) = (fx measurable for ever x € A C Zd. Call C/^, the set of Gibbs measures: it is easy to check that if d = 1 then Qv = 0, since if A = AN = {—N, —N + !,..., N} then /J.A is the law of a symmetric random walk on IN + 1 time steps, pinned at fixed values outside AN- In higher dimensions
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to stay in the realm of standard technics we have to resort to the very special case of quadratic V (note the Gaussian character of //A in this case) and one discovers that Q^ 7^ 0 if and only if d > 3 [20, Ch.13]: this result can be then extended (to a certain extent!) to the general case via the celebrated Brascamp-Lieb inequalities and other subtle arguments [5],[6]. This dimensional dependence has important consequences from the modelling viewpoint: from the point of view of a proof, it relies on the fact that in the quadratic/Gaussian case the random field has a representation in terms of a simple random walk and the existence of such a Gibbs measure requires the random walk to be transient (see §2.2 for more details). We will later present an extension of this representation to the case of general V (§2.2), that clarifies the connection massless measures - random walks and leads also to a different approach to the Brascamp-Lieb inequalities. However, in order to treat all the dimensions in a unified context, we will abandon (for now) the description of massless fields in terms of the variable V? (and we refer the reader to [6],[10],[13],[21]).
1.4. A random field of increments. As mentioned above, the d= 1 case is trivial because the random field defined by ^ is simply a symmetric random walk (at least if A is connected) and this follows from the fact that the Boltzmann weight in //^ is exponential and from the form of the Hamiltonian (1.5): if we change the variables via »7(X)y) = fy — px> x and y nearest neighbors, the Boltzmann weight factorizes, leading to a field of independent increments. This is not completely true, since, for example in the case A = AN we will have to consider the constraint Y^x=-N-i ^(x^+i) = 4>N+i — V'-JV-i) which however becomes weaker and weaker as N —t oo. Much more complicated is the situation in d > 2 where the number of constraints in the rj variable is now very large. Nonetheless one can show that a (in fact several) field of increments exists in every dimension. Let us therefore define the field of increments in a proper way. The set of Gibbs measures for the field of increments can be also approached in a DLR fashion. Let us denote by Xd the subset of Rzd consisting of {rn}b<=zd* (by •* we mean the directed bonds of the graph •) such that rifay) = —fl(y,x) and such that for every loop of lattice points XQ, xi,...,Xk = XQ (by loop we mean a chain, \x± — £j+i| = 1 for i = 0,1,..., k — 1, which is closed: XQ = x^) we have X)i=o rl(xi,xi+-i) = 0- It should be clear that given (p € M zd , then rj = V where the symbol 0 -4- x means that the sum is over an arbitrary chain of directed bonds leading from 0 to x. We say that /J.(dr]) is a Gibbs increment field (fi>(dri) € Q^) if JJL is a probability measure on Xd (the cr-algebra is the restriction of the product
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one) , E^ [T^ ] < oo and for every £ € Xd and every x
(£) oc exp -
Ffo)
^(dr,),
(1.7)
where Bx = {(x, y) :\y — x\ = 1} and ^(dr?) is the measure on Xd induced by the action of the map 77 = Vf on the measure dfx Hu^a: fyxd&Py) i f°r an arbitrary choice of ^ such that Vif) = £. Moreover f. denotes the cr-algebra generated by the canonical variables indexed by the bonds 6 € •. One can show, by using the Brascamp-Lieb inequalities, that Q^ is not empty for every d: note that the increment field does not enjoy the continuum symmetry and to two (f measures related by a continuum symmetry transformation corresponds a unique 77 measure. Nonetheless there are uncountably many meausures in Q^ and many of them are physically very relevant (see §2.1 for a precise statement and further explanations). Finally we remark that any fj, e Q^ is invariant (in fact reversible) for the evolution induced by (1.1) on the increments, that is ,:|,-v|=l V (%„)(*)) - £,S|M|=1 V (ri(x,z)(t))
dt+
+V2d(By(t) - Bx(t)) ,
(1.8) for every x, y such that \x — y\ = I. 1.5. A probabilistic analysis of random interfaces. The questions that
we will consider are • The structure of set of Gibbs measures: we will state and discuss (§2.1) a result in [19] that, by dynamical methods, classifies the shiftergodic measures in Q^ (here and below by shift we mean space translation). In this case the dynamics is used as a tool for investigating the properties of the equilibrium measures; • Deterministic large scale phenomena: we will review some of the law of large numbers obtained under space or space-time rescaling (§3, 4). • Large Deviations: we will present and discuss a pathwise LD principle and its consequences (§4). We will also briefly discuss other LD results on the model. • Fluctuation phenomena: we will consider also central limit theorems issues (§5) and discuss the limitations of the present approaches (§6). A central role in most of the results is played by the so called HS representation, that links random effective interfaces with symmetric random walks in random enviroment: this will be introduced and discussed in some detail (§2.2-2.5). We conclude (§6) with a quick overview of related directions of research and with a list of open problems.
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2. RECENT TECHNICAL DEVELOPMENTS FOR MASSLESS FIELDS 2.1. The Punaki-Spohn classification result. Understanding, even only partially, the structure of the set of Gibbs measures of a statistical mechanics model is typically a very challenging task. However classifying for example the measures that are ergodic under shifts opens the road to important consequences for the study of the equilibrium and the nonequilibrium dynamics (in a rather general context, see [28],[35]). In the context of Gibbs measures with massless interactions this very fruitful step has been accomplished by T. Funaki and H. Spohn [19]: their result plays a crucial role in most of the other results presented in this review. They succeeded in classifying all the shift ergodic increment Gibbs measures associated to the Hamiltonian (1.5). We will denote by rx the shift by x £ Rd of the r) variable, that is Tx"n(y,z) = il(y+x,z+x)- Call S^ the set of all DLR measures ^(drj) which are shift invariant. Recall that d is arbitrary and denote by ei the unit vector in the i-direction.
