorem 2.3.4 . Then V : Rd × Rd × R>0 −→ R+ , V (x, y; s) := inf I[0,s],x (f ) : f ∈ Cx ([0, s], Rd ) , f (s) = y denotes the cost of forcing the system X ε to connect x and y in time s (in the sense of theorem 2.4.3) . The function V : Rd × Rd −→ R+ , V (x, y) := inf V (x, y; s) s>0

is the quasipotential of y with respect to x ; it is considered as the cost of forcing the system X ε to connect x and y eventually (see theorem 2.4.6). For a point O ∈ Rd ,

2.4 Exit probabilities for non-degenerate systems

77

V : Rd −→ R+ , V (y) := V (O, y) denotes the quasipotential of the system X ε (with respect to O) . The meaning of “quasipotential” is clariﬁed in the following proposition which is cited from Freidlin and Wentzell [Fr-We 98, p.118f.]: Proposition 2.4.5. Let D be a bounded open domain in Rd , suppose that σ = idRd and let the drift b ∈ C(Rd , Rd ) derive from a potential U with an orthogonal component L on D and let b have a unique equilibrium in D, i.e. suppose that b(x) = −(∇U )(x) + L(x) (x ∈ D) , where U ∈ C 1 (D, R) and L ∈ C(Rd , Rd ) are functions such that (∇U )(x), L(x) = 0 (x ∈ D) and for some O ∈ D, one can state that U (O) = 0 as well as and (∇U )(x) = 0

U (x) > 0

(x ∈ D \ {O}) .

Then for all x ∈ D for which U (x) ≤ min U , the quasipotential V with respect ∂D

to O is given by:

V (x) ≡ V (O, x) = 2 U (x) .

If in addition U ∈ C 2 , L ∈ C 1 and x ∈ D is some point, then the rate function I(−∞,T ],O , deﬁned analogously to I[0,T ],O , has a unique extremal ϕ on the set d f ∈ C (−∞, T ], R : lim f (s) = O , f (T ) = x ; s→−∞

furthermore, this extremal ϕ is the solution of the ODE ϕ(s) ˙ = + ∇U (ϕ(s)) + L(ϕ(s)) (s ∈ (−∞, T ]) , ϕ(T ) = x . In the general case, σ( . ) ≡ idRd , the above proposition remains true, if D is endowed with the Riemannian metric ds2 :=

d

a(x)−1

ij

dxi dxj

i,j=1

and , as well as ∇ now denote the scalar product and the Riemannian gradient with respect to this metric, respectively; see Freidlin and Wentzell [We-Fr 72, Th.1].

78

2 Locality and time scales of the underlying non-degenerate system

Further properties of the quasipotential V are the following (see Freidlin and Wentzell [Fr-We 98, Ch.4]): V (O, . ) is Lipschitz continuous, but not necessarily diﬀerentiable ([Fr-We 98, p.108,119]). The function V (x, y, s) satisﬁes a Hamilton-Jacobi equation ([Fr-We 98, p.107]). In case that V (O, . ) is continuously diﬀerentiable, a corresponding Jacobi equation follows; from this Jacobi variational equation a decomposition for b as in proposition 2.4.5 can be deduced ([Fr-We 98, p.119]). Furthermore, in the situation of proposition 2.4.5 one obtains that V (O, x) > 0 if and only if x = O (see Day and Darden [Day-Dar 85, Cor.2]). Next the fundamental exit law for non-degenerate systems will be discussed. It concerns the noise induced ﬁrst exit of X ε,x , given by (2.1), √ dXtε,x = b(Xtε,x ) dt + ε σ(Xtε,x ) dWt , X0ε,x = x ∈ D from a bounded, open domain D ⊂ Rd with smooth boundary ∂D. For this purpose, it is not necessary to impose the set of requirements 2.1.1 to the full extent. Instead, the following assumptions are underlying: (A1) There exists a unique stable equilibrium7 point O ∈ D of the deterministic system dXt0,x = b(Xt0,x) dt ,

X00,x = x ,

(2.8)

to which D is attracted.8 (A2) All trajectories of the deterministic system starting in ∂D converge to O (as t → ∞). (A3) V := inf V (O, . ) < ∞ . ∂D

(A4) There exist K, ρ0 > 0 such that for all ρ ≤ ρ0 and all x, y ∈ Rd for which |x − z| + |y − z| ≤ ρ

for some z ∈ ∂D ∪ {O} ,

there is a function u ≡ uρ;x,y ∈ L2 ([0, Tρ ], Rd ) such that u∞ < K and k(Tρ ) = y , where

t

b(k(s)) ds +

k(t) := x + 0

t

σ(k(s)) u(s) ds , 0 ρ→0

and where Tρ ≥ 0 is a time such that Tρ −−−→ 0. 7 O is an (asymptotically) stable equilibrium point of the deterministic dynamical system, if for any neighborhood B1 of O there exists another neighborhood B2 ⊂ B1 such that all trajectories of the deterministic system starting in B2 converge to O (as t → ∞) without leaving B1 ; of course, b(O) = 0 . 8 D is attracted to O, if all trajectories of the deterministic system X 0,x starting in D converge to (the equilibrium position) O (as t → ∞) without leaving D .

2.4 Exit probabilities for non-degenerate systems

79

In the situation described by the assumptions 2.1.1 the above requirements (A1)-(A4) are fulﬁlled, if for some i ∈ {1, . . . , l}, the domain D satisﬁes that O := Ki ∈ D and D ⊂ Di , where t→∞

Di := {x ∈ Rd : Xt0,x −−−−→ Ki } denotes the domain of attraction of Ki under the deterministic motion X 0 . The fact that necessarily D ⊂ Di , is due to (A1) and (A2) which exclude any other equilibrium (i.e. any other element in {K1 , . . . , Kl } \ {Ki}) from being in D. (A4) is implied by (E), see Dembo and Zeitouni [De-Zt 98, p.224]. In general, (A2) prevents that b(x), N (x) = 0, ∀x ∈ ∂D, where N (x) is the outer normal to ∂D at x ; in this situation ∂D is a characteristic boundary; for studies on this case see Day [Day 90] and the references therein. Dembo and Zeitouni [De-Zt 98, Cor.5.7.16] investigate the situation when (A2) is skipped, but when the boundary is not necessarily characteristic. The above assumptions (A1)–(A4) are taken from Dembo and Zeitouni [De-Zt 98, p.221ﬀ.]; this reference is also underlying to the subsequent discussion. Since this section is intended to provide the argumentation in outlines, some of the proofs will only be sketched and the reader is referred to Dembo and Zeitouni [De-Zt 98, Sec.5.7] for details. These results are due to Freidlin and Wentzell; see [We-Fr 70, §3] and [Fr-We 98, §§4.2,4.4]. Theorem 2.4.6. Let the assumptions (A1)-(A4) be satisﬁed and let τ ε,x ≡ inf{t ≥ 0 : Xtε,x ∈ / D} , denote the ﬁrst exit time of X ε,x , given by (2.1), from a bounded, open domain D ⊂ Rd with smooth boundary ∂D. Then it follows 1) for the ﬁrst exit time: For all x ∈ D and δ > 0, lim Px e(V −δ)/ε < τ ε < e(V +δ)/ε = 1 , ε→0

and for all x ∈ D,

lim ε log Ex τ ε = V ;

ε→0

2) for the ﬁrst exit position: If N ⊂ ∂D is a closed set for which inf V (O, . ) N

> V , then for any x ∈ D lim Px {Xτεε ∈ N } = 0 ;

ε→0

thus, if V (O, . ) has a unique minimum z ∗ on ∂D, then for any x ∈ D and δ > 0, lim Px Xτεε − z ∗ < δ = 1 . ε→0

80

2 Locality and time scales of the underlying non-degenerate system

The proof of Theorem 2.4.6 relies on the following lemmas. Here, Bρ (O) and Sρ (O) will denote the closed ball and the sphere around O with radius ρ, respectively; furthermore, the radii of all balls and spheres appearing are chosen so small such that the balls and spheres are contained in D. Lemma 2.4.7 (Continuity of V given (A4)). Assume condition (A4). Then there exits for any δ > 0 a suﬃciently small ρ > 0 such that sup

inf V (x, y, t) < δ

(2.9)

x,y∈Bρ (O) t∈[0,1]

as well as

sup x,y∈Rd : inf (|x−z|+|y−z|)≤ρ

inf V (x, y, t) < δ . t∈[0,1]

(2.10)

z∈∂D

Proof. Given x, y ∈ Rd for which |x−z|+|y −z| ≤ ρ for some z ∈ ∂D∪{O}, let k, u and K, Tρ be the functions and constants made available by (A4). Due to theorem 2.3.4, 1 t I[0,t],x (k) = inf | g(s) ˙ |2 ds . { g∈H1 : k(s) = x + 0s b(k(r)) dr + 0s σ(k(r)) g(r) ˙ dr } 2 0 Hence, (A4) implies that V (x, y; Tρ ) ≡ inf I[0,Tρ ],x (f ) : f ∈ Cx ([0, Tρ ], Rd ) , f (Tρ ) = y ≤ I[0,Tρ ],x (k) 1 Tρ 2 | u(s) | ds ≤ 2 0 ≤ K 2 Tρ /2 , which can become arbitrarily small for an appropriate choice of ρ, again due to (A4). Next, ﬁve lemmas are formulated from which 2.4.6 then can be proved. Here, lemma 2.4.7 is needed for 2.4.8 and 2.4.10. The LDP for X ε will be used in terms of theorem 2.3.6 in the proofs of the ﬁrst three of these lemmas. In doing so, the boundedness conditions on b und σ in theorem 2.3.6 (see 2.3.4) are tacitly assumed to be satisﬁed. This is no restriction, since the system is only examined until its ﬁrst exit time τ ε from D which only9 depends on the values of b and σ on D.

9

See e.g. Hackenbroch and Thalmaier [Hb-Th 94, 6.22].

2.4 Exit probabilities for non-degenerate systems

81

Lemma 2.4.8 (Uniform lower bound on the exit probability for starts near O). Assume the set of conditions (A). For any η > 0 there is then a ρ0 > 0 such that for all ρ ∈ (0, ρ0 ], there exists T0 < ∞ for which lim inf ε log inf Px {τ ε ≤ T0 } > − V + η . ε→0

x∈Bρ (O)

Proof. Given η > 0, apply lemma 2.4.7 for δ := η6 , to get ρ0 > 0 such that (2.9) and (2.10) hold for ρ0 — and hence also for all ρ ∈ (0, ρ0 ]. Fix such a ρ and an arbitrary x ∈ Bρ (O): (2.9) provides a path ψ x and tx ∈ [0, 1] such that ψ x (tx ) = O ,

ψ x (0) = x ,

η ; 3

I[0,tx ],x (ψ x ) ≤ δ <

due to (A3) there are a path ψ 0 , t0 > 0 and z ∈ ∂D such that ψ 0 (0) = O ,

I[0,t0 ],O (ψ 0 ) ≤ V +

ψ 0 (t0 ) = z ,

η ; 6

for this choice of z, (2.10) yields a ψ z , tz ∈ [0, 1] and y ∈ / D for which dist(y, ∂D) = ρ and ψ z (0) = z ,

I[0,tz ],z (ψ z ) ≤ δ ≡

ψ z (tz ) = y ,

η ; 6

ﬁnally, let ψ y := X 0,y denote the solution curve of the deterministic system (2.8) started in y and considered until time ty := 2 − (tx + tz ), ψ y (0) = y ,

I[0,2−(tx +tz )],y (ψ y ) = 0 .

Juxtaposing ψ x , ψ 0 , ψ z and ψ y results in a path φx which is deﬁned on [0, T0 ], where T0 := tx + t0 + tz + ty ≡ t0 + 2 and for which I[0,T0 ],x (φx ) ≤ I(ψ x ) + I(ψ 0 ) + I(ψ z ) + I(ψ y ) < V + η . Using these functions φx , x ∈ Bρ (O), the set : ρ Ψ := ψ ∈ C([0, T0 ], Rd ) : ψ − φx ∞ < 2 x∈Bρ (O)

is open and { X ε,x ∈ Ψ } ⊂ { τ ε,x ≤ T0 }. Therefore it follows from theorem 2.3.6 that lim inf ε log ε→0

inf x∈Bρ (O)

Px {τ ε ≤ T0 } ≥ lim inf ε log ε→0

≥−

sup

inf x∈Bρ (O)

P {X•ε,x ∈ Ψ}

inf I[0,T0 ],x (φ)

x∈Bρ (O) φ∈Ψ

82

2 Locality and time scales of the underlying non-degenerate system

≥−

sup x∈Bρ (O)

I[0,T0 ],x (φx )

> −(V + η) . Lemma 2.4.9 (X ε cannot stay in D arbitrarily long without approaching O). Assume the set of conditions (A). Then we have for any ρ > 0, lim lim sup ε log sup Px σρε > t = −∞ , t→∞

where of O,

σρε

ε→0

x∈D

denotes the ﬁrst hitting time of either ∂D or a small neighborhood σρε,x := inf{t ≥ 0 : Xtε,x ∈ Bρ (O) ∪ ∂D} .

Sketch of Proof. For all x ∈ Bρ (O), σρε,x = 0 ; thus, only initial values x ∈ D \ Bρ (O) are of relevance. The set of functions which do not leave the closure of the latter set, Ψt := ψ ∈ C([0, t], Rd ) : ψ(s) ∈ D \ Bρ (O) for all s ∈ [0, t] (t > 0) is closed and { σ ε,x > t } ⊂ { X ε,x ∈ Ψt } for x ∈ D \ Bρ (O). Hence, theorem 2.3.6 yields lim sup ε log ε→0

sup x∈D\Bρ (O)

Px {σ ε > t} ≤ lim sup ε log ε→0

sup

Px {X•ε ∈ Ψt }

x∈D\Bρ (O)

≤ − inf I[0,t],φ(0) (φ) , φ∈Ψt

which reduces the claim of the lemma to proving that the right hand side diverges (t → ∞). Via (A2), a Gronwall argument (b is C ∞ , hence Lipschitz on D) and the compactness of D \ Bρ (O) one can get T < ∞ such that for all x ∈ D \ Bρ (O), the solution φx (t) := Xt0,x of the deterministic system (2.8) is contained in the ball B2ρ/3 (O) for all t ≥ T . In order to obtain a contradiction suppose the divergence inf φ∈Ψt t→∞ I[0,t],φ(0) (φ) − −−− → ∞ were wrong; so imagine that there exists an M < ∞ such that for all n ∈ N, there is some ψ n ∈ ΨnT for which I[0,nT ],ψn (0) (ψ n ) ≤ M ; merely considering times nT is no restriction, since I is additive. Dissecting ψ n into n pieces and using the additivity of I again, one obtains φn ∈ ΨT n→∞ such that I[0,T ],φn (0) (φn ) ≤ M/n −−−−→ 0 . As the rate function I is good, ΨT ∩ {I ≤ 1} is compact, providing a limit point φ∗ ∈ ΨT of (φn )n . I being lower semicontinuous, I[0,T ],φ∗ (0) (φ∗ ) = 0 follows and φ∗ is necessarily a solution curve of the deterministic system (2.8). Due to the deﬁnition of T this implies that φ∗ (T ) ∈ B2ρ/3 (O), contradicting the fact that φ∗ (T ) ∈ / Bρ (O)◦ , as an element of ΨT .

2.4 Exit probabilities for non-degenerate systems

83

Lemma 2.4.10 (Bound on the probability of leaving D before further approaching O). Assume the set of conditions (A). Then for all closed sets N ⊂ ∂D, lim sup lim sup ε log sup Py Xσερε ∈ N ≤ − inf V (O, . ) . ρ→0

ε→0

y∈S2ρ (O)

N

Sketch of Proof. Deﬁne VN,δ := min[inf N V (O, . ) − δ , 1δ ] for δ > 0. Due to the previous lemma 2.4.9 there exists T < ∞ such that lim sup ε log sup Py σρε > T < −VN,δ . ε→0

y∈S2ρ (O)

Now, one applies theorem 2.3.6 to the closed set Φ := φ ∈ C([0, T ], Rd ) : φ(t) ∈ N for some t ∈ [0, T ] to see that −VN,δ also bounds the exponential growth rate of supy∈S2ρ (O) Py {X•ε ∈ Φ} from above (as ε → 0), where ρ ≡ ρ(δ) derives from (2.9). The same bound on the exponential rate holds true for Py Xσερε ∈ N ≤ Py {X•ε ∈ Φ} + Py σρε > T . Finally, take δ → 0. The ﬁnal two lemmas are not based on the large deviations principle. Remarkably, the assertion of the next lemma is not uniform with respect to the initial point; in contrast, the other lemmas contain uniformity information. This is why theorem 2.4.6 does not hold uniformly on D (but only uniformly on compact subsets of D). Lemma 2.4.11 (The probability of approaching O without leaving D is large). Assume the set of conditions (A). Then it follows for any ρ > 0 and x ∈ D that lim Px Xσερε ∈ Bρ (O) = 1 . ε→0

Sketch of Proof. Since σρε,x = 0 for x ∈ Bρ (O), ﬁx x ∈ D\Bρ (O). Again, there is T < ∞ such that Xt0,x ∈ Bρ/2 (O) for t ≥ T . Now it holds for ∆ := min dist {φx (t) : t ∈ [0, T ]}, ∂D , ρ > 0 that Xσερε ∈ ∂D ⊂ X•ε − X•0 [0,T ] > ∆/2 and the probability of the latter event can be estimated from above by means of a Gronwall argument and the Burkholder-Davis-Gundy maximal inequality.10 This upper bound converges to 0 as ε → 0. By the deﬁnition of σρε this is the converse of the claim. See Dembo and Zeitouni [De-Zt 98, p.234f.] for details.

10

See e.g. Dembo and Zeitouni [De-Zt 98, E.3] or Hackenbroch and Thalmaier [Hb-Th 94, 4.63].

84

2 Locality and time scales of the underlying non-degenerate system

Lemma 2.4.12 (Upper bound on the distance of X ε from its starting point). Assume the set of conditions (A). For all ρ, c > 0 there exists T ≡ T (ρ, c) < ∞ such that ; lim sup ε log sup P ε→0

x∈D

sup |Xtε,x − x| ≥ ρ

< −c .

t∈[0,T ]

Sketch of Proof. Also in this case a Gronwall argument is applied to |Xtε,x − x| . By the time change theorem11 of martingale theory the upper bound, hence obtained, can be further modiﬁed such that the statement is seen to hold true. See Dembo and Zeitouni [De-Zt 98, p.235f.] for details. Proof of Theorem 2.4.6: 1) Let x ∈ D and δ > 0 be ﬁxed. First, the bounds in probability for τ ε,x , ε→0 Px τ ε ≥ e(V +δ)/ε −−−→ 0 and Px

τ ε ≤ e(V −δ)/ε

ε→0

−−−→ 0 ,

will be proved; afterwards, the assertion on Ex τ ε will then follow from these arguments. (a) τ ε,x < e(V +δ)/ε Fix η > 0. Lemma 2.4.8 yields ρ, ε0 > 0 and T0 < ∞ such that inf x∈Bρ (O)

η

Px {τ ε ≤ T0 } > e−(V + 2 )/ε

(ε < ε0 ) .

With this choice of ρ lemma 2.4.9 provides some T1 < ∞ such that η 1 sup Px σρε > T1 < e− 4 ε (ε < ε0 ) . x∈D

Furthermore, choose ε0 suﬃciently small such that eη/(2 ε) − eη/(4 ε) ≥ 1

(ε < ε0 ) .

Setting T := T0 + T1 , the deﬁnitions of σρε and τ ε , the strong Markov property and the previous string of estimates imply that for all ε < ε0 :

11

See e.g. Dembo and Zeitouni [De-Zt 98, E.2] or Hackenbroch and Thalmaier [Hb-Th 94, 5.24].

2.4 Exit probabilities for non-degenerate systems

qε

85

ε,X ε ε := inf Px {τ ε ≤ T } ≥ inf Px σρε ≤ T1 , τ σρ ≤ T0 x∈D x∈D ≥ inf Px σρε ≤ T1 · inf Px {τ ε ≤ T0 } x∈D x∈Bρ (O) η − V > 1 − e−η/(4 ε) e ( + 2 )/ε η

≥ e−η/(2 ε) e−(V + 2 )/ε

=

e−(V +η)/ε .

An iteration of the strong Markov property12 implies that sup Px {τ ε > kT } ≤ (1 − q ε )k

(k ∈ N) ;

x∈D

more precisely, for k = 1 this is just the deﬁnition of q ε ; for k > 1, ε

Px {τ ε > kT } = Px {τ ε > (k − 1)T , τ ε,X(k−1)T > T } 5

6 ε,x F(k−1)T H ◦ X(k−1)T +• , = E 1{τ ε,x >(k−1)T } · 1 − E where H is deﬁned on the path space by H := 1{τ ≤T } for τ (f ) := inf{t ≥ 0 : ft ∈ / D}, i.e. τ ε,x ≡ τ ◦X•ε,x . With (T H)(z) := E(H ◦X•ε,z ) the strong Markov property13 implies: ε,x EF(k−1)T H ◦ X(k−1)T +• ε,x ≡ E 1{τ ≤T } ◦ X•ε,z ; = (T H) ◦ X(k−1)T ε,x z=X(k−1)T

plugging this into the previous equation one gets : 5

6 ε ε,z Px {τ > kT } ≤ E 1{τ ε,x >(k−1)T } · 1 − inf E 1{τ ≤T } ◦ X• z∈D

≡ 1 − q ε Px {τ ε > (k − 1)T } and thus by the induction assumption (IA): sup Px {τ ε > kT } ≤ x∈D

IA k 1 − q ε sup Px {τ ε > (k − 1)T } ≤ 1 − q ε . x∈D

This induction result yields together with the previously obtained bound on q ε :

12

Such an iterative application of the strong Markov property will also appear in the last chapter which is the reason for us to calculate details explicitly here. 13 See e.g. Hackenbroch and Thalmaier [Hb-Th 94, 6.32 & 6.41].

86

2 Locality and time scales of the underlying non-degenerate system

sup Ex τ ε ≤ sup T x∈D

x∈D

Px {τ ε > kT } ≤ T

k∈N0

(1 − q ε )k

k∈N0

T = ε ≤ T e(V +η)/ε , q

(†)

the upper bound on the mean exit time. For η := inequality implies:

δ 2

, Chebyshev’s

sup Px {τ ε ≥ e(V +δ)/ε } ≤ e−(V +δ)/ε sup Ex τ ε ≤ T e−δ/(2ε) −−−→ 0 . ε→0

x∈D

x∈D

(b) τ ε,x > e(V −δ)/ε Fix ρ > 0 (not necessarily as above) such that S2ρ (O) ⊂ D and deﬁne θ0 ε,x τm ε,x θm+1

:= :=

0 , ε,x inf { t ≥ θm : Xtε,x ∈ Bρ (O) ∪ ∂D } (m ∈ N0 ) , ε,x ε,x ε,x inf { t ≥ τm : Xt ∈ S2ρ (O) } , Xτm ∈ Bρ (O) (m ∈ N0 ) . := ∞ , Xτε,x ∈ ∂D m

......................................................................................... ................................ ................ ....................... ............ .................... .......... ................... ......... ............ ........ . . . . . ...... .... . ..... . . ..... . . .... ... .... . .. .... . ... . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .......... ... 2ρ ....... . . . . .. . . . . . . ... . . . ....... . ..... . . . . . . .. . . . . . . ......... ............... ...... ... .. . . . ... ...... . . . . . . . ...... ..... .. ...... ........ .. . .... .... ... ... ........ ... . . . . . . . . . . . . ... ... ε,x .... ... ... .... .. . . . ... ... .. . .................. .. . . . . .. 0 . . . . . . . . . . ... ............................... ................. .................. ...... .. .. . . . . ... .......... .. .................. . .. .. . . . . . . . . ... . ... .. ....... .. . . . ... .... . .. . . .. ... ...... .... ... . . . .. ... ... .. .. ... . .. .... .... .. .. . . .... ... ... ... .... .. . .. .... ... ... ... ... ... ... .. .. ... .. ... .. ... ... .. ..... .. ... ... .............. .. ... .. ... ... .. .. .. .. .... ... ... . . . . . .... . ... ... ... .... . ... ... ... .... ........ ε,x ... .. ... .. ....... ... .... ... ... .. .. ..... . ... .. ... .... ....... .................... 1 ................ ... ... ..... . ... . . . . ........... . . . . . . . . . . . . ... ... . .. .. ... ..... ......... ... ... ... ... ε,x ..... ... ................ ... ..... ... ... ... ............... .. .... .... ... ... ..... .. ... ...............1 ..... ... . . ... . . . . . . . . . . . . . .......... ...... ... ... ... .. . ...... .. ... .. ........ .. ....... .. .. ....... ..... ... ......... .... ... .. .................. .......................... ... ... ... .......... ... .. ... . ... . . . . . . . . ............... ... ε,x .... .. ... ..... ... ..... ........ 2 ... ..... ........ ............... ... ........................................... ......... . . . . . . . . ................ . . . . . . ............. ........... . ............ ........................... ............................. ................. ................. ..... ....... .......... .................... ......................... ............... ....................... ........................................................................................................................

S

xq

τ

(0)

r

O

s

τr

rθ

r

D

θ

r

τ2ε,x

= τ ε,x

X•ε,x

ε,x ε,x Fig. 2.4 Sketch of the stopping times τm and θm+1 .

ε,x From these deﬁnitions, it follows that on {τ ε,x = τm } for m ≥ 1, X•ε,x hits Bρ (O) before it hits ∂D. Thus for all m ≥ 1, ε } ≤ sup Py Xσερε ∈ ∂D . Px {τ ε = τm y∈S2ρ (O)

2.4 Exit probabilities for non-degenerate systems

87

By applying lemma 2.4.10 with N := ∂D, ρ and ε0 > 0 can be chosen such that δ sup Py Xσερε ∈ ∂D < e(−V + 2 )/ε y∈S2ρ (O)

for all ε < ε0 . Consequently, ρ > 0 has been ﬁxed such that for all m ∈ N and ε < ε0 , ε } < e−(V − 2 )/ε . sup Px {τ ε = τm δ

(††)

x∈D

ε,x ε,x −τm−1 ≤ T2 (where From the above deﬁnitions it also follows that if θm T2 < ∞ is some parameter to be speciﬁed in a moment), then X•ε,x must cover the distance ρ between Bρ (O) and S2ρ (O) in time T2 , i.e. ε ε,y ε sup |Xt − y| ≥ ρ (m ∈ N). Px θm − τm−1 ≤ T2 ≤ sup P y∈D

t∈[0,T2 ]

Furthermore, using lemma 2.4.12 with the above choice of ρ and c := V + δ/2, one can choose T2 := T (ρ, c) < ∞ and modify ε0 > 0 such that for all ε < ε0 , δ ε,y sup P sup |Xt − y| ≥ ρ < e−(V − 2 )/ε . y∈D

t∈[0,T2 ]

Consequently, T2 < ∞ has been ﬁxed such that for all m ∈ N and ε < ε0 : δ ε ε sup Px θm − τm−1 ≤ T2 < e−(V − 2 )/ε . (†††) x∈D

Assume for the moment that τ ε,x ≤ k T2 for some k ∈ N : By deﬁnition ε,x there is a random variable m ∈ N0 such that τ ε,x = τm ; here, either ε,x m ≤ k or m > k, but in the latter case X• has already performed m > k hits of Bρ (O), (see ﬁgure 2.4); if the lengths of these excursion ε,x ] were ≥ T2 for all n = 0, . . . , m − 1 (≥ k), then time intervals [τnε,x , τn+1 k T2 < m T2 ≤ τ0ε,x +

m−1

ε,x τn+1 − τnε,x = τ ε,x ≤ k T2 ,

n=0

a contradiction. Therefore it follows for all k ∈ N0 that { τ ε,x ≤ k T2 } ⊂

k :

ε,x { τ ε,x = τm } ∪

m=0

k : m=1

ε,x ε,x θm − τm−1 ≤ T2

88

2 Locality and time scales of the underlying non-degenerate system

and thus Px { τ ε ≤ k T 2 } ≤

k

ε Px { τ ε = τm } +

m=0

k

ε ε P x θm − τm−1 ≤ T2

m=1

≤ Px { τ = ε

τ0ε

} + 2ke

−(V − δ2 )/ε

for all ε < ε0 , where in the last estimate, (††) and (†††) have been used. Now plug in = < k := T2−1 e(V −δ)/ε + 1 ( . denoting the integer part) to get: Px τ ε ≤ e(V −δ)/ε ≤ Px { τ ε ≤ k T2 } ≤ Px { τ ε = τ0ε } + 2 k e−(V − 2 )/ε δ

≤ Px { Xσερε ∈ / Bρ (O) } + 4 T2−1 e−δ/(2ε)

ε→0

−−−→ 0

by lemma 2.4.11. Altogether, the ﬁrst claim concerning the ﬁrst exit time is proven. lim ε log Ex τ ε = V

(c)

ε→0

The upper bound, lim sup ε log Ex τ ε ≤ V , has ε→0

already been veriﬁed in (†). For the lower bound note that Chebyshev’s inequality states that Px {τ ε > e(V −δ)/ε } ≤ e−(V −δ)/ε Ex τ ε and thus, as Px {τ ε > e(V −δ)/ε } → 1, it follows that for any δ > 0, ε→0 ε log Ex τ ε ≥ V − δ + ε log Px τ ε > e(V −δ)/ε −−−→ V − δ + 0 . 2) lim Px {Xτεε ∈ N } = 0 ε→0

Here, N ⊂ ∂D denotes a ﬁxed closed set such

that VN := inf N V (O, . ) > V ; if inf N V (O, . ) = ∞, then some VN ∈ (V , ∞) is ﬁxed instead. Again, for any ρ > 0 (to be speciﬁed later), Px {Xτεε ∈ N } ≤ Px Xσερε ∈ / Bρ (O) + sup Py {Xτεε ∈ N } . y∈Bρ (O)

By lemma 2.4.11 the ﬁrst summand on the right tends to 0 as ε → 0 for any choice of ρ. Thus it remains to be veriﬁed that also

2.4 Exit probabilities for non-degenerate systems

89 ε→0

Py {Xτεε ∈ N } −−−→ 0 .

sup y∈Bρ (O)

Fix η ∈ 0, VN3−V and choose ρ, ε0 > 0 according to lemma 2.4.10 such that sup z∈S2ρ (O)

Pz {Xσερε ∈ N } ≤ e−(VN −η)/ε

(‡)

ε,x and the strong Markov property it for all ε < ε0 . From the deﬁnition of τm follows that for all T3 ∈ R+ and κ ∈ N0 , ;

sup |Xtε,z − z| ≥ ρ

sup Pz {τκε ≤ κ T3 } ≤ κ sup P

z∈D

z∈D

;

t∈[0,T3 ]

on the other hand, by applying lemma 2.4.12 with the above choice of ρ and c := VN − η, one can choose T3 ≡ T3 (ρ, VN , η) < ∞ such that the latter term is further estimated as ; ≤ e−(VN −η)/ε

sup |Xtε,z − z| ≥ ρ

sup P z∈D

t∈[0,T3 ]

for all ε < ε0 . Consequently, there is T3 < ∞ such that for all ε < ε0 and κ ∈ N0 : (‡‡) sup Pz {τκε ≤ κ T3 } ≤ κ e−(VN −η)/ε . z∈D

ε,Xθε,y ε,y ε ε,y ε,y and τ ε,y ≡ θ ε,y + σ m , the Since14 τ ε,y > τm−1 ∈ Fτm−1 ⊂ Fθ m m m ρ strong Markov property implies here that for any y ∈ D : Py

ε ε ∈ N Xτεm ∩ {τ ε > τm−1 } =

=

dP EFθm 1 X ε,y ∈N ε,y

{

ε,y τ ε,y >τm−1

}

ε,y {τ ε,y >τm−1 }

ε

≤ Py {τ >

ε τm

dP P Xσε,z ε ∈ N ρ

ε τm−1 }

·

sup z∈S2ρ (O)

Pz

z=Xθε,y ε (.)

∈N . m

Xσερε

Now it follows for all y ∈ Bρ (O), κ ∈ N and ε < ε0 : Py {Xτεε ∈ N } ≤

κ

ε ε ε ∈ N , τ Py Xτεm > τm−1

m=1

≤κ

sup z∈S2ρ (O)

+ Py {τ ε > τκε }

Pz Xσερε ∈ N + Py {τ ε > κ T3 } + Py {τκε ≤ κ T3 }

≤ 2κ e−(VN −η)/ε + Py {τ ε > κ T3 } ; 14

See e.g. Hackenbroch and Thalmaier [Hb-Th 94, p.93f.].

90

2 Locality and time scales of the underlying non-degenerate system

the ﬁrst inequality is a general decomposition of the set {Xτεε ∈ N } in which it has been used that τ ε,y > τ0ε,y = 0 for all y ∈ Bρ (O) by deﬁnition; the second step follows from the previous application of the strong Markov property and the last inequality is due to (‡) and (‡‡). Further reducing ε0 (if needed) such that (†) holds true for all ε < ε0 , using Chebyshev’s inequality and plugging in the integer part of e(V +2η)/ε , < = κ := e(V +2η)/ε , and using that V − VN + 3η < 0 due to the choice of η, the previous estimate implies: lim sup

sup

ε→0

y∈Bρ (O)

Py {Xτεε ∈ N }

T (V +η)/ε ≤ lim sup 2κ e−(VN −η)/ε + e κ T3 ε→0 - . T ≤ lim sup 2e V −VN +3η /ε + ε→0 T eη/ε − e−(V +η)/ε

=

0.

3

As has been observed above, this concludes the proof of the claim, lim Px {Xτεε ∈ N } = 0. ε→0 lim Px Xτεε − z ∗ < δ = 1 follows hereby with N := {z ∈ ∂D : ε→0

|z − z ∗ | ≥ δ}.

Remark 2.4.13 (the corresponding Stratonovich SDE). Instead of proposing the Itˆ o-SDE (2.1) as above, it is also intriguing to consider the corresponding Stratonovich-SDE, √ X0ε,x = x , dXtε,x = b (Xtε,x ) dt + ε σ (Xtε,x ) ◦ dWt , where “◦” denotes the stochastic integral in the sense of Stratonovich.15 This equation is equivalent to the Itˆ o-SDE 0 1 √ ε dXtε,x = b (Xtε,x ) + σ (Xtε,x ) Dσ (Xtε,x ) dt + ε σ (Xtε,x ) dWt , 2 where the coordinates of σ(x) Dσ(x) are given by

σ(x) Dσ(x)

k

:=

d i,j=1

σij (x)

∂σkj (x) ∂xi

for k = 1, . . . , d . Hence, one also needs to cope with an ε-dependent drift 15

See e.g. Arnold [Ar 98, Ch.2] or Hackenbroch and Thalmaier [Hb-Th 94, Ch.5].

2.5 Sublimiting distributions: Metastability and quasi-deterministic behavior

bε (x) := b (x) +

91

ε σ (x) Dσ (x) 2

in the Itˆ o equation. Large deviation results on such equations can also be obtained: Freidlin and Wentzell (see [Fr-We 98, p.154f.] and [We-Fr 70, p.7ﬀ.]) prove that the action functional is the same as for (2.1), if the coordinates of bε converge to those of b uniformly in x as ε → 0. Therefore it would be conceptually equivalent to investigate the above Stratonovich-SDE instead of (2.1), as Freidlin [Fr 00] does indeed; also see Carmona and Freidlin [Cm-Fr 03] who assume the coeﬃcients to be (globally) bounded. In general, one then needs to assure that ε ε→0 Dσ (Xtε,x ) σ (Xtε,x ) −−−→ 0 2 uniformly with respect to x. It seems as if this assumption should be added to Freidlin’s [Fr 00] premises. However, the rationale of our work is to restrict the assumptions on b and σ to a minimum in order not to overburden the technical apparatus for installing locality into the treatment of linear diﬀerential systems: Since in most applications (see section 2.6), where σ = idRd , the Itˆ o-SDE and the Stratonovich-SDE coincide, it seems reasonable to stick to the Itˆo-SDE.

2.5 Sublimiting distributions: Metastability and quasi-deterministic behavior This subsection describes the concept of metastability, made precise by the convergence of the system to the sublimiting distributions on the respective time scales; the exposition here follows Freidlin [Fr 77], [Fr 00]; also see Freidlin and Wentzell [Fr-We 98, Ch. 6] as well as Carmona and Freidlin [Cm-Fr 03]. It is required throughout the section that the standing assumptions 2.1.1 are satisﬁed. For describing the behavior of X ε a hierarchy of “cycles” in the state space, together with their rotation and exit rate as well as their main state and the entrance point to the next cycle is needed. More precisely, the cycles will be deﬁned as certain subsets of L := {1, . . . , l} , equipped with a cyclic order which expresses the order of transition between the domains of (Ki )i∈L . In this construction the constants Vij := V (Ki , Kj )

(i, j ∈ L)

will play a central role. Due to (K) the entries of this matrix (Vij )i,j∈L are zero on the diagonal and strictly positive otherwise. In case that the sets Ki

92

2 Locality and time scales of the underlying non-degenerate system

do not consist of one point only, but are compact subsets of Rd , this deﬁnition of Vij would read Vij := V (x, y) for some x ∈ Ki and y ∈ Kj

(i, j ∈ L) ;

this deﬁnition then does not depend on the choices of x and y due to assumption (K3) in remark 2.1.2. Hence, the matrix (Vij )i,j∈L is well deﬁned also in this case. One more deﬁnition is needed (Freidlin and Wentzell [Fr-We 98, p.177]) now. Deﬁnition 2.5.1 (arrows, W-graphs on L). Let L := {1, . . . , l} be as above (or an arbitrary ﬁnite set whose elements are labeled as 1, . . . , l). (a) An arrow (α → β) is an ordered tuple (α, β) ∈ L2 where α and β are called the initial point and the endpoint of the arrow, respectively. (b) A set g of arrows is called a graph on L. (c) Let a subset W ⊂ L be ﬁxed; then a set g of arrows is called a W -graph on L, if 1) (α → β) ∈ g implies that α ∈ L \ W and α = β ; 2) for any α ∈ L\W there exists exactly one element of g whose initial point is α; 3) for any α ∈ L\W there exists a sequence in g leading from α into W (i.e. a sequence α → α1 → α2 → · · · → αm−1 → αm of arrows contained in g such that αm ∈ W ) . Given the initial two assumptions 1) and 2), 3) is equivalent to the following one: 4) There are no closed loops in g (i.e. for any sequence α0 → α1 → α2 → · · · → αm−1 → αm of arrows contained in g, it follows that α0 = αm ) . Let GW (L) := { g : g W -graph on L} denote the set of all W -graphs on L; if W consists of one element i only, one deﬁnes Gi (L) := G{i} (L) , the set of i-graphs on L. In other words, Gi (L) is the set of all sets g of arrows (α → β) such that α ∈ L \ {i}, to each initial point α exactly one element of g is attached and by forming sequences α0 → α1 → α2 → · · · → αm−1 → αm of arrows in g no closed paths are possible: α0 = αm .

2.5 Sublimiting distributions: Metastability and quasi-deterministic behavior

93

Example 2.5.2 (Simple cases of i-graphs). Let L ≡ {1, . . . , l}. Then one obtains for l = 1: G1 (L) = ∅ , l = 2: G1 (L) = {(2 → 1)} and G2 (L) = {(1 → 2)} , and l = 3: G1 (L) = {(2 → 1), (3 → 1)} , {(2 → 1), (3 → 2)} , {(3 → 1), (2 → 3)} . 1

r oS S 7 S Sr r 3 2

1

1

r

3 r

SS o S Sr 2

r

r 3

7 r2

Fig. 2.5 Visualization of the 1-graphs on {1, 2, 3}

The Hierarchy of cycles The hierarchy shall describe the transitions of X ε,x between the diﬀerent domains of attraction which belong to K1 , . . . , Kl . Successively the k-cycles will be deﬁned. The cycles of any rank k are subsets of L equipped with a cyclic order. Furthermore, to any cycle there will be assigned the main state, the rotation rate and the exit rate. The notation and terminology coined here will be illustrated in example 2.6.2 for the instructive situation of a two-well potential function. k = 0: A 0-cycle is an element of L, i.e. the set of all 0-cycles is deﬁned as C (0) := L. M : C (0) → L , M (i) := i deﬁnes the main state of a 0-cycle; the rotation rate R and the stationary distribution rate mi vanish on 0-cycles by deﬁnition, R(i) := 0

and

mi (i) := 0

(i ∈ C (0) ≡ L) ,

respectively. The exit rate E(i) of a 0-cycle i is given by the mapping ⎧ ⎨ min ViJ , l ≡ |L| ≥ 2 , (2.11) E(i) := J∈L\{i} E : C (0) → (0, ∞] , ⎩ ∞ , l =1. The meaning of the terminus “exit rate” (which will be also deﬁned for cycles of higher order) will be explained in theorem 2.5.3; it states that E yields the precise exponential rates for the exit time of the respective domains, analogously to the exit time law 2.4.6.

94

2 Locality and time scales of the underlying non-degenerate system

Now if l ≥ 2 (i.e. if the situation is non-trivial for what follows) and if i ∈ L is ﬁxed, one can choose J ≡ J(i) ∈ L \ {i} such that this minimum (2.11) is attained; by (G) it is actually postulated that this J is unique; thus there is a well-deﬁned mapping J : C (0) → C (0) , implicitly given by Vi,J(i) = E(i) ≡ min ViJ . J∈L\{i}

This deﬁnition of J is also expressed by saying that J(i) follows after i. k = 1: Applying the mapping J successively yields a sequence {J r (i)}r∈N0 in C (0) for each i ∈ C (0) . As C (0) ≡ L is a ﬁnite set, this sequence returns to itself at step r(i) := min{ r : J n (i) = J r (i) for some n < r }

(i ∈ C (0) ) .

At the step r(i) the sequence becomes periodic. Therefore it decomposes into 0-cycles i, J(i), . . . , J n(i)−1 (i) and the closed loop {J n(i) (i), J n(i)+1 (i), . . . , J r(i) (i) = J n(i) (i)}. If the cyclic order of the loop (which is simply a subset of L) shall be emphasized, it is written as succession of arrows, J n(i) (i) → J n(i)+1 (i) → · · · → J r(i) (i) = J n(i) (i) ; this loop does not contain i, J(i), . . . , J n(i)−1 (i) by deﬁnition. We conclude that each sequence {J r (i)}r∈N0 is characterized by the objects i, J(i), . . . , J n(i)−1 (i) , J n(i) (i) → J n(i)+1 (i) → · · · → J r(i) (i) = J n(i) (i) which are called the 1-cycles generated by i. Let C (1) denote the set of all 1-cycles generated by some i ∈ C (0) ; its elements are called 1-cycles. Now M : C (1) → L , implicitly given by the requirement VM(C),J(M(C)) = max Vi,J(i) i∈C

(2.12)

deﬁnes the main state of a 1-cycle C ; again, the uniqueness of such an element M (C) and hence the well-deﬁnedness of the main state mapping M is postulated in assumption 2.1.1(G). The rotation rate R of 1-cycles is given by R : C (1) → R+ ,

R(C) := max Vi,J(i) , i∈C

(2.13)

2.5 Sublimiting distributions: Metastability and quasi-deterministic behavior

95

the stationary distribution rate mC for a 1-cycle C is the map16 m C : C → R+ ,

mC (i) := R(C) − Vi,J(i)

and the exit rate E(C) of a 1-cycle C is deﬁned by ⎧ ⎨ min (mC (i) + Vij ) , C (1) ≥ 2, (1) / E : C → (0, ∞], E(C) := i∈C,j ∈C ⎩ ∞ , C (1) = 1.

(2.14)

If C (1) consists of one element only, this is equal to L, considered as a point if (1) set; in this case the recursive deﬁnition of cycles stops. Otherwise,(1) C ≥ 2, there is a cyclic order on these 1-cycles: Fix an arbitrary C ∈ C and deﬁne i∗ , j ∗ ∈ L by the requirement mC (i∗ ) + Vi∗ j ∗ =

min (mC (i) + Vij ) ≡ E(C) .

i∈C,j ∈C /

By (G) we assume that i∗ and j ∗ are unique, since they are uniquely determined by (2.14). Now let J(C) ∈ C (1) be the (unique) 1-cycle which contains j ∗ . This procedure provides a well-deﬁned mapping J : C (1) → C (1) and one says that the 1-cycle J(C) follows after the 1-cycle C. k > 1: Assume that the set C (k) of k-cycles has already been deﬁned and that the main state map M , the rotation rate R, the stationary distribution rate m, the exit rate E and the “follow”-map J have been extended to C (k) . Iterating J yields a sequence {J r (C)}r∈N0 in C (k) for each C ∈ C (k) . As C (k) is ﬁnite, this sequence returns to itself at step r(C) := min{ r : J n (C) = J r (C) for some n < r }

(C ∈ C (k) ) .

Therefore the sequence decomposes into C, J(C), . . . , J n(C)−1 (C) , J n(C) (C) → J n(C)+1 (C) → · · · → J r(C) (C) = J n(C) (C) which are called the (k + 1)-cycles generated by C. C (k+1) denotes the set of (k + 1)-cycles, i.e. of all (k + 1)-cycles generated by some C ∈ C (k) . For a

16

Freidlin [Fr 00, p.338] deﬁnes the stationary distribution rate mC for a 1-cycle C with the opposite sign, i.e. mC (i) := Vi,J(i) − R(C); however, this would result in a negative rate; here, we use the corrected deﬁnition of mC (equivalently of E(C)) for 1-cycles from Carmona and Freidlin [Cm-Fr 03, (2.2)].