Theorem 2.1. [19, Th.3.1 and Th.3.2] For every v € Rd there exists a unique p, € S^ which is shift ergodic and such that E/j[77(0,6;)] = Vi, i = l,...,d. The ergodic Gibbs state with mean v will be denoted by ^s and it will be referred to as Funaki-Spohn (F-S) state of tilt v. We recall that the shift ergodic states are precisely the extremal points of S^. Therefore any measure H € $r) can be written, in an essentially unique way, as linear superposition of F-S states. It is natural to ask whether such a richness of Gibbs states is an anomaly of the model. The answer is no: it can be traced back to the massless character of the field and hence to its very strong dependence on boundary conditions. The F-S state of tilt v is expected to be the limit of finite volume states like the measures fj^ defined in (1.4), with ij)x = v • x and A a (hyper) cube: even though to prove this seems a very challenging issue, it gives an immediate picture of the phenomenon that creates the F-S states. In short, F-S states describe flat interfaces in Rd+1 which are orthogonal to (—v, 1) e Zd+1. In proving Theorem 2.1 one has to establish existence of such a measure and its uniqueness. The proof of existence starts from the absolutely standard idea of approximating via finite volume systems defined on tori of increasing size: however the implementation of this idea is rather subtle for this model in particular for what concerns imposing the desired tilt v. We refer to the original paper [19] for any detail on this part: we pass instead to discuss some aspects of the proof of uniqueness, which uses dynamical coupling as key idea. Let us consider two solutions {o and {<^ }t>Q to (1.1) respectively with initial conditions f^ and 0 such that
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for every b
?j
This therefore implies that //W = //2) if v i = v%. One can however exploit (2.1) further: in [24, Lemma 3.2] it is proven that (2.1) implies also that i^s is time ergodic (see §2.5 for an application). 2.2. The Helffer-Sjostrand representation. A new idea for the analysis of Gibbs measures with continues spins can be found in a recent work of B. Helffer and J. Sjostrand [26]. It essentially allows to represent correlation functions of the spin system in terms of solutions of suitable elliptic PDEs. New proofs of established results as well as new results have been made possible by this representation. Among the novel proofs of known results it is worth mentioning the straightforward proof (see §2.4 below) of the basic Brascamp-Lieb variance inequality for arbitrary functions [5, Th.4.1]. Of course here we will outline this new method only in the case of massless fields, adding one step to the method: solutions to elliptic PDEs can be written in terms of expectations with respect to suitable diffusion processes. We refer to [2] , [25] , [26] and [33] as samples of further results achieved by the method in other contexts. 2.3. The HS representation in the massless context. The first application of the representation in the massless case can be found in [32]. We will follow however the probabilistic approach of [10] and [24]. Let us consider the reduced set up of zero boundary conditions outside a finite connected set A. In this case the dynamics is defined by (1.1) for x € A and
LH = G,
(2.2)
with G G (72(RA;R) such that E^[G] = 0 and with at most exponential growth as maxz \ipx\ —>• oo. A direct computation shows that, if we define the operator £ acting on suitably smooth functions / : A x RA —>• R as
e:|e|=l
(2.3)
= [Lf(x,-)](
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where OF : A x RA -> R is defined by dF(x, implies
Therefore (2.2)
jCdH(x,
(2.5)
A this point recall that G is centered and observe that L is symmetric in L 2 (dyu) to obtain
varM (G) = EM [(LH)2} = J^EM [dxHdxLH},= J^EM [OH(x, -)£dH(x, •)] X
X
(2.6) from where, using dH = C, ldG (cf.
(2.5)), we get to the formula
varM(G) = EM [dGC~ldG\ .
(2.7)
Formula (2.7) is what we call ITS representation. We will however give now a probabilistic version of (2.7). The basic idea, which is close in spirit to work by M. Freidlin (diffusions with a discrete component, see e.g. [14]), is that L is the generator of a Markov process {(p(t),X(t))}t>oThis process is defined starting from o> defined before: the discrete component {X(t)}t>o is a time inhomogeneous Poisson process with jump rates given by {F"(%(i))}6eA* and killing at d+ A (denote by T the killing time). Let us denote by Px,o, with initial condition y>(0) = and -X"(0) = x. Stochastic calculus [10, §2] yields
dH(x,
Uo
J
.
(2.8)
And therefore (2.7), which can obviously be extended to covariances by polarization, can be written in probabilistic terms as
( fT dG(X(s), v(s))ds}] , \Jo /J
cov^F, G) = £EM \dF(x, .)E*, x L
(2.9)
see [10, Prop.2.2]. Particularly enlightning is the expression for the covariances of two heights (spins) tpx and
cov^ (
\JQ
L
(2.10)
) J
Therefore the covariances between and
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2.4. FKG and B— L inequalities: novel proofs. Classical results can be recovered via the HS representation: we mention here the FKG inequality and the B-L inequalities. The FKG inequality for p, is immediate from (2.9): if F and G are non decreasing functions of each 0 and dG > 0 and therefore covn(F, G) > 0, (2.11) which is the FKG inequality. The B-L inequality instead follows directly from the fact that the lower bound in (1.2) implies that £ > — c_A (in the sense of operators, both viewed as acting on functions of x and (f). We refer to [10, §2] for a proof of a reversed B-L inequality, depending on c+. We observe also that one can extend the idea of the HS representation to recover also an anharmonic analog of formula (2.8), i.e. a representation for the mean and not only for the (co)variances [10]. 2.5. A space—time correlation formula. A further extension of the HS representation leads to a formula for space-time correlations: we will look at this case only in the infinite volume setting. Let us concentrate on the field of the increments, more precisely let us fix v € Kd, d = 1,2,..., and let us focus on /^(dr?) = /^5(dry). The H-S representation in this setting is introduced by defining the coupled process {(??(£), ^(£))}t>0> where ??(•) is solution to (1.8), with initial condition which is /^(d?7)-typical, and X(-) is the the time inhomogeneous random walk with Poisson jump rates given by {V"(ilb(t))}b£zd* t>o- IR wnat follows dx is still the derivative with respect of tpx' we apply it to local functions of 77 by using the fact that, given rj € Xd, up to a (for our purposes) irrelevant constant, we can reconstruct ip and we can therefore view 77 as a function of (p. Of course, the notion of regularity is with respect to dx. We have the following Proposition 2.2. [24, Prop.3.1] For every local smooth F,G£ L (x; //) and anyT>0 2
covt,(F(rj(0))-G(rj(T}))
= (2>12)
x, 77)EI,7? (dG(X(t + T), rj(t + T))] d*.