96

2 Locality and time scales of the underlying non-degenerate system

given C ∈ C (k+1) the requirement Vαβ = min min min g∈GM (C)

i∈C g∈Gi (C)

(α→β)∈g

Vαβ

(2.15)

(α→β)∈g

uniquely determines M ≡ M (C) ∈ C(⊂ L) thus extending the main state map M : C (k+1) → L on (k + 1)-cycles; recall that Gi (C) denotes the set of i-Graphs on C as in deﬁnition 2.5.1. The rotation rate R of (k + 1)-cycles is given by R : C (k+1) → R+ ,

R(C) := max{ E(C ) : C ∈ C (k) , C ⊂ C } ;

(2.16)

the stationary distribution rate mC for a (k + 1)-cycle C is the map mC (i) := min Vαβ − min m C : C → R+ , g∈Gi (C)

(α→β)∈g

g∈GM (C) (C)

Vαβ

(α→β)∈g

and the exit rate E(C) of a (k + 1)-cycle C is deﬁned by ⎧ ⎨ min (mC (i) + Vij ) , C (k+1) ≥ 2 , (k+1) / → (0, ∞] , E(C) := i∈C,j ∈C E :C ⎩ ∞ , C (k+1) = 1 . (2.17) If C (k+1) consists of one element only, this is equal to L as a point set and the recursive deﬁnition of cycles stops. Otherwise, if C (k+1) ≥ 2, there is a cyclic order on the (k + 1)-cycles again: Fix an arbitrary C ∈ C (k+1) and deﬁne i∗ , j ∗ ∈ L by the requirement mC (i∗ ) + Vi∗ j ∗ =

min (mC (i) + Vij ) ≡ E(C) ,

i∈C,j ∈C /

where mC (i) denotes the newly deﬁned stationary distribution rate for (k+1)cycles. By (G) we assume that i∗ and j ∗ are unique, since they are uniquely determined by (2.17). Now let J(C) ∈ C (k+1) be the unique (k + 1)-cycle which contains j ∗ ; i∗ is called the exit point of C and j ∗ is the entrance point of J(C). This procedure now provides a well-deﬁned map J : C (k+1) → C (k+1) and we say that the (k + 1)-cycle J(C) follows after the (k + 1)-cycle C. Example 2.6.2 shall give an illustration for the above deﬁnitions. There, a two well potential function is considered and the previous construction of the hierarchy of cycles is seen to terminate at k = 1. For a discussion of the three well situation which corresponds to a maximal degree k = 2 see Freidlin [Fr 00, p.344].

2.5 Sublimiting distributions: Metastability and quasi-deterministic behavior

97

Next, an explanation of the above terminology “exit rates” shall be cited. For this purpose let t→∞

Di := {x ∈ Rd : Xt0,x −−−−→ Ki }

(i ∈ L)

denote the domain of attraction of Ki under the deterministic motion X 0 as before which is an open subset of Rd . Furthermore, deﬁne the open set .◦ : Di D(C) := i∈C

for cycles C. The set D(C) is connected, since by taking the closures Di also the lower dimensional submanifolds are taken into account which are attracted to one of the points Kl+1 , . . . , Kl and which thus separate the domains D1 , . . . , Dl ; see remark 2.1.2 and assumption 2.1.1 (K1). Hence, > investigating the open set D(C) := i∈C Di instead17 would result in considering a disconnected set. Its topological components are the sets Di , separated by those lower dimensional submanifolds which are attracted to one of the points Kl+1 , . . . , Kl . Therefore, the corresponding exit time from D(C) coincides with the exit time from the domain Di containing the initial condition x. However, this is not the time of interest in the following discussion; rather one is concerned with the exit time from the whole cycle C in the sense that X ε,x exits the domain “spanned” by the domains Di for which i ∈ C and which also includes the separatrices between them. Theorem 2.5.3 (The exit rates). Let τCε,x := inf{ t > 0 : Xtε,x ∈ / D(C) } denote the ﬁrst exit time of X ε,x from D(C). Then for any x ∈ D(C), lim ε log Ex τCε = E(C)

ε→0

and for all δ > 0, lim Px e(E(C)−δ)/ε < τCε < e(E(C)+δ)/ε = 1 .

ε→0

This generalization of theorem 2.4.6 is taken from Freidlin [Fr 00, p.339], Carmona and Freidlin [Cm-Fr 03, p.61f.] and Freidlin and Wentzell [Fr-We 98, Th.6.6.2], respectively. Being technically very involved the proof of this theorem is beyond the scope of this book: It is necessary to consider balls around the points Ki for which stopping times are then deﬁned analogously as in the proof of theorem 2.4.6 above; ﬁgure 2.4 then depicts the case for l = 1. For 17

As Freidlin [Fr 00] and Carmona and Freidlin [Cm-Fr 03] do

98

2 Locality and time scales of the underlying non-degenerate system

the arguments of the proof see Freidlin and Wentzell [Fr-We 98, Ch.6]; also see Freidlin [Fr 77]. Note that the exit rates E( . ) considered here coincide with the rates C( . ) as used in the formulation of theorem 6.6.2 by Freidlin and Wentzell [Fr-We 98]: More precisely, let π be some cycle and deﬁne Vαβ , (2.18) C(π) := A(π) − min min i∈π g∈Gi (π)

where A(π) :=

min

g∈GL\π (L)

(α→β)∈g

Vαβ

(2.19)

(α→β)∈g

(see p.199 and p.201 in Freidlin and Wentzell [Fr-We 98]). Now GL\π (L) consists of those graphs whose arrows take their courses in π and whose last endpoint is in L \ π; this is the same as considering graphs in π terminating in i ∈ π (elements of Gi (π) in other words) and proceeding to some j ∈ / π subsequently. Hence, unwinding the deﬁnitions of graphs and main states yields that for any cycle π of order k > 1, Vαβ − min min Vαβ C(π) ≡ min g∈GL\π (L)

⎛

= min ⎝ min i∈π,j ∈π /

g∈Gi (π)

⎛ = min ⎝ min i∈π,j ∈π /

i∈π g∈Gi (π)

(α→β)∈g

g∈Gi (π)

(α→β)∈g

⎞

Vαβ + Vij ⎠ −

(α→β)∈g

(α→β)∈g

Vαβ −

min

min

g∈GM (π) (π)

g∈GM (π) (π)

(α→β)∈g

Vαβ ⎞

Vαβ + Vij ⎠

(α→β)∈g

≡ min (mπ (i) + Vij ) i∈π,j ∈π /

≡ E(π) . Therefore the above theorem coincides with Theorem 6.6.2 by Freidlin and Wentzell [Fr-We 98]. Furthermore, note that the compactness assumption on the state space which is underlying § 6.6 by Freidlin and Wentzell [Fr-We 98] is replaced here by assumption (V), i.e. by considering the compact set {y : V (0, y) ≤ V˜ } for some large V˜ instead which X ε leaves only with an exponentially small probability; see Freidlin [Fr 77].

The support of the sublimiting distribution Using the hierarchy of cycles we shall now deﬁne the metastable state Kµ(x,ζ) for an initial value x and a time scale eζ/ε . Again the presentation follows

2.5 Sublimiting distributions: Metastability and quasi-deterministic behavior

99

Freidlin and Wentzell ([Fr 77], [Fr 00] and [Fr-We 98, § 6.6]). An illustration will be provided in example 2.6.2 . Fix an initial value l : Di . x ∈ i=1

This set has full Lebesgue measure by assumption 2.1.1 (K). Choose i(x) ∈ L such that x ∈ Di(x) and let C (k) (x) denote the (unique) element of C (k) such that i(x) ∈ C (k) (x), i(x) ≡ C (0) (x) ⊂ C (1) (x) ⊂ · · · ⊂ C (κ−1) (x) ⊂ C (κ) = L , where κ < ∞ denotes the maximal rank in this sequence. This yields the ﬁnite sequences Ek (x) := E C (k) (x) and Rk (x) := R C (k) (x) (k = 0, 1, . . . , κ) of the corresponding exit and rotation rates, respectively. For these rates it follows that 0 ≡ R0 (x) < E0 (x) ≡ Vi(x),J(i(x)) ≤ R1 (x) ≤ E1 (x) ≤ · · · ≤ Rk (x) ≤ Ek (x) ≤ · · · ≤ Rκ (x) ≤ Eκ (x) ≡ ∞ Now ﬁx ζ ∈ R>0 \

κ :

{Rk (x), Ek (x)}

(2.20)

k=0

and deﬁne

m∗ ≡ m∗ (x) ∈ {−1, 0, 1, . . . , κ − 1}

by Em∗ (x) < ζ < Em∗ +1 (x) , where we put E−1 (x) := 0 for deﬁniteness. There are two cases to distinguish: ∗ ζ > Rm∗ +1 (x): In this case deﬁne µ(x, ζ) := M C (m +1) (x) . ∗

ζ < Rm∗ +1 (x): Since Rm∗ +1 (x) ≡ max{E(C ) : C ∈ C (m ) , C ⊂ C (m there is a ∗ ∗ Cˆ (m ) ∈ C (m )

such that

∗ ∗ Cˆ (m ) ⊂ C (m +1) (x)

∗

+1)

(x)},

∗

and E(Cˆ (m ) ) > ζ ;

∗ we also assume that Cˆ (m ) is the ﬁrst m∗ -cycle satisfying this statement ∗ ∗ which follows after C (m ) (x) in C (m +1) (x). Again there are two cases to distinguish: ∗

∗

ζ > R(Cˆ (m ) ): In this case deﬁne µ(x, ζ) := M (Cˆ (m ) ) .

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2 Locality and time scales of the underlying non-degenerate system

∗ ζ < R(Cˆ (m ) ): By the same argument as above there is a ∗ ∗ Cˆ (m −1) ∈ C (m −1)

∗ ∗ Cˆ (m −1) ⊂ Cˆ (m ) ,

such that

E(Cˆ (m

∗

−1)

)>ζ

∗

and Cˆ (m −1) is the ﬁrst (m∗ − 1)-cycle which follows after the (m∗ − 1)∗ cycle containing the entrance point of Cˆ (m ) . Again there are two cases to distinguish: ∗

∗

ζ > R(Cˆ (m −1) ): In this case deﬁne µ(x, ζ) := M (Cˆ (m −1) ) . ∗ ζ < R(Cˆ (m −1) ): We proceed successively as in the previous step till we ﬁnd ∗ ∗ Cˆ (m −n) ∈ C (m −n)

such that

E(Cˆ (m

∗

−n)

) > ζ > R(Cˆ (m

∗

−n)

).

Since R( . ) = 0 on 0-cycles, this condition is fulﬁlled after a ﬁnite number ∗ n ≤ m∗ of further steps; then deﬁne µ(x, ζ) := M (Cˆ (m −n) ). By the previous distinction of the possible cases one gets a well deﬁned index function - l . . κ : : µ : Di × R>0 \ {Rk (x), Ek (x)} −→ L . (2.21) i=1

k=0 l >

Deﬁnition 2.5.4 (Metastable states). Fix x ∈

Di and ζ > 0, where ζ

i=1

is not contained in a ﬁnite exceptional set depending on x (see (2.20)). Let T (ε) be a time scale which is logarithmically equivalent to eζ/ε , abbreviated as T (ε) ! eζ/ε in the sequel, i.e. let T be a function T : (0, ε0 ) → R>0 , where ε0 > 0, such that 0 < lim ε log T (ε) = ζ. ε→0

Then the metastable state for the initial value x and the time scale T (ε) is Kµ(x,ζ) , where µ(x, ζ) ∈ L denotes the index function deﬁned in (2.21).

2.5 Sublimiting distributions: Metastability and quasi-deterministic behavior

101

The following theorem clariﬁes the behavior of X ε on the time scales T (ε). Theorem 2.5.5 (Sublimiting distribution). Let the assumptions 2.1.1 be satisﬁed and consider the solution X ε of SDE (2.1). Fix an initial value >l x ∈ i=1 Di and a time scale T (ε) ! eζ/ε , where ζ > 0 is not contained in a ﬁnite exceptional set depending on x (see (2.20)). Then for all open sets B ⊂ Rd , / B, 0 , Kµ(x,ζ) ∈ ε→0 ε ε PT (ε) (x, B) ≡ Px XT (ε) ∈ B −−−→ 1 , Kµ(x,ζ) ∈ B . In particular, XTε,x (ε) converges in probability to the respective metastable state, P XTε,x − −−→ Kµ(x,ζ) . (ε) − ε→0

Furthermore, for all functions f ∈ C b (Rd , R), ε→0 uεf T (ε), x −−−→ f Kµ(x,ζ) , where uεf (t, x) ≡ Ex f (Xtε ) denotes the solution of the Cauchy problem (2.5) for (2.1). The ﬁrst and the third claim are a reformulation of Freidlin’s [Fr 77] theorem. The statement on the convergence in probability follows from the ﬁrst part simply by taking Bη := B Kµ(x,ζ) , η for arbitrary η > 0 and hence ε→0 − K | > η ≡ PTε (ε) (x, Bη ) −−−→ 0 . P | XTε,x µ(x,ζ) (ε) Note that in citing Freidlin’s [Fr 77] theorem the notation from (2.18) and (2.19) is used again: C(π) as deﬁned in (2.18) equals Freidlin’s [Fr 77] “c(π)” and A(π) as given by (2.19) coincides with Freidlin’s [Fr 77] “ (π)” and “A(π)”. Furthermore, note that Freidlin [Fr 77] formally assumes the coefﬁcient functions bi and aij of the SDE (of the generator G ε ) to be globally bounded. This assumption, however, is not used in the course of the paper and can be done away; see Freidlin [Fr 00]. For additional discussions see Freidlin and Wentzell [Fr-We 98, p.202f.]; further arguments and proofs can also be found in the work by Li and Qian [Li-Qi 99] and [Li-Qi 98]. This theorem in particular shows that the transition probabilities have diﬀerent limits for diﬀerent choices of the time scale and these limits do depend on the initial value x. Since the resultant convergence in probability implies weak convergence, one gets that the distribution of XTε,x (ε) converges weakly to the Dirac measure at the respective metastable state,

a

−1 w −− −−→ δKµ(x,ζ) . P ◦ XTε,x (ε) ε→0

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2 Locality and time scales of the underlying non-degenerate system

This fact is expressed by calling Kµ(x,ζ) the support of the sublimiting distributions for x and the time scale T (ε) ! eζ/ε , since in this sense δKµ(x,ζ) is the sublimiting distribution, in contrast to the limiting distribution ρε (section 2.2) for which it was obtained in equations (2.4) and (2.6) that t→∞

Ptε (x, B) −−−−→ ρε (B) and t→∞

uεf (t, x) −−−−→

f (y) ρε (dy) Rd

independently of x. Since the above hierarchy of cycles and the corresponding main states, rotation rates and exit rates — which all describe the behavior of X ε — do not depend on chance, the long-time evolution of this system has an intrinsic deterministic component; this is called the quasi-deterministic approximation of X ε (Freidlin [Fr 00]). More precisely, Freidlin’s [Fr 00] theorem 1 on the quasi-deterministic behavior of the solution X ε of the SDE (2.2) states the following, where L denotes the Lebesgue measure on R : Theorem 2.5.6 (Quasi-deterministic behavior). Let the assumptions l > 2.1.1 be satisﬁed. Fix x ∈ Di and a time scale T (ε) ! eζ/ε , where ζ > 0 i=1

is not contained in a ﬁnite exceptional set depending on x (see (2.20)). Then for any c, Γ > 0 P L t ∈ [0, Γ] : Xtε,x > c − K −−−−→ 0 . µ(x,ζ) T (ε) ε→0

Sketch of Proof. According to the deﬁnition of µ(x, ζ) in the distinction of cases preceding (2.21), there exist m∗ ≡ m∗ (x) ∈ N0 ∪ {−1} , a number s ∈ {0, . . . , m∗ + 1} of steps and a cycle ∗ ∗ Cˆ (m +1−s) ∈ C (m +1−s)

and

such that

E(Cˆ (m

∗

+1−s)

) > ζ > R(Cˆ (m

∗

+1−s)

)

∗ µ(x, ζ) = M (Cˆ (m +1−s) ) .

Applying theorem 2.5.5 to the time scale T(ε) := t0 T (ε) ! eζ/ε , for some t0 > 0, yields that for η > 0, lim Px XtεT (ε) ∈ / B(Kµ(x,ζ) , η) for all t ∈ [0, t0 ] = 0 . ε→0

The exit rate theorem 2.5.3 directly implies ∗ δ ∈ (0, E(Cˆ (m +1−s) ) − ζ) and all y ∈ B(Kµ(x,ζ) , η), lim Py

ε→0

ˆ (m∗ +1−s) )−δ)/ε

Γ T (ε) < e(E(C

< τCεˆ (m∗ +1−s)

that = 1 ,

for

all

2.5 Sublimiting distributions: Metastability and quasi-deterministic behavior

since E(Cˆ (m

∗

+1−s)

lim Px

ε→0

103

) > ζ ; in other words,

∗ XtεT (ε) ∈ D(Cˆ (m +1−s) ) for all t ∈ [t0 , Γ]

= 1.

By the deﬁnition of the rotation rate, it follows that E(C ) ≤ ∗ ∗ R(Cˆ (m +1−s) ) < ζ for all cycles C Cˆ (m +1−s) . The exit rate theorem ∗ 2.5.3 therefore implies that for all δ ∈ (0, ζ − R(Cˆ (m +1−s) )) , z ∈ D(C ) and t1 > 0, lim Pz τCε < e(E(C )+δ)/ε < t1 T (ε) = 1 ε→0

and hence, lim Pz

ε→0

XtεT (ε) ∈ D(J(C )) for some t ∈ (0, t1 )

= 1.

Now ﬁx η > c; then the closed ball Bc (Kµ(x,ζ) ) is contained in the open ball B(Kµ(x,ζ) , η). Piecing together the previous time scales t0 T (ε) > ∗ exp(R(Cˆ (m +1−s) )/ε) by using the Markov property, it can be seen that the system X ε,x mostly stays near Kµ(x,ζ) during the interval [0, ΓT (ε)]. The ∗ system also visits cycles C Cˆ (m +1−s) ; however, the lengths of these excursions is of exponential order smaller than ζ. Following this sketch of proof the claim can be deduced; see Freidlin [Fr 00, p.340]. Freidlin [Fr 00, Cor.1] also notes the following corollary. It provides the decisive connection between the LDP and the quasi-deterministic approximation deﬁned above. Corollary 2.5.7 (Boundedness in probability I). Let the assumptions l > Di and a time scale T (ε) ! eζ/ε , where ζ > 0 2.1.1 be satisﬁed. Fix x ∈ i=1

is not contained in a ﬁnite exceptional set depending on x (see (2.20)). Then for the set Fx,ζ := { y ∈ Rd : V (x, y) ≤ ζ } , it follows that for any Γ > 0 ε ∈ / F for some t ∈ [0, Γ] = 0 . lim Px XtT x,ζ (ε) ε→0

Furthermore, one obtains that for any p > 0 ε P X . T (ε) − Kµ(x,ζ) p −− −−→ 0 . ε→0 L ([0,Γ],L)

Sketch of Proof. The set Fx,ζ is compact (hence bounded) due to assumption (V). By ﬁxing a continuation of b and σ outside the open set Fδx,ζ := {y ∈ Rd : dist(y, Fx,ζ ) < δ} for some δ, the large deviation principle

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2 Locality and time scales of the underlying non-degenerate system

2.3.4 can be applied. This approach provides the asymptotic behavior for the exit time from Fδx,ζ in terms of the rate function and, hence also, in terms of the quasipotential as in theorem 2.4.3, since the ﬁrst exit time from Fδx,ζ does not depend on the speciﬁc continuation of b and σ on Fδx,ζ . This proves the ﬁrst statement. The second claim can be proven from the ﬁrst statement and theorem 2.5.6 by a similar argument as will be performed in theorem 2.5.10 below. The subsequent conclusion has also been drawn by Freidlin [Fr 00, Cor.2]: Corollary 2.5.8 (Boundedness in probability II). Let the assumptions l > Di and a time scale T (ε) ! eζ/ε , where ζ > 2.1.1 be satisﬁed. Fix x ∈ i=1

0 is not contained in a ﬁnite exceptional set depending on x (see (2.20)). Furthermore, suppose that ζ < Vµ(x,ζ),J(µ(x,ζ)) . Then for the set Hx,ζ := { y ∈ Rd : V (Kµ(x,ζ) , y) ≤ ζ } , it follows that for any Γ > 0 and c ∈ (0, Γ] ε lim Px XtT = 1 . (ε) ∈ Hx,ζ for all t ∈ [c, Γ] ε→0

Sketch of Proof. First note that also the set Hx,ζ is bounded due to assumption (V). As in the sketch of proof for theorem 2.5.6 µ(x, ζ) is the main state of the cycle corresponding to ζ. Hence, any neighborhood of Kµ(x,ζ) is hit before c T (ε) with probability converging to 1. However, as in corollary 2.5.7 above, the bound on the quasipotential on the respective set Hx,ζ (or Fx,ζ ) provides the time scale on which this set cannot be left. More precisely, the exit time law 2.4.6 for D := Hδx,ζ := {y ∈ Rd : V (Kµ(x,ζ) , y) < ζ + δ}, for small δ > 0, and O := Kµ(x,ζ) implies, since V := inf ∂Hδx,ζ V (Kµ(x,ζ) , . ) = ζ + δ , that for all y ∈ D and δ > 0, lim Py eζ/ε < τHε δ = 1 ; ε→0

x,ζ

in other words, lim Px

ε→0

XtεT (ε) ∈ Hδx,ζ for all t ∈ [c, Γ]

= 1.

The assumption that the claim is false can be led to a contradiction this way.

2.5 Sublimiting distributions: Metastability and quasi-deterministic behavior

105

Remark 2.5.9. As has already been remarked before, the property of Ki consisting of one single point might be relaxed: If Ki were a compact set, such that also (K3) and (K4) are satisﬁed, then the assertions would hold true, if suitably modiﬁed. For instance, theorem 2.5.6 would state the following: For any c, Γ > 0 ε,x P L t ∈ [0, Γ] : dist XtT > c , K −−−−→ 0 , µ(x,ζ) (ε) ε→0

where dist(x, B) ≡ inf y∈B |x − y|. Also see Freidlin [Fr 77, Remark 1] and [Fr 00]. Now we prove a consequence of the metastability which is not contained in the workings by Freidlin and Wentzell. This theorem is of central importance for the investigations of local Lyapunov exponents to come. Its ﬁnal assertion can be regarded as a “sublimiting ergodic theorem”. Theorem 2.5.10 (Consequences of the metastability of X ε,x ). Let the l > assumptions 2.1.1 be satisﬁed. Fix x ∈ Di and a time scale T (ε) ! eζ/ε , i=1

where ζ > 0 is not contained in a ﬁnite exceptional set depending on x (see (2.20)). Moreover, consider some continuous function f ∈ C(Rd , K). Then for all open sets B ⊂ K , / B, 0 , f (Kµ(x,ζ) ) ∈ ε→0 ε Px f XT (ε) ∈ B −−−→ 1 , f (Kµ(x,ζ) ) ∈ B . In particular, f (XTε,x (ε) ) converges in probability, P f XTε,x −− −−→ f (Kµ(x,ζ) ) . (ε) ε→0 Since the resultant convergence in probability implies weak convergence, one gets that the distribution of f (XTε,x (ε) ) converges weakly to the Dirac measure at the corresponding point, −1 w ) P ◦ f (XTε,x −− −−→ δf (Kµ(x,ζ) ) . (ε) ε→0 Furthermore, 1 T (ε)

0

T (ε)

f Xuε,x du =

0

1

P f Xtε,x − −−→ f (Kµ(x,ζ) ) . T (ε) dt − ε→0

Proof. The ﬁrst statement follows from theorem 2.5.5 together with the continuity of the function f . The second and the third statements follow immediately from the ﬁrst one. It remains to verify the fourth assertion:

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2 Locality and time scales of the underlying non-degenerate system

For this purpose we ﬁx an arbitrary η > 0; then ; 1 T (ε) ε,x P f Xu ( . ) du − f Kµ(x,ζ) > η T (ε) 0 1 ε,x = P > η f Xt·T ( . ) dt − f K µ(x,ζ) (ε) 0 1 ε,x ≤P f Xt·T (ε) ( . ) − f Kµ(x,ζ) dt > η 0 1 ε,x ε = P τR ( . ) ≤ 1 , f Xt·T (ε) ( . ) − f Kµ(x,ζ) dt > η 0 1 ε,x + P τRε ( . ) > 1 , f Xt·T (ε) ( . ) − f Kµ(x,ζ) dt > η 0

≡ Aε + B ε , where

Aε := P

1

τRε ( . ) ≤ 1 , 0

ε,x f X dt > η ( . ) − f K µ(x,ζ) t·T (ε)

and B := P ε

τRε ( . )

> 1, 0

and where τRε := inf

1

ε,x , f Xt·T (ε) ( . ) − f Kµ(x,ζ) dt > η

ε,x ≥ R t > 0 : Xt·T ( . ) − K µ(x,ζ) (ε)

(ε, R > 0)

ε,x denotes the ﬁrst exit time of Xt·T (ε) ( . ) from the open ball B Kµ(x,ζ) , R with center Kµ(x,ζ) and radius R. ε→0

ε→0

It needs to be shown that Aε −−−→ 0 and B ε −−−→ 0 which is done in the sequel: First, we have that ε→0

Aε ≤ P { τRε ≤ 1 } −−−→ 0 , if R is suﬃciently large; namely, choose R suﬃciently large such that Fx,ζ ⊂ B(Kµ(x,ζ) , R) and use that X ε is bounded in probability (corollary 2.5.7 with Γ = 1). For estimating B ε ﬁx ω ∈ {τRε > 1} and c > 0 in order to get:

ε,x f Xt·T (ε) (ω) − f Kµ(x,ζ) dt 0 ε,x = f Xt·T (ε) (ω) − f Kµ(x,ζ) dt 1

ε,x (ω)−Kµ(x,ζ) > c t∈[0,1] : Xt·T (ε)

2.5 Sublimiting distributions: Metastability and quasi-deterministic behavior

+

ε,x (ω)−Kµ(x,ζ) ≤ c t∈[0,1] : Xt·T (ε)

107

ε,x f Xt·T (ε) (ω) − f Kµ(x,ζ) dt

ε,x ≤ maxB(Kµ(x,ζ) ,R) f − f Kµ(x,ζ) · L t ∈ [0, 1] : |Xt·T (ε) (ω) − Kµ(x,ζ) | > c

+ maxB(Kµ(x,ζ) ,c) f − f Kµ(x,ζ)

;

now if c is chosen suﬃciently small such that the latter summand is bounded by η/2, η maxB(Kµ(x,ζ) ,c) f − f Kµ(x,ζ) < 2 (f is continuous by assumption) and if one deﬁnes m := maxB(Kµ(x,ζ) ,R) f − f Kµ(x,ζ) , then it follows from the previously obtained estimate that 1 ε,x ε ε ≡ P τR ( . ) > 1 , B f Xt·T (ε) ( . ) − f Kµ(x,ζ) dt > η 0 ≤ P ω : τRε (ω) > 1 ,

≤

η m · L t ∈ [0, 1] : > η − Kµ(x,ζ) | > c + 2 η ε,x P L{ t ∈ [0, 1] : | Xt·T (ε) ( . ) − Kµ(x,ζ) | > c } > 2 m ε,x | Xt·T (ε) (ω)

ε→0

−−−→ 0 by the stochastic convergence stated in theorem 2.5.6 with Γ = 1.

Remark 2.5.11 (Further concepts of metastability). The concept of metastability used here is the one as described by Freidlin [Fr 00]. Other authors also address related phenomena and oﬀer diﬀerent routes to metastability. We only mention a few of them. Cassandro et al. [Cs-Ga-Ol-Va 84] choose the “pathwise approach”: Here, the Curie-Weiss model (an Ising spin system) and the contact process (taking its values in the power set of Z), two Markov processes with discrete state spaces, are investigated; time averages are taken along each path and these averages are then shown to converge to measure valued jump processes. The same group, Galves et al. [Ga-Ol-Va 87], also applies this rationale to the gradient SDE (2.2) for a potential function with two wells. Surveys on this approach are given by Vares [Va 96] and Olivieri and Vares [Ol-Va 05]. Bovier et al. [Bv-Ec-Gd-Kn 04] and [Bv-Gd-Kn 05] develop a potential theoretic approach: By calculating capacities metastable exit times can be deduced. These authors are then able to obtain precise prefactors for the exit times, too.

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2 Locality and time scales of the underlying non-degenerate system

Huisinga et al. [Hui-Mey-Sch 04] study metastability using the “V -uniform ergodicity” of the process; this is a spectral theoretic assumption from which these authors deduce that the generator admits a spectral gap and the eigenfunctions provide a decomposition of the state space into “almost-absorbing subsets”. Remark 2.5.12 (Simulated annealing). A stochastic process which is closely related to the gradient SDE (2.2) is the “simulated annealing process”, i.e. the solution process X of the SDE (t ≥ 0) , dXt = − ∇U (Xt ) dt + ε(t) dWt where ε(t) → 0 is a time-varying noise-intensity. For the time dependent “temperature” ε(t) one can then consider cooling schedules ε(t) := εc (t) ≡

c log(2 + t)

(t ≥ 0)

for diﬀerent choices of c > 0. Hwang and Sheu [Hw-Sh 90, Th.3.3] prove that, under appropriate conditions, there is some cooling schedule such that X minimizes U ; more precisely, there is a constant d∗ > 0 such that for all c > d∗ , t→∞ P0,x {Xtc ∈ S} −−−− → 1 uniformly with respect to initial conditions x in compact sets, where d S := z ∈ R : U (z) = min U ( . ) Rd

and where X c denotes the solution of the above simulated annealing SDE for ε(t) = εc (t). There are many more interesting features of the simulated annealing process and interesting connections to the spectral theory of the generator of (2.2) which are beyond the scope of this book. Instead, the reader is referred to Hwang and Sheu [Hw-Sh 90], Royer [Roy 89], Chiang et al. [Cg-Hw-Sh 87] and Geman and Hwang [Gem-Hw 86] among others.

2.6 Sample systems The ﬁrst example, taken from Freidlin and Wentzell [Fr-We 98, 4.3.2], lies at the core of our investigations. On the one hand it is an illustrative example for the Freidlin-Wentzell theory; on the other hand it alludes to the manner in which the local Lyapunov exponents shall generalize the Lyapunov exponents of linear ODEs with constant coeﬃcients which are given by the real parts of the eigenvalues of the coeﬃcient matrix; see 1.5.2.

2.6 Sample systems

109

Example 2.6.1 (Multi-dimensional Ornstein-Uhlenbeck process). Consider the SDE (2.1) with linear drift b(x) := Ax and σ = idRd , √ dXtε,x = AXtε,x dt + ε dWt , X0ε,x = x ∈ Rd , where A ∈ Rd×d is a constant matrix. The origin K1 := 0 is an equilibrium of b and assuming that A is normal and A + A∗ is negative deﬁnite (i.e. its eigenvalues, which are real by symmetry, are strictly negative), then one gets for X 0 = eAt (the propagator) that ∗

| Xt0,x |2 = eAt x , eAt x = eA t eAt x , x ∗

= e(A+A

)t

x,x

A+A∗ t

≤ |x| e 2

t→∞

−−−−→ 0 ,

i.e. K1 attracts Rd , D1 = Rd . Now the vector ﬁeld b(x) ≡ Ax can be decomposed as b = −∇U + L , where U (x) := −

1 (A + A∗ )x, x 4

and

L(x) :=

1 (A − A∗ )x . 2

More precisely, let (ek )dk=1 denote the canonical basis of Rd ; then 1 ∇x (A + A∗ )x, x 4 1 ∂ = ek (aij + aji ) xj xi 4 ∂xk i,j

−(∇U )(x) ≡ +

k

1 = ek (aij + aji ) (δkj xi + δki xj ) 4 i,j,k

=

1 4

i,j

ej (aij + aji )xi +

1 1 ei (aij + aji ) xj = (A + A∗ )x . 4 i,j 2

The normality of A implies that this decomposition of b is orthogonal: For all x ∈ Rd ,

1 1 (A + A∗ )x , (A − A∗ )x −(∇U )(x) , L(x) = 2 2 1 Ax, Ax + 0 − A∗ x, A∗ x = 4 1 ∗ A Ax, x − AA∗ x, x = 0 . = 4

110

2 Locality and time scales of the underlying non-degenerate system

Proposition 2.4.5 thus implies for the quasipotential that V (K1 , x) = 2 U (x) for all x ∈ Rd . Theorem 2.5.6 yields that for all initial values x ∈ Rd , all time scales T (ε) ! eζ/ε for some parameter ζ > 0 and all c, Γ > 0, the following holds true: P > c L t ∈ [0, Γ] : Xtε,x −−−−→ 0 . T (ε) ε→0

Example 2.6.2 (Two well potential). Consider the gradient SDE (2.2) with a two well potential function U as sketched in the following ﬁgure 2.6, where v < V ; for a numerical example see (1.34). U

v V

K1

K3

K2

Fig. 2.6 A two well potential function U : Rd → R (cf. ﬁgure 1)

Here, the quasi-potential is calculated by means of proposition 2.4.5 as V (K1 , K3 ) = 2 (U (K3 ) − U (K1 )) ≡ 2v , V (K2 , K3 ) = 2 (U (K3 ) − U (K2 )) ≡ 2V and hence also V (K1 , K2 ) = 2v and V (K2 , K1 ) = 2V , since following the deterministic path X 0 amounts to a vanishing rate function. The assumption 2.1.1 (K) is satisﬁed with l = 2 and the stable points K1 , K2 ; the saddle point K3 (l = 3) is sorted out, since it violates the stability requirement (K2). In other words, we are concerned with . 0 2v (Vij )i,j∈L ≡ V (Ki , Kj ) i,j∈L = , 2V 0 where L := {1, 2}. Furthermore, the respective domain of attraction of Ki is Di , i ∈ L, where D1 denotes the shallow well (around K1 ) and where D2 is the deep well (around K2 ), respectively, both excluding K3 . The quasideterministic approximation is given by the following data here: The set of

2.6 Sample systems

111

0-cycles is C (0) = L ≡ {1, 2} . On the 0-cycles the main state, the rotation rate and the stationary distribution rate are trivial, i.e. M (i) = i, R(i) = 0 and mi (i) = 0, respectively, where i ∈ L. The exit rates are E(1) = V (K1 , K2 ) = 2v

and

E(2) = V (K2 , K1 ) = 2V .

The “follow-mapping” J is given by J(1) = 2

and

J(2) = 1 .

The possible 1-cycles are (1 → 2 → 1) and (2 → 1 → 2) which both describe the same cyclic order; hence, C (1) = (1 → 2 → 1) and the main state is

M (1 → 2 → 1) = 2

by (2.12). The rotation rate as deﬁned in (2.13) is R (1 → 2 → 1) ≡ max Vi,J(i) = V21 = 2V ; i∈{1,2}

furthermore, the stationary distribution rate m(1→2→1) for the 1-cycle (1 → 2 → 1) is given by m(1→2→1) (1) ≡ R (1 → 2 → 1) − V12 = 2V − 2v = 2(V − v) and m(1→2→1) (2) ≡ R (1 → 2 → 1) − V21 = 2V − 2V = 0 . The exit rate of the 1-cycle (1 → 2 → 1) is E (1 → 2 → 1) = ∞ due to (2.14), since C (1) = 1. The latter fact also means that C (1) is equal to L, considered as a point set. Hence, the recursive deﬁnition of cycles stops at k = 1 for the two well potential function. Next, we use the above ﬁndings to determine the support of the sublimiting distribution, i.e. we calculate the metastable states corresponding to initial values and time scales as given by deﬁnition 2.5.4 and the distinction of cases preceding it. Firstly, ﬁx x ∈ D1 . The sequence of cycles belonging to x is then 1 = i(x) ≡ C (0) (x) ⊂ (1 → 2 → 1) ≡ C (1) (x) = L

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2 Locality and time scales of the underlying non-degenerate system

and the corresponding exit and rotation rates are E0 (x) ≡ E(1) = 2v , E1 (x) ≡ E (1 → 2 → 1) = ∞ and R0 (x) ≡ 0 ,

R1 (x) ≡ R (1 → 2 → 1) = 2V ,

respectively, which we collect as 0 ≡ R0 (x) < 2v = E0 (x) ≤ 2V = R1 (x) < ∞ = E1 (x) . Due to (2.20) we ﬁx ζ ∈ R>0 \ {2v, 2V } and then we have −1 , if ζ < 2v , ∗ ∗ m ≡ m (x) = 0 , if ζ > 2v . By virtue of the distinction of cases leading to (2.21) we immediately get that µ(x, ζ) = 1, if m∗ = −1, i.e. if ζ < 2v, since ζ > 0 = R0 (x) . Otherwise, if m∗ = 0, i.e. if ζ > 2v, then µ(x, ζ) = 2, if also ζ > 2V = Rm∗ +1 (x) ; if m∗ = 0 and if, moreover, ζ ∈ (2v, 2V ), then we also get µ(x, ζ) = 2, since for ∗ ∗ ∗ Cˆ (m ) := 2 we have E(Cˆ (m ) ) = 2V > ζ and ζ > R(Cˆ (m ) ) ≡ 0 then. All in all, we get for x ∈ D1 and ζ ∈ R>0 \ {2v, 2V } that µ(x, ζ) = 1, if ζ < 2v and µ(x, ζ) = 2 otherwise. Secondly, ﬁx x ∈ D2 . The sequence of cycles corresponding to x is then 2 = i(x) ≡ C (0) (x) ⊂ (1 → 2 → 1) ≡ C (1) (x) = L and the respective exit and rotation rates are E0 (x) ≡ E(2) = 2V , and R0 (x) ≡ 0 ,

E1 (x) ≡ E (1 → 2 → 1) = ∞ R1 (x) ≡ R (1 → 2 → 1) = 2V ,

which we again collect as 0 ≡ R0 (x) < 2V = E0 (x) ≤ 2V = R1 (x) < ∞ = E1 (x) . On the basis (2.20) we ﬁx ζ ∈ R>0 \ {2V } and then we have −1 , if ζ < 2V , m∗ ≡ m∗ (x) = 0 , if ζ > 2V ; both cases directly yield that µ(x, ζ) = 2. Hence, we can summarize our above ﬁndings as follows: Fix some ζ ∈ R>0 \ {2V }, then

2.6 Sample systems

113

⎧ ⎪ ⎨2 µ(x, ζ) = 2 ⎪ ⎩ 1

, x ∈ D2 , x ∈ D1 and ζ > 2v , x ∈ D1 and ζ < 2v .

Now ﬁx an initial value x ∈ Rd \ {K3 } and a time scale T (ε) ! eζ/ε for such a parameter ζ ∈ R>0 \ {2V }. Then it follows from theorem 2.5.6 : 1) If x ∈ D1 and ζ < 2v , then for any c, Γ > 0, P > c − K L t ∈ [0, Γ] : Xtε,x −− −−→ 0 ; 1 T (ε) ε→0 2) in all other cases (x ∈ D1 and ζ > 2v ; x ∈ D2 ) for any c, Γ > 0 , L

P > c t ∈ [0, Γ] : Xtε,x − K −− −−→ 0 . 2 T (ε) ε→0

In the remainder of this section several real-world situations will be discussed which can be modeled by the SDEs (2.2) and (2.1). In doing so, the conceptual scope of these equations from an applications’ point of view shall be demonstrated. As in equation (2.2) the diﬀusion coeﬃcient of the SDE (2.1) will mostly be taken as σ( . ) = idRd . From an applications’ point of view this is not a severe restriction, but avoids technicalities; see e.g. 2.4.13.

Economic time series modeling Example 2.6.3 (Vasicek model for the interest rates with several wells). The classical Vasicek model [Vc 77] describes the short term interest rate18 as solution of the SDE √ dXtε,x = −β(Xtε − R) dt + ε dWt , X0ε,x = x , where β, R, x > 0 . This equation ﬁts into the framework of the SDEs (2.1) 2 x and (2.2) via b(x) := −β(x − R), d = 1 and U (x) := − 0 b(y)dy = β ( x2 − Rx) . U is a quadratic one-well potential and after a translation by R, X ε coincides with the Ornstein-Uhlenbeck process as described in example 2.2.3. Therefore, the transition densities are known and t √ Xtε,x = R + (x − R)e−βt + ε e−β(t−u) dWu 0

18

For an overview of interest rate models see e.g. Gibson et al. [Gs-Lh-Pr-Ta 99] and [Gs-Lh-Ta 01].

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2 Locality and time scales of the underlying non-degenerate system

is normally distributed, where the mean is given by the trajectory of the deterministic system Xt0,x . A characteristic feature of this process is the mean-reversion with respect to the one stable equilibrium point K1 := R. Hence, the assumptions 2.1.1 hold with l(= l ) = 1. Therefore, the previous large deviation considerations can be readily applied to the Vasicek interest rate model; see Callen et al. [Ca-Gv-Xu 00]. Using non-parametric tests, however, A¨ıt-Sahalia [A¨ı 96] rejects such linear drift speciﬁcations for the short term interest rate. In order to better capture the underlying economic eﬀects, regime switching models seem necessary (see A¨ıt-Sahalia [A¨ı 96, p.397]). In the literature such spot rate shifts have mostly been modeled by replacing the above mean parameter R by a Markov chain (Rt )t with ﬁnite state space (see Landen [Lan 00]): E.g. if (Rt )t can attain two possible values R(1) < R(2) , this amounts to a low-mean regime and an (exceptional) high-mean regime. However, the resulting drift b(t, x) := − β (1) x − R(1) 1{Rt =R(1) } − β (2) x − R(2) 1{Rt =R(2) } , β (1) , β (2) > 0, is discontinuous as time evolves. A¨ıt-Sahalia, [A¨ı 96, p.397f.] and [A¨ı 99], suggests to take a “two-regime potential drift” instead, i.e. to consider the gradient SDE (2.2) (in dimension d = 1) with a potential function U as in ﬁgure 2.6 of example 2.6.2. Here, the well at R(1) is supposed to be deeper than the well of the exceptional level R(2) . The Freidlin-Wentzell theory underlying the two-regime potential model (2.2) then allows to calculate the mean exit times from the regimes; see A¨ıt-Sahalia [A¨ı 96]). There is, however, one severe drawback of this model: As in the one-well case (the original Vasicek model) the interest rates may become negative. One possibility to overcome this disadvantage is to deﬁne the potential as ∞ on R≤0 ; see remark 2.2.4 above and the underlying references Meyer and Zheng [My-Zh 85] and Kunz [Kz 02]. Example 2.6.4 (Prices of energy commodities). Borovkova et al. [Br-Dh-Re-Tu 03] and Anderluh and Borovkova [Ad-Br 04] use the gradient SDE (2.2) to model prices of energy commodities (heating oil, gasoline) and agricultural commodities (coﬀee, cocoa, soybean). Focusing on oil prices these authors estimate the potential function U and ﬁt the model to the time series of daily closing prices of Brent North Sea oil from 1991 to 1998. As Borovkova et al. [Br-Dh-Re-Tu 03] note, the oil price was generally known to have several “preferred regions” at 14, 18 and 23 dollars per barrel during this period of time and most trading occurs there: The price “clusters” at these levels and deviates from them relatively brieﬂy. As in example 2.6.3 above this additive noise model can admit negative prices; again, see remark 2.2.4 above.

2.6 Sample systems

115

More examples for the application of “metastability” to economic (or more generally: socio-political) systems can be found in the literature: Weidlich [Wd 71] studies polarization phenomena in society (e.g. formation of opinion) akin to Ising spin models; in the continuous limit case this leads to Fokker-Planck equations on compact intervals. Bouchaud and Cont [Bd-Ct 98] use the gradient SDE (2.2) with a cubic potential to model the motion of the instantaneous return ut of a stock Xt . The exit of ut from the one well is then interpreted as “crash” of the stock. However, Bouchaud and Cont [Bd-Ct 98] deduce that in this situation the stock price Xt diverges to −∞. A more realistic model of crashes on ﬁnancial markets therefore has to investigate, more precisely, to which regime Xt moves in case of a crash. It seems reasonable to use a gradient ansatz (2.2) in which the potential function is time-dependent and mostly attains the onewell shape (“regular regime”). In exceptional situations the potential might feature a second well; the regime of the latter well then represents the (lower) price level to which the stock might crash. The depth of the regular well then determines the crash probability and expected crash time by the exit time law. Such an approach to the investigation of crashes is supported by oneperiod trading models; see Gennotte and Leland [Ge-Ll 90]. This is work in progress. Haag et al. [Hg-Wd-Mn 85] consider a macroeconomic potential function to study the structural change of an economy. This also leads to a timedependent potential function. However, Haag et al. [Hg-Wd-Mn 85] do not add white noise, but a (deterministic) periodic forcing to model business cycle eﬀects (e.g. Kondratieﬀ cycles, . . . ). Here, one might alternatively investigate a time-periodic potential function upon which additive white noise is imposed. In this situation it would then be interesting to investigate the phenomenon of “stochastic resonance”; see Pavlyukevich [Pv 02] and Freidlin [Fr 00].