Notice that, if T = 0, (2.12) reduces to a formula which is the natural infinite volume extension of (2.9). About the proof of (2.12) we stress only that it uses as a central ingredient a form of time-ergodicity Q,
(2.13)
which is not an immediate consequence of the shift ergodicity (i.e. the extremality in the set of shift invariant states S^), of p, = fJ-^ . It would rather follow from the extremality of p, in the whole set of Gibbs measures Qr). As mentioned at the end of §2.2 the coupling technique used in [19] and s
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in particular result (2.1) does work in tackling this problem too and (2.13) holds.
3. HYDRODYNAMIC LIMITS The original motivation to get the full characterization of shift ergodic Gibbs massless states in [19] was hydrodynamics. A result like Theorem 2.1 is a key step in most of the derivations of hydrodynamic equations as macroscopic limits from an underlying microscopic dynamics (see e.g. [28], [35]). Funaki and Spohn show in [19] that the hydrodynamic limit of massless fields is described by a nonlinear nondegenerate diffusion equation, which can be viewed as a motion by distorted mean curvature, the distortion coming from the fact that the interface is constrained to be a function. The nontrivial diffusion coefficient appearing in this equation is shown to be precisely the gradient of the surface tension of the model. We start by reviewing some important facts on surface tension. 3.1. The surface tension. Let us consider the finite box AAT = (—TV, N)dC\ Zd, TV a large integer. We impose boundary conditions i/jx = v • x outside Ajy and we set
log
exp
(-<») n
We call <7jv(w) finite volume surface tension at tilt v: it can be interpreted as price for bending an interface up to tilt v. Proposition 3.1. For every v e Kd the limit O/• oo exists and it will be denoted by &(v}. The function a : Rd —>• R is differentiate with Lipschitz derivatives and for every u, v € Kd -j-\u~ v\2 ^ a(v) ~ °(u) ~ Va(u) • (v - u) < ^-\u -v2,
(3.2)
where V denotes here the gradient in JRd. Therefore a is strictly convex: strict convexity does not play a central role in the hydrodynamic limit (convexity suffices), but it does play a role in other results (§4). Prop. 3.1 is taken from [19, Prop. 1.1], with the exception of the strict convexity statement which can be found, with two different proofs, both relying on the HS representation, in [10, Lemma 3.6] and [24, App. A]. 3.2. Motion by mean curvature. The set up in which the hydrodynamic limit is proven is not exactly the one we use in this review: the result is in fact proven with periodic boundary conditions. More precisely we still consider the coupled equations (1.1), but this time x runs in the torus IV = (ZmodTV)d, and T will be the unit d-dimensional torus. Note that, in spite of the fact that the finite volume Gibbs state on RTjv does not make sense (in
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the sense that it is not a probability measure), the dynamics is well defined since it is given by a finite set of SODEs with Lipschitz coefficients. Therefore let us consider the evolution defined by (1.1) with x £ TV and let us define the rescaled profile £N : T x R+ ->• R ),
(3.3)
where [q] denotes the point in TJV closest to q € NT. Note that the spacetime scaling is diffusive. Choose now any sequence of measures Vff on RTjv such that for some h0 € L 2 (T;R)
^lim^ [||/io(-) - 6v(-,0)||i2(T)] = 0.
(3.4)
Therefore we are imposing that £AT(-, 0) approximates a non random profile ho(-). With the notation
has a unique solution in C°([0,T],.L2(T)) n L 2 ( [ 0 , T ] , H l ( T ) ) , any T > 0. Moreover for every t > 0 = 0.
(3.6)
The result in Theorem 3.2 was certainly expected on physical grounds. The dynamics of the interface should [36] in fact be driven by the surface free energy
F(h)=
! (7 JT
(3.7)
in the sense that we expect dh/dt = —M(Vh)6F(h)/6h, 8/6h the L2 functional derivative. The function M goes under the name of mobility of the interface: comparison with (3. 5) shows that for V interface dynamics M = 1. Note that the true motion by mean curvature leads to o~(v) = y/1 + |v|2: the substantial difference with the massless effective model is that in the latter case a(v] = \v\2 for v large. The proof of (3.6) relies on the sophisticated machinery of hydrodynamic limits, on the line of the H~l technique developped by H.T. Yau in the context of the seminal work by M. Z. Guo, G. C. Papanicolau and S. R. S. Varadhan [28], [35].
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4. LARGE DEVIATIONS AND A SHAPE THEOREM The study of the asymptotics of rare events (Large Deviations) for both the static and dynamical model have been considered. In particular, it turns out that the Large Deviation functional in the static case is precisely the surface free energy: this has been proven in the case of zero boundary conditions and this result can be applied to study the Wulff shape of the anharmonic crystal. In the rotation invariant set up of §3 a Large Deviation principle for the hydrodynamic limit has been also established and the surface free energy as Large Deviation functional in the static (periodic) case is recovered. 4.1. A Large Deviation principle. Let consider /^v = ffy (d<£>), defined in (1.4), with DN = NDC\Zd, N a large positive integer and D a connected domain in Kd with piecewise smooth boundary. As in §3 we will be dealing with the rescaled interface £N '• D —> M, defined by ^AT(^) = (l/N)tprN if rN € Zd, and by a straightforward polilinear interpolation in the general case. B-L inequalities, via (1.2) and the symmetry of V, easily yield laws of large numbers: for example that for every e > 0
0.
(4.1)
We ask ourselves the question: what are the asymptotics of the (vanishing) probability that £N(-) is in the neighborhood of a fixed non zero trajectory h : Rd -> K. Let us consider the surface free energy ^u(h) = JD, that is for every measurable set E C L?(D) we have that
< liminf ±z logPMjv fa e E) < limsup -
logPMjv (&v € £ ? ) < - inf
where E° (resp. E) is the interior (resp. the closure) of E. The heart of the proof of Theorem 4.1 is in the computation of the logexponential moment generating function. In fact, as it may be clear from the result, the log-exponential moment generating function is strictly convex (this is directly implied by (3.2)) and therefore (4.2) follows from the idea in the Gartner-Ellis Theorem [11]. However this computation is rather involved and the proof uses arguments which are essentially hydrodynamic limit arguments. As a matter of fact one can prove much more than (4.2), namely that a form of local equilibrium holds for weakly perturbed Gibbs
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measures (see [10, Th.1.2]). This sheds some light on the nature of the fluctuations in the non equilibrium case (see §6 below). The presence of the boundary, as opposed to working on a torus, does create technical difficulties and some of the most involved estimates address the control of the height of the mean of the (perturbed) field close to the boundary: once the mean is controlled, one has a very good control on the field itself via B-L inequalities. 4.2. A model for a droplet on a wall. We apply the Large Deviation result of the previous subsection to investigate the asymptotic shape of the massless field with zero boundary conditions outside DN when we impose the constraints that the interface lies above the level 0 (hard-wall condition) and that the rescaled volume covered by the interface is strictly positive (volume condition). More precisely we study the droplet measure V.-\- f i
\
/i
I /~\~f-
,—x ITI
/C
T7"\\
yLijy (d
/ A o\
(4-3)
where
Q+ = {y?; 0 for every x € DN} ,
(4-4)
and
(
,
} (4.5)
with V and <5 > 0. Prom Theorem 4.1 one can easily extract the following statement, that we can view as a proof, for massless models, of the macroscopic Winterbottom construction. Theorem 4.2. [10, Th.1.4] If we fix the total volume V > 0, for every e > 0 we have that lim lim P v,+ Il I|S5 £AT - h
>e
=0,
(4.6)
where h is the unique minimizer of the variational problem inf JE D (/i) : h € H$(D), I h(x)dx = v\ .
(4.7)
Conditioning with respect to fi^ is far from affecting the probability we are computing, because PMJV(O^) > exp(—cN d ~ l ) for some c > 0 [9] and therefore it is much more probable than a large deviation event. If the conditioning on fi^ is removed, the result is then standard a standard consequence of Th. 5.1 (see [10, §5]). 4.3. Hydrodynamic limit and large deviations. It is certainly natural to investigate the Large Deviations also in a non equilibrium set up, more precisely the Large Deviations from the hydrodynamic limit. This has been considered in [17] where the authors prove a Large Deviation Principle associated to the law of large number in Theorem 3.2. We refer to the original paper for statements: here we point out only the fact that the expected
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connection between the dynamical LD functional and the static functional (or surface free energy) Si can be actually verified [16]. 5. HOMOGENIZATION AND CENTRAL LIMIT PHENOMENA
In this section we step back to the translation invariant equilibrium case and consider the problem of space—time fluctuations. 5.1. The fluctuation field and its scaling limit. Let us fix the dimension d(= 1, 2, 3, . . .: note however that the case d = 1 can be throughly studied via much more elementary tools). We choose a tilt v and consider ^(dr?) = ^5(d?y). For any test function / e C£°(Rd;R) = «S(Rd) and any i = 1, 2, . . . , d, we define
Cf (/, t) = N-W
f(x/N) [7?(^+ei)(7V2i) - vt] .
(5.1)
The fluctuation field C^ € C°(R+; (<S'(Rd))d) is defined by «f(*), /)S(R««) = <**(/> *)>with (•> ")s(Rd) the duality between S(Rd) and S'(Rd) (below we wil omit the subscript £). The main result of this section states that £N converges to a limit that we will denote by £. £ is a continuous (<S'(R
d (Ci(t), /) = - (C<(<), Bf) d* + V^d (W(t), Vtf) ,
(5.2)
for every test function / and every i. Wis the standard cylindrical Brownian motion, i.e. the (W, /) is a standard Brownian motion for every / G «S(R) such that JR / 2 (r)dr = 1. Moreover B is the second order elliptic operator — V-gV, and g is a strictly positive e?x d matrix (to be defined later). It is not too difficult to see that if £(0) is a centered Gaussian field with covariance
(5.3) with g one more test function and f ( k ) = (27r)~d/2 JKd elk'rf(r)dr, then the evolution 5.2 is stationary. Of course fig denotes the measure associated with this random field on (<S/(Rd))
'(5.4) where tt is any point in Rd, ^A : £ —>• R is local, smooth and bounded, Dii/>(ri) = il}(Teirj) — "4>(rf). It is clear that q is tilt (v) dependent and it is easy to show that q is strictly positive. We have the following:
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Theorem 5.1. [24, Th.2.1] As N tends to GO, (N € C°([Q,T\;(S'(Rd))d) converges in law to the Gaussian process £, with q given by (5.4). For fixed time this result reduces to the convergence of C^(0) to a continuum free field [32], [24]. 5.2. Homogenization for a random walk in random environment.