Models for climate systems Example 2.6.5 (Thermohaline circulation). The Thermohaline circulation (THC) describes the density driven circulation of ocean water: The respective components are the temperature (“thermo”), as cold water is denser than warm water, and the salinity (“haline”), as saltwater is denser than freshwater. The present state of the THC in the Atlantic Ocean is that warm surface water from the equator area ﬂows to the north, cools and sinks down near Greenland and Iceland; this cold deep water is then transported back to the south. An approximation of this phenomenon can be described by the use of box models: The classical examples by Stommel [Sto 61] and Cessi [Ce 94] use

116

2 Locality and time scales of the underlying non-degenerate system

two boxes for modeling the equatorial and polar basins of the North Atlantic Ocean, respectively19 : 6

evaporation: F/2

6

precipitation: −F/2

? mass exchange Q ≥ 0:

box “1” (low-latitude warm water): temperature T1 (t) [◦C] salinity S1 (t) [psu]20 kg density ρ1 (t) [m 3 ] depth H volume V

[m] [m3 ]

Q/2

Q/2

?

box “2” (high-latitude cold water):

-

temperature T2 (t) [◦C] salinity S2 (t) [psu] kg density ρ2 (t) [m 3] depth H volume V

[m] [m3 ]

Fig. 2.7 Schematic depiction of the thermohaline circulation

The mass exchange function Q [s−1 ] is supposed to depend only on the (non-dimensional) diﬀerence of the respective densities, ρ :=

ρ1 − ρ2 := αS ( S1 − S2 ) − αT ( T1 − T2 ) , ρ0

where ρ0 denotes a reference density and αS [psu−1 ] and αT [(◦C)−1 ] are constants. In the presence of a freshwater ﬂux F [m s−1 ], which stems from evaporation and precipitation and inﬂuences the salinities, and assuming a relaxational forcing for the temperatures, the time-dependent quantities T1 , T2 , S1 and S2 are then governed by the diﬀerential equations

1 1 θ d T1 (t) = − − Q ρ(t) T1 (t) − T2 (t) T1 (t) − dt tr 2 2

1 1 θ d T2 (t) = − − Q ρ(t) T2 (t) − T1 (t) T2 (t) + dt tr 2 2 F 1 d S1 (t) = S0 − Q ρ(t) S1 (t) − S2 (t) dt 2H 2 F 1 d S2 (t) = − S0 − Q ρ(t) S2 (t) − S1 (t) ; dt 2H 2 here, the following constants are used: tr [s] is the relaxation time for the temperatures towards ± θ2 [◦C] (if mass exchange were absent) and S0 [psu] is a reference salinity. 19 20

Square brackets are used here to indicate physical units (dimensions). “psu” denotes Practical Salinity Units, a scale for salinity.

2.6 Sample systems

117

From the above system of ODEs one gets that the salinity and temperature diﬀerences, and S := S1 − S2 , T := T1 − T2 are governed by the diﬀerential equations 1 d T (t) = − T (t) − θ − T (t) Q αS S(t) − αT T (t) dt tr F d S(t) = S0 − S(t) Q αS S(t) − αT T (t) . dt H Now the exchange function is speciﬁed as Q(ρ) :=

1 q 2 ρ , + td V

where td [s] is the “diﬀusive” time scale, q [m3 s−1 ] is proportional to the pressure diﬀerence (Poiseuille’s law) and V [m3 ] denotes the volume of one of the boxes. Note that Stommel [Sto 61], who pioneered the THC research by box models, used the exchange function QSto (ρ) :=

1 q |ρ| , + td V

which we will not take into account for convenience, since it exhibits a plane at which it is not diﬀerentiable. Reasonable choices for the constants in use are for example tr = 25 days, θ = 20◦C, F = 2.3 m year−1 , H = 4.5 · 103 m, S0 = 35 psu, αS = 0.75 · 10−3 psu−1 , td = 219 years ,

αT = 0.17 · 10−3 (◦C)−1 ,

q = 1.0 · 1012 m3 s−1 ,

ρ0 = 1029 kg m−3 ,

V = 1.1 · 1016 m3 ,

according to Cessi [Ce 94]. Introducing the dimensionless variables x1 :=

T1 − T2 T ≡ θ θ

and scaling time by td ,

and

x2 :=

αS (S1 − S2 ) αS S ≡ αT θ αT θ

tnew := td · told ,

the above ODEs are rewritten as 0 2 1 d x1 (t) = − α x1 (t) − 1 − x1 (t) 1 + ν 2 x1 (t) − x2 (t) (2.22) dt 1 0 2 d x2 (t) = p − x2 (t) 1 + ν 2 x1 (t) − x2 (t) , dt

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2 Locality and time scales of the underlying non-degenerate system

where α :=

td , tr

ν 2 :=

q td (αT θ)2 V

and

p :=

αS S0 td F . αT θ H

Using the above sample values for the parameters one would obtain the numerical approximations α = 3.20 · 103 ,

ν 2 = 7.3

and

p = 0.9 .

Note that the above ODE (2.22) can also be considered as deterministic slow-fast system, in which x1 represents the fast variable and x2 is the slow variable; see Berglund and Gentz [Bg-Gen 06]. Now we consider two ways of introducing stochastic dynamics in this system: a) Let b denote the two-dimensional drift vector of this ODE system (2.22), i.e.

2 2 1 + ν x ( x − x ) − α (x1 − 1) − 1 1 2 . b(x1 , x2 ) := p − x2 1 + ν 2 ( x1 − x2 )2 This drift ﬁts into the setting imposed on (2.1); adding additive random √ perturbations of intensity ε one gets the following SDE for the temperature and salinity diﬀerences: √ dXtε = b (Xtε ) dt + ε dWt . For appropriate choices of the parameters this THC-drift b has three equilibria, two of which are stable and one is not; the former states are then the metastable states K1 and K2 of the THC. Note that the above design of the THC-drift is due to Cessi [Ce 94]. If one uses Stommel’s choice [Sto 61] QSto (ρ) =

1 q |ρ| + td V

of the exchange function, the resulting drift after introducing dimension-less variables is

− α (x1 − 1) − x1 [ 1 + ν | x1 − x2 | ] bSto (x1 , x2 ) := p − x2 [ 1 + ν | x1 − x2 | ] which exhibits the same qualitative features, but also non-smoothness. b) Another route to a stochastic equation of motion for the THC is to consider the freshwater ﬂux as driving stochastic process, i.e. to add white noise to p , which will be described next: For this purpose we ﬁrst replace (2.22) by a one-dimensional equation, which can be justiﬁed since for a large α the ﬁrst summand in the x1 -equation forces the temperature diﬀerence x1 to stay near 1. Hence, by substituting x1 = 1 into (2.22) one gets the

2.6 Sample systems

119

one-dimensional equation for the salinity diﬀerence x2 : 0 2 1 d x2 (t) = p − x2 (t) 1 + ν 2 1 − x2 (t) dt = − U x2 (t) , where integrating the one-dimensional drift leads to the potential function ν2 4 2 ν2 3 (1 + ν 2 ) 2 x − x + x − px. 4 3 2 √ Adding small additive random perturbations of intensity ε of the freshwater ﬂux, as announced, this leads to the SDE for the salinity diﬀerence, √ dXtε = − U (Xtε ) dt + ε dWt , U (x) :=

where W is a Brownian motion in R. This SDE again ﬁts into the setting of (2.1); depending on p and ν 2 , the potential U has several local minima which are then the metastable states of the salinity diﬀerence of the THC. For further details we refer to Cessi [Ce 94]. More references on the THC are given by Imkeller and Monahan [Im-Mo 02]. Example 2.6.6 (El Ni˜ no-Southern Oscillation). El Ni˜ no is an anomaly of the sea surface temperature in the tropical Paciﬁc Ocean. This short term phenomenon recurs on time scales of several (6–10) years and has important consequences such as reduced ﬁshing in the eastern Paciﬁc Ocean, low grain yields in south Asia and Australia and high crop yields in the North American prairies (see Hansen et al. [Hn-Hd-Jo 98] and the references therein). Furthermore, the hurricane activity in the western Atlantic Ocean is reduced during the season following the El Ni˜ no event and returns to normal only in the second summer following an El Ni˜ no event (see Gray [Gy 84]). a) Wang et al. [Wg-Bc-Fg 99] describe the El Ni˜ no system by the SDE (2.1) in R2 with the drift function . a1 x1 + a2 x2 + a3 x1 (x1 − c x2 ) − 2 x31 b(x) := b (2x2 − x1 ) − 2 x32 and σ( . ) := idR2 . Here, the (nondimensional) variables x1 and x2 represent the anomalous sea surface temperature and the thermocline depth, respectively; a1 , a2 , a3 , b, c ∈ R are constants. Depending on the choice of the parameters this drift exhibits either a stable regime, a limit cycle regime or several equilibria. In the latter case the system can be adjusted such that there are three unstable ﬁxed points and two stable equilibria of b; one of the latter points (the “warm” state) represents the El Ni˜ no situation.

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2 Locality and time scales of the underlying non-degenerate system

For details such as typical values of the parameters see Wang et al. [Wg-Bc-Fg 99]. b) The multi-dimensional Ornstein-Uhlenbeck process of example 2.6.1 can also be used for describing the El Ni˜ no-Southern Oscillation, as has been carried out by Penland and Sardeshmukh [Pl-Sa 95b] and [Pl 96]. More precisely, consider the SDE (2.1) with linear drift b(x) := Ax , √ dXtε = AXtε,x dt + ε σ dWt , X0ε,x = x ∈ Rd , where A, σ ∈ Rd×d are constant matrices. Here, σ is additionally invertible, A again attracts D1 = Rd towards the origin K1 := 0 and the dimension is d := 15. If the eigenvalues (characteristic roots) are non-real, there are components of X 0 which show oscillatory behavior. In the randomly perturbed case ε > 0, X ε exhibits random periodicity. One can then calculate the respective rotation numbers as in Arnold [Ar 98, Sec.6.5] for the stationary situation (i.e. for ﬁxed ε > 0). This rotational behavior also implies random periodicity for the sea surface temperature which can be generated through X ε . For more references to the literature on stochastic (or deterministic) models of the El Ni˜ no phenomenon see Imkeller and Monahan [Im-Mo 02]. Example 2.6.7. A potential function with n wells is for example also given by

n 1 1 |x − zi |2 U (x) = |x|2 − , ki exp − 2 2 i=1 ci where x ∈ Rd ; ki > 0, ci > 0 and zi ∈ Rd are the parameters representing the depth, width and position of the i-th well. Teng et al. [Tg-Mo-Fy 04] use this potential as a model for the extratropical northern hemisphere atmosphere. The gradient SDE (2.2) is also used by Nicolis and Nicolis [Nc-Ni 81] and Sutera [Su 81] as a general model for climate transitions.

Further examples Finally, we brieﬂy list some more examples for the interested reader. Again, the goal is not to compile a complete list, but to hint at the various possibilities of application. Example 2.6.8 (Saksaul tree population). Freidlin and Svetlosanov [Fr-Sv 76] consider (2.2) with the potential function U (x) := −

a a l x + 3 arctan( κ · x ) + x2 , 2 κ κ 2

2.6 Sample systems

121

where a, κ, l are real parameters such that 0 < 2lκ a < 1. This potential function has a local minimum at K1 := 0, a global minimum at ⎛ ⎞ 3

2 a ⎝ 2lκ ⎠ K2 := 1+ 1− 2lκ2 a and a saddle at

⎛ ⎞ 3

2 a ⎝ 2lκ ⎠ K3 := 1− 1− . 2lκ2 a

Choose an initial condition x ∈ D2 , the domain of attraction of K2 . Then the process X ε,x represents the population of the Saksaul, a certain tree in the Gobi Desert which can best resist the drought there. Freidlin and Svetlosanov [Fr-Sv 76] then calculate the mean exit time from D2 by the exit time law. Such an exit and hence an approach of K1 = 0 means the extinction of the tree population. The parameters a, l and κ parametrize the sprout rate, the extinction rate and the inﬂuence of neighboring trees, respectively. Example 2.6.9 (Evolution). Newman et al [Nm-Co-Ki 85] use (2.2) to describe Neo-darwinian evolution: X ε depicts the population mean of some genetically determined character; the function −U is then interpreted as the mean ﬁtness of the population. Natural selection pushes the population √ mean X ε towards higher values of this ﬁtness landscape. The parameter ε measures the magnitude of the random genetic variations relative to that of natural selection. Example 2.6.10 (The Stochastic Disk Dynamo Model for the polarity of the earth’s magnetic ﬁeld). The stochastic disk dynamo is given by the SDE (2.1) in R2 for the drift function

− ν1 x1 + x1 x2 , b(x1 , x2 ) := − ν2 x2 + 1 − x21 where ν1 , ν2 > 0 are constants such that ν1 ν2 < 1. This drift has two stable equilibria at √ K1,2 := ± 1 − ν1 ν2 , ν1 and an unstable ﬁxed point at K3 := (0, 1/ν2 ). Ito and Mikami, [Ito-Mik 96] and [Ito 88], use this system as a model for the earth’s magnetic ﬁeld: The stable equilibria K1 and K2 then correspond to the two respective polarities.

122

2 Locality and time scales of the underlying non-degenerate system

The case of a compact state space Instead of considering a diﬀusion on Rd one could also consider a diﬀusion X ε on a d-dimensional compact manifold. In this case (V) is not necessary. Furthermore, a density satisfying the Fokker-Planck equation is always normalizable in the compact case. The following SDE provides an example of a system on the circle for which the stationary density can be calculated as in remark 2.2.1 due to the onedimensionality of its state space: Example 2.6.11 (Noisy north-south-ﬂow). The noisy north-south-ﬂow is the solution X ε := αε for the SDE √ d αεt = − cos αεt dt + ε dWt on the compact state space S 1 ≡ R/2πZ " [0, 2π). The state space being one-dimensional, the system can be thought of as the random perturbation of the north-south-ﬂow d αt = − U (αt ) dt

with the potential

U (α) := sin α

on S 1 ; this deterministic dynamical system has two equilibria: αN := π2 which is unstable and αS := 3π 2 which is stable. Calling αN and αS the north pole and the south pole of the “planet” S 1 , respectively, one gets an explanation of the name of this system. The assumptions (S) and (E) are clearly met; (K) holds true with l = 1, π K1 := αS ≡ 3π 2 ; additionally, l = 2 and K2 := αN ≡ 2 . Furthermore, also (G) is satisﬁed. The invariant measure of (Xtε )t≥0 ≡ (αεt )t≥0 can be also calculated explicitly: As in remark 2.2.1 on stationary measures, the invariant measure has a density pε (α) with respect to the Lebesgue measure on the circle, which needs to solve the Fokker-Planck-equation d ε d2 ε − cos(α) pε (α) = 0 p (α) − 2 2 dα dα for α ∈ (0, 2π), fulﬁll the normalization constraint

2π

pε (α) dα = 1

0

and additionally the continuity (periodicity) requirement pε (0) = pε (2π) . The solution to this problem is given by Khasminskii, [Kh 67] and [Kh 80, Sec.VI.8], as

2.6 Sample systems

1 p (α) = cε ε ε

where

123

5 1 +

W(2π) − 1 2π 0

W(s)ds

6

α

W(u)du

W(α)−1 ,

0

2 W(α) := exp + U (α) ε

and cε is the normalization constant; this can be seen by a direct calculation. Furthermore, the one peak of the stationary density pε is located at the south pole αS and in the small noise limit ε → 0 the invariant measure converges weakly to δαS . The latter property can be either directly veriﬁed from the above formula or be deduced from corresponding results by Freidlin and Wentzell, [We-Fr 69] and [Fr-We 98, Th.6.4.2], since the state space is compact. This noisy north-south-ﬂow system has been considered by Carverhill [Cv 85b, p.290ﬀ.]; note that Carverhill [Cv 85b, p.290] deﬁnes the noise free north-south-ﬂow as the stereographic projection of the ﬂow ηt (x) := xe−t in R to the unit circle; it can be calculated that this is equivalent to the above potential function U (α) := sin α on S 1 . A rotated version (a noisy westeast-ﬂow so to say) has been investigated and simulated by Carverhill et al. [Cv-Cl-Ew 86, p.54ﬀ.]. Carverhill [Cv 85b] calculates the (one) Lyapunov exponent λε for this system αεt , obtains a Furstenberg-Khasminskii-type formula for it and notes that in the small noise limit ε → 0, λε converges to −U (αS ) = sin(− π2 ) = −1. A more general investigation concerning the invariant measures of diﬀusion processes on a circle with small diﬀusion is given by Nevel’son [Nev 64]. The impetus for us mentioning this system on the circle here is twofold: On the one hand it illustrates how the results from the Freidlin-Wentzell theory also hold in the setting of compact state spaces. On the other hand — what is more — elliptic SDEs of the above type typically arise in the study of the angle (and hence of the Lyapunov exponent) of linear, two-dimensional stochastic systems with white noise. The crucial property when studying the SDE (1) is, however, that its angle satisﬁes a diﬀerential equation with real noise; see (1.6). Therefore, when considering the angle system of (1), the Freidlin-Wentzell results only hold for the noise process X ε , but not for the angle process; in particular, when examining (1) for (local) Lyapunov exponents, the Freidlin-Wentzell results cannot be applied to the angle process. For this reason we study exit probabilities of degenerate stochastic systems in the following chapter.

Chapter 3

Exit probabilities for degenerate systems

This chapter is devoted to the investigation of the exit probabilities for degenerate stochastic diﬀerential systems depending on small parameters. In doing so, the exposition follows Hern´ andez-Lerma [HL 80], [HL 81] and Fleming [Fl 80], [Fl 78], [Fl 77]. The object underlying all our considerations is the stochastic system (1). For obtaining information about the exponential growth rates of Z ε , the formulas (1.5) and (1.8) advise to investigate the movement of the direction ψ ε or the angle αε (n = 2) of Z ε , respectively; focusing on the two-dimensional case here this means, more precisely, that one needs assertions concerning the behavior of the system ¯ (X ε , αε ) dt , dαεt = h t t √ dXtε = b (Xtε ) dt + ε σ (Xtε ) dWt ; see (1.6). The results of the previous chapter imply that X ε mostly lingers near Kµ(x,ζ) . However, if A(Kµ(x,ζ) ) ∈ R2×2 has two diﬀerent eigenvalues, αε still can oscillate at two equilibrium points, A1 Kµ(x,ζ) and A2 Kµ(x,ζ) (see deﬁnition 1.2.4) — the former being attracting and the latter being repelling. Therefore, the results of this chapter will be applied (in section 4.4) to quantify the exit of αε from a neighborhood of A2 (Kµ(x,ζ) ); see theorem 4.4.6. As far as we are aware the only results available for exit probabilities of such degenerate systems are proven by Hern´ andez-Lerma and Fleming in the papers cited above, Dupuis and Kushner [Du-Ks 89], Barles and Blanc [Bar-Bln 97] and Freidlin [Fr 69]: Dupuis and Kushner [Du-Ks 89] assume that b ≡ b(x, u) also depends on the values of a control function u, and consider the problem of determining u as to minimize the exit probability of (X ε , αε ), in the above notation, over a given time interval as ε → 0. Barles

W. Siegert, Local Lyapunov Exponents. Lecture Notes in Mathematics 1963. c Springer-Verlag Berlin Heidelberg 2009

125

126

3 Exit probabilities for degenerate systems

and Blanc [Bar-Bln 97] consider the SDE (2.1) in which σ might degenerate and investigate the exit times from domains converging to a subset of ∂D, where D is a domain which the deterministic system X 0 uniformly exits. Freidlin [Fr 69] considers the associated Dirichlet problem for the SDE (2.1) under the condition that a ≡ σσ ∗ is just non-negative deﬁnite and constructs “generalized solutions”, i.e. solutions which converge to 0 towards the boundary of the domain. Note that the above stochastic equation for αε is a real noise system, i.e. the diﬀerential equation is not driven by white noise, but is a random diﬀerential equation due to its dependence on the diﬀusion X ε . Real noise systems and associated large deviation properties also have been investigated by Freidlin [Fr 73b], Nguyen Viet Phu [NgVP 74a] and [NgVP 74b] and Grin [Gn 75]; also see Freidlin and Wentzell [Fr-We 98, p.132ﬀ.]. However, these authors consider systems of the type αεt = b(αεt , εXt ) , where b is a diﬀerentiable drift function, X is a Gaussian stochastic process and ε becomes small. Since this is a diﬀerent setting, the corresponding results cannot be applied to the angle system αε of (1) which had been mentioned before. In the sequel, the ﬁndings by Hern´ andez-Lerma [HL 80] and [HL 81] shall be presented. Since these results are well documented (also see Fleming [Fl 80], [Fl 78] and [Fl 77]), we allow ourselves to sketch the arguments only and refer to these papers for more details.

3.1 Exit probabilities for degenerate systems depending on a small parameter The system under consideration is the stochastic process (Xtε , Ytε )t≥0 in Rd × Rm deﬁned by the SDE dYtε = F (t, Xtε , Ytε ) dt , √ dXtε = ˆb (t, Xtε ) dt + ε σ ˆ (t, Xtε ) dWt ,

(3.1)

where the coeﬃcients are smooth functions F : R+ × Rd × Rm → Rm , ˆb : R+ × Rd → Rd and σ ˆ : R+ × Rd → Rd×d ; ε ≥ 0 parametrizes the intensity of (Wt )t≥0 which again denotes a Brownian motion in Rd on a standard ﬁltered probability space (Ω, F , P, (Ft )t≥0 ) ; the initial condition is (Xsε , Ysε ) = (x, y) ∈ Rd × Rm

3.1 Exit probabilities for degenerate systems depending on a small parameter

127

for some initial time s ≥ 0 and the solution process is denoted by (Xtε,x , Ytε,y )t≥s . Furthermore, Ps,x,y denotes the law of this process conditioned on the above initial value. In the sequel, we will work with a ﬁxed bounded, open domain D ⊂ Rm and a time horizon T > 0. Furthermore, set a ˆ := σ ˆσ ˆ∗ , where ∗ denotes the transpose of the respective matrix as usual. Note that the notation in use here diﬀers from the one used by Hern´andezLerma [HL 80] and [HL 81] : The x- and the y-variable are interchanged in order to reserve “x” for the direction in which X ε (as deﬁned in chapter 2) is acting. Notation 3.1.1 (First exit time). Let D be a bounded, open domain in Rm with smooth boundary ∂D. Then the random variable T ε ≡ T ε (s, x, y) ≡ TDε (s, x, y) := inf { t ≥ s : Ytε,y ∈ / D} denotes the ﬁrst exit time of (Ytε,y )t≥s from D. Again, T ε is a stopping time with respect to the underlying standard ﬁltration (Ft )t≥0 . Notation 3.1.2 (Value function). Let u ∈ C 1 (R+ , Rd ) be a (“control”) function and let y 0 be the path deﬁned by dy 0 (t) = F (t, u(t), y 0 (t)) dt under the initial condition (u(s), y 0 (s)) = (x, y) ∈ Rd × D . Then

ϑ ≡ ϑD (s; x, y; u) := inf t ≥ s : y 0 (t) ∈ /D

denotes the corresponding exit time of y 0 from D and U ≡ UD (s; T ; x, y) := u ∈ C 1 (R+ , Rd ) : u(s) = x and ϑD (s; x, y; u) ≤ T is the (possibly empty) set of control functions for which an exit happens before T > 0. Using this notation, deﬁne the value function 0 (s; T ; x, y) I 0 ≡ I 0 (s, x, y) ≡ ID ⎧ ϑD (s;x,y;u) ⎪ ⎨ inf K t, u(t), u(t) ˙ dt := u∈UD (s;T ;x,y) s ⎪ ⎩ ∞

, U = ∅, , U = ∅,

where K : R + × Rd × Rd → R + ,

K(s, x, q) :=

0 1 2 1 ˆ(s, x)−1/2 q − ˆb(s, x) . a 2

128

3 Exit probabilities for degenerate systems

Here, a ˆ(s, x)−1/2 again denotes the (unique) d × d-matrix whose square is σ (s, x) σ ˆ (s, x)∗ ]−1 . Also a ˆ(s, x)−1/2 the positive deﬁnite matrix a ˆ(s, x)−1 ≡ [ˆ is a symmetric matrix; hence one can rewrite K as K(s, x, q) =

0 1 1 q − ˆb(s, x) , a ˆ(s, x)−1 q − ˆb(s, x) . 2

Assumption 3.1.3 (on the coeﬃcients of the SDE (3.1)). The data of the SDE (3.1) are supposed to satisfy: (a) F , ˆb and σ ˆ are continuous on their respective domains of deﬁnition and σ ˆ takes its values in the invertible matrices; furthermore, F ∈ C ∞ (R>0 × ˆ ∈ C ∞ (R>0 × Rd , Rd×d ) and Rd × Rm , Rm ) , ˆb ∈ C ∞ (R>0 × Rd , Rd ) , σ −1 ∞ d d×d σ ˆ ∈ C (R>0 × R , R ) are bounded with bounded ﬁrst derivatives (on the respective spaces). (b) The matrix a ˆ≡σ ˆσ ˆ ∗ is strictly positive deﬁnite, i.e. there is a constant c1 > 0 such that 2 c1 |x2 |2 ≤ a ˆ(t, x1 )x2 , x2 ≤ c−1 1 |x2 |

(t ∈ R+ ; x1 , x2 ∈ Rd ) .

(c) Let Lˆε (s, x, y) :=

d

d m ∂2 ∂ ˆbi (s, x) ∂ + ε a ˆij (s, x) + Fi (s, x, y) ∂x 2 ∂x ∂x ∂y i i j i i=1 i,j=1 i=1

denote the generator of (X ε , Y ε ) from (3.1); then the corresponding backward operator ∂ + Lˆε (s, x, y) ∂s is supposed to be hypoelliptic in Q0 := R>0 × Rd × Rm , i.e. for any distribution v ≡ v(s, x, y) we have that v is a C ∞ function in ∂ every open subset of Q0 in which ( ∂s + Lˆε )v is a C ∞ function (H¨ ormander [H¨ o 67, p.147]). (d) Let D be the ﬁxed bounded, open domain in Rm with smooth boundary ∂D, let N (y) denote the outer normal to ∂D at y ∈ ∂D and deﬁne G0 := {(t, x, y) ∈ R>0 × Rd × ∂D : F (t, x, y), N (y) = 0} ; then it is assumed that Ps,x,y (t, Xtε , Ytε ) ∈ G0 for some t ∈ [s, T ] = 0

3.1 Exit probabilities for degenerate systems depending on a small parameter

129

for all (s, x, y) in Q := (0, T ) × Rd × D , where T > 0 denotes a ﬁxed time horizon. (e) For all s < T and (x, y) ∈ Rd × D , UD (s; T ; x, y) = ∅ .

Remark 3.1.4 (on the set of assumptions 3.1.3). on (c): Given the non-degeneracy assumption (b), uniform ellipticity, the hypoellipticity (c) of a parabolic diﬀerential operator with smooth data F, ˆb and a ˆ is implied by the following condition:

∂Fi (t, x, y) rank = m ∂xj 1≤i≤m , 1≤j≤d for all (t, x, y) ∈ Q0 ; see Hern´andez-Lerma [HL 81] and H¨ormander [H¨ o 67, p.147-149]. For the arguments, which use the so-called “Levi method” of constructing a fundamental solution for the corresponding diﬀerential equation, we would like to refer to Ilin [Il 64], H¨ ormander [H¨ o 67, p.147– 149], Weber [Web 51] and Kolmogorov [Ko 34]. Ilin [Il 64] is using the stronger condition that m ≤ d and for some constant c2 > 0

∂Fi (t, x, y) 0 < c2 < det < c−1 2 ∂xj 1≤i,j≤m for all (t, x, y) ∈ R+ × Rd × Rm . The backward operator in (c) is ultraparabolic, i.e. its coeﬃcient matrix of the second derivatives is non-negative deﬁnite and has rank less than d + m, where d + m + 1 is the number of “independent” variables (Ilin [Il 64]). Also see Ichihara and Kunita [Ic-Ku 74] for further results concerning hypoellipticity. Following Hern´ andez-Lerma [HL 81] and [HL 80] we consider the case that the coeﬃcients F, ˆb and σ ˆ also depend on the time-parameter. Although the results of this section will be applied to the autonomous angle SDE (see theorem 4.4.6 in section 4.4)), we also present the more general case ∂ here. Note that calling ∂s + Lˆε (s, x, y) the backward operator to (3.1) is in complete agreement with the notation introduced in section 2.2 (see ∂ + Lˆε (s, x, y) = 0 is the backward parabolic p.61) according to which ∂s equation corresponding to (3.1) in the backward variables (s, x, y).

130

3 Exit probabilities for degenerate systems

on (d): From the law of motion (3.1) for Y ε it follows that Y ε cannot cross ∂D at points y where F (x, y), N (y) < 0 ; more formally, further deﬁne G+ := {(t, x, y) ∈ R>0 × Rd × ∂D : F (t, x, y), N (y) > 0} in correspondence to G0 ; then it is a priori known that Ps,x,y (T ε , XTε ε , YTε ε ) ∈ G+ ∪ G0 = 1 for all (s, x, y) ∈ Q ; see Stroock and Varadhan [Str-Vdh 72, Sec.7] and Fleming [Fl 80]. Hence the assumption (d) asserts that exits via G0 do not occur either. Furthermore, the smoothness assumption on the boundary ∂D can be relaxed to assuming ∂D to be of class C 2 ; see Hern´andez-Lerma [HL 80] and [HL 81]. 0 on (e): This assumption guarantees that ID is a well deﬁned function in [0, ∞).

Notation 3.1.5 (Exit and inclusion probabilities). Consider the SDE (3.1) under the above assumptions 3.1.3 and let T ε denote the ﬁrst exit time of (Ytε,y )t≥s from the bounded, open domain D ⊂ Rm as deﬁned in 3.1.1. Then Qε ≡ Qε (s, x, y) ≡ QεD (s; T ; x, y) := Ps,x,y {TDε ≤ T } denotes the exit probability corresponding to the ﬁxed time horizon T > 0 and ε PD (s; T ; x, y) := 1 − QεD (s; T ; x, y) = Ps,x,y {TDε > T } denotes the inclusion probability. Theorem 3.1.6 (PDE for the exit probability Qε ). Consider the SDE (3.1) under the assumptions 3.1.3. Then the exit probability Qε (s, x, y) is a solution of the boundary value problem

∂ ε ˆ + L (s, x, y) Qε (s, x, y) = 0 on Q ≡ (0, T ) × Rd × D (3.2) ∂s 1 on G+ T Qε (s, x, y) = 0 on {T } × Rd × D , + where all appearing partial derivatives are continuous functions and G+ T ⊂ G is deﬁned as d G+ T := {(s, x, y) ∈ (0, T ] × R × ∂D : F (s, x, y), N (y) > 0} .

3.1 Exit probabilities for degenerate systems depending on a small parameter

131

Furthermore, Qε is continuous on Q1 := Q ∪ ({T } × Rd × D) ∪ (s, x, y) ∈ G+ : s ∈ (0, T ) ≡ (0, T ] × Rd × D ∪ (s, x, y) ∈ (0, T ) × Rd × ∂D : F (s, x, y), N (y) > 0 .

Sketch of Proof. In addition to the process (Xt , Yt )t≥0 := (Xtε , Ytε )t≥0 from (3.1) also consider the non-degenerate system Xt , Ytδ t≥0 := ε ε,δ Xt , Yt t≥0 in Rd × Rm deﬁned by the SDE √ ?t , dYtε,δ = F (t, Xtε , Ytε ) dt + δ dW √ dXtε = ˆb (t, Xtε ) dt + ε σ ˆ (t, Xtε ) dWt ,

(3.3)

?t )t≥0 is a Brownian motion in Rm independent of (Wt )t≥0 , where δ > 0, (W ?• W• , and the same initial condition is fulﬁlled, W ε ε,δ Xs , Ys = (x, y) ∈ Rd × D for the initial time s ≥ 0. The solution process is now denoted by Xtx , Ytδ,y ≡ Xtε,x , Ytε,δ,y . t≥s

t≥s

In particular, the X ε -component of this process coincides with the one from (3.1), but the new coupled system is non-degenerate in Rd × Rm . In addition to the ﬁrst exit time T ε deﬁned in 3.1.1 let T δ ≡ T ε,δ (s, x, y) ≡ TDε,δ (s, x, y) := inf t ≥ s : Ytε,δ,y ∈ /D denote the ﬁrst exit time of (Ytε,δ,y )t≥s from D. Furthermore, for abbreviation set δ Y − Y := Y δ ( . ) − Y( . ) ≡ max Yuε,δ,y ( . ) − Yuε,y ( . ) , t [s,t] u∈[s,t]

where ( . ) indicates the dependence on chance and also set θδ := TDε,δ (s, x, y) ∧ T as well as θ := TDε (s, x, y) ∧ T , where ∧ denotes the minimum operation.

132

3 Exit probabilities for degenerate systems

Then one obtains for all (x, y) ∈ Rd × D and 0 ≤ s < t ≤ T that δ→0 i) Y δ − Y t −−−→ 0 , δ→0

ii) θδ −−−→ θ and δ→0

iii) Yθδδ −−−→ Yθ almost surely. Here, i) is proven by use of a Gronwall argument and the Lipschitz continuity of F (assumption 3.1.3(a)); ii) can be deduced from assumption 3.1.3(d) and iii) then follows from i) and ii); see Hern´ andez-Lerma [HL 81, p.42f.] for details. Now for proving that Qε (s, x, y) is a smooth solution of the proposed PDE (3.2),

∂ + Lˆε (s, x, y) v = 0 on Q , ∂s it must be veriﬁed that Qε (s, x, y) is a distribution solution of this PDE, due to the hypoellipticity assumption 3.1.3(c). The backward parabolic equation corresponding to the SDE (3.3) is

∂ δ + Lˆε (s, x, y) + ∆y v = 0 , ∂s 2 where ∆y denotes the Laplace-operator with respect to the variable y ∈ Rm . Furthermore, let ψ ∈ C(∂Q, R) be some continuous function1 deﬁned on the boundary of Q. Then the boundary value problem

δ ∂ ε ˆ + L (s, x, y) + ∆y v = 0 on Q ∂s 2 v = ψ on ∂ ∗ Q , where is solved by

∂ ∗ Q :=

{T } × Rd × D ∪ G+ T ,

. v δ (s, x, y) := Es,x,y ψ θδ , Xθδ , Yθδδ

This

theorem 2.2.5, since the noise coeﬃcient matrix √ follows from δ Im √ 0 of (3.3), Im denoting the m-dimensional unit matrix, 0 εσ ˆ (t, . ) satisﬁes the ellipticity condition (E) for a time-independent bound due to 1

This function ψ(s, x) is not to be confused with the stochastic process ψtε (ω) as deﬁned in equation (1.4). These two objects will not be considered simultaneously, so there is no ambiguity.

3.1 Exit probabilities for degenerate systems depending on a small parameter

133

assumption 3.1.3(b). Note, however, that theorem 2.2.5 cannot be applied directly, since the domain Rd × D is not bounded; this can be completed by a truncation argument. Now specify ψ by functions ψk ∈ C(Q, R), k ∈ N, such that the following conditions are also fulﬁlled: 1 , if (s, x, y) ∈ G+ T , ψk (s, x, y) = 0 , if (s, x, y) ∈ {T } × Rd × D and dist(y, ∂D) > k1 , ψk (s, x, y) ∈ [0, 1] , if (s, x, y) ∈ {T } × Rd × D and dist(y, ∂D) ≤ and such that ψk ( . ) − ψj ( . ) −k,j→∞ −−−−→ 0

1 k

,

uniformly on all compact subsets of Q .

The solution of the above PDE with boundary condition ψk is denoted by vkδ , i.e. vkδ (s, x, y) := Es,x,y ψk θδ , Xθδ , Yθδδ . δ Due to the properties i), ii) and iii) of θ , Yθδδ noted above, it follows from the dominated convergence theorem and the continuity of ψk that δ→0 vkδ (s, x, y) −−−→ Es,x,y ψk θ, Xθ , Yθ =: Qεk (s, x, y) for any k ∈ N. Now one veriﬁes that Qεk (s, x, y) is a distributional solution of the boundary value problem

∂ ε ˆ + L (s, x, y) v = 0 on Q ∂s v = ψk on ∂ ∗ Q . Then one proves that lim Qεk (s, x, y) = Qε (s, x, y)

k→∞

almost everywhere on Q, and this convergence then implies that also Qε (s, x, y) is a distributional solution of the proposed PDE (3.2),

∂ + Lˆε (s, x, y) v = 0 on Q . ∂s As has already been remarked before, the hypoellipticity assumption 3.1.3(c) then yields that Qε (s, x, y) is a smooth solution of this PDE. The second assertion of the theorem, i.e. the continuity of Qε on the set Q1 ≡ (0, T ] × Rd × D ∪ (s, x, y) ∈ (0, T ) × Rd × ∂D : F (s, x, y), N (y) > 0 ,

134

3 Exit probabilities for degenerate systems

is veriﬁed by use of the continuity properties of the Feller semigroup of the process (X ε , Y ε ); see Hern´andez-Lerma [HL 81, p.44] for details. The following theorem calculates the asymptotics of the exit probability QεD (s; T ; x, y) of the degenerate system. Note the similarities of this assertion 0 and the corresponding value function ID (s; T ; x, y) with theorem 2.4.3 for non-degenerate systems and its corresponding rate function I[0,T ] as deﬁned in theorem 2.3.4. Theorem 3.1.7 (Asymptotics of the exit probability Qε ). Consider the SDE (3.1) under the assumptions 3.1.3. Then it follows for the exit probability QεD (s; T ; x, y) ≡ Ps,x,y {TDε ≤ T } that 0 lim ε log QεD (s; T ; x, y) = − ID (s; T ; x, y)

ε→0

0 (s; T ; x, y) denotes the value for any s < T and (x, y) ∈ Rd × D, where ID function as deﬁned in 3.1.2,

0 (s; T ; x, y) ≡ I 0 ≡ ID

1 inf 2 u∈U

ϑ

0 1 2 ˆ(t, u(t))−1/2 u(t) ˙ − ˆb(t, u(t)) dt a

s

in which U ≡ UD (s; T ; x, y) ≡ u ∈ C 1 (R+ , Rd ) : u(s) = x and ϑD (s; x, y; u) ≤ T and

ϑ ≡ ϑD (s; x, y; u) ≡ inf t ≥ s : y 0 (t) ∈ /D

for the deterministic control system dy 0 (t) = F (t, u(t), y 0 (t)) dt

(t > s) ,

(3.4)

0

(u(s), y (s)) = (x, y) . Sketch of Proof. Deﬁne I ε as I ε ≡ I ε (s, x, y) := − ε log QεD (s; T ; x, y) for all s < T and (x, y) ∈ Rd × D. Then we have to prove that lim I ε = I 0 .

ε→0

1) The PDE for the exit probability Qε obtained in theorem 3.1.6 implies for I ε that

∂ + Lˆε (s, x, y) ∂s

I ε (s, x, y) =

d 1 ∂I ε (s, x, y) ∂I ε (s, x, y) a ˆij (s, x) 2 i,j=1 ∂xi ∂xj

3.1 Exit probabilities for degenerate systems depending on a small parameter

135

on Q ≡ (0, T ) × Rd × D , and the boundary behavior of I ε is given by = 0 on G+ T , ε I (s, x, y) s→T −−−→ ∞ for (x, y) ∈ Rd × D . This PDE for I ε can be rewritten as d ε ∂2I ε ∂I ε + a ˆij (s, x) + F (s, x, y), ∇y I ε m + H s, x, ∇x I ε = 0 R ∂s 2 i,j=1 ∂xi ∂xj

on Q, where ∇x I ≡ ε

∂I ε ∂I ε ∂x1 , . . . , ∂xd d

and ∇y I ≡ ε

∂I ε ∂I ε ∂y1 , . . . , ∂ym

(3.5)

denote the

gradient with respect to x ∈ R and y ∈ R , respectively, and H is deﬁned as H : R+ ×R ×R → R, d

d

m

d

d 1 ˆ H(s, x, p) := a ˆij (s, x)pi pj . bi (s, x) pi − 2 i,j=1 i=1

Note that the function K : R+ × Rd × Rd → R+ ,

K(s, x, q) ≡

0 1 2 1 ˆ(s, x)−1/2 q − ˆb(s, x) a 2

deﬁned in 3.1.2 is dual to H in the sense that K(s, x, q) = max H(s, x, p) − p, q p∈Rd

and

H(s, x, p) = min K(s, x, q) + p, q , q∈Rd

where . , . denotes the canonical scalar product in Rd ; see e.g. Freidlin and Wentzell [Fr-We 98, p.154].2

2 Freidlin and Wentzell [Fr-We 98, p.154,136f.] do not work with the duality deﬁned above but with

H(s, x, p) := − H(s, x, −p) ≡

d

d ˆbi (s, x) pi + 1 a ˆij (s, x) pi pj 2 i,j=1 i=1

(in our notation) and its Legendre transformation L(s, x, q) := sup p∈Rd

p, q − H(s, x, p)

=

0 1 2 1 ˆ(s, x)−1/2 q − ˆb(s, x) . a 2

The latter coincides with the function K used here, L(s, x, q) = K(s, x, q), as is immediately seen by a substitution p → −p in the maximization argument. For more details on duality and Legendre transformations see Rockafellar [Roc 97].

136

3 Exit probabilities for degenerate systems

2) Setting formally ε = 0 in (3.5) one obtains the PDE of dynamic programming corresponding to the control problem which deﬁnes the function 0 (s; T ; x, y) ≡ I 0 (s, x, y) ≡ ID

ϑD (s;x,y;u)

inf u∈UD (s;T ;x,y)

K t, u(t), u(t) ˙ dt

s

in 3.1.2, that is ∂I 0 + F, ∇y I 0 m + H s, x, ∇x I 0 = 0 ; R ∂s

(3.6)

see Fleming and Rishel [Fl-Ris 75, p.80-84,104]. 3) Next, one introduces a new stochastic control problem whose dynamic programming PDE is (3.5) for each ε > 0: Consider dηtε = F (t, Ξεt , ηtε ) dt √ d Ξεt = vt dt + ε σ(t, Ξεt ) dWt (Ξεs , ηsε ) = (x, y) ∈ Rd × D , where s ≤ t ≤ T and where v is a nonanticipative ((Ft )t≥0 -adapted) stochastic process taking its values in Rd such that

T

| vt |2 dt < ∞

E s

and such that the resulting process (Ξεt , ηtε )t∈[s,T ] fulﬁlls the assumption 3.1.3(d). A stochastic process v satisfying these properties is called admissible for the above control problem and the set of such admissible control processes is denoted by U ≡ U (s, x, y). Furthermore, ﬁx some bounded Lipschitz continuous function Φ : R+ × Rm −→ R+ such that Φ(s, y) = 0 for all (s, x, y) ∈ G+ and deﬁne 5 J (s, x, y) := min E v∈U

6

ϑε

K(t, Ξεt , vt ) dt

ε

ε

+ Φ(ϑ

, ηϑε ε )

,

s

where ϑε is the minimum of T and the ﬁrst exit time of η ε from D. Then one can prove that the minimum in the deﬁnition of J ε exists and that J ε (s, x, y) thus deﬁned satisﬁes the PDE (3.5) on Q ≡ (0, T ) × Rd × D together with the boundary condition J ε (s, x, y) = Φ(s, y) for all (s, x, y) ∈ ∂ ∗ Q ≡ {T } × Rd × D ∪ G+ T .

3.1 Exit probabilities for degenerate systems depending on a small parameter

137

The proof of this fact follows the same route as the one of theorem 3.1.6 and uses the so-called “veriﬁcation theorem” of dynamic programming (see Fleming and Rishel [Fl-Ris 75, p.159]); since this argument goes beyond the framework of this book, we would like to refer to Hern´ andez-Lerma [HL 81, p.46f.]. 4) lim supε→0 I ε ≤ I 0 : Consider the system y 0 (t) from (3.4) for some control function u ∈ U such that ϑ < T and v := u˙ ∈ U . Then take the system (Ξεt , ηtε ) from step 3) for this (deterministic) control process v , ﬁx T ∈ (ϑ, T ) and deﬁne T ε := T (η ε ) ∧ T , where T (η ε ) denotes the ﬁrst exit time of η•ε from D; in particular, T ε ≤ T < T. Now it follows that 5 ε d T ε ∂2I ε ∂I ε ε + I (s, x, y) = −E a ˆij + F, ∇y I ε m R ∂t 2 ∂x ∂x i j s i,j=1 6

◦ t, Ξεt , ηtε dt + E I ε T ε , ΞεT ε , ηTε ε + vt , ∇x I ε Rd

5

T

ε

=E

H t, x, ∇x I ε − vt , ∇x I ε ◦ t, Ξεt , ηtε dt

s

5

0 1 +E I ε T ε , ΞεT ε , ηTε ε

Tε

K

≤E

6

t, Ξεt , vt

6 0 1 dt + E I ε T ε , ΞεT ε , ηTε ε ,

s

where the ﬁrst equation is the Dynkin formula3 I ε (s, x, y) ≡ E[I ε ◦ Xs ] = E[I ε ◦ XT ε ] − E

Tε

(G I ε ) ◦ Xt dt s

for the process (Xt )t≥s := (t, Ξεt , ηtε )t≥s with generating operator G, the second equality is the PDE (3.5) for I ε found in step 1) and the estimate from above is immediate from the deﬁnition of K; note that “◦” denotes the composition of mappings here and is not to be confused with Stratonovich stochastic diﬀerentials. The second summand in the above bound can be seen to converge to 0 as ε → 0; the argument uses the fact that I ε = 0 on G+ T and an extension of theorem 1.1 by Freidlin and Wentzell [We-Fr 70]; see Hern´ andez-Lerma [HL 81, p.47f.] for details. 3

See e.g. Hackenbroch and Thalmaier [Hb-Th 94, p.131].