The proof of Theorem 5.1 follows roughly the standard procedure to prove central limit theorems for the fluctuations from the hydrodynamic limit [28] , even if some modifications are required. The basic tool to make the crucial estimates turns out to be the HS representation in infinite volume (2.12). The interesting side of the proof is that, to a certain extent, the fluctuation problem can be reduced to understanding the large scale behavior of the random walk in dynamic random environment X(-) (see [32] for a non probabilistic analog). Let us consider for example the covariances of the fluctuation field: by applying (2.12) we obtain that
f
Jo
where d- g and 9- / are discrete gradients of / and g on (l/N)Zd. It is therefore clear that the limit behavior of XN(-), with XN(t) = X ( [ t N 2 ] ) / N , plays a central role. What one can prove, following ideas in [29], is that if we view XN as an element of D([0, T]; R d ) (D is the Skorohod space of right continuous functions), then XN converges in law to X°° e C°([0,T];Rd) with X°° the d-dimensional Brownian motion characterized by
E[X°°(t)X?°(S)}=(t
A * )qitj,
(5.6)
with q given in (5.4). It is easy to verify that formally this leads to the covariance of the £ field defined via (5.2). However the proof of this fact is delicate, because the convergence of XN holds if t < T, any fixed T: we have therefore to truncate the integral in the right-hand side of (5.5) and estimate the remainder. The estimate of the remainder follows directly from the basic Nash inequality if d > 3, since this inequality guarantees that the integrand in the right-hand side of (5.5) is bounded by c||V/||i||Vp||ii~~<*/2, for fome c > 0. This clearly is not sufficient if d = 2, where a more articulate strategy has to be employed: roughly speaking one has to take advantage of the presence of the (discrete) gradient in front of / and g and of the fact that the heat kernel of X has some regularity properties (analog to the Holder continuity in the continuum) under the only assumption of strictly elliptic jump rates (see [24] for details). One can therefore obtain convergence of the covariance to its limit value: we are still quite far from proving that the limit process is Gaussian. In reality, knowing explicitly the covariance comes very handy (together with
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reversibility) in proving the so called Boltzmann-Gibbs principle of fluctuating hydrodynamics, which allows to identify the limit of the drift in the semimartingale equation for £N, and in showing that it is a linear functional of C- The limit is therefore Gaussian and, using the stationariety of the approximants £N, one identifies the limit.
6. OTHER DIRECTIONS OF RESEARCH AND OPEN PROBLEMS Here we just give a quick and partial sample of active directions of research and of some open problems. 6.1. Further directions. Particular attention has and is being devoted to • Entropic repulsion: static problems. By this we mean the reaction of the interface to the presence of a wall [7],[31]: it boils down to studying properties of p,^+ in the case in which 6 = oo (cf. 4.5), that is in absence of volume condition. We refer to [9] for the non Gaussian case and to [4] for the several results achieved in the Gaussian case. • Entropic repulsion: dynamical problems.
In this context one can
find a hydrodynamic limit on a torus [17], in all dimensions, and an equilibrium fluctuation result in d = 1 [18]. • Pinning and wetting. A rather complete picture of the localizing effect of a one-body attractive potential is given in [12] and the effect on decay of correlations in d > 3 is considered in [27]. For an analysis of the competition between entropic repulsion and attractive one-body potentials (wetting problem) we refer to [8] and references therein. We refer instead once again to [4] for analogous problems in the Gaussian context. • Exploiting further the HS representation. A way of closing the gap between Gaussian and non Gaussian results would certainly be understanding the inhomogeneous walk in great depth, possibly exploiting the fact that the random environment in which the particle hops is time and shift ergodic. In this line the most advanced results have been obtained by T. Delmotte and J.-D. Deuschel [private communication (2000)]. However some results should be achievable even regardless of any detail of the environment (see [22] for some steps in this direction). 6.2. Some open problems. In the opinion of the author the following issues are of great interest both from the point of view of modelization and
because they offer challenging mathematical questions. • What can be done if, in (1.2), c+ = oo? This is a challenging question on several levels. Let us observe that our basic assumption (1.2) is not satisfied by the classical example V(r}} = rj2 -f rf. It is remarkable that a more attractive potential, like this one, creates substantial mathematical problems even only at the level of existence
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and uniqueness of the dynamics [15]: these problems, on physical grounds, are rather unexpected, (observe for example that the associated Gibbs measure is well behaved [6]). • What can be done if c_ = 0 or even if c_ < 0? Surprisingly enough some answers (in the case c_ = 0 can be found already in [6]: but they are essentially the only ones for the moment. With an HS representation viewpoint, it seems quite clear that one should succeed in exploiting the presence of an evolving random environment to make up for the lack of convexity (see [2],[25],[33] for cases in which such an idea has been put to work: however the problem for massless fields seems more challenging). • Can one prove a Central Limit Theorem that applies to more general situations? Like for example to the hydrodynamic setting or to the droplet measure. This is a very interesting and challenging issue: despite nonequilibrium fluctuation results have been proven in particular models (see references in [23] and [35]), the problem of non equilibrium fluctuations is far from being understood.
ACKNOWLEDGMENTS I would like to thank the organizers of the conference on SPDEs held in Trento in January 2000 for the invitation of the support. The work that I reviewed has been supported in part by MURST (cofin99) and GNAFA.
REFERENCES [1] D.B. Abraham, Surface structures and phase transitions - exact results, Phase transitions and critical phenomena, Vol. 10, 1-74, Academic Press (1986). [2] V. Bach, T. Jecko, J. Sjostrand, Correlation asymptotics of classical lattice spin systems with nonconvex Hamilton function at low temperature, Ann. Henri Poincare I (2000), 59-100. [3] T. Bodineau, G. Giacomin and Y. Velenik, On entropic reduction fluctuations, J. Statist. Phys. 102 (2001), 1439-1445. [4] E. Bolthausen, Random walk representations and entropic repulsion for gradient models, Infinite Dimensional Stochastic Analysis, Koninklijke Nederlandse Akademie van Wetenschappen, Ph. Clement et al. eds., (2000), 5584. [5] H.J. Brascamp and E. Lieb, On extensions of the Brun-Minkowski and Prekopa-Leinler theorems, J. Fund. Anal. 22 (1976), 366-389. [6] H.J. Brascamp, E.H. Lieb and J.L. Lebowitz, The statistical mechanics of anharmonic lattices, Bulletin of the international statistical institute, Proceedings of the 40th session, Warsaw 1975, Book 1: invited papers (1976) 393-404. [7] J. Bricmont, A. el Mellouki and J. Prohlich, Random surfaces in statistical mechanics: roughening, rounding, wetting, J. Stat. Phys. 42 (1986), 743-798.
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[8] P. Caputo and Y. Velenik, A note on wetting transition for gradient Gelds,Stoch. Process. Appl. 87 (2000), 107-113. [9] J.-D. Deuschel and G. Giacomin, Entropic Repulsion for Massless Fields, Stock. Process. Appl. 89 (2000), 333-154. [10] J.-D. Deuschel, G. Giacomin and D. loffe, Large deviations and concentration properties for V interface models, Prob. Theory Rel. Fields 117 (2000), 49-111. [11] J.D. Deuschel and D.W. Stroock, Large Deviations, Academic Press, Series in Pure and Applied Mathematics 137 (1989). [12] J.D. Deuschel and Y. Velenik, Non-Gaussian surface pinned by a weak potential, Prob. Theory Rel. Fields 116 (2000), 359-377. [13] R. Fernandez, J. Froehlich and A. D. Sokal, Random walks, critical phenomena, and triviality in quantum field theory, Springer-Verlag, Berlin (1992). [14] M. I. Freidlin, Markov processes and differential equations: asymptotic problems, Lectures in Mathematics ETH Zurich. Birkhauser (1996). [15] J. Fritz, Infinite lattice systems of interacting diffusion processes, existence and regularity properties, Z. Wahrsch. Verw. Gebiete, 59 (1982), 291-309. • [16] T. Funaki and T. Nishikawa, Large Deviations for the Ginzburg-Landau V<)> Interface Model, Prob. Theory Rel. Fields 120 (2001), 535-568. [17] T. Funaki, T. Nishikawa and Y. Otobe, Hydrodynamic limit for V0interface model on a wall, preprint (2000). [18] T. Funaki and S. Olla, Fluctuations for V<|) interface model on a wall, Stock. Process. Appl. 94 (2001), 1-27. [19] T. Funaki and H. Spohn, Motion by mean curvature from the GinzburgLandau V interface model, Comm. Math. Phys. 185 (1997), 1-36. [20] H.-O. Georgii, Gibbs Measures and Phase Transitions, Studies in Mathematics, 9, W. de Gruyter ed. (1988). [21] G. Giacomin, Anharmonic lattices, random walks and random interfaces, Recent research developments in statistical physics, vol. I, Transworld research network (2000), 97-118. [22] G. Giacomin and G. Posta, On recurrent and transient sets of inhomogeneous symmetric random walks, Elect. Comm. Probab. 6 (2001), 39-53. [23] G. Giacomin, J. L. Lebowitz and E. Presutti, Deterministic and stochastic hydrodynamic equations arising from simple microscopic model systems, Stochastic partial differential equations: six perspectives, 107152, Math. Surveys Monogr., 64, AMS (1999). [24] G. Giacomin, S. Olla and H. Spohn, Space-time fluctuations for GinzburgLandau V(p model, Ann,, Probab., July 2001. [25] B. Helffer, Remarks on decay of correlations and Witten Laplacians, Brascamp-Lieb inequalities and semiclassical limit, J. Fund. Anal. 155 (1998), 571-586.
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[26] B. Helffer and J. Sjostrand, On the correlation for Kac-like models in the convex case, J. Stat. Phys. 74 (1994), 349-409. [27] D. loffe and Y. Velenik, A note on the decay of correlations under 6pinning, Prob. Theory Rel. Fields 116 (2000), 379-389. [28] C. Kipnis, C. Landim, Scaling limits of interacting particle systems, Grundlehren der Mathematischen Wissenschaften 320, Springer-Verlag, (1999). [29] C. Kipnis, S. R. S. Varadhan, Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions, Comm. Math. Phys. 104 (1986), 1-19. [30] G.F. Lawler, Intersections of Random Walks, Probability and its Applications, Birkhauser, (1991). [31] J.L. Lebowitz and C. Maes, The effect of an external field on an interface, entropy repulsion, J. Stat. Phys. 46 (1987), 39-49. [32] A. Naddaf and T. Spencer, On homogenization and scaling limit for some gradient perturbation of a massless free field, Comm. Math. Phys. 193 (1997), 55-84. [33] J. Sjostrand, Correlation asymptotics and Witten Laplacians, St. Petersburg Math. J. 8 (1997), 123-147. [34] F. Spitzer, Principles of random walks, Springer-Verlag, second edition (1976). [35] H. Spohn, Large scale dynamics of interacting particles, SpringerVerlag (1991). [36] H. Spohn, Interface motion in models with stochastic dynamics, J. Stat.
Phys. 71 (1993), 1081-1132.
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Second Order Hamilton-Jacobi Equations in Hilbert Spaces and Stochastic Optimal Control FAUSTO GOZZI Dipartimento di Matematica per le Decision! Economiche, Finanziarie e Assicurative, Universita di Roma "La Sapienza", Via del Castro Laurenziano n. 9, 00161 Roma, Italy
Abstract. This paper is devoted to a brief survey of known results on second order Hamilton-Jacobi (HJ from now on) equations in Hilbert spaces related to stochastic optimal control problems. More precisely the paper focus on: 1. stating results of existence and uniqueness of regular solutions (i.e. at least Frechet differentiable in the space variable) for semilinear second order HJ equation;
2. when the HJ equation is associated to a stochastic optimal control problem, applying these results to the associated stochastic optimal control problems to obtain verification theorems and feedback formulae for the optimal control strategies.