138

3 Exit probabilities for degenerate systems

The ﬁrst summand can be seen to converge to ε → 0, since v = u˙ in the system of step 3). Hence, this implies that

ϑ

lim sup I ≤ ε

ε→0

ϑ s

K t, u(t), u(t) ˙ dt as

K t, u(t), u(t) ˙ dt

s

and thus lim supε→0 I ε ≤ I 0 by the deﬁnition of I 0 (see 3.1.2). 5) As a preparation for proving that also lim inf ε→0 I ε ≥ I 0 , one considers the deterministic analogue of the system deﬁned in step 3), that is dηt0 = F (t, Ξ0t , ηt0 ) dt d Ξ0t = vt dt (Ξ0s , ηs0 ) = (x, y) ∈ Rd × D , where again s ≤ t ≤ T and where the set of admissible control functions U 0 ≡ U 0 (s, x, y) now consists of the continuous functions v ∈ C([s, T ], Rd ) such that T

| vt |2 dt < ∞ s

and such that possible exits of the generated trajectory (t, Ξ0t , ηt0 )t∈[s,T ] from R+ × Rd × D only occur through points in G+ . Let Φ be the function ﬁxed in step 3) and deﬁne J (s, x, y) := min0 v∈U

ϑ0

K(t, Ξ0t , vt ) dt + Φ(ϑ0 , ηϑ0 0 ) ,

0

s

where (s, x, y) ∈ [0, T ) × Rd × D and where ϑ0 is the minimum of T and the ﬁrst exit time of η 0 from D. Furthermore, deﬁne J 0 (s, x, y) = Φ(s, y) for (s, x, y) ∈ ∂ ∗ Q ≡ {T } × Rd × D ∪ G+ T which will play a role in the ﬁnal step. Then we have lim inf J ε (s, x, y) ≥ J 0 (s, x, y) ε→0

for all (s, x, y) ∈ Q ≡ (0, T ) × Rd × D. The proof of this fact works with an “optimal feedback control” for the system deﬁning J ε in step 3) and uses a stochastic version of the “principle of optimality in dynamic programming”; see Hern´andez-Lerma [HL 81, p.49f.] for details.

3.1 Exit probabilities for degenerate systems depending on a small parameter

139

6) lim inf ε→0 I ε ≥ I 0 : Fix some Lipschitz continuous function Ψ : D → R+ such that > 0 on D , Ψ = 0 on ∂D . Furthermore, for M > 0 deﬁne 0 1 QεM (s, x, y) := Es,x,y exp − M Ψ YTε ε ∧T /ε , where T ε denotes the ﬁrst exit time of Y ε from D as before; accordingly deﬁne ε (s, x, y) := − ε log QεM (s, x, y) . IM It follows immediately from the deﬁnitions that QεM (s, x, y) ≥ Es,x,y 1

Tε ≤T

≡ Qε (s, x, y)

ε and equivalently that IM (s, x, y) ≤ I ε (s, x, y). QεM (s, x, y) is seen to satisfy the same PDE (3.2) as Qε (s, x, y) together with the boundary condition 1 on G+ ε T QM (s, x, y) = exp [−M Ψ(y)/ε ] on {T } × Rd × D ; ε (s, x, y) satisﬁes the same PDE (3.5) as I ε (s, x, y) with the boundary IM condition 0 on G+ ε T IM (s, x, y) = M Ψ(y) on {T } × Rd × D ;

by remembering the boundary behavior of I ε , = 0 on G+ T , ε I (s, x, y) s→T −−−→ ∞ for (x, y) ∈ Rd × D , it becomes clear how I ε is being “truncated” near {T }×Rd ×D by considering ε IM . Now let Φ ∈ C(R+ × Rm , R+ ) be a Lipschitz function such that Φ(s, y) = 0 whenever (s, x, y) ∈ G+ and such that also 0 on G+ T Φ(s, y) = M Ψ(y) on {T } × Rd × D ; ε 0 and JM , the corresponding functions from steps 3) and 5) are denoted by JM respectively.

140

3 Exit probabilities for degenerate systems

ε ε 0 In particular, the functions IM , JM and JM just deﬁned are all equal to ∗ d Φ(s, y) whenever (s, x, y) ∈ ∂ Q ≡ ({T } × R × D) ∪ G+ T . By uniqueness it ε ε follows for the former two functions that IM = JM . Furthermore, comparing 0 the control problems in 3.1.2 and in step 5) deﬁning I 0 and JM , respectively, 0 0 one gets that I ≤ lim inf JM . M→∞

Also using the estimate mentioned at the end of step 5) and the fact that ε ≤ I ε , it follows altogether that IM 0 I 0 (s, x, y) ≤ lim inf JM (s, x, y) M→∞

ε ≤ lim inf lim inf JM (s, x, y) M→∞

ε→0

ε = lim inf lim inf IM (s, x, y) M→∞

ε→0

≤ lim inf I ε (s, x, y) ε→0

for all (s, x, y) ∈ Q ≡ (0, T ) × Rd × D.

3.2 Uniform consequence for the exit probability Next we prove a consequence of theorem 3.1.7 which is not contained in the paper by Hern´ andez-Lerma [HL 81]: This corollary will be used later in the proof of theorem 4.4.6. Corollary 3.2.1 (Uniform asymptotics of the exit probability Qε ). Consider the SDE (3.1) under the assumptions 3.1.3 and suppose that this SDE (3.1) is autonomous, i.e. F, ˆb and σ ˆ do not depend on t explicitly. 0 0 (x, y) := ID (0; T ; x, y). Furthermore, deﬁne ID Then it follows for the exit probability QεD (x, y) := QεD (0; T ; x, y) that 0 lim inf ε log min QεD (x, y) ≥ − max ID (x, y) ε→0

(x,y)∈G

(x,y)∈G

and 0 lim sup ε log max QεD (x, y) ≤ − min ID (x, y) ε→0

(x,y)∈G

(x,y)∈G

for all T > 0 and any compact set G ⊂ Rd × D. 0 Proof. First note that the functions under consideration, QεD and ID , are continuous with respect to (x, y); this follows from the respective PDEs in theorem 3.1.6 and in equation (3.6). More precisely, (s, x, y) → QεD (s, T, x, y) is continuous on (0, T] × Rd × D due to theorem 3.1.6 for any T > 0. Fixing

3.2 Uniform consequence for the exit probability

141

T := T + δ for given T and some δ > 0 this implies in particular, due to the time-homogeneity (i.e. the stationarity of the stochastic ﬂow4 ), that (x, y) → QεD (x, y) = QεD (δ, T + δ, x, y) 0 is continuous on Rd × D. The continuity of (x, y) → ID (x, y) follows from 0 the fact that ID satisﬁes the PDE (3.6) of dynamic programming. This PDE asserts diﬀerentiability in the sense that there exists a tangent plane at each point (see Fleming and Rishel [Fl-Ris 75, p.83f.]) and hence continuity. 0 are continuous and the proposed Therefore the functions QεD and ID maxima and minima are attained indeed.

First estimate: For brevity deﬁne 0

I D,G :=

0 max ID (x, y)

(x,y)∈G

and ﬁx some η > 0; it is to be shown that there is an ε0 > 0 such that for all ε < ε0 , 0 ε log min QεD (x, y) > − I D,G − η . (x,y)∈G

From theorem 3.1.7 it follows that for any (x, y) ∈ G, there exists ε(x,y) > 0 such that for all ε < ε(x,y) , 0 0 (x, y) − η ≥ − I D,G − η . ε log QεD (x, y) > − ID Due to the continuity of QεD it is hence possible to ﬁnd a suﬃciently small open ball B(x,y) and to choose ε(x,y) > 0 such that ε log

min (˜ x,˜ y)∈B(x,y)

0

QεD (˜ x, y˜) > − I D,G − η

for all ε < ε(x,y). Since G is compact, it can be covered by ﬁnitely many of k > such balls which are denoted by B(xi ,yi ) , i.e. G ⊂ B(xi ,yi ) . The bound i=1

for the parameter ε which corresponds to B(xi ,yi ) by virtue of the previous estimate is denoted by ε(xi ,yi ) , i ∈ {1, . . . , k}. Then one gets for all ε < ε0 := min ε(xi ,yi ) that

i=1,...,k

0

ε log min QεD (x, y) > − I D,G − η . (x,y)∈G

4

See e.g. Hackenbroch and Thalmaier [Hb-Th 94, p.335].

142

3 Exit probabilities for degenerate systems

Second estimate: The above compactness argument also works in this case. Deﬁne 0 (x, y) I 0D,G := min ID (x,y)∈G

and ﬁx an η > 0. By theorem 3.1.7 there is for any (x, y) ∈ G an ε(x,y) > 0 such that 0 (x, y) + η ≤ − I 0D,G + η ε log QεD (x, y) < − ID for all ε < ε(x,y) . Due to the continuity of QεD and the compactness of G, it is possible to cover G by ﬁnitely many open balls B(x,y) on which ε log

max (˜ x,˜ y)∈B(x,y)

QεD (˜ x, y˜) < − I 0D,G + η .

Thus the above claim holds true, if ε0 is again chosen as the minimum of the corresponding (ﬁnitely many) parameters ε(x,y) .

Chapter 4

Local Lyapunov exponents

In this chapter the goal of obtaining the “local Lyapunov exponents” as sublimiting exponential growth rates is tackled. As already described, the system under consideration is the real-noise driven stochastic system dZtε = A (Xtε ) Ztε dt √ dXtε = b (Xtε ) dt + ε σ (Xtε ) dWt ,

(1)

where A ∈ C(Rd , Kn×n ) is a continuous matrix function (K = R or C), d ∈ N and n ∈ N are the dimensions of the state spaces of X ε and Z ε , respectively, ε ≥ 0 parametrizes the intensity of (Wt )t≥0 which denotes a Brownian motion in Rd on a complete probability space (Ω, F , P) and X ε,x is a diﬀusion starting in x ∈ Rd , deﬁned by the SDE (2.1) such that the assumptions 2.1.1 hold. For Z ε , solving the random vector diﬀerential equation dZtε = A Xtε,x (ω) Ztε dt, Z0ε = z ∈ Kn , we will use the equivalent notations Z ε : R+ × Ω × Rd × Kn −→ Kn , (t, ω, x, z) → Z ε (t, ω, x, z) ≡ Z ε (t, ω, x) z ≡ Ztε (ω, x) z ≡ Ztε (ω, x, z), as before, where Ztε (ω, x) ≡ Z ε (t, ω, x, . ) solves the random matrix diﬀerential equation dZtε = A Xtε,x (ω) Ztε dt , Z0ε = idKn . 1 The object of interest is the exponential growth rate T (ε) log | ZTε (ε) (ω, x, z) | on the time scale T (ε). Any limit as ε → 0 of this rate will be called local Lyapunov exponent of Z ε .

W. Siegert, Local Lyapunov Exponents. Lecture Notes in Mathematics 1963. c Springer-Verlag Berlin Heidelberg 2009

143

144

4 Local Lyapunov exponents

4.1 Local Lyapunov exponents: upper and lower bound The n eigenvalues (roots of the characteristic polynomial) of A(x) ∈ Kn×n are denoted by Λj (x) := Λj (A(x)) ∈ C

(j ∈ {1, . . . , n} , x ∈ Rd )

in the sequel. In particular, Λj (Ki ) ≡ Λj (A (Ki )) lists the n eigenvalues of the matrix A(Ki ) at the equilibrium positions K1 , . . . , Kl as given by assumption 2.1.1(K); this enumeration is chosen such that Re (Λ1 (Ki )) ≥ Re (Λ2 (Ki )) ≥ · · · ≥ Re (Λn (Ki )) for all i = 1, . . . , l and the distinct numbers in this list are denoted as λ1 (Ki ) > λ2 (Ki ) > · · · > λpi (Ki ) , where the quantity pi = p A (Ki ) also depends on the space point Ki . Note that if ε = 0, theorem 1.4.3 provides the Lyapunov exponent of the system Zt0 (x, z) as deterministic limit as one of the numbers λ1 (Ki ) , . . . , λpi (Ki ), where i ∈ L ≡ {1, . . . , l} is chosen such that x is in the domain of attraction of Ki according to assumption 2.1.1(K). Fixing the metastable state Kµ(x,ζ) for a speciﬁc time scale T (ε) ! eζ/ε and initial value x = X0ε,x of the noise process (see 2.5.4 for the deﬁnitions), the SDE for Z ε in (1) can be rewritten as dZtε = [ Aµ(x,ζ) + G(Xtε,x ) ] Ztε dt , where the ﬁxed matrix Aµ(x,ζ) is deﬁned as Aµ(x,ζ) := A(Kµ(x,ζ) ) and G( . ) := A( . ) − Aµ(x,ζ) is considered as stochastic perturbation of the deterministic coeﬃcient matrix Aµ(x,ζ) . This perturbation is “small” on the chosen time scale. More precisely, we note the following consequences of section 2.5 on the metastability of X ε,x . Proposition 4.1.1 (Consequences of the metastability of X ε,x ). Consider and let X ε satisfy the assumptions 2.1.1. Fix >l the above situation ζ/ε x ∈ i=1 Di and T (ε) ! e , where ζ > 0 is not contained in a ﬁnite exceptional set depending on x (see (2.20)). Furthermore, let C ∈ Kn×n be some invertible matrix. Then for all open sets B ⊂ R, ε −1 0, 0 ∈ / B, ε→0 ∈ B −−−→ C G XT (ε) C Px 1, 0 ∈ B .

4.1 Local Lyapunov exponents: upper and lower bound

145

−1 In particular, C G(XTε,x converges in probability to zero, (ε) ) C

P C G X ε,x C −1 −−− −→ 0. T (ε) ε→0 Since the resultant convergence in probability implies weak convergence, one −1 gets that the distribution of C G(XTε,x converges weakly to the Dirac (ε) ) C measure at zero, −1 w −1 P ◦ C G(XTε,x −− −−→ δ0 . (ε) ) C ε→0 Furthermore, T (ε) 1 −1 1 P C G Xuε,x C −1 du = C G X ε,x dt −−− −→ 0. t T (ε) C ε→0 T (ε) 0 0 (4.1) Proof. This is theorem 2.5.10 for the (continuous) function f := C G( . ) C −1 ≡ C A( . ) − Aµ(x,ζ) C −1 which vanishes at Kµ(x,ζ) , f (Kµ(x,ζ) ) ≡ C G(Kµ(x,ζ) ) C −1 = 0. Theorem 4.1.2 (Local Lyapunov exponents: upper and lower bound). Consider a mapping A ∈ C(Rd , Kn×n ) and let X ε satisfy the >l assumptions 2.1.1. Fix x ∈ i=1 Di and T (ε) ! eζ/ε , where ζ > 0 is not contained in a ﬁnite exceptional set depending on x (see (2.20)). Furthermore, ﬁx z ∈ Kn \{0} and let Ztε (ω, x, z) be the corresponding solution process of (1). Then the exponential growth rate of Ztε (ω, x, z) on the time scale T (ε), 1 log ZTε (ε) ( . , x, z) , T (ε) is bounded from above by Re Λ1 (Kµ(x,ζ) ) and from below by Re Λn (Kµ(x,ζ) ) , where Kµ(x,ζ) denotes the metastable state corresponding to x and ζ as given by deﬁnition 2.5.4. More precisely, for any η > 0, 1 log ZTε (ε) ( . , x, z) P Re Λn (Kµ(x,ζ) ) − η ≤ T (ε) ε→0 ≤ Re Λ1 (Kµ(x,ζ) ) + η −−−→ 1. Proof. The SDE (1) under consideration for Z ε is rewritten as above as dZtε = [ Aµ(x,ζ) + G(Xtε,x ) ] Ztε dt ,

146

where and

4 Local Lyapunov exponents

Aµ(x,ζ) ≡ A(Kµ(x,ζ) ) G( . ) ≡ A( . ) − Aµ(x,ζ)

which is “small” on the chosen time scale in the sense of proposition 4.1.1. In order to simplify the exposition, we also set (j = 1, . . . , n) Λj := Λj A Kµ(x,ζ) for the eigenvalues of Aµ(x,ζ) ≡ A(Kµ(x,ζ) ) and choose an enumeration of these eigenvalues such that Re(Λ1 ) ≥ Re(Λ2 ) ≥ · · · ≥ Re(Λn ), as before. We are going to view the SDE for Z ε as stochastic perturbation of d the linear ODE dt zt = Aµ(x,ζ) zt with constant coeﬃcient matrix Aµ(x,ζ) ; x and ζ specify the metastable state Kµ(x,ζ) which remains unchanged during the whole treatment. This notation resembles the one used in the proof of the Hartman-Wintner-Perron theorem 1.4.3. It is the purpose of this theorem to investigate to which extent their arguments provide information in the stochastic case. In short, the Gronwall argument and proposition 4.1.1 will imply the upper and lower bounds: As in the proof of 1.4.3, we can assume that after a constant, invertible coordinate transformation, Aµ(x,ζ) is given in (complex) Jordan canonical form, ⎛ ⎞ Λ1 0 0 ⎜ a Λ 0 ⎟ 2 ⎜ ⎟ ⎜ 0 a Λ ⎟ 3 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ Λ ⎜ ⎟, 4 Aµ(x,ζ) = ⎜ ⎟ ⎜ ⎟ .. ⎜ ⎟ . ⎜ ⎟ ⎜ ⎟ ⎝ Λn−1 0 ⎠ a Λn where in this example Λ1 = Λ2 = Λ3 and Λn−1 = Λn ; again, the boxes depict lower Jordan blocks of the following qualitative shape: ⎞ ⎛ Λ0 ⎟ ⎜a Λ 0 ⎟ ⎜ ⎟ ⎜ a Λ 0 ⎟ ⎜ ⎟ ⎜ .. .. .. ⎟; J a := ⎜ . . . ⎟ ⎜ ⎟ ⎜ ⎜ a Λ 0 ⎟ ⎟ ⎜ ⎝ a Λ 0⎠ a Λ

4.1 Local Lyapunov exponents: upper and lower bound

147

here, a > 0 is a parameter and all matrix entries not mentioned explicitly are zero. As in 1.4.3, we have taken the “ordinary” lower Jordan blocks J 1 and carried out another transformation tε := CZtε , Z where C := diag(1, a, . . . , an−1 ) ; then pathwise (“ω-wise”) diﬀerentiation with respect to t yields ε = C dZ ε = CAµ(x,ζ) Z ε + CG(X ε,x )Z ε dZ t t t t t = CAµ(x,ζ) C −1 Z˜tε + CG(Xtε,x )C −1 Z˜tε ; hence, we can redeﬁne Aµ(x,ζ) as CAµ(x,ζ) C −1 and G(Xtε,x ) as CG(Xtε,x )C −1 ; after this transformation the Jordan-blocks of Aµ(x,ζ) are of the form Jja . The “smallness” of G(Xtε,x ) on the chosen time scale in the sense of proposition 4.1.1 is maintained under this change of the coordinate sys1 log |ZTε (ε) | is tem, as already noted in 4.1.1. The investigated behavior of T (ε) invariant under this (invertible) transformation of variables. More precisely, the norm |C . | on Kn is equivalent to the “old” norm | . | as Kn is ﬁnite dimensional; but factors from converting these norms into each other end up ε for as vanishing summands, since T (ε) → ∞; therefore, we redeﬁne Z ε as Z the following considerations. Hence, the underlying random diﬀerential equation for Z ε from (1) is of the form Λ1 Z ε;1 + (Gεt Z ε )1 dt d Z ε;1 = d Z ε;i = ai−1 Z ε;i−1 + Λi Z ε;i + (Gεt Z ε )i dt (i = 2, . . . , n) , where the superscript i in ( . )i again indicates the i-th coordinate, in particular, (i = 1, . . . , n) Z ε;i := (Z ε )i for the coordinates of Z ε ; furthermore, we used the notation Gεt := Gεt (ω) := G (Xtε,x (ω)) and ai := Aµ(x,ζ) i+1,i ∈ {0, a}

(i = 1, . . . , n − 1)

denoting the n − 1 entries of Aµ(x,ζ) below the diagonal. Also put a0 := 0 and Z ε;0 := 0 for deﬁniteness. Furthermore, again let λ1 > λ2 > · · · > λp

148

4 Local Lyapunov exponents

denote the distinct numbers in the list Re(Λ1 ) ≥ Re(Λ2 ) ≥ · · · ≥ Re(Λn ) ; in particular, it again follows from these deﬁnitions that i := min{ i : Re(Λi ) = λk } =⇒ ai−1 = 0

(4.2)

for all k = 1, . . . , p. As in the proof of the Hartman-Wintner-Perron theorem 1.4.3, we now deﬁne auxiliary processes describing the norms of the projection of Z ε onto the subspaces belonging to the diﬀerent real part of eigenvalues; more precisely, we deﬁne the following stochastic processes pathwise as ε;i := | Ztε;i | , t

i = 0, . . . , n ,

and hereby

Lε;k t (ω) :=

2 ε;i (ω) t

, k = 1, . . . , p ,

{i : Re(Λi )=λk }

Mtε;k (ω) :=

Ntε;k (ω) :=

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

0

{i : Re(Λi )>λk } ⎪ n ⎪ 2 ⎪ ⎪ ε;i ⎪ (ω) ⎪ t ⎩ i=1

⎧ ⎪ ⎨

⎪ ⎩

{i : Re(Λi )≤λk }

0

,k = 1 ,

2 ε;i (ω) t ≡ | Ztε (ω) | 2 ε;i t (ω)

, k = 2, . . . , p , 2

,k = p + 1 , , k = 1, . . . , p , ,k = p + 1 ,

such that ε;p | Ztε |2 = Mtε;k + Ntε;k = Lε;1 . t + · · · + Lt

(4.3)

The above random diﬀerential equation for Z ε implies that the calculations as performed in the proof of Hartman-Wintner-Perron theorem 1.4.3 can be directly applied pathwise, i.e. for any realization ω ∈ Ω, in order to get the ε;k ε;k and Ntε;k . Since we are going to discuss what diﬀerentials of ε;i t , Lt , Mt can and what cannot be carried over from theorem 1.4.3 to the stochastic situation here and since we are also going to investigate the more tractable case that A( . ) only takes its values in the diagonal matrices (theorem 4.3.1), it seems indicated to recall the calculations in the notation used here such that these cases can be compared in full detail. Therefore (1.22) reads for the stochastic process ε;i t due to the above form of the random diﬀerential equation for Z ε pathwise as

4.1 Local Lyapunov exponents: upper and lower bound

149

d ¯ ε;i ε;i d ε;i 2 ( ) = (Z Z ) dt t dt

d ¯ ε;i d ε;i ε;i ε;i ¯ = Z Z +Z + 0 Z dt dt

d ε;i Z = 2 Re Z¯ ε;i dt ε;i ai−1 Z ε;i−1 + Λi Z ε;i + (Gεt Z ε )i = 2 Re Z¯ = 2ai−1 Re(Z¯ ε;i Z ε;i−1 ) + 2 Re(Λi ) ( ε;i )2

(4.4)

t

+ 2 Re(Z¯ ε;i (Gεt Z ε )i )

(4.5)

for all i = 1, . . . , n, where as usual ¯ denotes complex conjugation; hence, d( ε;i )2 ε;i 2 − 2 Re(Λi ) ( ) ≤ 2ai−1 ε;i ε;i−1 + 2 ε;i (Gεt Z ε )i . dt (4.6) Thus it follows for Lε;k that ε;k dL ε;k dt − 2λk L ≡

+

{i:Re(Λi )=λk }

(4.6)

≤ 2

≤ 2⎝

( ε;i )2 ⎠

{i:Re(Λi )=λk }

⎛

+2⎝

ε;i |(Gεt Z ε )i |

{i:Re(Λi )=λk }

⎞1/2 ⎛

C.-S.

ai−1 ε;i ε;i−1 + 2

{i:Re(Λi )=λk }

⎛

, d( ) − 2λk ( ε;i )2 dt ε;i 2

⎝

⎞1/2

a2i−1 ( ε;i−1 )2 ⎠

{i:Re(Λi )=λk }

⎞1/2 .1/2 n ε;i 2 ⎠ ε ε i 2 ( ) |(Gt Z ) |

{i:Re(Λi )=λk }

⎛

i=1

⎞ 1/2

= 2(Lε;k )1/2 ⎝

(4.2)

a2i−1 ( ε;i−1 )2 ⎠

{i:Re(Λi )=λk ,i−1≥i}

+ 2(L

ε;k 1/2

)

⎛

≤ 2a(Lε;k )1/2 ⎝

|(Gεt Z ε )|

⎞1/2 ( ε;i )2 ⎠

+ 2(Lε;k )1/2 Gεt |Z ε |

{i:Re(Λi )=λk } (4.3)

= 2aLε;k + 2Gεt (Lε;k )1/2 (M ε;k + N ε;k )1/2 . / M ε;k + N ε;k ε;k ε 2a + 2Gt =L Lε;k

(4.7)

150

4 Local Lyapunov exponents

for all k = 1, . . . , p, where “C.-S.” is an abbreviation for the Cauchy-Schwarz inequality. Furthermore,

d M ε;k = dt

{i : Re(Λi )>λk }

(4.5)

=

d ε;i 2 ( ) dt

2ai−1 Re(Z¯ ε;i Z ε;i−1 ) + 2 Re(Λi ) ( ε;i )2

{i : Re(Λi )>λk }

≥ −2

+ 2 Re(Z¯ ε;i (Gεt Z ε )i )

ai−1 ε;i ε;i−1 + 2λk−1

{i : Re(Λi )>λk }

−2

( ε;i )2

{i : Re(Λi )>λk }

ε;i | (Gεt Z ε )i |

{i : Re(Λi )>λk } (4.2)

≥ − 2 a M ε;k + 2 λk−1 M ε;k − 2 Gεt (M ε;k )1/2 |Z ε | √ (4.3) = 2 (λk−1 − a) M ε;k − 2 Gεt M ε;k M ε;k + N ε;k

(4.8)

for k = 2, . . . , p + 1; and d N ε;k = dt

{i : Re(Λi )≤λk }

(4.5)

=

d ε;i 2 ( ) dt

{i : Re(Λi )≤λk }

≤ 2

2ai−1 Re(Z¯ ε;i Z ε;i−1 ) + 2 Re(Λi ) ( ε;i )2 + 2 Re(Z¯ ε;i (Gεt Z ε )i ) ai−1 ε;i ε;i−1 + 2λk

{i : Re(Λi )≤λk }

+2

( ε;i )2

{i : Re(Λi )≤λk }

ε;i | (Gεt Z ε )i |

{i : Re(Λi )≤λk } (4.2)

≤ 2 a N ε;k + 2 λk N ε;k + 2 Gεt (N ε;k )1/2 |Z ε | √ (4.3) = 2 (λk + a) N ε;k + 2 Gεt N ε;k M ε;k + N ε;k for k = 1, . . . , p. From (4.8), (4.9) and (4.3) one gets for |Z ε |2 ≡ M ε;p+1 ≡ N ε;1 that for any ω, d |Z ε |2 ≤ 2 λ1 + a + Gεt |Ztε |2 2 λp − a − Gεt |Ztε |2 ≤ dt t

(4.9)

4.1 Local Lyapunov exponents: upper and lower bound

151

so that integration with respect to t ≥ 0 yields pathwise: exp (λp − a)t −

t 0

Gεu du |z| ≤ |Ztε | ≤ exp (λ1 + a)t +

t 0

Gεu du |z| .

Furthermore, due to equation (4.1) in proposition 4.1.1, 1 T (ε)

T (ε) 0

ε P Gu ( . ) du −−− −→ 0 . ε→0

Now if η > 0 is given, then ﬁx the small parameter a as a ∈ (0, η) and assume without loss of generality that |z| = 1 (otherwise the summand ε→0 1 −−→ 0 deterministically vanishes). Then the above two statements T (ε) log |z| − which had been lastly obtained yield that 0 1 1 ε log ZT (ε) ( . , x, z) ∈ P / Re Λn (Kµ(x,ζ) ) − η, Re Λ1 (Kµ(x,ζ) ) + η T (ε) 1 1 ε ε log | ZT (ε) | < λp − η + P log | ZT (ε) | > λ1 + η =P T (ε) T (ε) ; T (ε) 1 Gεu du < λp − η ≤ P (λp − a) − T (ε) 0 ; T (ε) 1 ε Gu du > λ1 + η + P (λ1 + a) + T (ε) 0 ; T (ε) 1 Gεu du > η − a =2P T (ε) 0 ε→0

−−−→ 0. This is a reformulation of what had been claimed.

Corollary 4.1.3 (p = 1). Consider the situation of theorem 4.1.2. If p A Kµ(x,ζ) = 1, i.e. if

Re Λ1 (Kµ(x,ζ) ) = Re Λn (Kµ(x,ζ) ) =: λ ,

then

1 P log ZTε (ε) ( . , x, z) −−− −→ λ ε→0 T (ε)

for all z ∈ Kn \{0}. Remark 4.1.4 (n = 1). Consider the situation of corollary 4.1.3. Then the assumption p = 1 is trivially fulﬁlled if n = 1. In this case, A =: a is just a

152

4 Local Lyapunov exponents

continuous function on Rd taking its values in the scalar ﬁeld K (R or C). The random vector diﬀerential equation for Z ε in (1) is now a scalar diﬀerential equation dZtε = a Xtε,x (ω) Ztε dt , Z0ε = z ∈ K which can be explicitly integrated pathwise as t

a Xuε,x (ω) du . Ztε (ω, x, z) = z · exp 0

Therefore the convergence assertion of corollary 4.1.3 can also be directly deduced by applying (the ﬁnal statement of) theorem 2.5.10 to the continuous function f (x) := Re[a(x)] in the one-dimensional case n = 1. Due to corollary 4.1.3 it is left to consider the case that p ≡ p A Kµ(x,ζ) > 1 . More precisely, one needs to investigate the convergence behavior of the 1 random variable T (ε) log | ZTε (ε) ( . , x, z) | as ε → 0 in the case that there are several possible “candidates” λ1 , . . . , λp for this growth rate. Theorem 4.1.2 states that λ1 and λp serve as upper and lower bound in the sense of convergence in probability, respectively. In the light of the Multiplicative Ergodic Theorem 1.3.1 and the deterministic theorem 1.4.3 which enframe the sublimiting situation at hand, one might conjecture here (in the situation of theorem 4.1.2 with p > 1) that for any z ∈ Kn \{0}, there is an index J ∈ {1, . . . , p} such that 1 P log | ZTε (ε) ( . , x, z) | −−− −→ λJ ; ε→0 T (ε)

(4.10)

another conjecture, motivated by the law of large numbers (1.16), is that under some hypoellipticity condition for any z ∈ Kn \{0}, 1 P log | ZTε (ε) ( . , x, z) | −−− −→ λ1 . ε→0 T (ε)

(4.11)

For this purpose, we ﬁrst continue the investigations of the proof of theorem 4.1.2 as in 1.4.3: Excluding the trivial case that Ztε = 0, we can consider the auxiliary processes vtε;k :=

Mtε;k Mtε;k ≡ ∈ [0, 1] | Ztε |2 Mtε;k + Ntε;k

for k = 1, . . . , p + 1; in particular v ε;1 ≡ 0 and v ε;p+1 ≡ 1. These processes further satisfy the pathwise relations d d ε;k N ε;k dt M ε;k − M ε;k dt N d ε;k v = , ε;k ε;k 2 dt (M + N )

4.1 Local Lyapunov exponents: upper and lower bound

153

where M ε;k , N ε;k and Z ε have rectiﬁable paths, and v ε;k ( 1 − v ε;k ) =

M ε;k N ε;k . (M ε;k + N ε;k )2

Due to (4.8) and (4.9), it follows for the pathwise derivatives with respect to t that d ε;k v ≥ 2 (λk−1 − λk − 2a) v ε;k ( 1 − v ε;k ) − 2 Gεt dt for any k = 1, . . . , p + 1; note that this estimate only contains information for k = 2, . . . , p, since it is trivially satisﬁed by v ε;1 ≡ 0 and v ε;p+1 ≡ 1. The full calculations can be found on p. 38f., where — in the current terminology — the pathwise derivatives are estimated for every ﬁxed ε and ω. In other words, deﬁning v ε := v ε;k and b := 2 (λk−1 − λk − 2a) which is strictly positive, b > 0, if we choose a > 0 suﬃciently small, 2a <

min

k=1,...,p−1

λk − λk+1 ,

as p > 1, then one arrives at the random Riccati-type diﬀerential inequality d ε vt ≥ b vtε 1 − vtε − 2 Gεt . dt

(4.12)

For this situation, the following statement provides the stochastic analogue of 1.4.2: Lemma 4.1.5 (Random Riccati-type diﬀerential inequality (4.12)). Consider the situation of theorem 4.1.2 and the matrix function G( . ) := A( . ) − Aµ(x,ζ) as investigated in 4.1.1. Let (vtε )t≥0 be a family of stochastic processes on (Ω, F, P), ε > 0, taking their values in the interval [0, 1] such that the paths of (vtε )t≥0 are diﬀerentiable and satisfy the random Riccati-type diﬀerential inequality (4.12), i.e. d ε vt (ω) ≥ b vtε (ω) 1 − vtε (ω) − 2 G Xtε,x (ω) (t > 0 , ω ∈ Ω) , dt where b > 0 and Xtε,x is the underlying diﬀusion driving (1). Then it follows for any initial value v0 = v0ε and any η1 , η2 > 0 that 1 P L t ∈ [0, T (ε)] : vtε ∈ / [η1 , 1 − η2 ] −−− −→ 1 . ε→0 T (ε) Proof. For brevity, we deﬁne gtε (ω) := 2 G Xtε,x (ω) ≡ 2 Gεt (ω)

154

4 Local Lyapunov exponents

and β : [0, 1] → [0, b/4] ,

β(v) := b v (1 − v) .

Then one can rewrite the proposed inequality pathwise (for any ω ∈ Ω) as ε vT (ε) = v0 +

≥ v0 +

T (ε)

0 T (ε) 0

d ε v dt t

dt

β (vtε ) dt −

T (ε) 0

≥ v0 +

gtε dt β (vtε ) dt −

T (ε) 0

gtε dt

{ t∈[0,T (ε)] : vtε ∈[η1 ,1−η2 ] }

min β · L { t ∈ [0, T (ε)] : vtε ∈ [η1 , 1 − η2 ] } − ≥ v0 + = v0 +

[η1 ,1−η2 ]

min

[η1 ,1−η2 ]

β

· T (ε) · L

T (ε) 0

ε s ∈ [0, 1] : vs·T (ε) ∈ [η1 , 1 − η2 ]

Rearranging and dividing by T (ε) as well as by

min

gtε dt

−

T (ε) 0

gtε dt .

β > 0 now yields that

[η1 ,1−η2 ]

1 ε L {t ∈ [0, T (ε)] : vtε ∈ [η1 , 1 − η2 ]} = L s ∈ [0, 1] : vs·T ∈ [η , 1 − η ] 1 2 (ε) T (ε) .

−1 T (ε) ε vT (ε) − v0 1 ε min β gt dt + . ≤ T (ε) 0 T (ε) [η1 ,1−η2 ] 1 (v0 − However, v ε taking its values in [0, 1] is bounded and therefore T (ε) ε vT (ε) ) → 0 surely as ε → 0 (i.e. in the deterministic sense for any path); thus, as the time average T (ε) 1 gtε dt T (ε) 0

tends to zero in probability as ε → 0 by 2.5.10 and 4.1.1, the same holds true for the whole right hand side of the last inequality. Therefore, this estimate implies that also 0 ≤

1 P L { t ∈ [0, T (ε)] : vtε ∈ [η1 , 1 − η2 ] } −−− −→ 0 . ε→0 T (ε)

Hence the lemma is proven.

One might think that the proof of lemma 1.4.2 could be transferred to the stochastic situation above by means of a subsequence argument; this would entail to ﬁx a sequence (tn )n∈N along which (gtε )t converges almost surely. However, the continuity of t → v(t) had been crucially used in 1.4.2; so by restricting this argument to the discrete time set {tn : n ∈ N}, it turns out that one cannot deduce a more reﬁned convergence assertion on vtε here.

4.1 Local Lyapunov exponents: upper and lower bound

155

Now we consider the stochastic analogue of remark 1.4.6: Remark 4.1.6 (Kn = R2 ). Consider the two-dimensional real situation, Kn = R2 , for illustration. As before, p ≡ p(A(Kµ(x,ζ) )). If p = 1, i.e. if Re(Λ1 (Kµ(x,ζ) )) = Re(Λ2 (Kµ(x,ζ) )), then A(Kµ(x,ζ) ) either has two complex (conjugate) characteristic roots or it has one real eigenvalue with geometric multiplicity 1. This situation is covered by corollary 4.1.3. If p = 2, i.e. if Re(Λ1 (Kµ(x,ζ) )) > Re(Λ2 (Kµ(x,ζ) )), the situation is not included in corollary 4.1.3. Here, λ1 := Re Λ1 (Kµ(x,ζ) ) > Re Λ2 (Kµ(x,ζ) ) =: λ2 are two real eigenvalues of the matrix Aµ(x,ζ) =

λ1 0 0 λ2

(after the coordinate transformation C) and the canonical unit vector ei is the eigenvector corresponding to λi , i = 1, 2. Then vtε := vtε;2 :=

2 Mtε;2 ( ε;1 t ) ε 2 ≡ ε;2 2 = (cos αt ) , 2 | Ztε |2 ( ε;1 t ) + ( t )

where αεt again denotes the angle of Ztε in R2 . Figure 1.1 of remark 1.4.6 illustrates this situation for ﬁxed ε and ω, if zt , vt and αt are replaced by Ztε , vtε and αεt , respectively. It follows from the Riccati diﬀerential inequality (4.12) for v ε and from lemma 4.1.5 that for any η1 , η2 > 0, P 1 L t ∈ [0, T (ε)] : vtε ∈ / [η1 , 1 − η2 ] −−− −→ 1 ; ε→0 T (ε) in other words, for any η1 , η2 > 0, P 1 L t ∈ [0, T (ε)] : αεt ∈ B(0, η1 ) or αεt ∈ B(π/2, η2 ) −−− −→ 1 , ε→0 T (ε) where (αε )t is again considered as process in [0, π), i.e. as the angle of the projective line Z ε ∈ P 1 ; more precisely, the angles 0 and π are identiﬁed as belonging to the same projective residue class as are π2 and 3π 2 . The preceding convergence in probability reﬂects the fact that the drift ¯ . , Kµ(x,ζ) ) of the angle process αε has two equilibria, namely at 0 and π . h( t 2 In section 4.3 we assume even stronger that A( . ) only takes its values in the diagonal matrices which means that ¯ h( . , x) has the same equilibria simultaneously for all x; in other words, the above coordinate transformation C simultaneously diagonalizes all values of A; vtε is then seen to obey a Verhulsttype diﬀerential equation (4.22) (Verhulst-type diﬀerential inequality (4.18)

156

4 Local Lyapunov exponents

in arbitrary dimension n) which is more precise than the previously obtained Riccati diﬀerential inequality (4.12). Note that by retransforming the coordinates via C −1 it generally follows for the angle process (αε )t from the preceding convergence in probability that P 1 L t ∈ [0, T (ε)] : αεt ∈ B(α1 , η1 ) or αεt ∈ B(α2 , η2 ) −−− −→ 1 , ε→0 T (ε) where α1 = α2 are the angles of the two eigendirections in [0, π) corresponding to the eigenvalues λ1 > λ2 of the (not necessarily diagonal) matrix Aµ(x,ζ) .

4.2 The local growth rate of the determinant In the situation of theorem 4.1.2 one can also obtain a local growth rate for the determinant of Z ε . This is due to the Jacobi equation and the consequence 2.5.10 of the metastability of X ε . Theorem 4.2.1. Consider a mapping A ∈ C(Rd , Kn×n ) and let X ε satisfy >l the assumptions 2.1.1. Fix x ∈ i=1 Di and T (ε) ! eζ/ε , where ζ > 0 is not contained in a ﬁnite exceptional set depending on x (see (2.20)). Furthermore, let Ztε (ω, x) be a matrix solution of (1) started in a ﬁxed matrix Z0 ∈ GL(n, K). Then the exponential growth rate of det Ztε (ω, x) on the time scale T (ε) is trace A(Kµ(x,ζ) ) , where Kµ(x,ζ) again denotes the metastable state corresponding to x and ζ as given by deﬁnition 2.5.4 and the exposition preceding it. More precisely, 1 P log det ZTε (ε) ( . , x) −−− −→ trace A(Kµ(x,ζ) ) . ε→0 T (ε) Proof. From the Jacobi (Liouville) equation (1.2), it follows that t

det(Ztε (ω, x)) = (det Z0 ) · exp trace A Xuε,x (ω) du . 0

Therefore, log | det Z0 | 1 1 log det ZTε (ε) (ω, x) = + T (ε) T (ε) T (ε)

-

T (ε) 0

trace A Xuε,x (ω) du

.

and the claim follows from theorem 2.5.10, for the continuous function f := trace A. Note that the above theorem provides the sublimiting counterpart to the trace formula (1.13) for the sum of the Lyapunov exponents in the situation

4.3 Local Lyapunov exponents in the diagonal case

157

of Oseledets’ theorem, 1 log | det Ztε (ω, x) | t 1 t trace A Xuε,x ( . ) du = lim t→∞ t 0 trace A (x) pε (x) dx , =

Λε1 + · · · + Λεn = lim

t→∞

Rd

where the convergence holds almost surely.

4.3 Local Lyapunov exponents in the diagonal case In this section a special case is to be considered in detail: The conjecture (4.10) is to be veriﬁed in the situation that in the underlying diﬀerential system, dZtε = A (Xtε ) Ztε dt √ dXtε = b (Xtε ) dt + εσ (Xtε ) dWt ,

(1)

the continuous matrix function A ∈ C(Rd , Rn×n ) now solely takes its values in the diagonal, real matrices (n ∈ N); as before, d ∈ N denotes the dimension of the state space of X ε , ε ≥ 0 parametrizes the intensity of (Wt )t≥0 , a Brownian motion in Rd on a complete probability space (Ω, F , P), and X ε,x is a diﬀusion starting in x ∈ Rd , deﬁned by the underlying SDE (2.1) such that the assumptions 2.1.1 hold. Again, Λj (x) := Λj (A(x))

(j = 1, . . . , n)

will denote the eigenvalues of the n×n matrix A(x), where x ∈ Rd . Since A(x) is assumed to be diagonal, the canonical j-th unit vector ej is an eigenvector corresponding to the (real) eigenvalue Λj (x) = (A(x))jj for each x. Theorem 4.3.1 (Local Lyapunov exponents). Let the assumptions 2.1.1 be satisﬁed and suppose that A ∈ C(Rd , Rn×n ) only takes its values in the diagonal matrices on Fx,ζ , where Fx,ζ denotes the subset of Rd as deﬁned in corollary 2.5.7. >l Fix x ∈ i=1 Di and T (ε) ! eζ/ε , where ζ > 0 is not contained in a ﬁnite exceptional set depending on x (see (2.20)). Furthermore, ﬁx z ∈ Rn \ {0}. Then the stochastic process Ztε (ω, x, z), given by (1), has the following exponential growth rates on the time scale T (ε): 1 P log ZTε (ε) ( . , x, z) −−− −→ Λj Kµ(x,ζ) ε→0 T (ε)

158

4 Local Lyapunov exponents

for some j ∈ {1, . . . , n}, where Kµ(x,ζ) denotes the metastable state corresponding to x and ζ as given by deﬁnition 2.5.4 and the exposition preceding it. This stochastic limit Λj Kµ(x,ζ) will be called the local (j th ) Lyapunov exponent of Z ε with respect to x, ζ and z. Proof. As before, ﬁrst rewrite the RDE for Z ε in (1) as dZtε = [ Aµ(x,ζ) + Gεt ] Ztε dt , where Aµ(x,ζ) ≡ A(Kµ(x,ζ) ) and Gεt := Gεt (ω) := G(Xtε,x (ω)) := A(Xtε,x (ω)) − Aµ(x,ζ) . For brevity, again deﬁne Λj := Λj Kµ(x,ζ) , decreasingly indexed as Λ1 ≥ Λ2 ≥ · · · ≥ Λn , and the distinct numbers in this list are again denoted by λ1 > λ2 > · · · > λp . In the case that p = 1 the claim follows directly from corollary 4.1.3. Therefore, it is assumed that p ≥ 2 without loss of generality in the following analysis. Deﬁne ΩεΓ;F := ω ∈ Ω : Xtε (ω) ∈ Fx,ζ for all t ∈ [0, Γ · T (ε)] for some parameter Γ > 0, for which it is known from corollary 2.5.7 that lim P ΩεΓ;F = 1 . ε→0

Then it follows from the diagonality assumption on A that the RDE under consideration now becomes d ε;i Zt (ω) = Λi Xtε,x (ω) Ztε;i (ω) dt i = Λi Ztε;i (ω) + Gεt (ω) Ztε (ω) ε;i = Λi Ztε;i (ω) + Gε;i t (ω) Zt (ω)

for all ω ∈ ΩεΓ;F and t ∈ (0, Γ T (ε)), where the superscript i ∈ {1, . . . , n} indicates the i-th coordinate as before and Gεt also takes its values in the

4.3 Local Lyapunov exponents in the diagonal case

diagonal matrices,

159

⎛

⎞ Gε;1 0 t ··· ⎜ ⎟ .. Gεt = ⎝ ⎠ . . 0 · · · Gε;n t

(4.13)

Hence, due to the diagonality assumption, Z ε can be explicitly calculated as t

ε,x ε;i i Λi Xu (ω) du (i = 1, . . . , n) Zt (ω) = z exp 0

which by theorem 2.5.10 implies that 1 P log ZTε;i(ε) −−− Λi Kµ(x,ζ) ≡ Λi − → ε→0 T (ε)

(i = 1, . . . , n) .