1
Introduction
Second order HJ equations are a well-studied subject in the finite dimensional case, mainly for their relationship to stochastic optimal control problems since they can be useful to provide the verification and the synthesis of optimal control strategies. The infinite dimensional case is less studied but still can have interesting applications in studying SPDB driven stochastic optimal control problems. Such applications usually needs some regularity of the solution (i.e. at least one time differentiable in space). This paper try then to give a survey of known results in this direction, more precisely: 255
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1. existence and uniqueness of regular solutions (i.e. at least Frechet differentiable in the space variable) fo semilinear second order HJ equation; 2. when the HJ equation is associated to a stochastic optimal control
problem in infinite dimension, application of these results to the associated stochastic optimal control problems to obtain verification theorems and feedback formulae for the optimal control strategies. We shall be concerned mainly with the case of semilinear equations (which correspond to stochastic optimal control problems driven by additive noise) since this is the most studied subject. More precisely the HJ equations that we will study are of the following type:
7 = T* [Qvxx}+ < Ax, ux > +H(t, ot 2
x, ux),
t€ [0, T[, x € D(A)
v(0, x) = (x), x € X, (1)
(S)
Xu + -Tr [Quxx}+ < Ax, ux > -H(x, ux) = 0, Zi
a; € D(A)
(2) where X is a separable Hilbert space, Q : X i-> X is a linear bounded
selfadjoint operator, A : D(A) C X 1-4- X generates a strongly continuous semigroup, tf> : X i-» IR is a measurable bounded function and the so called "Hamiltonian" H : [0, T\ x X x X\ (X\ C X that means possible unboudedness of H in third variable) is a nonlinear function. When these equations are related to some stochastic optimal control problem then the function H is a supremum of linear functions in the third variable (ux) and the HJ equation is called Hamilton-Jacobi-Bellman (HJB from now on) equation: this will be mainly the case treated in this work. The above equations are a special case of the more general second order HJ equations
-
di
= H(t, x, ux, uxx),
t 6 [0, T[, x€X1 (3)
u(0, x) = (x), x € X,
Xu = H(x,ux,uxx) = Q,
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x€D(A).
(4)
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To explain better the problem we first consider the case of finite dimensional X. It is well known that in such case existence and uniqueness of regular solutions is false for a wide class of second order HJ equations, (see e.g. [37]) while a more general existence and uniqueness results holds if we consider a weaker concept of solution, the so-called viscosity solutions introduced by Crandall and Lions (see [24] for a survey on this topic). However when regular solutions exist and are unique, much stronger results of verification and optimal synthesis hold (e.g. the existence and uniqueness of optimal feedbacks) for the associated stochastic control problem. This is one of the reasons (of course not the only one) why it seems worth to develop a theory for the existence and uniqueness of regular solutions for a class of HJB equation as wide as possible: many efforts in this direction have been made, finding a lot of of cases where regularity of solution holds (in particular in the case when H is uniformly elliptic a huge amount of work in this direction has been done, see e.g. [11, 23, 25, 58, 71]). The picture for the infinite dimensional case goes along the same lines but with some important differences mainly due to the fact that much less work has been done in this direction, so much less results are available in the finite dimensional case and the results obtained are quite far from being complete. • The theory of viscosity solutions (see [62, 63] for first results in this direction, and also [45, 46]) has been extended to this case covering a large class of HJ equations with general assumptions on the data but with a hole: it covers only the case when the operator Q is nuclear while many examples coming from applications calls for a study of the case when Q = Identity or non nuclear. This fact is due to the so called Alexandroff theorem (see [24, Appendix]) which is a key tool for the theory of viscosity solution in finite dimension that does not hold in infinite dimension. The infinite dimensional theory then is developed by approximating infinite dimensional problems with finte dimensional ones. But then strong approximation results are needed (namely convergence of projections) that holds only when Q is nuclear. Only recently, with the paper [45] an extension of the theory to the case of nonnuclear Q has been started. • On the other side a theory of existence and uniqueness of regular solutions has been started by Barbu and Da Prato (see [1, 2, 3]) and
then developed by others (including the author of this paper), see e.g. [10, 12, 13, 19, 20, 22, 27, 28, 29, 30, 38, 48, 40, 39, 41, 42, 44, 60]. This theory is based on the deep study of the linear part of HJ equations like (1) and (2) when H = 0 (the so called Kolmogorov equations studied e.g. in ([33, Chapter 9])) and then on the use of perturbation
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methods for the treatment of the nonlinear part.
In this paper we try to give an idea of the results available with this second approach. By simplifying a bit such results could be divided in two families depending on the spaces where the solution is found. • Spaces of continuous functions (see e.g. [12, 13, 19, 20, 27, 28, 29, 30, 32, 48, 39, 41, 42, 44, 60]. This approach has been developed first and allows to get stronger regularity results, but with some limitations due to the non reflexivity of the spaces of continuous functions.
• Spaces of integrable functions with respect to a given reference measure (usually the invariant measure associated to the linear part of the HJ equation, see e.g.[10, 22, 38, 40]). This approach has been developed later and still few results are available. Less regularity is found with this approach, however, some useful results for the control problem can still be proved. Moreover, working in L2 spaces allows to cover cases that do not fit in the first case like the control of stochastic delay equations (see [40]) Since both approaches are based on a perturbation approach, we will describe here only the approach in spaces of continuous functions. The results in this direction are not of course complete and many things still need to be improved and clarified. However it seems to us that
• some interesting example can be covered, like some boundary control problems; • also cases when Q = I (and in general Q is non nuclear), where the theory of viscosity solutions does not work (except for the case studied in [45]), can be treated; • strong results on the associated control problems (like existence and uniqueness of optimal feedbacks) can be obtained. We are aware of the fact that this is a new and developing area where still many interesting and challenging problems coming from applied problems remain open. It is enough to think to partially observed optimal control problems (see e.g. [57, Part II] and [46]), adaptive control (see e.g. [35, 39]), financial applications (e.g. the robustness of the Musiela model of interest rates, see [47] for the problem of robustness of interest rate derivatives) and so on. In particular we note that almost all the results we mention on existence and uniqueness of regular solution hold in the case of additive noise. It is of course very interesting to look at the case of multiplicative
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noise (in particular in the case of control dependent diffusion, which gives rise to fully nonlinear HJB equations, see on this [72]), but this is mainly to be done (see [32] for a first result in this direction). The plan of the paper is the following: first we describe a motivating problem in Section 2, then in Section 3 we describe a tipical result on existence and uniqueness for semilinear HJ equations and in Section 5 its application to the model problem of Section 2. Section 6 is devoted to describe two examples. In Section 7 we give some more details on the literature on HJ equations in infinite dimension.