However, this does not imply the claim yet and hence we aim at a more reﬁned analysis. For this purpose, we again consider the auxiliary processes as in the proofs of the Hartman-Wintner-Perron theorem 1.4.3 and theorem 4.1.2. These processes now describe the norms of the projection of Z ε onto the subspaces spanned by the canonical unit vectors e1 , . . . , en of Rn which are the eigenvectors of Aµ(x,ζ) corresponding to Λ1 , . . . , Λn . Recalling the deﬁnitions, we consider more precisely the following stochastic processes ε;i ε;i i ε;i t (ω) ≡ | Zt (ω) | , starting in 0 = z

(i = 0, . . . , n),

and hereby

Lε;k t (ω) ≡

2 ε;i t (ω)

, k = 1, . . . , p ,

{i : Λi =λk }

Mtε;k (ω) ≡

Ntε;k (ω) ≡

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

0

2

ε;i t (ω)

{i : Λi >λk } ⎪ n ⎪ 2 ⎪ ⎪ ε;i ⎪ (ω) ⎪ t ⎩ i=1

⎧ ⎪ ⎨

⎪ ⎩

{i : Λi ≤λk }

0

≡ | Ztε (ω) |2

2 ε;i t (ω)

,k = 1 , , k = 2, . . . , p , ,k = p + 1 , , k = 1, . . . , p , ,k = p + 1 ,

ε;p such that |Ztε |2 = Mtε;k + Ntε;k = Lε;1 t + · · · + Lt , as has already been noted in (4.3). In contrast to the general calculations (4.5)–(4.9), the above form of the RDE for Z ε;i in the diagonal case then implies on ΩεΓ;F for t ∈ (0, Γ T (ε)) and i ∈ {1, . . . , n} that

160

4 Local Lyapunov exponents

d ε;i 2 d Ztε;i t = 2 Ztε;i dt dt

=

2 2 2Λi ε;i ε;i + 2 Gε;i . (4.14) t t t

Hence for k ∈ {1, . . . , p}, ε;k + , dL d( ε;i )2 ε;k ε;i 2 − 2λ − 2λ = L ( ) k k dt dt {i : Λi =λk } ε;i 2 ε;i (4.14) = 2 G {i : Λi =λk }

≤ 2 Lε;k Gε ,

(4.15)

and for k = 2, . . . , p + 1, d M ε;k dt

=

{i : Λi >λk }

(4.14)

=

{i : Λi >λk }

≥ 2λk−1

d ε;i 2 ( ) dt

2Λi ( ε;i )2 + 2 Gε;i ( ε;i )2

( ε;i )2 − 2 Gε

{i : Λi >λk }

ε;i 2

{i : Λi >λk }

= 2 ( λk−1 − G ) M ε

ε;k

,

(4.16)

as well as for k = 1, . . . , p + 1, d N ε;k dt

=

{i : Λi ≤λk }

(4.14)

=

d ε;i 2 ( ) dt

2Λi ( ε;i )2 + 2 Gε;i ( ε;i )2

{i : Λi ≤λk }

≤ 2λk

( ε;i )2 + 2 Gε

{i : Λi ≤λk }

= 2 ( λk + G ) N ε

ε;i 2

{i : Λi ≤λk } ε;k

.

(4.17)

Now we need to ﬁnd for each solution Ztε ≡ 0 an index J ∈ {1, . . . , p} such that 1 P log | ZTε (ε) ( . , x, z)| −−− −→ λJ ε→0 T (ε) which had been formulated as conjecture (4.10).

4.3 Local Lyapunov exponents in the diagonal case

161

As already mentioned on p.152, consider the auxiliary processes vtε;k :=

Mtε;k Mtε;k ≡ ∈ [0, 1] | Ztε |2 Mtε;k + Ntε;k

for k = 1, . . . , p + 1 for which in particular v ε;1 ≡ 0 and v ε;p+1 ≡ 1. Then one gets altogether on ΩεΓ;F for the pathwise derivatives with respect to t ∈ (0, Γ T (ε)) that d ε;k v dt

= (4.16),(4.17)

≥

= =

d d N ε;k ( dt M ε;k ) − M ε;k ( dt N ε;k ) (M ε;k + N ε;k )2 ε;k 1 N 2 ( λk−1 − Gε ) M ε;k ε;k ε;k 2 (M + N ) − M ε;k 2 ( λk + Gε ) N ε;k 1 2 M ε;k N ε;k (λk−1 − λk − 2 Gε ) ε;k ε;k 2 (M + N )

2 v ε;k ( 1 − v ε;k ) (λk−1 − λk − 2 Gε ) .

Hence, lemma 4.3.4 (see below) applies to the random Riccati-/Verhulsttype diﬀerential inequality d ε;k v ≥ 2 v ε;k ( 1 − v ε;k ) (λk−1 − λk − 2 Gε ) , dt

(4.18)

with v ε := v ε;k , b1 := 2 (λk−1 − λk ) > 0 and b2 := 4; note that the strict positivity of b1 is due to the assumption that p > 1. Therefore for all k ∈ {1, . . . , p + 1}, either v•ε;k (ω) ≡ 0 (∀ ε)

or

v•ε;k (ω) ≡ 1 (∀ ε)

or

P vTε;k(ε) −−− −→ 1. ε→0

Since v ε;1 ≡ 0 and v ε;p+1 ≡ 1, the following deﬁnition of the index J is thus well-deﬁned and only depends on z ≡ Z0ε : J := max k = 1, . . . , p : v•ε;k ≡ 0 P = min k = 2, . . . , p + 1 : vTε;k(ε) −−− −→ 1 − 1 . ε→0

Zε

This index displays the direction eJ to which the projective line |Ztε | cont verges in probability; 0 and 1 are ﬁxed points of the random Riccati-/ ε Z Verhulst-type diﬀerential inequality which means that |Ztε | is starting and t staying in the respective hyperplane of eigendirections of Aµ(x,ζ) ; any other P initial direction, i.e. any other v0ε;k ∈ (0, 1), implies that vTε;k(ε) −−− −→ 1; this

means that within the corresponding plane

Ztε |Ztε |

ε→0

is attracted towards the

162

4 Local Lyapunov exponents

eigendirection corresponding to the largest eigenvalue represented in this plane. Note the analogy to the deﬁnition of the J in theorem 1.4.3. This implies, as M ε;k = Lε;1 + · · · + Lε;k−1 and as k → vtε;k is increasing, that ⎧ ⎪ ε;k+1 ε;k ⎨ P = 0, k < J ; Lε;k − M M T (ε) T (ε) T (ε) ε;k+1 ε;k −− −−→ 1 , k = J ; = = vT (ε) − vT (ε) ε→0 ⎪ |ZTε (ε) |2 |ZTε (ε) |2 ⎩ P 0, k > J . −−−−→ ε→0

(4.19) Now (4.19) reduces the study of |ZTε (ε) | to

2

ε;J Lε;J T (ε) ; namely on {LT (ε) = 0},

2 Lε;J 1 T (ε) ε;J P ε log ε 2 −−− log LT (ε) − log |ZT (ε) | = −→ 0 ε→0 2 |ZT (ε) | and hence the above claim, (4.10), concerning the local Lyapunov exponent reduces to proving that 1 P log Lε;J − −−→ 2 λJ . T (ε) − ε→0 T (ε) But considering (4.15) for k = J , ε;J dL ε;J dt − 2λJ L

≤ 2 Lε;J Gε ,

ε;J yields on {Lε;J T (ε) = 0} after dividing by LT (ε) that for any ﬁxed η > 0:

1 ε;J P log LT (ε) − 2λJ > η T (ε) 5 ;

6 T (ε) 1 d ε;J ε;J = P log Lu − 2λJ du > η log L0 + T (ε) du 0 5 6 ; T (ε) 1 ε;J d log Lε;J du > η − 2λ ≤ P log L0 + J u du T (ε) 0 6 ; 5 T (ε) (4.15) 1 ε;J ≤ P Gεu du > η log L0 + 2 T (ε) 0 ; T (ε) log(|z|2 ) 2 ε Gu du > η ≤ P + T (ε) T (ε) 0 ε→0

−−−→ 0,

due to proposition 4.1.1.

4.3 Local Lyapunov exponents in the diagonal case

163

For proving the central lemma 4.3.4 on Riccati-type diﬀerential inequalities we will use the following comparison theorem from ODE theory; a proof of this theorem is e.g. given by Bainov and Simeonov [Bn-Si 92, p.57]. Proposition 4.3.2 (Comparison theorem). Let R(t, v) be a continuous, real-valued function on an open set O ⊂ R2 containing the point (t0 , w0 ), and suppose that the initial value problem r(t) ˙ = R(t, r(t)) , r(t0 ) = w0 has a maximal (minimal) solution r(t) with domain [t1 , t2 ] containing t0 . Let w(t) be a diﬀerentiable function on [t1 , t2 ] such that (t, w(t)) ∈ O for all t ∈ [t1 , t2 ] and (≥)

≤ R(t, w(t)) w(t) ˙

(t ∈ [t1 , t2 ]) ,

w(t0 ) = w0 . Then (≥)

w(t) ≤ r(t)

(t ∈ [t1 , t2 ]) .

In the proof of 4.3.1 the problem is reduced to considering a random diﬀerential inequality for the processes v ε;k which is of the type of the logistic Verhulst ODE. It is interesting to observe that this kind of diﬀerential equation is used to model randomly inﬂuenced population genetics (without “epistasis”); see the respective papers by Kliemann and Arnold: [Kl 80, p.112], [Kl 83b, p.491] and [Ar-Kl 83, p.29]. In these references the timedependent prefactor φ is chosen as (stationary) Ornstein-Uhlenbeck process. A pathwise solution can be given explicitly: Lemma 4.3.3 (Verhulst ODE). Consider the (logistic) Verhulst ODE w(t) ˙ = φ(t) w(t) 1 − w(t) (t ≥ t0 ) , w(t0 ) = w0 , where φ ∈ C([t0 , ∞), R) and w0 ∈ [0, 1]. Then if φ ≡ 0, any w ∈ [0, 1] is an equilibrium; if φ ≡ 0, w = 0 and w = 1 are the equilibria (ﬁxed points). For all w0 ∈ (0, 1] the solution is given explicitly by

w(t) = 1+

1 w0

1

− 1 exp −

t t0

φ(u) du

(t ≥ t0 ) .

164

4 Local Lyapunov exponents

Proof. Computing the derivative of the speciﬁed function w(t) yields that t − φ(t) w10 − 1 exp − t0 φ(u) du w(t) ˙ =− 0 12 t 1 + w10 − 1 exp − t0 φ(u) du t 1 − 1 exp − φ(u)du w0 t0 φ(t) = t t 1 1 1 + w0 − 1 exp − t0 φ(u)du 1 + w0 − 1 exp − t0 φ(u)du = φ(t) w(t) 1 − w(t) and the initial condition is obviously fulﬁlled. The uniqueness of the solution follows from the local Lipschitz continuity of the drift R(t, w) := φ(t) w(1 − w). Figure 4.1 sketches the tendencies of the drift vector ﬁeld R(φ, w) := φ w(1 − w) as considered in the Verhulst-type lemma 4.3.3; also see Kliemann [Kl 80, p.112]. w 6

r(0, 1)

?

?

6

?

6

6

r (0, 0)

-φ

Fig. 4.1 The tendencies of the drift vector ﬁeld R(φ, w) := φ w(1 − w)

The following lemma is the crucial ingredient for the proof of theorem 4.3.1. It allows to sharpen the statement of lemma 4.1.5, the reason being that the diagonality assumption on A produced a random Riccati diﬀerential inequality which is also of Verhulst-type; see (4.18). In contrast, one had to deal with a less speciﬁc Riccati diﬀerential inequality in the previous situation of lemma 4.1.5. Also note that the diagonal structure of A is not going to be used explicitly in the subsequent lemma; it had come into play to deduce the random Riccati-/Verhulst-type diﬀerential inequality (4.18) which is now further treated: Lemma 4.3.4 (Random Riccati-/Verhulst-type diﬀerential inequality). Consider the situation of theorem 4.3.1 and the matrix function Gεt := Gεt (ω) := G(Xtε,x (ω)) := A(Xtε,x (ω)) − Aµ(x,ζ)

4.3 Local Lyapunov exponents in the diagonal case

165

as above for Xtε,x being the underlying diﬀusion driving (1) under the assumptions 2.1.1. Furthermore, let (vtε )t≥0 be a family of stochastic processes on the ﬁxed probability space (Ω, F, P), indexed by ε > 0, taking their values in the interval [0, 1] and starting in a common ﬁxed initial value v0ε = v0 ∈ [0, 1]. Suppose that the paths (vtε )t>0 are diﬀerentiable for any ε > 0 and satisfy the random Riccati-/Verhulst-type diﬀerential inequality (RVDI) d ε vt (ω) ≥ vtε (ω) 1 − vtε (ω) b1 − b2 G Xtε (ω) , dt for all (t, ω) such that Xtε,x (ω) ∈ Fx,ζ ; here, b1 > 0 and b2 ≥ 0 are constants. Then v ≡ 0 and v ≡ 1 are ﬁxed points of this diﬀerential equation and for any v0 ∈ (0, 1], the corresponding family satisﬁes P vTε (ε) −−− −→ 1 . ε→0

Proof. The statements for v0 ∈ {0, 1} follow immediately from the diﬀerential equation. Hence we choose v0 ∈ (0, 1); this open interval is invariant under the v ε motion (as will also follow from the arguments to come). If b2 = 0, then the statement is deterministic and follows from lemma 4.3.3 by the comparison assertion of proposition 4.3.2. Therefore, we will also suppose that b2 > 0 in the sequel. Fx,ζ denotes the subset of Rd as deﬁned in corollary 2.5.7. As usual, let B(Kµ(x,ζ) , c) denote the open ball around Kµ(x,ζ) with radius c > 0. Since g(y) := G(y) is continuous on Rd and vanishes at Kµ(x,ζ) , also gt := g(Xtε,x ) ≡ Gεt depends continuously on Xtε,x . Therefore, theorem 2.5.6 implies that for any δ, Γ > 0 1 P L { t ∈ [0, Γ · T (ε)] : Gεt ≤ δ } −−− −→ Γ . ε→0 T (ε) In other words, it follows from theorem 2.5.6 that for any δ, Γ, η > 0, lim P ΩεδΓη = 1 , ε→0

where ΩεδΓη

:=

ω∈Ω :

1 L { t ∈ [0, Γ · T (ε)] T (ε) : b1 − b2 gt (ω) ∈ / [b1 − b2 δ, b1 ] } ≤ η

.

166

4 Local Lyapunov exponents

Since b1 , b2 > 0, we can choose δ > 0 suﬃciently small such that b1 − b2 δ > 0 . Deﬁning Cx,ζ := max g(y) ≡ max G(y) y∈Fx,ζ

y∈Fx,ζ

we get for any Γ > 0 that ΩεΓ;F ≡ ⊂

ω ∈ Ω : Xtε (ω) ∈ Fx,ζ for all t ∈ [0, Γ · T (ε)]

ω ∈ Ω : b1 − b2 Gεt (ω) ∈ [b1 − b2 Cx,ζ , b] for all t ∈ [0, Γ · T (ε)]

,

where it is known from corollary 2.5.7 that lim P ΩεΓ;F = 1 . ε→0

First case: b1 − b2 Cx,ζ ≥ 0, i.e. g(y) ≤

b1 b2

for all y ∈ Fx,ζ : In this case the

right hand side v(1 − v){b1 − b2 g(y)} of the proposed RVDI is non-negative for all (v, y) ∈ (0, 1) × Fx,ζ and hence the drift Rω (t, v) := v(1 − v){b1 − b2 gt (ω)} is nonnegative for any ω ∈ ΩεΓ;F . Therefore, for any ω ∈ ΩεΓ;F the trajectory t → vtε (ω) starting in v0 ∈ (0, 1) is non-decreasing. Fixing δ < bb12 and ω ∈ Ωεδ1η implies (due to the continuity of gt ) that there are intervals [p, q] ⊂ [0, T (ε)] such that for all t ∈ [p, q] : b1 − b2 gt (ω) ≥ b1 − b2 δ (> 0) and the Lebesgue measure L of the union of such intervals is greater than (1 − η) T (ε). Such time intervals [p, q] will be called “good” in the remainder of this proof. Now we ﬁx an ω ∈ Ωεδ1η ∩ Ωε1;F where δ < bb12 and such an interval [p, q] with the described property. Then we have ˜ ω (v) > 0 Rω (t, v) ≥ v (1 − v){b1 − b2 δ} =: R

(t ∈ [p, q], v ∈ (0, 1)) .

Hence, the comparison proposition 4.3.2 together with lemma 4.3.3 (for φ(t) := b1 − b2 δ > 0) implies that vtε (ω) ≥

1+

1 vpε (ω)

1

− 1 exp {−(b1 − b2 δ)(t − p)}

(t ∈ [p, q]) ,

4.3 Local Lyapunov exponents in the diagonal case

167

since the function on the right hand side solves the comparison ODE w(t) ˙ = ˜ ω (w(t)) started in w0 := v ε (ω). Rewriting this estimate yields in other R p words (as has already been calculated in the proof of lemma 4.3.3) that 1 − 1 exp {−(b1 − b2 δ)(t − p)} ε vp (ω) 1 − vtε (ω) ≤ (t ∈ [p, q]) . 1 + vε1(ω) − 1 exp {−(b1 − b2 δ)(t − p)} p

Since the Lebesgue measure of the union of intervals [p, q] where this estimate holds true is larger than (1 − η) T (ε) and t → vtε (ω) is at least non-decreasing outside these intervals, it follows that for any κ > 0, 1 − 1 exp {−(b1 − b2 δ) (1 − η) T (ε)} v 0 < κ, 1 − vTε (ε) (ω) ≤ 1 + v10 − 1 exp {−(b1 − b2 δ) (1 − η) T (ε)} b1 b2

where the latter inequality is valid for ε small enough (for η < 1 and δ < ﬁxed). Thus, we can conclude altogether that for any ﬁxed κ > 0, ε→0 P 1 − vTε (ε) ≤ κ ≥ P Ωεδ1η ∩ Ωε1;F −−−→ 1 . Second case: b1 − b2 Cx,ζ < 0, i.e. Cx,ζ ≡ maxy∈Fx,ζ g(y) > case we ﬁx ω ∈ Ωεδ1η ∩ Ωε1;F for some δ < small enough such that (b1 − b2 δ)

b1 b2 .

b1 b2 :

Also in this

Furthermore, we choose η

1−η > − (b1 − b2 Cx,ζ ) (>0) . η

(†)

On the “good” time intervals [p, q] (where the notation is as above) the same estimate vtε (ω) ≥

1+

1 vpε (ω)

1

− 1 exp {−(b1 − b2 δ)(t − p)}

(t ∈ [p, q])

follows. However, in this case the right hand side of the RVDI (i.e. the factor b1 − b2 gt (ω)) might become negative on certain intervals [h, k]; such time intervals will be called “bad” in the sequel. On “bad” time sets we have the following bound from below: ¯ ω (v) Rω (t, v) ≡ v(1 − v){ b1 − b2 gt (ω) } ≥ v(1 − v){b1 − b2 Cx,ζ } =: R AB C @ <0, t∈[h,k]

for t ∈ [h, k]. Figure 4.2 qualitatively sketches the tendencies of the comparison drift vector ﬁeld

168

4 Local Lyapunov exponents

R(y, v) := [b1 − b2 g(y)] v(1 − v) given the sublimiting statistics of X ε,x . Note that this vector ﬁeld is just deﬁned such that R(Xtε,x (ω), v) = Rω (t, v) for each ω. The “bad” time intervals as deﬁned above are precisely those time sets during which X ε,x sojourns in the subset of Rd which is depicted on the left of the ordinate in ﬁgure 4.2. v 6

r(0, 1)

?

?

?

6

6

6

- y ∈ Rd

r (0, 0)

@

Kµ(x,ζ) AB

C

B(Kµ(x,ζ) ,c)

@

AB

Fx,ζ

C

Fig. 4.2 The tendencies of the drift vector ﬁeld R(y, v) := [b1 − b2 g(y)] v(1 − v) given the sublimiting statistics of X ε,x .

Hence, the comparison proposition 4.3.2 together with lemma 4.3.3 implies that vtε (ω) ≥

1+

1 ε (ω) vh

1

− 1 exp {−(b1 − b2 Cx,ζ )(t − h)}

(t ∈ [h, k])

for the “bad” time intervals, since the function on the right hand side solves ¯ ω (w(t)) starting in w0 := v ε (ω). Since ω ∈ the comparison ODE w(t) ˙ =R h ε Ωδ1η by assumption, this means that min t∈[0,T (ε)]

vtε (ω) ≥

1+

1 v0

1

− 1 exp {−(b1 − b2 Cx,ζ ) η T (ε)}

=: v¯ > 0 .

This estimate describes the worst case possible. More precisely, due to the fact that ω ∈ Ωεδ1η and due to the deﬁnition of this set, the “bad” time set can at most constitute an interval of overall length η T (ε). Furthermore, the “bad” time set can be supposed to be connected and thus to be an interval indeed, since the two comparison ODEs, on the “good” and on the “bad” time sets, respectively, are autonomous. The latter fact implies that there is no other conﬁguration of the “bad” time set which could result in a “worse” case, i.e. which would produce a value of v ε (ω) smaller than v¯.

4.3 Local Lyapunov exponents in the diagonal case

169

In other words, the value v¯ can only be attained by the trajectory t → vtε (ω), if the “bad” time set is [h, k] = [0, η T (ε)] and we then have v¯ = vkε (ω). In this case, however, we get the “good” interval [p, q] = [k, T (ε)] = [η T (ε), T (ε)] by the deﬁnition of Ωεδ1η and therefore, estimating t → vtε (ω) on the latter interval from below, we can conclude altogether that 1 4.3.3 {−(b1 − b2 δ)(1 − η)T (ε)} ε v ¯ − 1 exp 1 − vT (ε) (ω) ≤ 1 1 + v¯ − 1 exp {−(b1 − b2 δ)(1 − η)T (ε)} 1 = exp {+(b1 − b2 δ)(1 − η)T (ε)} 1+ 1 −1 v¯ 1 Def.¯ v = exp {+(b . 1 − b2 δ)(1 − η)T (ε)} 1+ 1 − 1 exp {−(b1 − b2 Cx,ζ )ηT (ε)} v0 = 1+

v0 1−v0

1 exp T (ε) (b1 − b2 δ)(1 − η) + (b1 − b2 Cx,ζ )η

(†)

< κ,

where the latter inequality is true for ε small enough, since the parameters are related via (†). Therefore, we can conclude that given η and δ as in (†), then for any ﬁxed κ > 0, ε→0 P 1 − vTε (ε) ≤ κ ≥ P Ωεδ1η ∩ Ωε1;F −−−→ 1 .

Hence the lemma is proved.

In the situation of lemma 4.3.4 the theorem 2.5.6 on the quasi-deterministic behavior of X ε can be “lifted” to a corresponding assertion concerning the proportion of the sojourn time of v ε . This subsequent corollary is not going to be used in the rest of the exposition. Corollary 4.3.5 (Proportion of the sojourn time of v ε• in 4.3.4). Consider the situation of 4.3.4. Then it follows, furthermore, for any v0 ∈ (0, 1] and Γ, β > 0 that P L t ∈ [0, Γ] : vtε T (ε) < 1 − β −−−−→ 0 . ε→0

Proof. The statement is trivial for v0 = 1, β ≥ 1 or b2 = 0; hence, ﬁx v0 ∈ (0, 1), β ∈ (0, 1) and b2 > 0. We retain the same notation as before in 4.3.4. In particular, choose δ > 0 such that b1 − b2 δ > 0 and ﬁx the set

170

4 Local Lyapunov exponents

ΩεδΓη ≡

ω∈Ω :

1 L { t ∈ [0, Γ · T (ε)] T (ε)

/ [b1 − b2 δ, b1 ] } ≤ η : b1 − b2 gt (ω) ∈ for which it is known from theorem 2.5.6 that for any δ, Γ, η > 0, limε→0 P(ΩεδΓη ) = 1. For Cx,ζ ≡ max g(y) ≡ max G(y) , y∈Fx,ζ

y∈Fx,ζ

one knows that for any Γ > 0, ΩεΓ;F ≡ ω ∈ Ω : Xtε (ω) ∈ Fx,ζ for all t ∈ [0, Γ · T (ε)] ⊂ ω ∈ Ω : b1 − b2 Gεt (ω) ∈ [b1 − b2 Cx,ζ , b] for all t ∈ [0, Γ · T (ε)] , where limε→0 P(ΩεΓ;F ) = 1 due to corollary 2.5.7. First case: b1 − b2 Cx,ζ ≥ 0, i.e. g(y) ≤

b1 b2

for all y ∈ Fx,ζ : Fix ω ∈ ΩεδΓη ∩

ΩεΓ;F . In this case v•ε (ω) is greater than or equal to the trajectory vt which is exposed to the zero drift until time ηΓ T (ε) and is exposed to the drift v(1 − v){b1 − b2 δ} during the time interval [ηΓ T (ε), Γ T (ε)]. In other words, it is bounded below by ⎧ ⎨v0 , t ∈ [0, ηΓ T (ε)] , vt := 1 , t ∈ [ηΓ T (ε), Γ T (ε)] . ⎩ 1 1+

v0

−1 exp −(b1 −b2 δ) t−ηΓ T (ε)

Hence, it follows from solving for t that vtε (ω) ≥ vt ≥ 1 − β , (at least) if

t ≥ ηΓ T (ε) −

log

β v0 1−β 1−v0

b1 − b2 δ

.

Dividing the right hand side by T (ε) the resulting number converges to ηΓ as ε → 0. This proves that L t ∈ [0, Γ] : vtε T (ε) (ω) ≥ 1 − β β v0 log 1−β 1−v 1 0 ε→0 ≥ Γ − ηΓ + −−−→ Γ (1 − η) T (ε) b1 − b2 δ

4.3 Local Lyapunov exponents in the diagonal case

171

for all ω ∈ ΩεδΓη ∩ΩεΓ;F . On the other hand, we also have that P(ΩεδΓη ∩ΩεΓ;F ) → 1 as ε → 0. Altogether, the desired convergence in probability follows in the ﬁrst case. Second case: b1 − b2 Cx,ζ < 0, i.e. Cx,ζ ≡ maxy∈Fx,ζ g(y) > a ﬁxed ω ∈

ΩεδΓη

∩

ΩεΓ;F .

In this case we estimate

v•ε

b1 b2 :

Again consider

by (compare with)

v˙ = v(1 − v){b1 − b2 δ} on a time interval of length (1 − η)Γ T (ε) and with v˙ = v(1 − v){b1 − b2 Cx,ζ } on a time interval of length ηΓ T (ε). Since both these comparison ODEs are autonomous, i.e. their drifts only depend on the state variable and not on time, we can again assume without restriction that the “bad” time set of length ηΓ T (ε), on which the second comparison ODE is valid, is connected and at the beginning of the whole time set under consideration. Therefore, vtε (ω) ≥ vt ≡

1+

1 v0

1

− 1 exp {−(b1 − b2 Cx,ζ )t}

for t ∈ [0, ηΓ T (ε)] and vtε (ω) ≥

1+

1 v0

1 − 1 exp {−(b1 − b2 Cx,ζ )ηΓ T (ε)} exp {−(b1 − b2 δ)(t − ηΓ T (ε))}

for t ∈ [ηΓ T (ε), Γ T (ε)]. Again we solve for t which amounts to requiring that !

1−β ≤

1+

1 v0

1 − 1 exp {−(b1 − b2 Cx,ζ )ηΓ T (ε)} exp {−(b1 − b2 δ)(t − ηΓ T (ε))}

or

exp − b1 − b2 Cx,ζ η Γ T (ε) exp − b1 − b2 δ t − η Γ T (ε)

−1 ! 1 1 −1 ≤ −1 1−β v0 β v0 = ; 1 − β 1 − v0 in other words, ! exp − b1 − b2 δ t − ηΓ T (ε) ≤ exp + b1 − b2 Cx,ζ ) η Γ T (ε)

β v0 1 − β 1 − v0

172

4 Local Lyapunov exponents

or, as −(b1 − b2 δ) < 0, equivalently !

t ≥ ηΓ T (ε) −

(b1 − b2 Cx,ζ )ηΓ T (ε) + log

β v0 1−β 1−v0

b1 − b2 δ

Again this implies for ω ∈ ΩεδΓη ∩ ΩεΓ;F that Tε L t ∈ [0, Γ] : vtε T (ε) (ω) ≥ 1 − β ≥ Γ − T (ε)

=: Tε .

v0 β log (b1 − b2 Cx,ζ ) 1 − β 1 − v0 ηΓ + = Γ(1 − η) + b1 − b2 δ (b1 − b2 δ)T (ε) which tends to

+

(b1 − b2 Cx,ζ ) η Γ (1 − η) + b1 − b2 δ

,

as ε → 0. Since η > 0 can be chosen arbitrarily small, the claim is proved as it again holds true that P(ΩεδΓη ∩ ΩεΓ;F ) → 1 as ε → 0, too. Remark 4.3.6 (n = 2). Consider the situation of theorem 4.3.1 and assume in addition that n = 2. Due to the above theorem, 1 P log ZTε (ε) ( . , x, z) −−− (j = 1, 2) . −→ Λj Kµ(x,ζ) ε→0 T (ε) This stochastic limit Λj Kµ(x,ζ) is called the local (j th ) Lyapunov exponent of Z ε with respect to x, ζ and z. The crucial step in the course of the proof had been to deduce the RVDI (4.18) d ε vt (ω) ≥ vtε (ω) 1 − vtε (ω) b1 − b2 G Xtε (ω) dt in the case that p > 1 which had then been dealt with in lemma 4.3.4. In the two-dimensional situation we would like to remark how this information can be sharpened; more precisely, one can deduce a random Riccati-/Verhulsttype diﬀerential equation (RVDE); see (4.22). For this purpose, we keep all notation as above; also see remark 4.1.6. Furthermore, we assume that p > 1, i.e. that p = 2 which entails that λ1 := Λ1 > Λ2 =: λ2 . In this case the auxiliary processes are given by Lε;k ≡ ( ε;k )2 , k ∈ {1, 2} ⎧ ⎪ ⎨0 ε;k M ≡ ( ε;1 )2 ⎪ ⎩42 ε;i 2 ε 2 i=1 ( ) ≡ | Z | ⎧42 ε;i 2 ε 2 ⎪ ⎨ i=1 ( ) ≡ | Z | ε;k N ≡ ( ε;2 )2 ⎪ ⎩ 0

,k = 1 ,k = 2 ,k = 3 ,k = 1 ,k = 2 ,k = 3

4.3 Local Lyapunov exponents in the diagonal case

173

and describe the norms of the projection of Z ε onto the subspaces spanned by the canonical unit vectors e1 and e2 of R2 which are the eigenvectors of Aµ(x,ζ) corresponding to λ1 and λ2 ; also see remark 1.4.6. (4.14) and (4.15) now are expressed by ε;k dL d ε;k 2 ε;k ε;k 2 dt − 2λk L ≡ dt ( ) − 2λk ( ) 2 ε;k G = 2 ε;k ≡ 2 Lε;k Gε;k ; (4.16) for M ε := M ε;2 becomes d Mε dt

d ε;1 2 dt (4.14) = 2 ( ε;1 )2 λ1 + Gε;1 ≡ 2 M ε λ1 + Gε;1 ≡

(4.20)

and (4.17) for N ε := N ε;2 can be changed into d Nε dt

d ε;2 2 dt (4.14) = 2 ( ε;2 )2 λ2 + Gε;2 ≡ 2 N ε λ2 + Gε;2 . ≡

(4.21)

For Ztε ≡ 0, we again consider the auxiliary process vtε := vtε;2 ≡

2 Mtε;2 ( ε;1 t ) ≡ ≡ (cos αεt )2 ; ε ε;1 2 2 | Zt | ( t )2 + ( ε;2 ) t

see remark 4.1.6 and ﬁgure 1.1 in remark 1.4.6. The processes v ε;1 :=

Mtε;1 ≡ 0 | Ztε |2

and

v ε;3 :=

Mtε;3 ≡ 1 | Ztε |2

deﬁned analogously do not contain any speciﬁc information. Then we get altogether from (4.20) and (4.21) that d d N ε ( dt M ε ) − M ε ( dt N ε) d ε v = dt (M ε + N ε )2 ε 1 N 2 M ε λ1 + Gε;1 = ε ε 2 (M + N ) − M ε 2 N ε λ2 + Gε;2

174

4 Local Lyapunov exponents

2 Mε Nε ( λ1 − λ2 ) + Gε;1 − Gε;2 (M ε + N ε )2 = 2 v ε (1 − v ε ) ( λ1 − λ2 ) + Gε;1 − Gε;2 , =

(4.22)

the RVDE which had been announced above. Of course, lemma 4.3.4 also applies here and therefore one gets the following deﬁnition P 1 , if vTε (ε) −−− −→ 1 , ε→0 J := 2 , else for the index function J such that the claim (4.10), 1 P log | ZTε (ε) | −−− −→ λJ , ε→0 T (ε) holds true. This means by lemma 4.3.4 that J = 2, if v0 = 0 and J = 1, if v0 ∈ (0, 1]. Again, this index J displays the direction eJ to which the Zε projective line |Ztε | converges in probability. t

Example 4.3.7. Consider for example the potential function from (1.34), U1 (x, y) :=

3 4 2 3 x − x3 − 3 x2 + c x y + y 4 , 2 3 2

where c ∈ R which had been discussed in 1.5.3 and 2.2.2: A sketch of the potential function U1 is given in ﬁgure 2.1 for c = 1. Note that in this example we use the less clumsy notation (x, y) for elements of R2 , instead of (x1 , x2 ) or (x1 , x2 ). Let X ε be the diﬀusion given by the SDE √ dXtε = −∇U (Xtε ) dt + ε σ dWt , where σ ∈ R2×2 is supposed to be a constant, invertible matrix, ε ≥ 0 and W is a Brownian motion in R2 . Considering the negative of the Hesse matrix of U1 ,

−c − 18x2 + 4 x + 6 , A1 (x, y) := −D∇U (x) ≡ − HU1 (x, y) = −c − 18 y 2 yields the linearized (“variational”) equation dZtε = A1 (Xtε ) Ztε dt √ dXtε = b (Xtε ) dt + ε σ dWt , √ since the diﬀusion coeﬃcient ε σ does not depend on the state variable in this example; see examples 1.5.1 and 1.5.3.

4.3 Local Lyapunov exponents in the diagonal case

175

Assume that c = 0. Also in this case, the function U1 is a two well potential as qualitatively described by ﬁgure 1; see example 2.6.2. However, the Hessian matrix now is everywhere diagonal and therefore remark 4.3.6 (or theorem 4.3.1 in other words) is applicable to this Jacobian system: The function U1 has a local minimum at K1 := (−0.847, 0), a global minimum at K2 := (1.180, 0) and a saddle point at K3 := (0, 0). Let Λ1 (Ki ) ≥ Λ2 (Ki ) denote the eigenvalues of A1 (Ki ) ≡ − HU1 (Ki ), i ∈ {1, 2}, which are non-positive due to the properties of the curvature at the minima. In this numerical example at hand (c = 0), Λ1 (K1 ) = Λ1 (K2 ) = 0 and −10.301 , i = 1 , Λ2 (Ki ) = −14.343 , i = 2 . Now we deﬁne the time scales T (ε) := eζ/ε which means that we connect the large parameters t = T (ε) and ε(t) := T −1 (t) =

1 ε

by

ζ log t

for a scaling parameter ζ > 0 in order to approach the sublimit distributions δK1 and δK2 . The object of our interest is thus the random variable ε(t) 1 1 log ZTε (ε) ( . , x, z) = log Zt ( . , x, z) T (ε) t and the assertion of 4.3.6 and 4.3.1 implies that it converges in probability to Λ0J (Ki ) as t → ∞ which is equivalent to ε → 0, where i ∈ {1, 2} depends on the ε,(x,y) and on the time scale parameter ζ of Xtε . initial position (x, y) = X0 More precisely, if (x, y) is in the K1 -regime and if also ζ < 2v, where v := 1 2 V (K1 , K3 ) denotes the height of the potential barrier (see example 2.6.2), then i = µ(x, ζ) = 1; otherwise, i = µ(x, ζ) = 2. Note that generally we also need to exclude the case that ζ = 2V , where V := 12 V (K2 , K3 ) is the height of the deep potential barrier (see example 2.6.2). The index J ∈ {1, 2} depends on the initial direction of Z ε . Then we call the possible diﬀerent limits for the exponential growth rate of Z ε the local Lyapunov exponents of X ε . A ﬁnal remark concerning the presumed diagonality of A1 ≡ − HU1 seems necessary: One might conjecture that the above procedure could also be

176

4 Local Lyapunov exponents

applied in case that c = 0 (which we assume from now on), if A1 only is transformed into diagonal form. However, the point here is that one has to simultaneously diagonalize A1 (x, y) for all (x, y) ∈ R2 . As is known from linear algebra, two diagonalizable matrices are simultaneously diagonalizable, if and only if they commute. Denoting the entries of A1 (x, y) by aij (x, y), this criterion implies the condition that

a11 (x, y) − c a11 (˜ x, y˜) − c x, y˜) − c a22 (x, y) − c a22 (˜

a11 (x, y) − c x, y˜) − c a11 (˜ = x, y˜) − c a22 (˜ − c a22 (x, y) for all (x, y), (˜ x, y˜) ∈ R2 . One then arrives at the requirement that a11 (x, y) − a22 (x, y) = a11 (˜ x, y˜) − a22 (˜ x, y˜) for all (x, y), (˜ x, y˜) ∈ R2 ; in other words, we get the constraint for the matrix function that the mapping R2 $ (x, y) → a11 (x, y) − a22 (x, y) needs to be constant. However, plugging in the speciﬁc values of A1 yields that a11 (x, y) − a22 (x, y) = − 18x2 + 18 y 2 + 4 x + 6 which clearly violates this condition. More precisely, A1 cannot be simultaneously diagonalized on any subset of R2 with non-void interior—such as Fx,ζ . Note that this fact can also be viewed from another perspective: As had been calculated in example 1.5.3, the angle αεt satisﬁes the pathwise ¯ ε , αε ), where the drift is given by diﬀerential equation α˙ εt = h(X t t ¯ (x, y), α = c ( sin2 α − cos2 α ) − a11 (x, y) − a22 (x, y) sin α cos α h ≡ c ( sin2 α − cos2 α ) + ( 18x2 − 18 y 2 − 4 x − 6 ) sin α cos α = − c cos 2α + ( 9x2 − 9 y 2 − 2 x − 3 ) sin 2α . If A1 was simultaneously diagonalizable, there would exist two linearly independent directions of eigenvectors, the angles of which are by deﬁnition ﬁxed ¯ (x, y), . for all (x, y). However, the zero set of h ¯ (x, y), . is points of h characterized by the requirement tan 2α = −

2c a11 (x, y) − a22 (x, y)

≡ c ( 9x2 − 9 y 2 − 2 x − 3 )−1 ,

4.4 The two-dimensional, general case

177

where equality “=” is to be understood in R∪{±∞}. Hence, this presentation of the problem yields the same result as above; more precisely, we again get the constraint that (x, y) → a11 (x, y) − a22 (x, y) be constant. Thus, for (x, y) in a subset of R2 with non-void interior, such a ﬁxed angle α cannot be found unless c = 0, since the right hand side of the condition last obtained is non-constant on any such subset of R2 . Therefore, the case c = 0 features a behavior of Z ε which is not captured by the situation c = 0 and the hitherto existing theorem 4.3.1 cannot be applied here. Thus a diﬀerent argument is necessary which is the target of the following section.

4.4 Local Lyapunov exponents in the two-dimensional, general case In this section conditions are to be given which allow to deduce the claim (4.11), 1 P log | ZTε (ε) ( . , x, z) | −−− −→ λ1 , ε→0 T (ε) in the general case p ≡ p(A(Kµ(x,ζ) )) = 2 in which the system matrix, however, also takes non-diagonal values, as opposed to the situation of the previous section. We investigate dZtε = A (Xtε ) Ztε dt √ dXtε = b (Xtε ) dt + ε σ (Xtε ) dWt

(1)

where A ∈ C ∞ (Rd , R2×2 ) now denotes a 2 × 2-matrix function with smooth entries; in contrast to the hitherto investigations of this chapter, A is not supposed to be merely continuous, but diﬀerentiable the reason being that the results of the previous chapter are to be applied here. Furthermore, this section will exclusively deal with the two-dimensional situation, i.e. n = 2. All other data of (1) remain unchanged. As has already been remarked in 4.1.6, the above claim follows from corollary 4.1.3, if p = 1, i.e. if Re(Λ1 (Kµ(x,ζ) )) = Re(Λ2 (Kµ(x,ζ) )). Hence, we consider the case p = 2 here, i.e. the case that λ1 := Λ1 (Kµ(x,ζ) ) > Λ2 (Kµ(x,ζ) ) =: λ2

(4.23)

are two real eigenvalues of the matrix Aµ(x,ζ) ≡ A(Kµ(x,ζ) ); note that these values are indeed characteristic roots (not just real parts of such), since n = 2. Let the coordinate system be chosen such that

λ1 0 . Aµ(x,ζ) = 0 λ2

178

4 Local Lyapunov exponents

Then it is known from remark 4.1.6 that for any η1 , η2 > 0, P 1 L t ∈ [0, T (ε)] : αεt ∈ B(0, η1 ) or αεt ∈ B(π/2, η2 ) −−− −→ 1 , ε→0 T (ε) where αεt denotes the angle of Ztε in R2 as measured with respect to the coordinate system chosen. As had been seen, this behavior is due to the fact ¯ µ(x,ζ) , . ) of the angle process αε has two equilibria, namely that the drift h(K t π at 0 and 2 . For establishing the above claim (4.11), it needs to be shown that αεt has the preferential state 0 corresponding to the eigenvector e1 of Aµ(x,ζ) with respect to (the greater eigenvalue) λ1 ; cf. ﬁgure 1.1 of remark 1.4.6 which sketches this situation for ﬁxed ε and ω, if zt , vt and αt are replaced by Ztε , vtε and αεt , respectively. The latter claim concerning the preferential state of the angle process is considered as an assertion on the sublimiting behavior of αεt and will be made precise in theorem 4.4.6. The eﬀect on the local growth rate of Z ε , the local Lyapunov exponent, will be noted in theorem 4.4.8. In doing so, our reasoning leads via a sublimiting version of the FurstenbergKhasminskii formula: The coupled process (Xtε , αεt ) determines which values ¯ are sampled and which weight they get. Note that the ﬁnal step in the of Q proof of the Hartman-Wintner-Perron theorem 1.4.3 cannot be transferred to the stochastic situation here to prove (4.11), since the corresponding estimate from above might become singular in the stochastic case when starting near π2 ; therefore, the bound from above thus obtained (see p.40) would be useless. Since all these issues pose the question of how the real noise driven angle process behaves in the sublimiting situation ε → 0, they can be considered more generally as problems concerning the qualitative theory of a certain nonlinear real noise driven system on time scales; these problems are tackled in the following subsection and will then be applied to αεt . Note that the terminology refers to the “qualitative theory of nonlinear stochastic [in the sense of ‘real noise driven’] systems” by Kliemann; see [Kl 80], [Kl 79] and [Kl 83b].

4.4.1 Qualitative theory of nonlinear real noise systems on time scales First, we investigate the situation in which the real noise system taking its values in a compact set is attracted solely to one switching curve (see deﬁnition 1.2.3). Such real noise driven systems have been prototypically investigated by many authors: Although the physical signiﬁcance of this setting is “rather tenuous” (Horsthemke and Lefever [Hh-Lf 80, p.242]), it constitutes one of the sparse model classes of degenerate systems in which the probability distribution of the perturbed system can be calculated explicitly given that

4.4 The two-dimensional, general case

179

the underlying noise is stationary; see Hongler [Hr 79], Horsthemke and Lefever [Hh-Lf 85, Sec.8.3], Arnold et al. [Ar-Hh-Lf 78], Kliemann [Kl 80, p.31,37f.,51ﬀ.,82f.,132], Kliemann [Kl 83b, p.494ﬀ.], Arnold and Kliemann [Ar-Kl 83, p.72] and Sancho and San Miguel [Sn-SM 80, p.362]. In these references the real noise case with one attracting switching curve is used to describe (noise induced) phase transitions (“bifurcations”); more precisely, if the stationary noise input is a stationary Ornstein-Uhlenbeck process, then the invariant probability distribution of the real noise system is unimodal or bimodal depending on the variance of the input noise. Another model class of real noise driven systems in which the stationary probability distribution can be calculated explicitly is discussed by Kitahara et al. [Kk-Hh-Lf 79] and [Kk-Hh-Lf-Ia 80]; also see Horsthemke and Lefever [Hh-Lf 85, p.259ﬀ.]. These authors use stationary, dichotomous noise. However, all the results and references mentioned above which crucially rely on stationarity cannot be used here due to the special statistics of the driving noise process X ε , i.e. due to its sublimiting behavior on time scales. In order to clearly distinguish between the initial point of X ε and the space variable in Rd , the former will be denoted by x0 in the sequel while the latter is referred to by x from now on. Furthermore, since the results of this subsection shall be applied to the angle process αε of Z ε , the state space of the real noise driven system, called Y ε in the sequel, is taken to be one-dimensional. Theorem 4.4.1 (Qualitative sublimiting behavior in case of one attracting switching curve: Quasi-deterministic behavior of Y ε ). Consider the stochastic process (Ytε )t≥0 given as the solution of the RDE Y˙ tε = F (Xtε , Ytε ) ,

Y0ε = y0 ∈ R ,

where F : Rd × R → R is a diﬀerentiable function admitting pathwise unique solutions and where Xtε,x0 is given by the SDE (2.1) under the assumptions >l 2.1.1 with initial condition x0 . Suppose that x0 ∈ i=1 Di and ﬁx T (ε) ! eζ/ε , where ζ > 0 is not contained in a ﬁnite exceptional set depending on x0 (see (2.20)). Suppose that F has exactly one attracting switching curve (see deﬁnition 1.2.3) on Fx0 ,ζ , where the latter set had been deﬁned in corollary 2.5.7; more precisely, suppose that there is a continuous function Ysc : Fx0 ,ζ → R such that Ysc (x) is the only zero of F (x, . ) for any x ∈ Fx0 ,ζ and such that ∂F ∂y (x, y) < 0 at all points (x, y) ∈ graph(Ysc ). Then it follows for any initial value y0 ∈ R and all Γ, β > 0 that P L t ∈ [0, Γ] : YtεT (ε) − Ysc Kµ(x0 ,ζ) > β −−−−→ 0 ε→0

and

P YTε(ε) −−− −→ Ysc Kµ(x0 ,ζ) . ε→0

180

4 Local Lyapunov exponents

Proof. For the sake of brevity, we set Yµ(x0 ,ζ) := Ysc (Kµ(x0 ,ζ) ) . Due to the continuity of Ysc ( . ) we ﬁx c > 0 such that Ysc (x) − Yµ(x ,ζ) < β 0 2

( ∀ x ∈ B(Kµ(x0 ,ζ) , c) ) .

Furthermore, deﬁne M ∗ := max Ysc (x) x∈Fx0 ,ζ

and

M∗ :=

min Ysc (x)

x∈Fx0 ,ζ

which are well deﬁned real numbers, since Fx0 ,ζ is compact and Ysc is continuous. Also ﬁx δ > 0 such that Yµ(x0 ,ζ) ± β ∈ (M∗ − δ, M ∗ + δ) . See ﬁgure 4.3 for a sketch of the situation at hand.

Fig. 4.3 The tendencies and the attracting switching curve of the drift vector ﬁeld F (x, y) given the sublimiting statistics of X ε,x0

It is assumed in the sequel that y0 ∈ [M∗ − δ, M ∗ + δ]; this does not restrict generality, since there is no other switching curve of F on Fx0 ,ζ ; therefore, for any y0 ∈ [M∗ − δ, M ∗ + δ], there is some (deterministic) time t0 > 0 such that Ytε,y0 ∈ [M∗ − δ, M ∗ + δ] for all t ≥ t0 , where we already used the fact that only noise paths X•ε,x0 ⊂ Fx0 ,ζ will be considered due to corollary 2.5.7; see below. In the course of the proof we will make use of the following preliminary remark: The following quantities are well deﬁned and not equal to zero: m1 := max F (x, y) : x ∈ B(Kµ(x0 ,ζ) , c) , y ∈ [Yµ(x0 ,ζ) + β, M ∗ + δ] < 0 and

4.4 The two-dimensional, general case

m2 := min

181

F (x, y) : x ∈ B(Kµ(x0 ,ζ) , c) , y ∈ [M∗ − δ, Yµ(x0 ,ζ) − β]

> 0;

in particular, for a path Xt := Xtε,x0 (ω) which takes its values only in B(Kµ(x0 ,ζ) , c) we have for y0 ∈ [M∗ − δ, M ∗ + δ] , ≤ m1 < 0 , if y0 ≥ Yµ(x0 ,ζ) + β ε ε Y˙ t = F (Xt , Yt ) ≥ m2 > 0 , if y0 ≤ Yµ(x0 ,ζ) − β / B(Yµ(x0 ,ζ) , β) due to comparison 4.3.2. By virtue of these as long as Ytε ∈ bounds m1 and m2 for the function F on B(Kµ(x0 ,ζ) , c) × B(Yµ(x0 ,ζ) , β), comparison of Ytε,y0 with these two constant velocity ODEs yields that Ytε,y0 being subject to a driving trajectory X• ≡ X•ε,x0 (ω) ⊂ B(Kµ(x0 ,ζ) , c) hits B(Yµ(x0 ,ζ) , β) in ﬁnite time; due to the uniformity of the bounds m1 and m2 also the maximum of these hitting times with respect to the initial condition y0 ∈ [M∗ −δ, Yµ(x0 ,ζ) −β]∪[Yµ(x0 ,ζ) +β, M ∗ +δ] is ﬁnite; calling this maximum t1 further implies that Ytε ∈ B(Yµ(x0 ,ζ) , β)

(t ≥ t1 ) .

Let y˜ ∈ [M∗ − δ, M ∗ + δ] \ B(Yµ(x0 ,ζ) , β) denote an initial value for which this maximal time t1 is needed in order to approach B(Yµ(x0 ,ζ) , β). After this preliminary remark we now use the “preferences” of X ε as described by the Freidlin-Wentzell theory: Theorem 2.5.6 implies that for any Γ, η > 0, lim P ΩεcΓη = 1 , ε→0

where

ΩεcΓη

:=

ω∈Ω : Γ −

1 L { t ∈ [0, Γ · T (ε)] T (ε) :

Xtε,x0

∈ B(Kµ(x0 ,ζ) , c) } ≤ η

.

Fixing ω ∈ ΩεcΓη implies that there are intervals [p, q] ⊂ [0, Γ T (ε)] such that for all t ∈ [p, q] : Xtε,x0 (ω) ∈ B(Kµ(x0 ,ζ) , c) and the Lebesgue measure L of the union of such intervals is greater than or equal to (Γ − η) T (ε). Such intervals [p, q] will be called “good” in the remainder of the proof; note that this terminology is consistent with the one used in the proof of lemma 4.3.4. Furthermore, corollary 2.5.7 implies that for any Γ > 0, lim P ΩεΓ;F = 1 , ε→0

where we again use the notation ΩεΓ;F := { ω ∈ Ω : Xtε,x0 (ω) ∈ Fx0 ,ζ for all t ∈ [0, Γ · T (ε)] } .

182

4 Local Lyapunov exponents

Now ﬁx η > 0 and ω ∈ ΩεcΓη ∩ ΩεΓ;F . In order to calculate L t ∈ [0, Γ] : YtεT (ε) (ω) − Yµ(x0 ,ζ) > β , we compare Y•ε with the “constant velocity” system Y˜ deﬁned in the following: Let Y˜0 = y0 , if 0 is in a “good” time interval and let Y˜0 = y˜ otherwise, where the default point y˜ ∈ [M∗ − δ, M ∗ + δ] \ B(Yµ(x0 ,ζ) , β) had been ﬁxed in the preliminary remark. On a “good” time interval let m1 if y˜ > Yµ(x0 ,ζ) + β , d ˜ Yt = dt m2 if y˜ < Yµ(x0 ,ζ) − β ; on all other (“bad”) times let Y˜t = y˜. These two comparison cases for “good” and “bad” time sets capture the possible “worst case scenarios”. Since both these comparison systems are autonomous, i.e. their velocities m1 , m2 and 0 do not depend on the time variable t themselves, we can again assume without restriction that the “bad” time set is of length η T (ε), connected (an interval) and at the beginning of the whole time set under consideration. Altogether it follows therefore that L t ∈ [0, Γ] : YtεT (ε) (ω) − Yµ(x0 ,ζ) > β ≤ η + L t ∈ [η, Γ] : Y˜ty˜T (ε) − Yµ(x0 ,ζ) > β 1 L t ∈ [ηT (ε), ΓT (ε)] : Y˜ty˜ − Yµ(x0 ,ζ) > β ≡η+ T (ε) t1 , =η+ T (ε) where the preliminary remark has been used in the ﬁnal step. Since ω ∈ ΩεcΓη ∩ ΩεΓ;F and the probability of this set tends to 1 as ε → 0, this proves in other words that P L t ∈ [0, Γ] : YtεT (ε) − Yµ(x0 ,ζ) > β −−−−→ 0 . ε→0

The second claim concerning the stochastic convergence, P YTε(ε) −−− −→ Yµ(x0 ,ζ) , ε→0

follows in the same manner: As before ﬁx an arbitrary β > 0, the corresponding parameter c ≡ c(β) > 0 and set Γ := 1; again let Y˜t (ω) denote the system which equals y˜ on the “bad” time set on which Xtε (ω) ∈ / B(Kµ(x0 ,ζ) , c) and which follows the ODE

4.4 The two-dimensional, general case

d ˜ Y = m1 dt

183

or

d ˜ Y = m2 dt

depending on the position of Y˜ = y˜ on the “good” time set on which Xtε (ω) ∈ B(Kµ(x0 ,ζ) , c). Due to the preliminary remark, it follows that 1B(Yµ(x0 ,ζ) ,β) (Ytε ) ≥ 1B(Yµ(x0 ,ζ) ,β) (Y˜t ) and therefore P YTε(ε) − Yµ(x0 ,ζ) < β ≥ P Y˜T (ε) − Yµ(x0 ,ζ) < β ≡ P Xtε ∈ B(Kµ(x0 ,ζ) , c) for all t ∈ [T (ε) − t1 , T (ε)] ; however, the latter must tend to 1 as ε → 0, due to the sublimiting distribution characteristics of theorem 2.5.5 as X ε has continuous trajectories and t1 is a ﬁxed number (not depending on ε). In other words, one can apply theorem 2.5.5 with the time scale T(ε) := T (ε) − t1 and observe that the deterministic lag of YTε(ε) is at most t1 . The following sample drift is taken from Kliemann [Kl 80, p.31,37f.,51ﬀ., 82f.,132] : Example 4.4.2. Consider the system 1 − Ytε + Ytε (1 − Ytε ) Xtε =: F (Xtε , Ytε ) , Y˙ tε = 2

Y0ε = y0 ,

in which Xtε also takes its values in R, i.e where d = 1. The drift (right hand side) 1 − y + y (1 − y) x F (x, y) ≡ 2 of this RDE vanishes at the points (x, y) ∈ R2 for which x = (y − 12 )[ y (1 − y) ]−1 ; solving for y yields that (0, 12 ) is a zero, and for x = 0: 1 1 − ± y = y(x) = 2 2x

√

1 + x2 2x

.

Since F (x, 1) = − 12 < 0 for all x ∈ R, there is no trajectory of Y ε which can traverse the set {(x, y) : y = 1}, if started in y0 ≤ 1. In the controltheoretic terminology by Kliemann [Kl 80, Ch.6] and [Kl 79, p.461f.], this fact is summarized by calling the hypersurface {(x, y) : y = 1} strongly nonpenetrable from below or a strong no return plane from above; analogously, since F (x, 0) = 12 > 0, the set {(x, y) : y = 0} is strongly non-penetrable from above or a strong no return plane from below. In other words, the interval [0, 1] is invariant under any Ytε (irrespective of the time horizon) which is expressed by calling

184

4 Local Lyapunov exponents

y ∈R

1 x∈R

−2

2 −1

Fig. 4.4 The tendencies and switching curves of the drift F (x, y) =

1 2

−y +y(1−y)x

)0, 1( := R × [0, 1] a strong no return set. Hence, we restrict the state space of Y ε to [0, 1]. Note that this restriction is also reasonable from an applications’ point of view: The above diﬀerential equation can be interpreted as the kinetic equation of a chemical reaction in which “Y ” represents the reactant’s portion of the overall particle number; see Arnold et al. [Ar-Hh-Lf 78] for details who consider the system F (x, y) := 12 − y − y (1 − y) x which is basically a reﬂection of F at the ordinate. In the following, the switching curve in (0, 1) which is uniquely determined will be denoted by Ysc . It is this curve Ysc which has already been plotted in ﬁgure 4.3. If y0 ∈ [0, 1], theorem 4.4.1 is applicable to the system Y ε,y0 . Although 4.4.1 requires a global condition on F , namely that there is no other switching curve, the theorem is valid here, since Y ε,y0 only takes its values in [0, 1]. Either one redeﬁnes F on R2 \ R × [0, 1] such that the conditions of 4.4.1 are formally satisﬁed, where the behavior of Y ε,y0 and hence the result does not depend on the speciﬁc choice of this continuation of F ; or one recalls the proof of 4.4.1 which requires only a compact subset of R itself. The situation of theorem 4.4.1 and the previous example can be heuristically characterized by stating that the real noise driven system Y ε is “trapped in the neighborhood” of an attracting switching curve in each case. Besides, there is no other switching curve present near this neighborhood which might “interfere” in the sense that Y ε could be transferred to another domain . The following deﬁnition is motivated by this loose description:

4.4 The two-dimensional, general case

185

Deﬁnition 4.4.3 (Separated switching surfaces). Let I ⊂ R denote some (open or closed) interval and consider a continuous function F : Rd ×I → R. Such a function F is thought of as a vector ﬁeld of an RDE Y˙ t = F (Xt , Yt ) ,

Y0 = y0 ,

for a stochastic process (Yt )t≥0 , where X is a driving process (e.g. given by (2.1) under the assumptions 2.1.1). Suppose that there is a set M ⊂ Rd such that all the zero sets of F (x, . ), x ∈ M, can be precisely written as two 1 2 : M → R and Ysc : M → R , where continuous functions Ysc 1 2 Ysc (x) ≥ Ysc (x)

(x ∈ M)

1 2 and y → F (x, y) changes its sign at each Ysc (x) and Ysc (x). Then the function F (or the above RDE, respectively) is said to have separated switching surfaces on M (or separated switching curves on M, if d = 1), 1 2 ≥ supM Ysc . If the latter inequality is strict, if inf M Ysc 1 2 inf Ysc > sup Ysc , M

M

the system is said to have strictly separated switching surfaces (curves) on M. If a continuous function F : Rd × R → R has a restriction F : Rd × I → R which exhibits (strictly) separated switching surfaces on M in the above sense for some interval I ⊂ R and if in addition F (x, . ) is p-periodic for all x ∈ M, where p := L(I) denotes the length of I, then F is also said to have (strictly) separated switching surfaces on M. y 1 Ysc (x)

2 (x) Ysc

x ∈ Rd

Fig. 4.5 Strictly separated switching surfaces on Rd

Figure 4.5 illustrates strictly separated switching curves. Also see Arnold and Kliemann [Ar-Kl 83, p.38]; further note that the conditions which

186

4 Local Lyapunov exponents

Wihstutz [Wh 75, p.111ﬀ.] assumes for the harmonic oscillator imply strict separatedness of the switching curves. For applying corollary 3.2.1 in the following theorems 4.4.6 and 4.4.7 to the real noise angle system αε we need to assure that the hypoellipticity assumption 3.1.3(c) can be met. Remark 3.1.4 yields that this can be achieved by means of a rank condition on the velocity vector ﬁeld. Since our considerations are local in the sense of chapter 2, i.e. the relevant diﬀusive behavior takes place on M ∈ {Fx0 ,ζ , Hx0 ,ζ }, we are led to work with a localized version of 3.1.3(c). This is the contents of the new concept of “strong hypoellipticity on a set M” which we are now going to introduce. In order to compare the latter with a localized version of the hypoellipticity (1.15) for n = 2, we will also establish “hypoellipticity on a set M” . However, note that only “strong hypoellipticity on M” will come into play. Deﬁnition 4.4.4 (Local hypoellipticity). Let F : Rd × R → R be a diﬀerentiable function which is again considered as the velocity of (Yt )t≥0 given by the RDE Y˙ t = F (Xt , Yt ) ,

Y0 = y0 ,

where X is a driving process (e.g. given by (2.1) under the assumptions 2.1.1). Furthermore, let M ⊂ Rd be a ﬁxed set. (a) The drift function F will be called hypoelliptic on M, if for any y ∈ R there is some x ∈ M such that F (x, y) = 0 . (b) The drift function F will be called strongly hypoelliptic on M, if for all (x, y) ∈ M × R ,

∂F (x, y) = 1. rank ∂xj 1≤j≤d

4.4.2 The local Lyapunov exponent Deﬁnition 4.4.5 (Value function of the angle process). The angle αε of the stochastic process Z ε is given by the SDE ¯ X ε , αε dt dαεt = h t t √ dXtε = b Xtε dt + ε σ Xtε dWt in the case that the state space of Z ε is two-dimensional, n = 2. Here, the ¯ which formally is a real-valued function on Rd ×[0, 2π) can be drift function h ¯ : Rd × R → R which is π-periodic with respect equivalently read as function h to the second argument. Therefore, αε can also be considered as process in R. Hence, it is in accordance with notation 3.1.2 (s := 0) to deﬁne α0 as the

4.4 The two-dimensional, general case

187

solution of the ODE ¯ dα0 (t) = h(u(t), α0 (t)) dt , where u is a “control” function u ∈ C 1 (R+ , Rd ) and where the initial condition is (u(0), α0 (0)) = (x, α) ∈ Rd × R ; furthermore, /D ϑ ≡ ϑD (0; x, α; u) := inf t ≥ 0 : α0 (t) ∈ denotes the corresponding exit time of α0 from some set D ⊂ R and U ≡ UD (0; T ; x, α) := u ∈ C 1 (R+ , Rd ) : u(0) = x and ϑD (0; x, α; u) ≤ T is the set of control functions for which an exit happens before a prescribed time horizon T > 0. As in assumption 3.1.3(e), UD (0; T ; x, α) needs to be nonempty. Then the value function of αε on Fx0 ,ζ is deﬁned as 0 ID,F (0; T ; x, α) x0 ,ζ

:=

ϑD (0;x,α;u)

inf

u∈UD (0;T ;x,α)∩{u• ∈Fx0 ,ζ }

K u(t), u(t) ˙ dt ,

0

where x ∈ Fx0 ,ζ and {u• ∈ Fx0 ,ζ } denotes the set of control functions which only take their values in the set Fx0 ,ζ which had been deﬁned in corollary 2.5.7 and where 2 1 K(x, v) := a(x)−1/2 [v − b(x)] . 2 In the same manner, the value function of αε on Hx0 ,ζ is deﬁned as 0 ID,H (0; T ; x, α) x0 ,ζ

:=

inf

u∈UD (0;T ;x,α)∩{u• ∈Hx0 ,ζ }

ϑD (0;x,α;u)

K u(t), u(t) ˙ dt ,

0

where x ∈ Hx0 ,ζ and {u• ∈ Hx0 ,ζ } now denotes the set of control functions which only take their values in the set Hx0 ,ζ which had been deﬁned in corollary 2.5.8 under the assumptions imposed there. Note, in particular, that the value functions of αε on Fx0 ,ζ and Hx0 ,ζ only ¯ b, σ on the respective set. depend on the behavior of h, The following theorem contains our fundamental result concerning the asymptotic behavior of the angle process αε on time scales. As will become clear from the proof, a further condition is needed to hold involving the value function as introduced in deﬁnition 4.4.5. This is requirement (4.24); see below. Intuitively it says that the cost of controlling the angle system away from the unstable switching curve A2 needs to be smaller than the scale parameter ζ. This constraint is plausible, since on the one hand the cost

188

4 Local Lyapunov exponents

of controlling the angle system is calculated by means of the L2 -deviation ϑ K(u(t), u(t))dt ˙ of a control function u from X 0 in deﬁnition 4.4.5 . On the 0 other hand, ζ is a cost parameter for the diﬀusion X ε itself; see corollary 2.5.7; it provides a bound on the values of the quasipotential which X ε can overcome on the time scale T (ε) ! eζ/ε and the quasipotential is also calculated via the L2 -deviation which had been called the “rate function” I[0,T ],x0 (u) ≡ T K(u(t), u(t))dt ˙ ; see theorem 2.3.4 . 0 As will be proven afterwards in remark 4.4.12 this condition (4.24) is satisﬁed, if the SDE for X ε is of gradient type (2.2). Theorem 4.4.6 (Proportion of the occupation time of the angle process αε near A1 (K µ(x0 ,ζ) ) = 0 becomes full on Fx0 ,ζ ). Consider the stochastic process Z ε solving the SDE (1), where X ε is given by the SDE (2.1) under the assumptions 2.1.1. Furthermore, suppose that n = 2 and that A ∈ C ∞ (Rd , R2×2 ). Let αε be the angle process of Z ε which is given as the solution of the random diﬀerential equation (1.6), ¯ ε , αε ) , α˙ εt = h(X t t

αε0 = α0 .

>l Fix an initial value x0 ∈ i=1 Di of X ε and a time scale T (ε) ! eζ/ε , where ζ > 0 is not contained in a ﬁnite exceptional set depending on x0 ; see (2.20). Let Fx0 ,ζ be the set deﬁned in corollary 2.5.7 according to the underlying diﬀusion X ε,x0 and the time scale T (ε) ! eζ/ε . ¯ has strictly separated switching surfaces Suppose that the drift function h A1 and A2 on Fx0 ,ζ ; see deﬁnitions 4.4.3 and 1.2.4; according to the latter deﬁnition, A1 is supposed to be attracting whereas A2 is repelling. ¯ is strongly hypoelliptic on Fx0 ,ζ ; see deﬁnition Furthermore, suppose that h 4.4.4. If α0 ∈ A2 (Fx0 ,ζ ), then it is assumed in addition that there exists T¯ > 0 such that 0 ¯ ; x, α) : (x, α) ∈ Fx0 ,ζ × A2 (Fx0 ,ζ ) , (4.24) (0; T ζ > max ID,F x0 ,ζ 0 (0; T ; x, α) is where D ⊃ A2 (Fx0 ,ζ ) denotes some open set and where ID,F x0 ,ζ ε the value function for α as deﬁned in 4.4.5. Then it follows for any η > 0 that

P 1 L t ∈ [0, T (ε)] : αεt ∈ B A1 (Kµ(x0 ,ζ) ), η −− −−→ 1 . ε→0 T (ε) Proof. If α0 ∈ / A2 (Fx0 ,ζ ), the statement follows from theorem 4.4.1 in the same way as example 4.4.2. Hence, we consider the case that α0 ∈ A2 (Fx0 ,ζ ) for which the additional assumption (4.24) is valid. Note that the separatedness assumption on the switching curves necessarily implies that λ1 > λ2 , for otherwise λ1 = λ2 ¯ µ(x ,ζ) , α) = 0 for all α. Thus the separatedness of the would imply that h(K 0

4.4 The two-dimensional, general case

189

switching curves automatically enforces the condition p = 2 which characterizes the case yet to be solved; see the discussion at the beginning of this section. Deﬁne α1 := A1 (Kµ(x0 ,ζ) )

α2 := A2 (Kµ(x0 ,ζ) )

and

for brevity. Note that no confusion with the values of the stochastic process (αεt )t≥0 is possible, since the latter object is always written with its superscript ε; furthermore, this notation is consistent with deﬁnition 1.2.4 as αi ≡ Ai (Kµ(x0 ,ζ) ) = arctan ξi (A(Kµ(x0 ,ζ) )) for i ∈ {1, 2}, if the numeration is chosen such that α1 is attracting and α2 is repelling. However, this decomposition by means of ξ-variables will not be used in the sequel; for a discussion see remark 4.4.9 nonetheless. E.g. if A is diagonal at the metastable state,

λ1 0 , Aµ(x0 ,ζ) ≡ A(Kµ(x0 ,ζ) ) = 0 λ2 then α1 ≡ A1 (Kµ(x0 ,ζ) ) = 0 and α2 ≡ A2 (Kµ(x0 ,ζ) ) = π2 ; considering Aµ(x0 ,ζ) in diagonal form here amounts to applying a constant transformation of the system A which leaves all its properties unchanged; note that the separatedness of the switching curves is not necessarily preserved under such a coordinate transformation; however, the diagonality of Aµ(x0 ,ζ) is not crucial but only used for illustrative purposes, since it has been used in the previous proofs of this chapter. In the general case the eigendirections of Aµ(x0 ,ζ) are represented by angles α1 = α2 in [0, π) corresponding to λ1 > λ2 . From 4.1.6 it is already known that P 1 L t ∈ [0, T (ε)] : αεt ∈ B(α1 , η1 ) or αεt ∈ B(α2 , η2 ) −−− −→ 1 ; ε→0 T (ε) hence, it is left to be shown that P 1 L t ∈ [0, T (ε)] : αεt ∈ B(α2 , η) −−− −→ 0 ε→0 T (ε) for arbitrary η > 0. In the sequel we will prove the following slightly stronger claim: P 1 L t ∈ [0, T (ε)] : αεt ∈ A2 (Fx0 ,ζ ) −−− −→ 0 . ε→0 T (ε) ¯

∂h , the strict separatedness of the Due to the continuity of A1 , A2 and ∂x j switching curves and the strong hypoellipticity necessarily also hold true on an open enlargement of Fx0 ,ζ , i.e. on some open set

D x ,ζ ⊃ Fx ,ζ , F 0 0

190

4 Local Lyapunov exponents

D x ,ζ \ Fx ,ζ can be arbitrarily small. For proving the where the diﬀerence F 0 0 D x0 ,ζ ) , i.e. an open preceding claim we ﬁx an open neighborhood D of A2 (F set D ⊂ R such that D x ,ζ , D ⊃ A2 F 0 for which we also have

Dx ,ζ > 0 , dist D, A1 F 0

where dist( . , . ) denotes the distance function. The former condition concerning the choice of D can be trivially satisﬁed, whereas the latter requirement can be met, since the separation of the switching surfaces A1 and A2 is D x0 ,ζ as well). After taking the interassumed to be strict on Fx0 ,ζ (and on F section, we can assume that this set has the prescribed property (4.24) and it will be denoted by D further on for the investigations to come. The rest of the proof is organized in ﬁve steps: In the ﬁrst step the coeﬃcients of the angle SDE are modiﬁed such that they satisfy the assumptions 3.1.3. This allows to apply 3.2.1 in the second step. Afterwards this result is used to estimate the exit time of the coupled process from the set Fx0 ,ζ × A2 (Fx0 ,ζ ) “around” the repelling switching curve by means of the strong Markov property. In the fourth step this is used to derive that, due to the control-theoretic structure of the posed situation, the system αε cannot reenter D after leaving this set on the event ΩεΓ;F on which the paths X•ε take their values exclusively in Fx0 ,ζ ; hence, the exit time of αε from A2 (Fx0 ,ζ ) as calculated before can be used to estimate the length of the time interval until αε leaves A2 (Fx0 ,ζ ); this step yields that the decisive probability can be estimated from above by a function of ε of exponential type. The latter bound is then veriﬁed to converge (in the deterministic sense) to zero in the ﬁnal step. ¯ of the angle SDE be modiﬁed on 1) Let the coeﬃcient functions b, σ and h D x ,ζ and on (F D x ,ζ ) × R, respectively, as smooth functions ˆb, σ F ˆ and F such 0 0 that the set of assumptions 3.1.3 on the SDE (3.1) is satisﬁed; more precisely, the functions ˆb, σ ˆ , F are chosen to satisfy 3.1.3 and ˆb, σ ˆ shall coincide with ¯ on F D x0 ,ζ × R : D b, σ on Fx0 ,ζ as well as F coincides with h In particular, the (global) hypoellipticity assumption 3.1.3(c) on F can be ¯ is strongly hypoelliptic on F Dx0 ,ζ ; see deﬁnition 4.4.4 and remark met, since h D ¯ can 3.1.4. Furthermore, since dist D, A1 Fx0 ,ζ > 0 , the continuation F of h be chosen such that F has strictly separated switching surfaces (also called A1 and A2 ) on Rd which do not intersect ∂D. Especially this implies that 3.1.3(d) holds true. Finally, 3.1.3(e) is met since A2 is repelling (deﬁnition 1.2.4) and non-constant due to the strong hypoellipticity assumption; hence, there are trajectories which can leave D within a prescribed time horizon; also see below.

4.4 The two-dimensional, general case

191

ˆ ε, α In order to avoid any confusion, the new system is denoted by (X ˆ ε ), i.e. ε ε ˆ ,α dˆ αεt = F X t ˆ t dt ε √ ε ε ˆ t = ˆb X ˆ t dt + ε σ ˆ t dWt , dX ˆ X satisfying the ﬁxed initial condition ε ε ε ε ˆ ,α X 0 ˆ 0 = (X0 , α0 ) ≡ (x0 , α0 ) ; for abbreviation we also denote this system by ε,(x0 ,α0 )

ˆ α (X, ˆ )t

0 ˆ tε,x0 , α := (X ˆ ε,α ) . t

As in 3.1.1 (s := 0), let 0 T ε ≡ T ε (0, x0 , α0 ) ≡ TDε (0, x0 , α0 ) := inf { t ≥ 0 : α ˆ ε,α ∈ / D} t 0 denote the ﬁrst exit time of (ˆ αε,α )t≥s from the set D which had been ﬁxed t previously. According to 3.1.2, let u ∈ C 1 (R+ , Rd ) be a control function and let α0 be the path deﬁned by dα0 (t) = F (u(t), α0 (t)) dt

under the initial condition (u(0), α0 (0)) = (x, α) ∈ Rd × R (s := 0). As before, ϑ ≡ ϑD (0; x, α; u) := inf t ≥ 0 : α0 (t) ∈ /D denotes the corresponding exit time of α0 from D and U ≡ UD (0; T ; x, α) := u ∈ C 1 (R+ , Rd ) : u(0) = x and ϑD (0; x, α; u) ≤ T is the set of control functions for which an exit happens before a prescribed time horizon T > 0. Note that UD (0; T ; x, α) is nonempty for all ¯ (x, α) ∈ Rd × R, since A2 is repelling (deﬁnition 1.2.4) and non-constant (h is strongly hypoelliptic) as has been already mentioned above in connection with assumption 3.1.3(e). More formally, here we use the control-theoretic fact that Rd × D is a weak control set for the above control system α˙ 0 (t) = F (u(t), α0 (t)); see Kliemann [Kl 80, p.31f.].

192

4 Local Lyapunov exponents

Using this notation, the value function according to 3.1.2 is deﬁned as 0 (0; T ; x, α) := I 0 ≡ ID

where

ϑD (0;x,α;u)

inf u∈UD (0;T ;x,α)

K u(t), u(t) ˙ dt ,

0

0 1 2 1 ˆ(x)−1/2 v − ˆb(x) . a 2 we have

K(x, v) := Note that for x ∈ Fx0 ,ζ

0 ID (0; T ; x, α) ≤

ϑD (0;x,α;u)

inf

u∈UD (0;T ;x,α)∩{u• ∈Fx0 ,ζ }

0 = ID,F x

0

K u(t), u(t) ˙ dt

0

(0; T ; x, α) , ,ζ

where {u• ∈ Fx0 ,ζ } again denotes the set of control functions which take 0 their values in Fx0 ,ζ only. Furthermore, the latter quantity ID,F (0; T ; x, α) x0 ,ζ which had been deﬁned in deﬁnition 4.4.5 only depends on the behavior of 0 ¯ b, σ (and thus also of F, ˆb, σ h, ˆ ) on Fx0 ,ζ ; in other words, ID,F (0; T ; x, α) x0 ,ζ ¯ does not depend on how h, b, σ are redeﬁned outside Fx0 ,ζ . ˆ α) 2) Applying corollary 3.2.1 to the (open) set D and the process (X, ˆ ε deﬁned in step 1), it follows that for any ﬁxed η˜ > 0, there is an ε0 > 0 such that for all ε ∈ (0, ε0 ], the exit probability QεD (0; T ; x, α) ≡ P0,x,α {TDε ≤ T } ≡ P TDε (0, x, α) ≤ T satisﬁes

1 0 − max ID (0; T ; x, α) − η˜ > exp ε (x,α)∈G

1 0 − max ID,Fx ,ζ (0; T ; x, α) − η˜ ≥ exp 0 ε (x,α)∈G 0 I (T ) + η˜ , ≡ exp − max ε

min (x,α)∈G

QεD (0; T ; x, α)

where G denotes the compact set G := Fx0 ,ζ × A2 (Fx0 ,ζ ) ⊂ Rd × D and 0 Imax (T ) :=

0 max ID,F x

(x,α)∈G

0 ,ζ

(0; T ; x, α)

for abbreviation. Due to assumption (4.24) there is some time horizon T¯ > 0 such that 0 (T¯ ) ; in other words, it is possible to ﬁx η˜ suﬃciently small such ζ > Imax that

4.4 The two-dimensional, general case

193

0 ζ > Imax (T¯) + η˜

for the investigations to come. In other words, it follows for the inclusion probability ε PD (0; T ; x, α) ≡ 1 − QεD (0; T ; x, α) ≡ P TDε (0, x, α) > T from 3.2.1 and by assumption (4.24) that 0 Imax (T¯ ) + η˜ < 1 − exp − ε ζ . < 1 − exp − ε

max (x,α)∈G

ε PD (0; T¯ ; x, α)

3) Let ε,(x,α)

ε τG ≡ τG

:= inf

ε ˆ ε, α t ≥ 0 : (X /G t ˆt ) ∈

ε,(x,α) ˆ α from G; since G is closed and the denote the ﬁrst exit time of (X, ˆ )t ε underlying ﬁltration is complete, τG is a stopping time.1 Since G ⊂ Rd × D, it follows from step 2) that for any ε ≤ ε0 , ε,(x,α) max P τG > T¯ ≤ max P TDε (0, x, α) > T¯ (x,α)∈G

(x,α)∈G

≡

ε max PD (0; T¯; x, α)

(x,α)∈G

< 1 − exp

−

< 1 − e−ζ/ε

0 Imax (T¯ ) + η˜ ε

.

By induction, this implies that for any ε ≤ ε0 , k I 0 (T¯ ) + η˜ ε,(x,α) max P τG > k T¯ < 1 − exp − max (x,α)∈G ε

(k ∈ N) .

More precisely, one obtains for any initial value (x, α) ∈ G and ε ≤ ε0 that ε,(x,α) P τG > k T¯ ; ˆ ε,x ε, X ˆ ε,α ¯ ,α ¯ ε,(x,α) (k−1) T (k−1) T > (k − 1) T¯ , τG > T¯ = P τG

1

See e.g. Hackenbroch and Thalmaier [Hb-Th 94, 3.12].

194

4 Local Lyapunov exponents

+ = E 1 τ ε,(x,α) >(k−1)T¯ · G

1 − EF(k−1)T¯

1 τ

ˆ ε,x X ˆ ε,α ¯ +• , α ¯ +• (k−1)T (k−1)T

≤T¯

⎤

⎥ ⎦

, + ε,(x,α) ˆ α) ˆ (k−1)T¯ +• , ≡ E 1 τ ε,(x,α) >(k−1)T¯ · 1 − EF(k−1)T¯ H ◦ (X, G

where H is given on the path space C(R+ , Rd+1 ) by H := 1{τ ≤T¯} and τ is deﬁned on the paths h ∈ C(R+ , Rd+1 ) by τ (h) := inf{t ≥ 0 : ht ∈ / G} ε,(x,α)

such that τG

ˆ •ε,x , α ≡ τ ◦ (X ˆ ε,α • ). With

ˆ α (T H)(p) := E(H ◦ (X, ˆ )ε,p • )

(p ∈ Rd × R)

the strong Markov property2 for the stochastic system ε,(x,α)

ˆ α (X, ˆ )t

ˆ ε,x , α ≡ (X ˆ ε,α t t )t

implies that ε,(x,α) ε,(x,α) ˆ α) ˆ α EF(k−1)T¯ H ◦ (X, ˆ (k−1)T¯ +• = (T H) ◦ (X, ˆ )(k−1)T¯ ε,p ˆ ≡ E 1{τ ≤T¯} ◦ (X, α ˆ )•

.

ε,(x,α) ˆ α) p=(X, ˆ (k−1)T¯

Plugging this in the previous string of equations, we get (as we look only at ε,(x,α) ˆ α the case that p = (X, ˆ )(k−1)T¯ is still in G): ε,(x,α) P τG > k T¯ 5 = E 1 τ ε,(x,α) >(k−1)T¯ G

5 =E

2

1

ε,(x,α)

τG

>(k−1)T¯

6 ε,(x,α) ˆ α) ˆ (k−1)T¯+• · 1 − EF(k−1)T¯ H ◦ (X, ˆ α · 1 − E 1{τ ≤T¯} ◦ (X, ˆ )ε,p •

6 ε,(x,α) ˆ α) p=(X, ˆ (k−1)T¯

See e.g. Hackenbroch and Thalmaier [Hb-Th 94, 6.32 & 6.41].

4.4 The two-dimensional, general case

5 ≤E

1

5 ≡E

1

ε,(x,α)

τG

ε,(x,α)

τG

>(k−1)T¯

>(k−1)T¯

ˆ α · 1 − inf E 1{τ ≤T¯} ◦ (X, ˆ )ε,p •

195

6

p∈G

6 ε,p ¯ · sup P τG > T p∈G

0 Imax (T¯ ) + η˜ ε,(x,α) · P τG > (k − 1)T¯ < 1 − exp − ε k 0 Imax (T¯ ) + η˜ , < 1 − exp − ε

where in the last two steps the case “k = 1” shown above and the induction assumption have been used. 4) Now, ﬁx an arbitrary δ > 0 and deﬁne I H δ · T (ε) −1 , k ≡ k(ε) := T¯ where . denotes the integer part function. Furthermore, for the set ε,x0 Ωε := Ωε1;F ≡ ω ∈ Ω : XtT (ω) ∈ F for all t ∈ [0, 1] , x0 ,ζ (ε) it is known from corollary 2.5.7 (Γ := 1) that lim P ( Ωε ) = 1 .

ε→0

D x ,ζ × ˆ ε, α ˆ ε ) coincide on the open set F Since the SDEs for (X ε , αε ) and (X 0 R, it follows that these processes are necessarily equal to each other (nonD x ,ζ × R. More precisely3 , there is a unique distinguishable) before exiting F 0 Dx0 ,ζ × R ; maximal solution of the SDE with respect to the “state space” F this fact will be used in the subsequent calculation. Furthermore, it will now be used that on Ωε , i.e. on the event that the path map X•ε only takes its values in Fx0 ,ζ , the system αε cannot reenter A2 (Fx0 ,ζ ) after leaving this set. Formally, this fact is expressed in control-theoretic terminology by noting that Fx0 ,ζ × ([0, π) \ A2 (Fx0 ,ζ )) is a strong control ¯ set of the control system α˙ 0 (t) = h(u(t), α0 (t)) with control functions u ∈ d C(R+ , R ) ∩ {u• ∈ Fx0 ,ζ } ; see Kliemann [Kl 80, p.31f.]. Due to our previous construction, this is equivalent to the fact that the set Rd × ([0, π) \ A2 (Rd )) is a strong control set of the control system α˙ 0 (t) = F (u(t), α0 (t)) with control functions u ∈ C(R+ , Rd ) . The reason behind this argument is that 3

See e.g. Hackenbroch and Thalmaier [Hb-Th 94, 6.22]. The maximality of the solution mentioned above refers to the maximal choice of its life time, i.e. the ﬁrst exit D x ,ζ ×R ; see e.g. Hackenbroch and Thalmaier [Hb-Th 94, 6.19]. What will time from F 0 D x ,ζ × R . ˆ ε, α ˆ ε ) coincide before exiting F be used here is the fact that (X ε , αε ) and (X 0

196

4 Local Lyapunov exponents

¯ has strictly separated switching curves A1 and A2 by assumption. More h precisely, in the following it will be used that for all ω ∈ Ωε , 0 L t ∈ [0, T (ε)] : αε,α (ω) ∈ A2 (Fx0 ,ζ ) ≥ (k(ε) + 1)T¯ t if and only if 0 αε,α (ω) ∈ A2 (Fx0 ,ζ ) ; (k(ε)+1)T¯

we denote this argument by (∗); of course the time (k(ε) + 1)T¯ which is relevant for the subsequent estimate can be replaced by any other element of [0, T (ε)] here. Thus it follows altogether for any initial condition (x0 , α0 ) of (Xtε , αεt ) that

1 0 ∈ A (F ) ≥ δ L t ∈ [0, T (ε)] : αε,α 2 x0 ,ζ t T (ε)

ε ε ε,α0 ≤ P Ω + P Ω ∩ L t ∈ [0, T (ε)] : αt ∈ A2 (Fx0 ,ζ ) ≥ δT (ε)

0 ¯ ∈ A (F ) ≥ (k(ε) + 1) T ≤ P Ωε + P Ωε ∩ L t ∈ [0, T (ε)] : αε,α 2 x ,ζ t 0

(∗) 0 = P Ωε + P Ωε ∩ αε,α ∈ A2 (Fx0 ,ζ ) (k(ε)+1)T¯

0 Xtε,x0 ∈ Fx0 ,ζ for all t ∈ [0, T (ε)] ∩ αε,α ∈ A (F ) ≡ P Ωε + P 2 x ,ζ ¯ 0 (k(ε)+1)T ε ε,x0 ε,α0 ∈ G for all t ∈ 0, (k(ε) + 1)T¯ ≤ P Ω + P Xt , αt 0 ˆ tε,x0 , α = P Ωε + P X ˆ ε,α ∈ G for all t ∈ 0, (k(ε) + 1)T¯ t ε,(x ,α ) ≡ P Ωε + P τG 0 0 ≥ (k(ε) + 1)T¯ ε,(x ,α ) ≤ P Ωε + P τG 0 0 > k(ε)T¯ k(ε) I 0 (T¯ ) + η˜ , < P Ωε + 1 − exp − max ε

P

where in the ﬁnal estimate the inductive consequence of 3.2.1 obtained in step 3) has been used. As already mentioned, the ﬁrst summand of the hence obtained upper bound tends to zero as ε → 0 due to corollary 2.5.7. 5) Therefore, the claim has been reduced in step 4) to verifying that

1 − exp

Since

−

k(ε) 0 Imax (T¯ ) + η˜ ε→0 −−−→ 0 . ε

δ δ T (ε) − 1 ≥ k(ε) > ¯ T (ε) − 2 , T¯ T

4.4 The two-dimensional, general case

197

it is clear that also k(ε) ! eζ/ε ; in particular, as 0 ζ > Imax (T¯ ) + η˜ ,

which had been assured by assumption (4.24), see step 2), it is possible to ﬁx an arbitrary 0 η˜ ˜ ∈ 0 , ζ − Imax (T¯ ) − η˜ , for which we have

k(ε) > K exp

ζ − η˜˜ ε

for all ε ≤ ε0 and some constant K > 0 (depending only on η˜˜ and ε0 ). Using this preliminary note, the auxiliary function h(ε) :=

1

1 − exp −

0 Imax (T¯ ) + η ˜ ε

−1 =

exp +

1 0 Imax (T¯ ) + η ˜ ε

−1

∈ (0, ∞)

and the Bernoulli inequality4 , one gets altogether that

1 − exp

−

0 (T¯) + η˜ Imax ε

k(ε) ≡ ≤ <

<

=

1 k(ε) 1 + h(ε) 1 1 + k(ε) · h(ε) 1 k(ε) h(ε) ¯) + η I0 (T ˜ exp + max ε ˜ K exp ζ−ε η˜ 0 ζ − Imax (T¯) − η˜ − η˜ ˜ 1 exp − K ε

ε→0

−−−→ 0 .

Therefore, the claim is proven.

Note that step 5) of the previous proof which involves the auxiliary function h and the Bernoulli inequality adapts the familiar argument for proving that rn → 0 as n → ∞, for a ﬁxed r ∈ (−1, 1), to our situation involving the time scales at hand. Next we consider the situation, when the use of corollary 2.5.7 and its corresponding set Fx0 ,ζ is replaced by corollary 2.5.8 and the respective smaller set Hx0 ,ζ . For an illustration of Hx0 ,ζ consider e.g. the two well potential 4

For all h ≥ −1 and k ∈ N, it follows that 1 + kh ≤ (1 + h)k .

198

4 Local Lyapunov exponents

of example 2.6.2 and suppose that the diﬀusion X ε starts in the shallow well, i.e. x0 ∈ D1 ; furthermore choose ζ ∈ (2v, 2V ). Then it follows that µ(x0 , ζ) = 2 and ζ < 2V = V21 = Vµ(x0 ,ζ),J(µ(x0 ,ζ)) ; in other words, in this sample situation we have Hx0 ,ζ = { y ∈ D2 : 2[U (y) − U (K3 )] ≤ ζ } D2 . As in theorem 4.4.6 one more condition concerning the value function from deﬁnition 4.4.5 is necessary; see (4.25) below. Just like (4.24) it will be proven afterwards in remark 4.4.12 that (4.25) is satisﬁed, if the SDE for X ε is of gradient type (2.2). Theorem 4.4.7 (Proportion of the occupation time of the angle process αε near A1 (K µ(x0 ,ζ) ) = 0 becomes full on Hx0 ,ζ ). Consider the stochastic process Z ε solving the SDE (1), where X ε is given by the SDE (2.1) under the assumptions 2.1.1. Furthermore, suppose that n = 2 and that A ∈ C ∞ (Rd , R2×2 ). Let αε be the angle process of Z ε which is given as the solution of the random diﬀerential equation (1.6), ¯ ε , αε ) , α˙ εt = h(X t t

αε0 = α0 .

>l Fix an initial value x0 ∈ i=1 Di of X ε and a time scale T (ε) ! eζ/ε , where ζ > 0 is not contained in a ﬁnite exceptional set depending on x0 ; see (2.20). Assume that ζ < Vµ(x0 ,ζ),J(µ(x0 ,ζ)) , where again V denotes the matrix of mutual quasipotential values and J is the map expressing the “following-behavior” of 0-cycles; let Hx0 ,ζ be the set hence deﬁned in corollary 2.5.8 according to the underlying diﬀusion X ε,x0 and the time scale T (ε) ! eζ/ε . ¯ has strictly separated switching surfaces Suppose that the drift function h A1 and A2 on Hx0 ,ζ ; see deﬁnitions 4.4.3 and 1.2.4. ¯ is strongly hypoelliptic on Hx ,ζ ; see deﬁnition Furthermore, suppose that h 0 4.4.4. Assume in addition that there exists T¯ > 0 such that 0 ¯ ; x, α) : (x, α) ∈ Hx0 ,ζ × A2 (Hx0 ,ζ ) , (4.25) (0; T ζ > max ID,H x0 ,ζ 0 (0; T ; x, α) is where D ⊃ A2 (Hx0 ,ζ ) denotes some open set and where ID,H x0 ,ζ ε the value function for α as deﬁned in 4.4.5. Then it follows for any α0 and for any η > 0 that P 1 L t ∈ [0, T (ε)] : αεt ∈ B A1 (Kµ(x0 ,ζ) ), η −− −−→ 1 . ε→0 T (ε)

Proof. As in the proof of the previous theorem 4.4.6, let α1 := A1 (Kµ(x0 ,ζ) )

and

α2 := A2 (Kµ(x0 ,ζ) )

4.4 The two-dimensional, general case

199

¯ µ(x ,ζ) , . ) in [0, π) corresponding to denote the two distinct zeros of h(K 0 λ1 > λ2 . Recall from 4.1.6 that P 1 L t ∈ [0, T (ε)] : αεt ∈ B(α1 , η1 ) or αεt ∈ B(α2 , η2 ) −−− −→ 1 ε→0 T (ε) which is why it needs to be shown that P 1 L t ∈ [0, T (ε)] : αεt ∈ B(α2 , η) −−− −→ 0 ε→0 T (ε) for arbitrary η > 0 also in this situation. In the sequel we will prove the following slightly stronger claim: P 1 L t ∈ [0, T (ε)] : αεt ∈ A2 (Hx0 ,ζ ) −−− −→ 0 . ε→0 T (ε) In the same manner as in the proof of the previous theorem 4.4.6, the strict separatedness of the switching curves and the strong hypoellipticity necessarily also hold true on an open enlargement of Hx0 ,ζ , i.e. on some open set D x0 ,ζ ⊃ Hx0 ,ζ H ¯ ∂h D x ,ζ \ Hx ,ζ due to the continuity of A1 , A2 and ∂x ; again, the diﬀerence H 0 0 j might be arbitrarily small. For verifying the preceding claim ﬁx an open D x0 ,ζ ) in the situation at hand, i.e. an open set D ⊂ R neighborhood D of A2 (H such that D x0 ,ζ , D ⊃ A2 H

for which we also have

D x ,ζ > 0 . dist D, A1 H 0

As in the previous proof, the former condition concerning the choice of D can be trivially satisﬁed; the latter requirement can be met since the separation of the switching surfaces A1 and A2 is assumed to be strict on Hx0 ,ζ (and D x0 ,ζ ). After taking the intersection, this set also has the prescribed also on H property (4.25) and it will also be denoted by D for the further investigations. As has already become clear from these introductory remarks, this proof follows the same lines as the one of the previous theorem 4.4.6; nonetheless, all arguments will be provided in the sequel for deﬁniteness. This proof is organized in four steps: In the ﬁrst step the coeﬃcients of the angle SDE are modiﬁed such that they satisfy the assumptions 3.1.3. This allows to apply 3.2.1 in the second step. Afterwards this result is used to estimate the exit time of the coupled process from the set Hx0 ,ζ × A2 (Hx0 ,ζ ) around the repelling switching curve by means of the strong Markov property. In the fourth step this bound is used together with the control-theoretic

200

4 Local Lyapunov exponents

structure of the posed situation; more precisely, the system αε cannot reenter D after leaving this set on the event Ωεc,Γ;H on which the paths X•ε take their values exclusively in Hx0 ,ζ during the time interval [c T (ε), Γ T (ε)] ; hence, the exit time of αε from A2 (Hx0 ,ζ ) can then be used to estimate the length of the time interval until αε leaves A2 (Hx0 ,ζ ); this step yields that the decisive probability can be estimated from above by a function which is exponential in ε. The resulting bound then converges to zero by what has been shown in step 5) in the proof of theorem 4.4.6. ¯ of the angle SDE be modiﬁed on 1) Let the coeﬃcient functions b, σ and h D D Hx0 ,ζ and on (Hx0 ,ζ ) × R, respectively, as smooth functions ˆb, σ ˆ and F such that the set of assumptions 3.1.3 on the SDE (3.1) is satisﬁed; more precisely, the functions ˆb, σ ˆ , F are chosen to satisfy 3.1.3 and ˆb, σ ˆ shall coincide with ¯ on H D x ,ζ × R : D x ,ζ as well as F coincides with h b, σ on H 0 0 ¯ is The hypoellipticity assumption 3.1.3(c) on F can be assured, since h D strongly hypoelliptic on Hx0 ,ζ ; again see deﬁnition 4.4.4 and remark 3.1.4. ¯ can be chosen such D x0 ,ζ As dist D, A1 H > 0 , the continuation F of h that F has strictly separated switching surfaces (also denoted by A1 and A2 ) on Rd which do not intersect ∂D. In particular, this implies that 3.1.3(d) holds true. Finally, 3.1.3(e) is met, since A2 is repelling and non-constant ¯ is strongly hypoelliptic) and hence, there are trajectories which can leave (h D within a prescribed time horizon; also see below. The resulting system is ε,(x,α) ˆ ε,x , α ˆ α ≡ (X ˆ ε,α again denoted by (X, ˆ )t t t ), i.e. ˆ tε , α dˆ αεt = F X ˆ εt dt √ ε ˆ tε = ˆb X ˆ tε dt + ε σ ˆ t dWt , dX ˆ X satisfying the ﬁxed initial condition ε ε d ˆ ,α X 0 ˆ 0 = (x, α) ∈ R × R . ε,(x,α) ˆ α Note that neither this process (X, ˆ )t≥0 nor the set D (nor the terms to be 0 ε ε derived thereof in the sequel such as TDε , UD , ID , QεD , PD , G, τG , H and Ωε ) in the present proof do coincide with the homonymous objects of the previous proof (theorem 4.4.6). However, since each deﬁnition is made precise and no confusion is possible, there is no need to introduce a clumsy notation such as adding tildes. Furthermore, using the respective symbols again highlights the parallels between the two proofs. As in 3.1.1 (s := 0), let

T ε ≡ T ε (0, x, α) ≡ TDε (0, x, α) := inf { t ≥ 0 : α ˆ ε,α ∈ / D} t denote the ﬁrst exit time of (ˆ αε,α t )t≥s from D.

4.4 The two-dimensional, general case

201

According to 3.1.2 (s := 0), let u ∈ C 1 (R+ , Rd ) be a control function and let α0 be the path deﬁned by dα0 (t) = F (u(t), α0 (t)) dt under the initial condition (u(0), α0 (0)) = (x, α) ∈ Rd × R . As before, ϑ ≡ ϑD (0; x, α; u) := inf t ≥ 0 : α0 (t) ∈ /D denotes the corresponding exit time of α0 from D and U ≡ UD (0; T ; x, α) := u ∈ C 1 (R+ , Rd ) : u(0) = x and ϑD (0; x, α; u) ≤ T is the set of control functions for which an exit happens before a prescribed time horizon T > 0. The set UD (0; T ; x, α) is nonempty for all (x, α) ∈ Rd × R also in this case, since A2 is repelling and non-constant; see the above discussion of assumption 3.1.3(e) and Kliemann [Kl 80, p.31f.]. Using this notation, the value function according to 3.1.2 is 0 I 0 ≡ ID (0; T ; x, α) :=

where K(x, v) :=

ϑD (0;x,α;u)

inf u∈UD (0;T ;x,α)

K u(t), u(t) ˙ dt ,

0

0 1 2 1 ˆ(x)−1/2 v − ˆb(x) . a 2

For x ∈ Hx0 ,ζ we have 0 ID (0; T ; x, α)

≤

u∈UD (0;T ;x,α)∩{u• ∈Hx0 ,ζ }

0 = ID,H x

ϑD (0;x,α;u)

inf

0

K u(t), u(t) ˙ dt

0

(0; T ; x, α) , ,ζ

where {u• ∈ Hx0 ,ζ } is the set of control functions which take their values in 0 (0; T ; x, α) which had been deﬁned in 4.4.5 only depends Hx0 ,ζ only; ID,H x0 ,ζ ¯ b, σ (and thus also of F, ˆb, σ on the behavior of h, ˆ ) on Hx ,ζ . 0

2) In this situation apply corollary 3.2.1 to the (open) set D and the process ˆ α) (X, ˆ ε deﬁned in step 1). Then it follows that for any ﬁxed η˜ > 0, there is an ε0 > 0 such that for all ε ∈ (0, ε0 ], the exit probability QεD (0; T ; x, α) ≡ P0,x,α {TDε ≤ T } ≡ P TDε (0, x, α) ≤ T

202

4 Local Lyapunov exponents

satisﬁes

1 0 − max ID (0; T ; x, α) − η˜ ε (x,α)∈G

1 0 − max ID,H (0; T ; x, α) − η ˜ ≥ exp x0 ,ζ ε (x,α)∈G I 0 (T ) + η˜ , ≡ exp − max ε

min QεD (0; T ; x, α) > exp

(x,α)∈G

where in the present situation, G denotes the compact set G := Hx0 ,ζ × A2 (Hx0 ,ζ ) ⊂ Rd × D and accordingly 0 Imax (T ) :=

0 max ID,H x

(x,α)∈G

0 ,ζ

(0; T ; x, α) .

Due to assumption (4.25) one can ﬁx T¯ > 0 and η˜ > 0 such that 0 (T¯ ) + η˜ . ζ > Imax

In other words, it follows for the inclusion probability ε PD (0; T ; x, α) ≡ 1 − QεD (0; T ; x, α) ≡ P TDε (0, x, α) > T from 3.2.1 and by assumption (4.25) that 0 Imax (T¯ ) + η˜ < 1 − exp − ε ζ . < 1 − exp − ε

max (x,α)∈G

ε PD (0; T¯ ; x, α)

3) Let ε,(x,α)

ε τG ≡ τG

:= inf

ε ˆ ε, α t ≥ 0 : (X /G t ˆt ) ∈

ε,(x,α) ˆ α from the set G as deﬁned in step 2). denote the ﬁrst exit time of (X, ˆ )t d Since G ⊂ R × D, it follows from step 2) that for any ε ≤ ε0 , ε,(x,α) max P τG > T¯ ≤ max P TDε (0, x, α) > T¯ (x,α)∈G

(x,α)∈G

ε max PD (0; T¯; x, α) I 0 (T¯ ) + η˜ < 1 − exp − max ε −ζ/ε <1 − e .

≡

(x,α)∈G

4.4 The two-dimensional, general case

203

By induction, this implies that for any ε ≤ ε0 , k I 0 (T¯ ) + η˜ ε,(x,α) max P τG > k T¯ < 1 − exp − max ε (x,α)∈G

(k ∈ N) .

The proof of this fact again uses the strong Markov property: More precisely, the notation has been chosen such that the argument given in step 3) of the proof of theorem 4.4.6 literally holds true in the situation at hand. We refrain from repeating it verbatim. 4) Now, ﬁx an arbitrary δ > 0, some c ∈ (0, δ ∧ 1) and deﬁne I H (δ − c) · T (ε) −1 . k ≡ k(ε) := T¯ First, it follows for the set ε,x0 Ωε := Ωεc,1;H := ω ∈ Ω : XtT (ω) ∈ H for all t ∈ [c, 1] x0 ,ζ (ε) from corollary 2.5.8 (Γ := 1) that lim P ( Ωε ) = 1 .

ε→0

ˆ ε, α Secondly, by construction the SDEs for (X ε , αε ) and (X ˆ ε ) again coinD x0 ,ζ × R here; hence, it follows5 cide on the relevant open set which is H D that when started in a point (x, α) ∈ Hx0 ,ζ × R these processes are nondistinguishable before exiting this set. Furthermore, it will be used that on Ωε , i.e. on the event that the paths X•ε take their values in Hx0 ,ζ on the time interval [c T (ε), T (ε)] , the system αε cannot reenter A2 (Hx0 ,ζ ) during the time interval [c T (ε), T (ε)] after leaving this set A2 (Hx0 ,ζ ). Again, the control-theoretic reason is that Hx0 ,ζ × ([0, π) \ ¯ A2 (Hx0 ,ζ )) is a strong control set of the control system α˙ 0 (t) = h(u(t), α0 (t)) with control functions u ∈ C(R+ , Rd ) ∩ {u• ∈ Hx0 ,ζ }; see Kliemann [Kl 80, D x0 ,ζ × p.31f.]. Due to the continuation F of the coeﬃcient function ¯h outside H d d R, this is equivalent to saying that the set R × ([0, π) \ A2 (R )) is a strong control set of the control system α˙ 0 (t) = F (u(t), α0 (t)) with control functions ¯ has strictly separated u ∈ C(R+ , Rd ). Here we made use of the fact that h switching curves A1 and A2 on Hx0 ,ζ by assumption. More precisely, in the following this remark will be used in the form that for all ω ∈ Ωε , 0 L t ∈ [c T (ε), T (ε)] : αε,α (ω) ∈ A (H ) ≥ (δ − c) T (ε) 2 x ,ζ 0 t

5

See e.g. Hackenbroch and Thalmaier [Hb-Th 94, 6.22].

204

4 Local Lyapunov exponents

if and only if 0 αε,α (ω) ∈ A2 (Hx0 ,ζ ) for all t ∈ [c T (ε), δ T (ε)] ; t

again we denote this control-theoretic conclusion by (∗). D be given on the path space C(R+ , Rd+1 ) by Finally, let H D := 1 H

τ <(δ−c)T (ε)

for the mapping τ again being deﬁned on paths h ∈ C(R+ , Rd+1 ) by τ (h) := inf{t ≥ 0 : ht ∈ / G} ; with

D (p) := E H D ◦ (X, α)ε,p TH •

(p ∈ Rd × R) ,

the strong Markov property6 for the stochastic system ε,(x0 ,α0 )

(X, α)t

0 := (Xtε,x0 , αε,α )t t

now implies that D ◦ (X, α)ε,(x0 ,α0 ) = T H D ◦ (X, α)ε,(x0 ,α0 ) EFcT (ε) H cT (ε)+• cT (ε) ε,p ≡ E 1{τ <(δ−c)T (ε)} ◦ (X, α)•

ε,(x ,α0 )

0 p=(X,α)cT (ε)

Let this equation be abbreviated by (∗∗). Altogether it follows that 1 0 P L t ∈ [0, T (ε)] : αε,α ∈ A (H ) ≥ δ 2 x0 ,ζ t T (ε) 1 0 L t ∈ [c T (ε), T (ε)] : αε,α ∈ A (H ) ≥ δ − c ≤ P 2 x0 ,ζ t T (ε) 0 ∈ A2 (Hx0 ,ζ ) ≤ P Ωε + P Ωε ∩ L t ∈ [c T (ε), T (ε)] : αε,α t

≥ (δ − c) T (ε)

(∗) 0 = P Ωε + P Ωε ∩ αε,α ∈ A (H ) for all t ∈ [c T (ε), δ T (ε)] 2 x ,ζ t 0 ε 0 ∈ G for all t ∈ [c T (ε), δ T (ε)] ≤ P Ω + P Xtε,x0 , αε,α t 6

See e.g. Hackenbroch and Thalmaier [Hb-Th 94, 6.32 & 6.41].

.

4.4 The two-dimensional, general case

205

5

= P Ωε + E 1

ε,x

ε,α

0 ,α 0 Xc T (ε) ∈G c T (ε)

Fc T (ε)

1−E

1 τ

5

≡ P Ωε + E 1

ε,x

ε,α

Fc T (ε)

1−E (∗∗)

= P Ω

ε

·

ε,x

ε,α

0 0 Xc T (ε)+• , αc T (ε)+•

0 0 Xc T (ε) ,αc T (ε) ∈G

6

< (δ−c) T (ε)

·

D ◦ (X, α)ε,(x0 ,α0 ) H c T (ε)+•

6

5 + E 1 -

ε,x

ε,α

0 ,α 0 Xc T (ε) ∈G c T (ε)

·

ε,p 1 − E 1{τ <(δ−c)T (ε)} ◦ (X, α)•

.6 ε,(x ,α0 )

0 p=(X,α)cT (ε)

+ ≤ P Ωε + E 1 ε,x0 ε,α0 · Xc T (ε) ,αc T (ε) ∈G , 1 − inf E 1{τ <(δ−c)T (ε)} ◦ (X, α)ε,p • p∈G + = P Ωε + E 1 ε,x0 ε,α0 · Xc T (ε) ,αc T (ε) ∈G , ˆ α 1 − inf E 1{τ <(δ−c)T (ε)} ◦ (X, ˆ )ε,p • p∈G + , ε ε,p · sup P τG ≥ (δ − c)T (ε) ≡ P Ω + E 1 ε,x0 ε,α0

ε

ε

≤ P Ω ≤ P Ω

Xc T (ε) ,αc T (ε) ∈G

+ sup P p∈G

ε,p τG

p∈G

≥ (k(ε) + 1)T¯

ε,p + sup P τG > k(ε)T¯

< P Ωε +

p∈G

1 − exp

−

0 Imax (T¯ ) + η˜ ε

k(ε) ,

where the ﬁnal estimate is due to the inductive consequence of 3.2.1 deduced in step 3). As had been mentioned previously, the ﬁrst summand of this upper bound tends to zero as ε → 0 due to corollary 2.5.8. As k(ε) ! eζ/ε and since assumption (4.25) assured that the parameters could be chosen 0 such that ζ > Imax (T¯ ) + η˜ in step 2), it follows immediately from step 5) in the proof of theorem 4.4.6 that also the second summand converges to zero as ε → 0 .

206

4 Local Lyapunov exponents

The preceding theorems now enable us to obtain the local Lyapunov exponent also in the case that Λ1 Kµ(x0 ,ζ) > Λ2 Kµ(x0 ,ζ) , for the two eigenvalues of A Kµ(x0 ,ζ) . Note that for the case that Λ1 Kµ(x0 ,ζ) = Λ2 Kµ(x0 ,ζ) , the existence of the local Lyapunov exponent had already been proven in corollary 4.1.3 . The following theorem containing the convergence to the local Lyapunov exponent in the strongly hypoelliptic case can be considered as the main result of this book. Theorem 4.4.8 (Local Lyapunov exponent). Fix an initial value (x0 , z0 ) l > of (X ε , Z ε ), x0 ∈ Di , z0 ∈ Rn \ {0} , and let the situation either of i=1

theorem 4.4.6 or of theorem 4.4.7 be given. Then the stochastic process Z ε deﬁned by (1) has the following exponential growth rate on the time scale T (ε): 1 P log ZTε (ε) ( . , x0 , z0 ) −−− −→ Λ1 Kµ(x0 ,ζ) , ε→0 T (ε) where Kµ(x0 ,ζ) denotes the metastable state corresponding to x0 and ζ as given by deﬁnition 2.5.4 andthe exposition preceding it. This stochastic limit Λ1 Kµ(x0 ,ζ) shall be called the local Lyapunov exponent of Z ε with respect to x0 , ζ and z0 . 0 Proof. Again, let (αε,α )t≥0 denote the angle process corresponding to t ε,z0 )t≥0 , where of course α0 is the angle of z0 . Applying theorem 2.5.6 (Z together with either theorem 4.4.6 or theorem 4.4.7 yields for the process 0 ) that for any c > 0, (Xtε,x0 , αε,α t

1 0 L t ∈ [0, T (ε)] : (Xtε,x0 , αε,α ) ∈ B Kµ(x0 ,ζ) , c × B A1 (Kµ(x0 ,ζ) ), c t T (ε) P −− −−→ 1 ; ε→0

in other words, one obtains for any c > 0 that P L Tεc ( . ) −−− −→ 1 , ε→0

4.4 The two-dimensional, general case

where

207

ε,x0 ε,α0 Tεc (ω) := t ∈ [0, 1] : Xt·T (ω) , α (ω) (ε) t·T (ε) ∈ B Kµ(x0 ,ζ) , c × B A1 (Kµ(x0 ,ζ) ), c .

It needs to be proven that this implies that 1 T (ε)

0

T (ε)

P 0 ¯ Xtε,x0 , αε,α dt −−− Q −→ Λ1 Kµ(x0 ,ζ) , t ε→0

since the convergence claimed above then follows from (1.8): 1 log |z0 | 1 log |ZTε (ε) | = + T (ε) T (ε) T (ε)

0

T (ε)

P 0 ¯ Xtε,x0 , αε,α dt − Λ1 Kµ(x0 ,ζ) . Q → t ε→0

Deﬁne Kµ,i :=

≡ Kµ(x0 ,ζ) , αi ∈ Rd+1 Kµ(x0 ,ζ) , Ai Kµ(x0 ,ζ)

for i = 1, 2. Since by construction, ¯ µ,i ) ≡ Q ¯ Kµ(x ,ζ) , Ai Kµ(x ,ζ) Q(K = Λi Kµ(x0 ,ζ) 0 0

(i = 1, 2),

we have to show that T (ε) 1 P 0 ¯ Xtε,x0 , αε,α ¯ µ,1 ) dt −−− Q −→ Q(K t ε→0 T (ε) 0 in this notation. A statement of this type without the angle coordinate has already been veriﬁed in theorem 2.5.10; the same rationale now also applies here. For an arbitrary η > 0 one gets that P

; 1 T (ε) ¯ (Kµ,1 ) > η ¯ X ε,x0 , αε,α0 dt − Q Q t t T (ε) 0 1 ¯ (Kµ,1 ) > η ¯ X ε,x0 ( . ) , αε,α0 ( . ) dt − Q = P Q t·T (ε) t·T (ε) 0 1

≤P

0

≤ P Ωε

¯ ε,x0 ε,α0 ¯ (Kµ,1 ) dt > η (.) − Q Q Xt·T (ε) ( . ) , αt·T (ε)

1

¯ ε,x0 ε,α0 ¯ (Kµ,1 ) dt > η (.) − Q , + P Ωε ∩ Q Xt·T (ε) ( . ) , αt·T (ε) 0 AB C @ =:B ε

where the set Ωε is taken as ε,x0 Ωε := Ωε1;F ≡ ω ∈ Ω : XtT (ω) ∈ F for all t ∈ [0, 1] , x ,ζ 0 (ε)

208

4 Local Lyapunov exponents

for which it is known from corollary 2.5.7 (which holds true in either of the posed situations) that lim P Ωε = 0 . ε→0

ε→0

Therefore, it needs to be shown that B ε −−−→ 0 , which is done in the sequel: For estimating B ε , we ﬁx ω ∈ Ωε and c > 0 for which we get

¯ ε,x0 ε,α0 ¯ (Kµ,1 ) dt (ω) − Q Q Xt·T (ε) (ω) , αt·T (ε) 0 ¯ ε,x0 ε,α0 ¯ (Kµ,1 ) dt = (ω) − Q Q Xt·T (ε) (ω) , αt·T (ε) 1

[0,1]\Tε c (ω)

+

¯ ε,x0 ε,α0 ¯ (Kµ,1 ) dt (ω) − Q Q Xt·T (ε) (ω) , αt·T (ε)

Tε c (ω)

¯ ¯ (Kµ,1 ) : (x, α) ∈ Fx ,ζ × [0, 2π] α) − Q ≤ L [0, 1] \ Tεc (ω) · max Q(x, 0 ¯ ¯ (Kµ,1 ) : (x, α) ∈ B(Kµ(x,ζ) , c) × B(A1 (Kµ(x,ζ) ), c) α) − Q + max Q(x,

¯ and its periodicity in the angle varidue to the continuity of the function Q able, as well as due to the deﬁning properties of the sets Ωε and Tεc (ω). Now if we choose c suﬃciently small such that the latter summand is bounded by ¯ is continuous), η/2 (Q ¯ α) − Q ¯ (Kµ,1 ) : (x, α) ∈ B(Kµ(x,ζ) , c) × B(A1 (Kµ(x,ζ) ), c) < η max Q(x, 2 and deﬁne

¯ α) − Q ¯ (Kµ,1 ) : (x, α) ∈ Fx0 ,ζ × [0, 2π] , m := max Q(x,

then it follows from the previously obtained estimate that 1

¯ ε,x0 ε,α0 ε ε ¯ ≡ P Ω ∩ B Q Xt·T (ε) ( . ) , αt·T (ε) ( . ) − Q (Kµ,1 ) dt > η 0 η ε ε > η ≤ P ω ∈ Ω : m · L [0, 1] \ Tc (ω) + 2 η ≤ P L [0, 1] \ Tεc ( . ) > 2m ε→0

−−−→ 0 , since the stochastic convergence stated in the beginning of this proof says in P other words that L [0, 1] \ Tεc ( . ) − → 0 as ε → 0.

4.4 The two-dimensional, general case

209

This theorem and its proof provide the counterpart to the FurstenbergKhasminskii formula (1.17) in the situation that X ε has sublimiting distributions on time scales: Under the prescribed assumptions, we obtained ¯ α) δKµ,1 (dx, dα) = Λ1 Kµ(x ,ζ) as the limit in the subQ(x, d 0 R ×[0,π) limiting situation; hence, we consider the above result as a sublimiting Furstenberg-Khasminskii formula. Further note the analogy with the law of large numbers (1.16): The statement of (1.16) is often described by saying that “one only observes the top Lyapunov exponent Λε1 in the hypoelliptic case”. In contrast, our ﬁndings of theorem 4.4.8 can be paraphrased by stating that “one only observes the top eigenvalue Λ1 (Kµ(x0 ,ζ) ) in the sublimiting, strongly hypoelliptic, strictly separated case”. In the following the assumptions of this section are going to be discussed. Remark 4.4.9 (Strictly separated switching curves entail distinct eigenvalues (4.23)). As in the previous theorems, consider a coeﬃcient matrix A ∈ C ∞ (Rd , R2×2 ) of (1) and the thus deﬁned angle velocity function ¯ h(A(x), α) ≡ − a12 (x) sin2 α + a21 (x) cos2 α + (a22 (x)−a11 (x)) sin α cos α . ¯ has strictly separated switching curves A1 and A2 on If it is assumed that h M := Fx0 ,ζ (theorem 4.4.6) or on M := Hx0 ,ζ (theorem 4.4.7), respectively, it follows in particular that the characterizing inequality (4.23) of this section (see p.177) is automatically satisﬁed; moreover, it does not only hold at Kµ(x0 ,ζ) , but on all of M, Λ1 (x) > Λ2 (x)

for all x ∈ M ;

the reason is that due to deﬁnitions 4.4.3 and 1.2.4 there are exactly two switching curves: A1 (A(x)) is the attracting and A2 (A(x)) is the repelling branch of arctan ξ1,2 (A(x)) , where ξ1,2 (A(x)) =

(a22 − a11 )2 + 4a12 a21 , 2a12 0 and (a22 − a11 )2 > −4a12 a21 , =

(a22 − a11 ) ± if a12

and ξ1 (A(x)) = ∞ , ξ2 (A(x)) =

a21 , if a12 = 0 and a11 = a22 . a11 − a22

The eigenvalues of A(x) are calculated as Λ1,2 (x) =

1 1 0 (a11 + a22 ) ± (a11 − a22 )2 + 4a12 a21 2

and are distinct in either of the two cases for the matrix A(x).

210

4 Local Lyapunov exponents

Next, we would like to take a closer look at the assumptions of theorems 4.4.6 and 4.4.7 in order to discuss criteria under which the conditions of theorem 4.4.8 are valid. First, we pose the question in which situation strict separatedness of the switching curves A1 and A2 can be read oﬀ the matrix function A. As Kliemann [Kl 80, p.150–153] and [Kl 79, p.467f.] shows, there are two “strong ¯ has exactly two switching control sets” for the angle motion in case that h curves and a12 is constant. We translate Kliemann’s ﬁndings to our situation for which we assume for convenience that a12 ( . )

and

a21 ( . )

are both constants, again denoted by a12 and a21 , respectively, which also satisfy that a12 = 0 ,

a21 = 0

and

sign(a12 ) = sign(a21 ) .

The latter line of conditions is e.g. fulﬁlled in the symmetric case a12 = a21 = 0 ; furthermore note that the case of constant entries a12 = a21 = 0 obviously yields strictly separated switching curves 0 and π2 and had been solved in section 4.3 . Remark 4.4.10 (Strictly separated switching curves of the angle ¯ in (1)). Let A ∈ C ∞ (Rd , R2×2 ) be the coeﬃcient matrix of (1) and drift h suppose that its secondary diagonal elements a12 and a21 are both constant such that a12 = 0 ,

a21 = 0

and

sign(a12 ) = sign(a21 ) .

¯ are strictly separated on any compact set. Then the switching curves of h Proof. As follows from deﬁnition 1.2.4 (the relevant part being recalled above in remark 4.4.9) under these assumptions there are exactly two switching surfaces represented by the zeros 2 [a22 (x) − a11 (x)] ± [a22 (x) − a11 (x)]2 + 4a12 a21 ξ1,2 (A(x)) = 2a12 1 −b(x) ± b(x)2 + a = a12 of the drift of tan αεt , where b(x) :=

a11 (x) − a22 (x) 2

and

a := a12 a21 > 0 .

4.4 The two-dimensional, general case

211

Now deﬁne ξ1,2 (A(x)) := a12 · ξ1,2 (A(x)) ≡ −b(x) ± b(x)2 + a , where the positive root is deﬁned as ξ1 and the negative root is deﬁned as ξ2 ; note that this enumeration does not necessarily correspond to the one of A1 (attracting) and A2 (repelling). Furthermore, let M ⊂ Rd be some compact set as for example M := Fx0 ,ζ in theorem 4.4.6 or M := Hx0 ,ζ in theorem 4.4.7, respectively. We have to prove that inf ξ1 (A(x)) > sup ξ2 (A(x)) ,

x∈M

x∈M

since it then follows from the continuity of arctan on R that ¯h has strictly separated switching surfaces A1and A2 on M in the sense of deﬁnition 4.4.3. However, since a > 0, |b(x)| < b(x)2 + a and therefore ξ1 (A(x)) ≡ −b(x) + b(x)2 + a >0 > −b(x) − b(x)2 + a ≡ ξ2 (A(x)) ; the strict inequality asserted above now follows from the continuity of ξ1 and ξ2 as well as from the compactness of M. Remark 4.4.11. In the light of the previous remark 4.4.10 one is tempted to look for generic examples of theorem 4.4.8 in the class of coeﬃcient matrices A ∈ C ∞ (Rd , R2×2 ) of (1) with constant nonzero secondary diagonal elements a12 and a21 of the same sign. However, suppose that some coeﬃcient matrix function A( . ) with constant elements a12 , a21 ∈ R is given. Then it follows from remark 1.2.1 that the angle-drift ¯ h is calculated as ¯ α) = h ¯ A(x), α h(x, = − a12 sin2 α + a21 cos2 α + a22 (x) − a11 (x) sin α cos α 1 a22 (x) − a11 (x) sin 2α = − a12 sin2 α + a21 cos2 α + 2 implying that ¯ α) 1 ∂ h(x, = ∂xj 2

∂ a11 (x) ∂ a22 (x) − ∂xj ∂xj

sin 2α

(j = 1, . . . , d),

212

4 Local Lyapunov exponents

since a12 and a21 do not depend on x. As this term vanishes for α ∈ ¯ cannot be strongly hypoelliptic on any set M k π2 : k ∈ Z it follows that h in the sense of deﬁnition 4.4.4 . In other words, the assumption of the previous remark 4.4.10 that A ∈ C ∞ (Rd , R2×2 ) shall have constant (nonzero) secondary diagonal elements a12 and a21 (of the same sign) prevents the set of conditions in the theorems 4.4.6 and 4.4.7 and in theorem 4.4.8, respectively, from being satisﬁed. What would remain to be done here in order to obtain such a result nevertheless is ¯ outside the to truncate and appropriately redeﬁne the coeﬃcient function h D D set M × A2 (M) in the proof of 4.4.6 and 4.4.7 instead of only redeﬁning it D × R , where M D denotes the respective set F D x ,ζ or H D x ,ζ ; see the outside M 0 0 beginnings of the steps 1) in the proofs of 4.4.6 and 4.4.7, respectively. Note in particular that the above class of coeﬃcient matrices of (1) with constant nonzero secondary diagonal elements of the same sign contains the linearized gradient SDEs, A = −HU , as discussed in 1.5.3 with potential functions U ∈ C ∞ (R2 , R) for which there is a constant c such that ∂U (x) = c = 0 ∂x1 ∂x2

(for all x ∈ R2 )

due to the symmetry of the Hesse matrix. An example for this situation is given by the potential function U1 (x) =

3 4 2 3 x − x31 − 3 x21 + c x1 x2 + x42 2 1 3 2

(x ∈ R2 )

as deﬁned in (1.34) in case that c = 0; again see ﬁgure 2.1 for c = 1. Here, the coeﬃcient matrix of the linearized system had been calculated as

−c − 18x21 + 4 x1 + 6 (x ∈ R2 ) . A1 (x) ≡ − HU1 (x) = −c − 18 x22 The reason for concentrating on the case of nonzero secondary elements above is that if the matrix has constant zero secondary elements, a12 = a21 = 0, then our ﬁndings of section 4.3 already apply. The target of the following remark is to prove that in case that b derives from a potential function the conditions (4.24) and (4.25) are redundant given the other assumptions of theorems 4.4.6 and 4.4.7 (and hence also of theorem 4.4.8), respectively. If the underlying set of assumptions is changed, e.g. if the switching curves are not supposed to be strictly separated, the arguments can be suitably modiﬁed. Remark 4.4.12 (on the assumptions (4.24) and (4.25)). As before let the function A ∈ C ∞ (Rd , R2×2 ) be the coeﬃcient matrix of the SDE (1) and ¯ of the angle process αε of Z ε in the again suppose that the drift function h RDE (1.6), ¯ ε , αε ) , α˙ εt = h(X αε0 = α0 , t t

4.4 The two-dimensional, general case

213

¯ is has strictly separated switching surfaces A1 and A2 on M and that h d strongly hypoelliptic on M, where the compact set M ⊂ R either represents >l M := Fx0 ,ζ (theorem 4.4.6) or M := Hx0 ,ζ (theorem 4.4.7), x0 ∈ i=1 Di being the initial value of X ε in SDE (2.1) (under the assumptions 2.1.1) and ζ > 0 parametrizing the time scale T (ε) ! eζ/ε . Additionally assume for the diﬀusion X ε that its SDE (2.1) is of gradient form (2.2), √ dXtε,x0 = − ∇U (Xtε,x0 ) dt + ε dWt , X0ε,x0 = x0 ∈ Rd , i.e. that the drift b = −∇U is given by a potential U ∈ C ∞ (Rd , R), and where σ = idRd . Then the conditions (4.24) and (4.25) are already satisﬁed, i.e. there exists T¯ > 0 such that 0 (0; T¯ ; x, α) : (x, α) ∈ M × A2 (M) , ζ > max ID,M 0 where D ⊃ A2 (M) denotes some open set and where the terms ID,F x0 ,ζ 0 ε (0; T ; x, α) and ID,Hx ,ζ (0; T ; x, α) are the value functions of α as deﬁned in 0 4.4.5 corresponding to the respective choice of M ∈ Fx0 ,ζ , Hx0 ,ζ .

Proof. As in the proofs of theorems 4.4.6 and 4.4.7 let D ⊂ R be some open set such that A2 (M) ⊂ D and D ∩ A1 (M) = ∅ ; due to the latter two properties, the separatedness of the switching curves A1 and A2 and the π¯ α) it follows that L(D) < π . This set D will be ﬁxed periodicity of α → h(x, throughout the following argument and can also be used in the proofs of the previous theorems. According to deﬁnition 4.4.5 it is to be proven that there are T¯ > 0 and η ∈ (0, ζ) such that for all (x, α) ∈ M × A2 (M), 0 ID,M (0; T¯ ; x, α) ≡

ϑD (0;x,α;u)

inf

u∈UD (0;T¯ ;x,α)∩{u• ∈M}

K u(t), u(t) ˙ dt < ζ −η;

0

unwinding all relevant notation, the latter assertion means that there exists some function u ∈ C 1 (R+ , Rd ) taking all its values in M (which is abbreviated as {u• ∈ M}), starting in u(0) = x such that ϑD (0; x, α; u) ≤ T¯ , where

ϑ ≡ ϑD (0; x, α; u) ≡ inf t ≥ 0 : α0 (t) ∈ /D

denotes the ﬁrst exit time of the solution α0 of the controlled initial value problem d 0 ¯ α (t) = h(u(t), α0 (t)) , α0 (0) = α , dt

214

4 Local Lyapunov exponents

from D and such that

ϑ

K u(t), u(t) ˙ dt < ζ − η ,

0

where K(q1 , q2 ) =

1 2 | q2 + ∇U (q1 ) | , 2

due to the special gradient form (2.2) of the SDE (2.1). For proving this claim we will use the following notation: Given two continuous functions Pi : [0, Zi ] −→ Rd

(i ∈ {1, 2})

(“paths”), where Z1 , Z2 ≥ 0 are two time horizons and where P1 (Z1 ) = P2 (0), we deﬁne the juxtaposition of P1 and P2 as the continuous function , t ∈ [0, Z1 ], P1 (t) d P1 P2 : [0, Z1 +Z2 ] −→ R , (P1 P2 )(t) := P2 (t − Z1 ) , t ∈ [Z1 , Z1 + Z2 ]. The proof is now organized in six steps. Firstly, after ﬁxing the relevant notation we construct an absolutely continuous path, called g2x , which connects x with one of the local minima Ki of U such that the “cost function” K(g2x (t), g2x (t))dt can be reasonably estimated. In the second step this function g2x is used as a control input for the angle system. Two cases now arise: In the ﬁrst situation the resulting angle, called α , is bounded away from the ¯ can be uniunstable equilibrium A2 (Ki ) and hence the velocity function h formly bounded away from 0 which forces the controlled angle process to leave D; this is elaborated in step 3). In the second case α can be arbitrarily close ¯ can become arbitrarily small; this situato A2 (Ki ) and hence the function h tion is coped with in 4). The ﬁfth step concludes from 3) and 4) that we have altogether constructed a control input g which has all the desired properties with the exception that g is absolutely continuous instead of being continuously diﬀerentiable. The ﬁnal step then explains that the latter drawback can be overcome by means of a Sobolev argument. 1) According to the set of assumptions 2.1.1, there are ﬁnitely many points K1 , . . . , Kl which are the local minima of U in the gradient case and points Kl+1 , . . . , Kl which are the saddles and the local maxima of U (see remark 2.1.2 on (K) ). Let k ∈ N denote the number of all critical points {K1 , . . . , Kl } in M and let k ≤ k denote the number of local minima {K1 , . . . , Kl } in M ; we relabel these points such that {K1 , . . . , Kk } are the local minima in M and such that {Kk+1 , . . . , Kk } are all the other critical points in M .

4.4 The two-dimensional, general case

215

Let U ⊂ Rd denote the union of all those lower dimensional subspaces which are attracted to {Kk+1 , . . . , Kk } under the deterministic system (2.8), X˙ 0 = −∇U (X 0 ); further deﬁne Uκ := M ∩ v ∈ Rd : dist(v, U) < κ (κ > 0) , the set of points in M whose distance to U is at most κ, where the parameter κ > 0 will be ﬁxed later; see step 5). Fix ρ > 0 suﬃciently small such that B(Ki , ρ) ⊂ M\Uκ for all i = 1, . . . , k , where B(Ki , ρ) again denotes the open ball around Ki with radius ρ. Then there is a constant T1 ≡ T1 (ρ) < ∞ such that for all x ∈ M \ Uκ , : B(Ki , ρ) (t ≥ T1 ) , f1x (t) := Xt0,x ∈ i=1,...,k

where as usual X 0,x denotes the solution of the deterministic diﬀerential equation (ε = 0) starting in x. As in the proof of lemma 2.4.9, this follows since all trajectories starting in M \ Uκ are attracted towards a unique element of {K1 , . . . , Kk }; the universality of the constant T1 follows from the Gronwall argument (b ≡ −∇U > is Lipschitz on the compact set M \ Uκ ) and the compactness of (M \ Uκ ) \ i=1,...,k B(Ki , ρ). Moreover, if x ∈ Uκ , let y ≡ y(x) denote its proximum in M \ Uκ , i.e. a (not necessarily unique) element y ∈ M \ Uκ such that dist(x, M \ Uκ ) = |y − x| =: T0 (x) ≤ κ ; such an element y exists due to the compactness of M \ Uκ ; in particular, y(x) the previous discussion yields a well-deﬁned function f1 , since y ≡ y(x) ∈ M \ Uκ . Moreover, for f0x (t) := x + t

y−x , |y − x|

it follows that f0x (0) = x , f0x (T0 (x)) = y and 0

T0 (x)

K f0x (t), f˙0x (t) dt ≡ I[0,T0 (x)],x (f0x ) 1 ≡ 2

0

T0 (x)

2 ˙x f0 (t) + ∇U (f0x (t)) dt

2 1 1 + max |∇U | ≤ T0 (x) ¯κ 2 U ≤ LM κ , where LM :=

2 1 1 + max |∇U | M 2

216

4 Local Lyapunov exponents

and where I denotes the rate function of X ε ; see theorem 2.3.4 . This argument is an adapted and corrected version of Lemma 4.2.3 by Freidlin and Wentzell [Fr-We 98, p.112]. Now juxtapose f0x and f1y , if necessary; more precisely, deﬁne y(x) , if x ∈ Uκ , f0x f1 x d x g1 := g1 : [0, T1 (ρ) + κ] −→ R , x f1 , if x ∈ M \ Uκ ; y(x)

note that formally f0x has been deﬁned till time T0 (x) ≤ κ ; therefore, f1 is supposed to have the domain [0, T1 (ρ) + κ − T0 (x)] in order to make the juxtaposition well-deﬁned; furthermore, f1x is taken as deﬁned on [0, T1 (ρ) + κ] ; since both the latter intervals contain [0, T1 (ρ)] , g1x hits one of the balls y(x) B(Ki , ρ) by deﬁnition of f1x and f1 . All in all, for any x ∈ M we have a piecewise diﬀerentiable (absolutely continuous, in particular) path g1x which is deﬁned on [0, κ + T1 (ρ)] such that : B(Ki , ρ) g1x κ + T1 (ρ) ∈ i=1,...,k

and 0

κ+T1 (ρ)

K (g1x (t), g˙ 1x (t)) dt ≡ I[0,κ+T1 (ρ)],x (g1x ) ≤ I[0,T0 (x)],x (f0x ) + 0 ≤ LM κ ; y(x)

are trajectories of the this estimate holds true, since the paths f1x and f1 deterministic dynamical system X 0 and hence do not contribute to the rate function. By (2.9) of lemma 2.4.7, applied with O := Ki for any i ∈ {1, . . . , k} , there exists some absolutely continuous function f2 : [0, 1] → Rd such that f2 (0) = g1x κ + T1 (ρ) , f2 (1) = Ki and I[0,1],f2 (0) < δ ; here, δ > 0 denotes an arbitrary parameter only determining the speciﬁc choice of ρ. Just as κ, we will choose δ (and hence ρ in other words) in step 5). Now we prolong g1x by means of f2 . For this purpose deﬁne the absolutely continuous function g2x := g1x f2 : [0, κ + T1 (ρ) + 1] −→ Rd . Then g2x (0) = x , g2x κ + T1 (ρ) + 1 = Ki for some i ∈ {1, . . . , k} and (again due to the additivity of I)

4.4 The two-dimensional, general case

κ+T1 (ρ)+1 0

217

K g2x (t), g˙ 2x (t) dt ≡ I[0,κ+T1 (ρ)+1],x (g2x ) ≤ LM κ + δ .

2) Using g2x as input function for the controlled initial value problem under investigation, d 0 ¯ g x (t), α0 (t) , α (t) = h 2 dt

α0 (0) = α ,

this system reaches some angle α := α0 (κ + T1 (ρ) + 1) ∈ [0, 2π) within the prescribed time interval [0, κ + T1 (ρ) + 1]. We need to distinguish two cases: Firstly, we will consider the situation in which α has some pre¯ i , . ) is bounded scribed distance to A2 (Ki ); this is the easier case, since h(K away from 0 here. Secondly, we investigate the more intricate situation that ¯ i , . ) has its (unstable) zero. More precisely, α is “near” A2 (Ki ), where h(K let γ > 0 be some small parameter. The ﬁrst case is deﬁned by the condition that α ∈ D \ A2 (Ki ) − γ, A2 (Ki ) + γ which will be considered in step 3). The second case is given by α ∈ (A2 (Ki ) − γ, A2 (Ki ) + γ) which is the object of step 4). 3) Assume that α ∈ D \ A2 (Ki ) − γ, A2 (Ki ) + γ . ¯ i , α) Since A2 is the repelling switching surface, the mapping α → h(K changes its sign from “−” to “+” at A2 (Ki ); see deﬁnition 1.2.4. Since furthermore D ∩ A1 (M) = ∅ (see the beginning of this proof) and A1 and A2 ¯ on any interval of length π, it follows that describe all the zeros of h ¯ i , α) > 0 max h(K {α∈D : α ≥ A2 (Ki )+γ }

and

¯ i , α) < 0 . min h(K {α∈D : α ≤ A2 (Ki )−γ }

Due to the compactness of α ∈ D : α ≥ A2 (Ki ) + γ ∪ α ∈ D : α ≤ A2 (Ki ) − T2 ≡ T2 (γ) > 0 such that for any γ}, there is hence some α ∈ D \ A2 (Ki ) − γ, A2 (Ki ) + γ , the system d 0 ¯ f3 (t), α0 (t) , α (t) = h dt

α0 (0) = α

218

4 Local Lyapunov exponents

exits D within time T2 , where f3 (t) := Ki for all t ≥ 0. Applying this fact to the original system by considering the control function g3x := g2x f3 : [0, κ + T1 (ρ) + 1 + T2 (γ)] −→ Rd , then

α0 κ + T1 (ρ) + 1 + T2 (γ) ∈ D

for the system α0 with control input g3x . In particular, ϑD (0; x, α; g3x ) ≤ T¯1 , where

T¯1 ≡ T¯1 κ, ρ, γ := κ + T1 (ρ) + 1 + T2 (γ) .

Hereby we have 0

T¯1

K g3x (t), g˙ 3x (t) dt ≡ I[0,κ+T1 (ρ)+1+T2 (γ)],x(g3x ) ≤ LM κ + δ ,

since f3 remains in the equilibrium Ki . 4) Now we concentrate on the second case that α ∈ (A2 (Ki ) − γ, A2 (Ki ) + γ) . ¯ is strongly hypoelliptic on M meaning that By assumption the function h

¯ ∂ h(x, α) = 1 rank ∂xj 1≤j≤d for all (x, α) ∈ M × R due to deﬁnition 4.4.4. In particular, applying this property at x = Ki , it follows that it is possible to choose some v1 ≡ vi1 ∈ B(Ki , ρ) \ {Ki } such that ¯ r1 (t) > 0 for all t ∈ (0, |v1 − Ki |] , h where r1 (t) := Ki + t and such that

v1 − Ki |v1 − Ki |

(t ∈ [0, |v1 − Ki |] )

¯ h Xt0,v1 > 0 for all t ≥ 0 .

Note that for the last assertion the gradient assumption b = −∇U is needed which implies that there are no rotations of X 0 around Ki . Correspondingly, it follows that it is possible to choose some v2 ≡ vi2 ∈ B(Ki , ρ) \ {Ki } such

4.4 The two-dimensional, general case

that

219

¯ r2 (t) < 0 for all t ∈ (0, |v2 − Ki |] , h

where r2 (t) := Ki + t and such that

v2 − Ki |v2 − Ki |

(t ∈ [0, |v2 − Ki |] )

¯ h Xt0,v2 < 0 for all t ≥ 0 .

The points v1 ≡ vi1 and v2 ≡ vi2 will be chosen suﬃciently close to Ki for all i ∈ {1, . . . , k} in step 5). Consider the juxtaposition rj X 0,vj for j ∈ {1, 2} . By comparison it follows that these two input functions achieve to force the angle to leave D in some ﬁnite time T3 > 0 when started at α = A2 (Ki ) (“worst case scenario”). Again by comparison and choosing r1 X 0,v1 or r2 X 0,v2 as control function (depending on α ≥ A2 (Ki ) or α < A2 (Ki ) ), this implies that for any α ∈ A2 (Ki ) − γ, A2 (Ki ) + γ , there is some control function called r3 , r3 ∈ r1 X 0,v1 , r2 X 0,v2 , such that the controlled system d 0 ¯ r3 (t), α0 (t) , α (t) = h dt

α0 (0) = α

has left D at the time T3 > 0 and the corresponding “costs” can be estimated by the previous argument and the additivity of I as 0

T3

K r3 (t), r˙3 (t) dt ≡ I[0,T3 ],Ki (r3 ) (j ∈ {1, 2}) = I[0,|vj −Ki |],Ki (rj ) |vj −Ki | 1 2 ≡ | r˙j (t) + ∇U (rj (t)) | dt 2 0 .2 1 1 + max |∇U | ≤ |vj − Ki | 2 B(Ki ,ρ) ≤ LM |vj − Ki | .

g2x

Juxtaposing the paths g2x and r3 yields an absolutely continuous path g4x := r3 on the time interval [0, T¯2 ], where

220

4 Local Lyapunov exponents

T¯2 ≡ T¯2 κ, ρ, γ, (vij ) := κ + 1 + T1 (ρ) +

max

i=1,...,k ; j=1,2

|vij − Ki | ;

here, the dependence on γ is implicitly given via the distinction of cases in the second step; furthermore, note that formally the domain of r3 is not bounded; this deﬁnition of g4x by juxtaposition is hence to be understood in the sense that r3 ≡ rj X 0,vj follows the trajectory X 0,vj for a suﬃciently long time such that g4x is well-deﬁned on [0, T¯2 ]. By the previous construction it follows that ϑD (0; x, α; g x ) ≤ T¯2 4

for the controlled initial value problem d 0 ¯ g x (t), α0 (t) , α (t) = h 4 dt

α0 (0) = α .

The associated “costs” are T¯2 K g4x (t), g˙ 4x (t) dt ≡ I[0,T¯2 ],x (g4x ) 0

= I[0,κ+T1 (ρ)+1],x (g2x ) + I[0,T3 ],Ki (r3 ) ≤ LM κ + δ + LM

max

i=1,...,k ; j=1,2

|vij − Ki | .

5) Using the deﬁnitions and results of the previous two cases 3) and 4) we can deﬁne T¯ ≡ T¯ κ, ρ, γ, (vij ) := max T¯1 , T¯2

and g :=

g3x g4x

in case 3) in case 4)

for which ϑD (0; x, α; g) ≤ T¯ by deﬁnition and the following “worst case estimate” holds

ϑD (0;x,α;g)

K g(t), g(t) ˙ dt

-

0

≤ max

T¯1

K

0

≤ L M κ + δ + LM

g3x (t), g˙ 3x (t) max

dt ,

i=1,...,k ; j=1,2

T¯2

K

0

g4x (t), g˙ 4x (t)

. dt

|vij − Ki |

<ζ −η , where the latter inequality can be accomplished by choosing κ, δ and |vj −Ki | suﬃciently small.

4.4 The two-dimensional, general case

221

More precisely, the last estimate shows the following. We have constructed an absolutely continuous control function g such that g has all necessary properties mentioned at the beginning of this proof (where “u” is replaced by “g”) except being an element of C 1 ([0, T¯], Rd ) ; in particular, the above ϑ estimate shows that g can be chosen such that 0 D K g(t), g(t) ˙ dt is bounded by any positive number; due to our requirements we ﬁxed this number as ζ−η, but this choice is arbitrary, since ζ − η had not been used at any place in the argument before. In other words, this arguments shows that in the gradient case (2.2) for the underlying diﬀusion X ε the calculation of the value function is “decoupled” from the time scales of X ε . 6) It is now left to prove that the previous absolutely continuous (piecewise diﬀerentiable) function g can be replaced by a control function u ∈ C 1 such that the previous properties of g carry over to u. Now g is an element of the Sobolev space W 1,2 (0, T¯; Rd ) with the norm g2W 1,2 :=

T¯

|g(t)|2 + |g(t)| ˙ 2

dt ;

0

see Evans [Ev 02, p.286]. Therefore, g can be approximated even by C ∞ functions un which are dense in all the Sobolev spaces. This approximation ¯ is diﬀerentiable maintains that the exit of α0 from D occurs before T¯ , since h 0 and the solution α then depends continuously on the right hand side of its ODE. Furthermore, the approximation of g by un with respect to . 2W 1,2 also implies that the “distance” with respect to the cost function I[0,T¯] of un to g converges to zero; more precisely, using the auxiliary functions An (t) := u˙ n (t) + ∇U (un (t))

and

B(t) := g(t) ˙ + ∇U (g(t))

we get ¯ 2 2 1 T I[0,T¯] (un ) − I[0,T¯] (g) = ˙ + ∇U (g(t)) dt u˙ n (t) + ∇U (un (t)) − g(t) 2 0 T¯ 2 1 n 2 A (t) − B(t) dt ≡ 0 2 T¯ 1 n An (t) − B(t) , An (t) − B(t) = A (t) − B(t) , B(t) + dt 2 0 2 1 n A − B L2 ([0,T¯],Rd ) ; = An − B , B + 2 d ¯ 2 L ([0,T ],R ) such an estimate had already been used in the proof of theorem 2.4.3; also see Freidlin and Wentzell [Fr-We 98, p.89]. However, note that the convergence in the Sobolev space, n→∞

un − g2W 1,2 −−−−→ 0 ,

222

4 Local Lyapunov exponents n→∞

also implies that An − B −−−−→ 0 in L2 ([0, T¯], Rd ), since ∇U is Lipschitz continuous on any compact set containing g and (un )n . Hence, the claim is proven by taking u := un for some suﬃciently large n. Now we return to our motivating examples 1.5.3 and 1.5.4 for which we formulate our ﬁndings of theorems 4.4.6, 4.4.7 and 4.4.8 by means of remark 4.4.12: Example 4.4.13 (Linearized gradient SDE). Consider the SDE (2.2) √ dXtε = − ∇U (Xtε ) dt + ε dWt in dimension d = 2, where U ∈ C ∞ (R2 , R). The linearized system (1.33) is then given by dZtε = − HU Xtε Ztε dt in dimension n = d = 2, where the Hesse matrix ⎞ ⎛ ∂2 U ∂2 U (x) (x)⎟ ⎜ ⎟ ⎜ ∂x21 ∂x1 ∂x2 HU (x) ≡ ⎜ ⎟ 2 ⎠ ⎝ ∂2 U ∂ U (x) (x) 2 ∂x1 ∂x2 ∂x2 determines the coeﬃcient matrix A(x) ≡ −HU (x) in the system (1). Suppose that the >l drift b := −∇U satisﬁes the assumptions 2.1.1. Fix an initial value x0 ∈ i=1 Di of Xtε and a time scale T (ε) ! eζ/ε , where ζ > 0 is not contained in a ﬁnite exceptional set depending on x0 ; see (2.20). Let M ⊂ R2 either denote the set Fx0 ,ζ as deﬁned in corollary 2.5.7 or the set Hx0 ,ζ as given in corollary 2.5.8 under the additional condition that ζ < Vµ(x0 ,ζ),J(µ(x0 ,ζ)) , where the quasipotential values V can be calculated as twice the potential barriers. Assume that the drift

∂2U 1 ∂2U ∂2U ¯ h(x, α) = − sin 2α cos 2α + − ∂x1 ∂x2 2 ∂x21 ∂x22 ¯ ε , αε ) dt , has strictly separated switching for the angle αε of Z ε , dαεt = h(X t t surfaces A1 and A2 on M in the sense of the deﬁnitions 4.4.3 and 1.2.4 and ¯ is strongly hypoelliptic on M meaning that suppose that h ¯ ¯ α)

∂ h(x, α) ∂ h(x, = 1 , rank ∂x1 ∂x2

4.4 The two-dimensional, general case

223

for all (x, α) ∈ M × R ; see deﬁnition 4.4.4. Then it follows for any initial value z0 ∈ Rn \ {0} of Z ε that P 1 logZTε (ε) ( . , x0 , z0 ) −−− Λ1 Kµ(x0 ,ζ) − → ε→0 T (ε) ¯ Kµ(x ,ζ) , A1 Kµ(x ,ζ) ≤ 0, =Q 0 0 where 2 2 2 ¯ α) = − ∂ U cos2 α − ∂ U sin2 α − 2 ∂ U sin α cos α Q(x, 2 2 ∂x1 ∂x2 ∂x1 ∂x2

and where Kµ(x0 ,ζ) denotes the metastable state corresponding to x0 and ζ ; see deﬁnition 2.5.4. This limit Λ1 Kµ(x0 ,ζ) is the local Lyapunov exponent of Z ε with respect to x0 , ζ and z0 . This number is non-positive here, since the curvatures of U at the local minimum Kµ(x0 ,ζ) are non-negative; in other words, the eigenvalues of the Hesse matrix of U at Kµ(x0 ,ζ) are non-negative and hence the eigenvalues Λi Kµ(x0 ,ζ) , i ∈ {1, 2}, of the negative Hessian A(Kµ(x0 ,ζ) ) ≡ −HU (Kµ(x0 ,ζ) ) are non-positive at Ki . 2 Example 4.4.14 (Diagonal matrix plus small skew-symmetric perturbation). Consider the system (1) with coeﬃcient matrix of example 1.5.4, . . . 0 1 λ1 x λ1 0 +x = , A(x) = 0 λ2 −x λ2 −1 0 where λ1 > λ2 , n = 2, d = 1 and with X ε being a one-dimensional diﬀusion deﬁned by (2.2), √ dXtε = − U (Xtε ) dt + ε dWt . > Again suppose that b = −U satisﬁes 2.1.1; ﬁx an initial value x0 ∈ li=1 Di of Xtε and a time scale T (ε) ! eζ/ε , where ζ > 0 is not contained in a ﬁnite exceptional set depending on x0 ; see (2.20). Let M ⊂ R either denote the set Fx0 ,ζ as deﬁned in corollary 2.5.7 or the set Hx0 ,ζ as given in corollary 2.5.8 under the additional condition that ζ < Vµ(x0 ,ζ),J(µ(x0 ,ζ)) . Assume that the metastable state corresponding to x0 and ζ is Kµ(x0 ,ζ) = 0

and that M ⊂

λ1 − λ2 λ2 − λ1 , 2 2

224

4 Local Lyapunov exponents

Fig. 4.6 The drift vector ﬁeld ¯ h(x, α) = − x + 12 (λ2 −λ1 ) sin 2α and the sublimiting statistics of X ε,x0

entailing that the drift ¯ α) = − x + λ2 − λ1 sin 2α h(x, 2 for the angle αεt has strictly separated switching curves A1 and A2 on M; see ¯ is a priori strongly hypoelliptic, since ﬁgure 4.6 . Note that h ¯ α) ∂ h(x, = −1 . ∂x As ¯ α) = λ1 + (λ2 − λ1 ) sin2 α Q(x, and hence ¯ Kµ(x ,ζ) , A1 Kµ(x ,ζ) = Q ¯ 0, A1 (0) = Q(0, ¯ 0) = λ1 , Λ1 Kµ(x0 ,ζ) ≡ Q 0 0 it follows altogether that for any initial condition z0 ∈ Rn \ {0} of Z ε , 1 P log ZTε (ε) ( . , x0 , z0 ) −−− −→ λ1 . ε→0 T (ε)

4.4 The two-dimensional, general case

225

This limit is the local Lyapunov exponent of Z ε with respect to x0 , ζ and z0 . 2 Remark 4.4.15. The stochastic convergence P 1 ¯ Kµ(x ,ζ) , A1 Kµ(x ,ζ) log ZTε (ε) (., x0 , z0 ) −−− Λ1 Kµ(x0 ,ζ) = Q − → 0 0 ε→0 T (ε) which had been veriﬁed under certain assumptions in theorem 4.4.8 and which had been applied to the previous two examples can again be rewritten as in section 4.3: Let the time scales be explicitly given by T (ε) := eζ/ε and then deﬁne ε(t) := T −1 (t) =

ζ log t

(t > 0)

for a scaling parameter ζ > 0 which connects the large parameters t = T (ε) and 1ε . The above convergence towards the local Lyapunov exponent can thus be rewritten as ε(t) 1 P log Zt ( . , x0 , z0 ) −−− Λ1 Kµ(x0 ,ζ) − → t→∞ t for any initial value z0 ∈ Rn \ {0} of Z ε .

2

Remark 4.4.16 (Stability). In this exposition we are concerned with the asymptotic behavior of the linear system Z ε solving (1) and the conclusion of our work is that on certain time scales T (ε) ! eζ/ε this system realizes growth rates (local Lyapunov exponents) given by the spectrum of A(Kµ(x0 ,ζ) ). It is therefore natural to call Z ε (un)stable at x0 with respect to the time scale parameter ζ, if the real parts of the eigenvalues of A(Kµ(x,ζ) ) are strictly negative (positive). Due to the standing assumption (K) the local Lyapunov exponents of the linearized systems (1.33) dZtε = Db (Xtε ) Ztε dt √ dXtε = b (Xtε ) dt + ε σ dWt are non-positive as has already been mentioned in the penultimate example of a linearized two well potential. Therefore, such systems cannot be unstable with respect to a time scale parameter ζ. Note that this concept of stability is diﬀerent from the measure of stability as introduced by Freidlin and Wentzell [Fr-We 98, p.364] who do not consider the linearized system, but measure stability by means of a quasipotential barrier thus calculating the exit time asymptotics.

226

4 Local Lyapunov exponents

4.5 Concluding remarks Finally we would like to comment on possible generalizations, extensions and limitations of the previously obtained results: Remark 4.5.1. It might be interesting to consider instead of dZtε = A (Xtε ) Ztε dt √ dXtε = b (Xtε ) dt + ε σ (Xtε ) dWt

(1)

the more general system dZtε = A (Xtε ) Ztε dt +

√ i ε B (Xtε ) Ztε ◦ dWti i

dXtε = b (Xtε ) dt +

√ ε σ (Xtε ) ◦ dWt

(4.26)

where B i are further n × n matrices; note that such a system arises for example when linearizing the SDE (2.1) for X ε as in (1.33) in case that σ is not a constant matrix. An appropriate assumption on B i might be that all these mappings are bounded and the question is whether the previous results are still valid then. Certainly our argumentation fails, since it crucially uses and copes with the degenerate SDE (1). In the more general case there would appear more stochastic terms instead of “0” in equation (4.4), which need to be taken care of; this “0” is a consequence of the rectiﬁability of Z ε , due to (1). It is remarkable that if b ≡ 0 and also σ ≡ 0 in this case, then the “time-scale-diﬀusion-part” of the original stochastic dynamical system is not observable any more, but only produces the constant solutions X ε,x = x, and setting A := A(x) and B := B(x) the system reduces to the linear white noise SDE √ i dZt = A Zt dt + ε B Zt ◦ dWti . i

This is the SDE which has been investigated e.g. by Khasminskii [Kh 80], Nishioka [Nk 76], B¨ ohme [Bm 80], Auslender and Milshtein [Al-Mi 82], Pardoux and Wihstutz [Pd-Wh 88] and Imkeller and Lederer [Im-Ld 99] and [Im-Ld 01]. These authors mostly consider cases where A is given in normal form and expand the Lyapunov exponent around the exponent of the t corresponding deterministic system dz dt = Azt in orders of ε. In this setting it seems challenging to investigate, if there is a “wide band limit of local Lyapunov exponents”: More precisely, it is known that white noise SDEs can be considered as limit in law, the wide band noise limit, of a suitably scaled sequence of solutions of a real noise SDE; see e.g. Pardoux [Pd 86], Blankenship and Papanicolaou [Bls-Pp 78], Kushner [Ks 82b] and [Ks 82a] and the papers by Wong and Zakai. The question then is what can

4.5 Concluding remarks

227

be inferred in this limit for the growth rate on time scales, if such an object exists in the respective setting. Remark 4.5.2 (exponential time scales). In the preceding exposition we exclusively used exponential time scales T (ε) ! eζ/ε . It would be interesting to consider the case of polynomial scales T (ε) := ε−p ,

p>0;

here, one gets that ζ := limε→0 ε log T (ε) = 0 which is not an admissible parameter ζ; see (2.20). Hence, the results by Freidlin and Wentzell are not available. Furthermore, the ﬁnal step 5) in the proofs of theorems 4.4.6 and 4.4.7 fails in case that “ζ = 0”. Remark 4.5.3. A worthwhile task would be to try to get rid of the assump¯ shall have strictly separated switching curves tion that the drift function h on M (= Fx0 ,ζ or Hx0 ,ζ ). E.g. consider ﬁgure 4.6 for the situation of example 1.5.4 and suppose that

λ1 − λ2 λ2 − λ1 M ⊂ , . 2 2 Then X ε can leave M on the time scales under consideration thus enabling αε to perform rotations. It is not certain what happens in this case, but one then needs to prove generalizations of theorems 4.4.6 and 4.4.7 for the sojourn times of αε . At best this would lead to results on sublimiting distributions for the angle process. However, such assertions are not in sight for two reasons: Firstly, it is not sure how Hern´ andez-Lerma’s theorem 3.1.7 can then be applied, since one additionally needs to take care of its assumption 3.1.3 (d) which had been eluded by appropriately truncating the drift in the proofs of theorems 4.4.6 and 4.4.7. This problem might be tackled by means of a Markov chain argument. But secondly, also the step 4) in the proofs of theorems 4.4.6 and 4.4.7 break down: Here, the crucial argument is to identify the length of the sojourn time of αε in the set A2 (M) with its exit time from D ⊃ A2 (M); this reasoning is not true, if αε can reenter A2 (M) after leaving this set. Furthermore, note that the condition of separated switching curves resembles the controltheoretic ﬁndings of Kliemann [Kl 80]7 , if one assumes that the noise process X takes its values exclusively in M: For strictly separated switching curves A1 and A2 , the set A2 (M) is a “weak control set” implying transience and A1 (M) is a “strong control set” entailing recurrence. Therefore, one might also interpret the assertions of our theorems 4.4.6 and 4.4.7 as transience of αε on the time scale T (ε). Likewise we call αε 7 Kliemann [Kl 80, Ch.6 & 14] focuses on linear systems in dimension n = 2 as we do in this exposition; control-theoretic investigations of linear systems in dimension n = 3 are undertaken by Sommer [So 81].

228

4 Local Lyapunov exponents

recurrent on the time scale T (ε), if the switching curves are not separated. However, even in the familiar context of recurrence (as t → ∞) it is not possible to calculate the expected average of the sojourn time of a process in T a certain set S, T1 0 P (t, x, S)dt, since the latter integral diverges as T → ∞ without further information on the speed of this divergence available; see e.g. the description of the dichotomy of recurrence and transience by Kliemann [Kl 87, Prop.3.1] and [Kl 83a] and also cf. Arnold and Kliemann [Ar-Kl 87b] as well as Roynette [Ry 75]. Therefore, one is also tempted to be pessimistic, if such an average of the sojourn time can be calculated on time scales for a process αε which is recurrent on the time scale in the above sense. Hence, a generalization to the case of non-separated switching curves seems to be an intricate task. It might be instructive to start with performing numerical simulations in this general situation. For a discussion of the invariant measure for degenerate diﬀusions and the emerging problems see Stettner [Ste 89]. The invariant measure of the harmonic oscillator with real Markovian noise in the domain where the oscillator has no switching curve at all is calculated by R¨ umelin [Rm 78] and [Rm 79]. Remark 4.5.4. Consider the real noise driven linear diﬀerential system dZˆt = A (Xt ) Zˆt dt dXt = − ∇U (Xt ) dt +

ε(t) dWt ,

where A ∈ C(Rd , Rn×n ), ε(t) → 0 is a time dependent noise intensity, W is a Wiener process in Rd and U is a potential function. In other words, the process X ε in (1) has been replaced by a simulated annealing process; see remark 2.5.12. The task here is to give conditions under which the exponential ˆ growth rate of Z, 1 log Zˆt ( . , x0 , z0 ) t converges (e.g. in probability) to Λ1 A(S) , where S denotes the global minimum of U . Since S is necessarily a metastable state for the process X ε deﬁned by (2.2) for this choice of the potential U , on might then prove that “the local Lyapunov exponent is the Lyapunov exponent of Zˆ ”. Remark 4.5.5. This exposition has been concerned with exponential growth rates on time scales. A diﬀerent task is to investigate the convergence of the Lyapunov spectrum Λε1 , Λε2 , . . . , Λεn as given by the Multiplicative Ergodic Theorem 1.3.1 in the limit as ε → 0. Such a convergence result for Lyapunov exponents has been obtained by Ledrappier and Young [Le-You 91] for the system (4.26) in the linearized i i case, A = Db and B = (Dσ) , under the assumptions that the εvector ﬁelds b( . ) and σ( . )ei i=1,...,d are divergence free, the state space of X is compact

4.5 Concluding remarks

229

and the equation for Z ε in the system (4.26) is elliptic. It should be interesting to bridge the gap to their arguments. However, diﬃculties will likely appear, since our derivation of the local Lyapunov exponent is not based on a modiﬁcation of the proof of the Multiplicative Ergodic Theorem.

Notations

K Z R+ R>0 N N0 s∧t d x x˙ = dt U =

d dx

U

the set R of real numbers or the set C of complex numbers set of integers {t ∈ R : t ≥ 0} {t ∈ R : t > 0} set of strictly positive integers N ∪ {0} min(s, t) ; minimum of two real numbers s and t diﬀerentiation of a function x with respect to the time variable t diﬀerentiation of a function U (x) with respect to the space variable x ⎛ ∂U ⎞ ∂x1

∇U , ∇x U

⎜ .. ⎟ ⎝ . ⎠; gradient of the function U with respect to the ∂U ∂xd

∆f , ∆x f

a := b , b =: a a≡b f (x) ≡ c !

multi-dimensional space variable x = (x1 , . . . , xd ) ∂2 f ∂2f + · · · + ∂x 2 ; Laplacian of the real-valued function f ∂x21 d with respect to the multi-dimensional space variable x = (x1 , . . . , xd ) a is deﬁned by b a equals b by deﬁnition the function f attains the constant value c for all x T (ε) ! eζ/ε denotes logarithmic equivalence of a function T (ε) (which is deﬁned for small values of ε, i.e. T : (0, ε0 ) → R>0 ) to eζ/ε in the sense that lim ε log T (ε) = ζ; see p.100 ε→0

stochastic independence of two objects with respect to a prescribed measure P

−−−−→ w

−−−−→

convergence in probability weak convergence (convergence in distribution) 231

232

(α → β) , | | ∞ , [T1 ,T2 ]

1M ∂M M M◦ t

Notations

arrow with initial point α and endpoint β; see p.92 scalar product of the underlying space Kn (Rn or Cn ) norm of the underlying space Kn (Rn or Cn ) operator norm supremum (maximum) norm on the space of (continuous) functions f : [T1 , T2 ] → Rn ; f ∞ ≡ f [T1 ,T2 ] := supt∈[T1 ,T2 ] |f (t)| indicator function of the set M topological boundary of the set M topological closure of the set M topological interior of the set M max{k ∈ N0 : k ≤ t}; integer part of t ∈ R+ (“Gaußbracket” or “ﬂoor” of t)

Im z Re z z¯

imaginary part of the complex number z real part of the complex number z complex conjugate of the number z ¯ of αε ) (exception: drift function h t

a

σ σ ∗ ; εa is the covariance matrix of X ε in the SDE (2.1) σ ˆσ ˆ ∗ corresponding to the SDE (3.1) for X ε small parameter in the Jordan-form matrix calculations; see the proofs of theorems 1.4.3 and 4.1.2 angle of the solution Ztε of the linear diﬀerential system (1); characterized by the diﬀerential equation (1.6) attracting switching curve of the angle processes under consideration; see p.18f. repelling switching curve of the angle processes under consideration; see p.18f. system matrix of the SDE (1) given by a continuous function A : Rd → Kn×n adjoint operator (matrix) for an operator A

a ˆ a>0 αεt A1 A2 A A∗ b ˆb B(X) B(x, r) Br (x)

drift coeﬃcient of the SDE (2.1) for (Xtε )t≥0 (component of the) drift coeﬃcient of the SDE (3.1) for (Xtε , Ytε )t≥0 Borel-σ-algebra on the topological space X { y ∈ Rd : |x − y| < r}; open ball with center x and radius r { y ∈ Rd : |x − y| ≤ r}; closed ball with center x and radius r

Notations

M C k (M, N )

C(M, N ) C b (Rd , R) C c (Rd , R) Cx (J, M ) C (k) d D D Di δij δx (B) det(A) dist(x, B) div b e 1 , . . . , ed ε≥0 E(f ) EF Ex E(C) F F FX Fx0 ,ζ

233

E \ M ; set-theoretical complement of M ⊂ E {f : M → N continuous, all derivatives of f of order up to k exist and are continuous } for diﬀerentiable manifolds M, N and k ∈ N0 C 0 (M, N ) {f ∈ C(Rd , R) : f bounded} {f ∈ C(Rd , R) : supp(f ) compact} {f : J → M continuous, f (0) = x} for an interval J ⊂ R, 0 ∈ J, a diﬀerentiable manifold M and x ∈ M set of k-cycles; see p.93ﬀ. dimension of the state space of (Xtε )t≥0 as deﬁned in (2.1) bounded, open domain in Rd to which the exit time investigations in the non-degenerate case apply; see 2.4.1ﬀ. bounded, open domain in Rm to which the exit time investigations in the degenerate case apply; see 3.1.1ﬀ. t→∞ {x ∈ Rd : Xt0,x −−−−→ Ki }; the domain of attraction of Ki under the deterministic motion X 0 ; see p.79 1 if i = j and 0 otherwise; Kronecker symbol 1 if x ∈ B and 0 otherwise; Dirac measure at x determinant of the matrix A inf y∈B |x − y| ; distance between the point x ∈ Rd and the set B ⊂ Rd ∂bd ∂b1 ∂x1 + · · · + ∂xd ; divergence of the vector-valued function b canonical unit vectors in Rd parameter of the noise intensity in SDEs (1), (2.1) and (3.1) f dP; expected value of the function f with respect to the probability measure P conditional expectation with respect to the σ-algebra F expected value with respect to the probability measure Px exit rate of the cycle C; see p.93ﬀ. (component of the) drift coeﬃcient of the SDE (3.1) for (Xtε , Ytε )t≥0 continuous mapping between separable metric spaces “pushing forward” the LDP; see p.68 σ-algebra generated by the random variable X { x ∈ Rd : V (x0 , x) ≤ ζ }; the set to which X ε is asymptotically constrained on the time scale T (ε); see corollary 2.5.7

234

g GW (L) Gi (L) Γ Gε G, Gε G0 , G+ , G+ T GL(n, K)

h(x, ψ) ¯ α) h(x,

Hx0 ,ζ

HU

H1 ≡ H1 ([0, T ], Rd)

H(s, x, p)

Notations

graph on L, i.e. a set of arrows; see p.92 the set of W -graphs on L; see p.92f. the set of i-graphs on L; see p.92f. parameter for the (scaled) time horizon; see p.102ﬀ. 4d 4d ε ∂ ∂2 i=1 bi ∂xi + 2 i,j=1 aij ∂xi ∂xj ; generator of the diﬀusion process (Xtε )t≥0 ; see p.53 perturbations of the system matrix A point sets corresponding to the data of the SDE (3.1); see p.128ﬀ. general linear group, consisting of the invertible n × n matrices with entries in K A(x)ψ − A(x)ψ , ψ ψ ; drift function of the direction ψtε of the solution Ztε of the linear diﬀerential system (1); see RDE (1.4) 2 2 −a 12 (x) sin α + a 21 (x) cos α + a22 (x) − a11 (x) sin α cos α ; drift function of the angle αεt of the solution Ztε of the linear diﬀerential system (1); see RDE (1.6) { x ∈ Rd : V (Kµ(x0 ,ζ) , x) ≤ ζ } ; the set where X ε gets asymptotically stuck on the time scale T (ε); see 2.5.8 corollary ∂2 U ∂xi ∂xj

; Hesse matrix of U , i.e. the i,j=1,...,d

symmetric matrix of second derivatives of the potential function U ∈ C 2 (Rd , R) T 0

g(s) ds : g ∈ L2 ([0, T ], Rd) ; the space of absolutely continuous functions starting in 0 with L2 -derivative 4d ˆ 4d 1 ˆij (s, x) pi pj ; i=1 bi (s, x) pi − 2 i,j=1 a see p.135

In = idKn I : E → [0, ∞]

identity operator on Kn (unit matrix) rate function, deﬁned on a separable metric space (mostly a function space) E; see p.68ﬀ.

J

index singling out the (local) Lyapunov exponent in the list of eigenvalues; see p.38ﬀ. and p.160ﬀ. cycle following after the cycle C; see p.93ﬀ.

J(C) K 1 , . . . , Kl Kl+1 , . . . , Kl Kµ(x0 ,ζ)

stable attractors of X 0 unstable attractors of X 0 metastable state for the initial value x0 and the time scale T (ε) ! eζ/ε ; see deﬁnition 2.5.4

Notations

K(s, x, q)

l L L L(x) Lp ([a, b], Rd )

Lε Lˆε

Λ1 (A) , . . . , Λn (A)

M (C) m(C) µ

n N (x), N (y) O (Ω, F , P) p ≡ p(A) Ps,z

235

2 [ q − ˆb(s, x)] ; dual function of H(s, x, p) ; see 3.1.2 and p.135 1 ˆ(s, x)−1/2 2 a

number of the stable attractors Ki of X 0 {1, . . . , l} ; enumeration representing the set of stable attractors {K1 , . . . , Kl } of X 0 ; see p.91 Lebesgue measure vector ﬁeld such that b(x) = −∇U (x) + L(x) and L(x) ⊥ ∇U (x) , where such a decomposition exists space of (equivalence classes of the) Rd -valued, Lebesgue-measurable functions on [a, b] such that their norm is p−integrable with respect to the Lebesgue measure L ∂ G ε + h ∂s ; generator of the coupled (Markov) process (Xtε , ψtε ) ; see p.14 4m 4d ˆ ∂ 2 ε 4d ∂ ˆij ∂x∂i ∂xj + i=1 Fi ∂y ; i=1 bi ∂xi + 2 i,j=1 a i generator of the coupled (Markov) process (Xtε , Ytε ) ; see p.128 eigenvalues (characteristic roots, i.e zeros of the characteristic polynomial in the complex plane) of the n × n-matrix A main state of the cycle C; see p.93ﬀ. stationary distribution rate of the cycle C; see p.93ﬀ. the index function which assigns the respective metastable state Kµ(x0 ,ζ) to the initial value x0 and to the time scale parameter ζ ; see (2.21) dimension of the state space of (Ztε )t≥0 as deﬁned in (1) outer normal vector to ∂D at x or to ∂D at y, respectively stable attractor of X 0 in the ﬁrst exit time investigations (element of {K1 , . . . , Kl }); see p.78ﬀ. underlying probability space number of distinct values in the set of real parts of eigenvalues {Re Λ1 , . . . , Re Λn } of the matrix A law of the stochastic process Z conditioned to start in z at time s ≥ 0; Ps,z {Z ∈ M } = P{Zs = z ∧ Z ∈ M }

236

Pz P (t, x, A), Ptε (x, A) pεt (x, y) pε (x) P n−1 pr[0,T ] φ Φε ψtε

Ψ P ε (s, x, y) Qε (s, x, y) Q 0 , Q , Q1 Q(x, ψ) ¯ α) Q(x,

Notations

P0,z ; law of Z conditioned to start in z at time 0 Markov transition probabilities density of Ptε (x, A) density of ρε ; see section 2.2 {s ∈ Rn : |s| = 1}/{s = −s}; projective space in Rn pr[0,T ] : (Rd )[0,∞) → (Rd )[0,T ] , pr[0,T ] (f ) := f [0,T ] ; restriction map on the function space prefactor of the Verhulst comparison ODE; see 4.3.3f. cocycle solution of (1.1); see section 1.3 Ztε n−1 spherical component (direction) of the |Ztε | ∈ S ε solution Zt of the linear diﬀerential system (1); characterized by the diﬀerential equation (1.4) set on which large deviation estimate is supposed to hold; see p.68, p.74 inclusion probability; see p.130 exit probability; see p.130 for the SDE (3.1); see p.128ﬀ. parameter domains A(x)ψ , ψ ; integral function identifying the modulus of the solution Ztε of the linear diﬀerential system (1); see (1.3) and (1.5) a11 (x)cos2 α + a22 (x) sin2 α + a12 (x) + a21 (x) sin α cos α ; integral function identifying the modulus of the solution Ztε of the linear diﬀerential system (1) in dimension n = 2; see (1.7) and (1.8)

R(C) ρε εt

rotation rate of the cycle C; see p.93ﬀ. stationary distribution of X ε ; see section 2.2 | Ztε | ∈ (0, ∞) radial component (modulus) of the solution Ztε of the linear diﬀerential system (1); characterized by the diﬀerential equation (1.3)

s S n−1 Sr (x)

time parameter { ψ ∈ Rn : |ψ| = 1}; unit sphere of Rn { y ∈ Rd : |x − y| = r}; sphere with center x and radius r

σ σ ˆ

noise coeﬃcient of the SDE (2.1) for (Xtε )t≥0 noise coeﬃcient of the SDE (3.1) for (Xtε , Ytε )t≥0

Notations

237

σ(M) σρε,x

σ-algebra generated by a family M of sets or functions inf{t ≥ 0 : Xtε,x ∈ Bρ (O) ∪ ∂D}; ﬁrst hitting time of X•ε,x of either ∂D or a small neighborhood of O; see 2.4.9

t ε,x τ ε,x ≡ τD

trace(A)

time parameter inf{t ≥ 0 : Xtε,x ∈ / D}; ﬁrst exit time of X•ε,x from a bounded, open domain D in Rd with smooth boundary ∂D; see 2.4.1 inf{t ≥ 0 : Ytε,x ∈ / D}; ﬁrst exit time of Y•ε,y from a bounded, open domain D in Rm with smooth boundary ∂D; see 3.1.1 stopping times in the proof of the exit time law; see p.86 T ≥ 0; ﬁxed time horizon T (ε) ! eζ/ε ; time scale on which the respective sublimiting distribution is observed; see theorem 2.5.5 trace of the matrix A

U

U ∈ C ∞ (Rd , R); potential function

(Wt )t≥0

Brownian motion in Rd over a complete probability space (Ω, F , P)

x, x0

ξtε

space variable (element in Rd ) ; initial point of the stochastic processes X, X ε and X ε ; for X ε in (2.1), the >l initial condition is taken from i=1 Di , a set of full Lebesgue measure, in order to have the metastability results available tan αεt ; characterized by the diﬀerential equation (1.9)

y0

initial point of the stochastic processes Y and Y ε

ζ

parameter which determines the time scale T (ε) ! eζ/ε ; for the index function µ(x0 , ζ) to be well deﬁned, ζ must not be contained in a ﬁnite exceptional set depending on x0

X ≡ (Xt )t≥0 X ε ≡ (Xtε )t≥0

real noise process diﬀusion process of Freidlin-Wentzell type as deﬁned in (2.1) diﬀusion process driving the SDE (3.1) solution process of a real noise SDE driven by X solution process of the real noise SDE (3.1) driven by X ε

T ε,y ≡ TDε,x ε,x ε,x , θm+1 τm T T (ε)

X ε ≡ (Xtε )t≥0 Y ≡ (Yt )t≥0 Y ε ≡ (Ytε )t≥0

238

Z ε ≡ (Ztε )t≥0 X•ε , Z•ε , . . .

Notations

solution process of the linear real noise SDE (1) driven by X ε t → Xtε or Ztε , . . . ; path mappings of the respective processes

LDP ODE PDE RDE RDS RVDE RVDI SDE

large deviation principle ordinary diﬀerential equation partial diﬀerential equation random diﬀerential equation random dynamical system random Riccati-/Verhulst-type diﬀerential equality random Riccati-/Verhulst-type diﬀerential inequality stochastic diﬀerential equation

cf. e.g. et al. i.e. p.

confer for example (exempli gratia) and others (et alii) that is (id est) page(s)

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Index

action functional 55 arrow 92 asymptotic integration (Hartman and Wintner) 29 asymptotically autonomous ODE 44 asymptotically constant, linear ODE 28 attracting switching surface 20 backward operator 128f. backward (parabolic) equation 61 boundedness in probability 103f. cocycle 22 commodity prices 114 comparison 163 contraction principle 68 cycle 93ﬀ. diagonal matrix plus small skewsymmetric perturbation 47, 223f. duality correspondence 135 dynamic programming PDE 136 El Ni˜ no-Southern Oscillation 119 ellipticity 56, 129 entrance point of a cycle 96 ergodic dynamical system 22 evolution model 121 exit point of a cycle 96 exit position 79 exit probability in the degenerate case 125ﬀ., 130ﬀ., 134ﬀ., 140f. in the non-degenerate case 72ﬀ. exit rate of a cycle 93ﬀ., 97

exit time in the degenerate case 127ﬀ. in the non-degenerate case 72ﬀ., 79ﬀ. ﬁrst exit position see exit position ﬁrst exit time see exit time Fokker-Planck equation see Kolmogorov forward equation forward (parabolic) equation 61 Freidlin-Wentzell theory 53ﬀ. Furstenberg-Khasminskii formula 27, 209 graph 92 harmonic oscillator 50 Hartman-Wintner-Perron theorem 33 Hern´ andez-Lerma theorems 130ﬀ., 134ﬀ. hierarchy of cycles 93ﬀ. hypoellipticity 26, 128, 186 i-graph 92 inclusion probability 130 invariant (with respect to Markov semigroup) probability measure 62 Jacobi equation 11, 22f., 156 Jacquot condition 64 Kolmogorov backward equation 62f. forward equation 62f. large deviation principle (LDP) 68ﬀ. law of large numbers for the top Lyapunov exponent 27 Legendre transformation 135

253

254 linearized SDE 2, 44ﬀ., 222 Liouville equation see Jacobi equation local growth rate of the determinant 156 local hypoellipticity 186 local Lyapunov exponent 4ﬀ., 45, 143ﬀ. − upper and lower bound 145ﬀ. − limit in probability 151 in the diagonal case 157ﬀ., 175 in the general case 206ﬀ. locality 53 logarithmic equivalence 100 logistic ODE see Verhulst ODE Lyapunov exponent in the deterministic case 30, 33ﬀ. in the stochastic case 2, 23ﬀ. main state of a cycle 93ﬀ. Markovian noise 9 mean-reversion 114 metastability 1, 53, 91, 107, 144 metastable state 3, 100f. metric (measure preserving) dynamical system 21 multiplicative ergodic theorem 23 noisy north-south-ﬂow 122 occupation time of the real noise system 179ﬀ., 188ﬀ., 198ﬀ. of the underlying diﬀusion 102 Ornstein-Uhlenbeck process 45, 65, 109 Oseledets see multiplicative ergodic theorem parabolic equations see backward equation, forward equation parametrically excited system 9 projection method 12ﬀ., 20 quasi-deterministic approximation 53, 102 quasi-deterministic behavior of the non-degenerate diﬀusion 102 of the real noise system 179ﬀ., 188ﬀ., 198ﬀ. quasipotential 56, 76ﬀ. random dynamical system 22 random Riccati-type diﬀerential inequality 153 random Riccati-/Verhulst-type diﬀerential inequality 164

Index rate function 68ﬀ. real noise system 9 recurrence on time scales 228 repelling switching surface 20 rheolinear system 9 Riccati − Riccati-type diﬀerential inequality 31 − random Riccati-type diﬀerential inequality 153 − random Riccati-/Verhulst-type diﬀerential inequality 164 rotation rate of a cycle 93ﬀ. Saksaul see tree population model Schilder’s Theorem 70 separated switching surfaces (curves) 185 simulated annealing 108, 228 stability on time scales 225 stationary distribution rate of a cycle 93ﬀ. stationary probability distribution 21, 62f. stochastic disk dynamo model 121 strictly separated switching surfaces (curves) 185, 209f. strong hypoellipticity 186 sublimiting distribution 3, 101f. sublimiting Furstenberg-Khasminskii formula 209 switching surfaces (curves) 18, 185 thermohaline circulation 115ﬀ. trace formula for the sum of the Lyapunov exponents 24, 156 transience on time scales 227 tree population model 120 two well potential 110, 175 ultraparabolic operator 129 value function 127, 187 variational equation see linearized SDE Vasicek model for the interest rates 113 Verhulst − Verhulst-ODE 163 − random Riccati-/Verhulst-type diﬀerential inequality 164 W -graph 92 white noise system 10, 226 wide band noise limit 226

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