2
Description of a model problem
We start by describing a typical stochastic control problem we will deal with and which is our model problem. Consider a stochastic controlled dynamical system modelled by the stochastic differential equation in a separable Hilbert space X (the state space)
dy(s) = [Ay(s) + F(y(s)) + Bz(s}\ ds + ^dW(s),
t<s
(5) y(t) = x,
x€X
where A : D(A) C X M- X and B : D(B) C U i-4 X are linear operators, F : D(F) C X i-4- X is nonlinear, Q is a symmetric nonnegative operator on X and W is a cylindrical Wiener process on a given probability space (fl, F, IP). The function z : [t,T\ \-t U (where U is another separable Hilbert space: the control space) belongs to M^-(t,T;U) which is the space of all stochastic processes with value in U that are square integrable and progressively measurable with respect to W and represents the control strategy while the function y(-) represents the state trajectory. The horizon of the problem is the fixed number T > 0, possibly infinite. Under suitable assumptions, that we will precise later, the above equation (5) has a unique solution that will be denoted by y(-- 1, x, z] or, for short by ?/(•). Consider then the optimal control problem of minimizing the cost
functional
( rT
J(t, x; z) = IB \ / \g(y(s; t, x, z)) + h(z}]ds +
](6)
}
Here g and y : X —>• 1R are Borel
measurable and bounded function, and h : X —¥ IR is convex and lower semicontinuous .
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The value function of this problem is defined as
V(t, x) = V '
inf
J(t, x; z) V
(7) V
'
'
and by a classical argument of control theory (see e.g. [37, p. 137] in the finite dimensional case and [64, 56] for the infinite dimensional one) , we expect that V satisfies the following functional equation
V(t,x)=
inf
IE
which is well known as the Bellman's Dynamic Programming Principle. On the basis of the Dynamic Programming Principle above (simply assuming V regular enough, dividing by T — t and then letting r —y t) we expect that the function V is a solution (in a suitable sense) of the following Hamilton-Jacobi-Bellman equation
dt
+
2
Tr [Qvxx]+ -H0(vx) + g(x) = 0, (g)
t € [0,T[, x € D(A)v(T,x) =
= h*(-B*p) = sup{-(Bz,p) - h(z)} z&U
SupH0,CV(z,p).
(9)
Here HQ^CV is the so-called Current Value Hamiltonian and h* is the Legendre transform of h (see e.g. [59]). Remark 2.1 If we consider more general control problems where the nonlinear part is given by a function Fi(t, x, z) (instead of F(x) + Bz) and the current cost is given by l(t,x,z) (instead of g(x) + h(z}}, then the related HJB equation would be exactly like 1 with the Hamiltonian H given by
H(t,x,p) = sup{-(Fi(t,x,z),p) -l(t,x,z)} d= su.pHCv(t,x,p;z).
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The so-called dynamic programming approach to these kind of problems consists in solving first equation (8) and second to prove a "verification theorem" , namely that the solution v of (8) is the value function V defined in (7). This can be done by proving the following fundamental identity for
v(t, or)+E //
[Ho(Dxv(s, y(s; t,x,z))) — HQJCV(Z(S), Dxv(s, y(s; t, x, z)))} ds =
Jt
J(t,x;z). When the solution v is sufficiently regular (i.e. at least C1 in the space variable) and H is differentiable, the above identity allows us to prove the equality v = V and that the existence of an optimal control z* given by the feedback formula *{ \ —
R* c ^°(r>
I
*( \\\
(^^\
where y* is the optimal state given by the solution of the closed loop equation
dy(s) = [Ay(s) + F(y(s)) - Bz*(s)} ds + ^/QdW(s),
t<s
y(t) = x,
x € X.
This procedure of finding a feedback formula for the optimal control is called the "synthesis of the optimal control" and is very important for applications (see e.g. the books [37, 55] for comment on this). In the infinite horizon case (i.e. when T = +00) we generally set 0)
J(x-z} = IE { / ° ° e - X s [ g ( y ( s - 0 , x , z ) ) + h(z)}ds\
(.Jo
>
over all controls z € M^(0, oo; U) (note that we suppress the dependence on t here) and define the value function as before V(x)=
inf
J(x;z)
The procedure summarized above for finding a PDE satisfied by the value function above leads in this case to the following stationary equation
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Gozzi
\v + i Tr [Qvxx] + -H0(vx) + g(x) = 0,
x£ D(A) (13)
which is the elliptic analogue of (8).
3
Existence and uniqueness results for HJB equations
In this section we study Hamilton-Jacobi equations of the kind (8) as a PDE in a Hilbert space without developing its connection to optimal control problems. We study existence and uniqueness of regular solutions (i.e. at least Cl in the space variable) by using a perturbation approach which has been introduced first in [12] and then developed in [13, 41, 42, 19]. Here we present this approach using a more general and complete setting (partly suggested in [73]) that is the subject of a forthcoming paper [43]. The proofs are similar to the ones of the paper quoted above. We will not give then for brevity, referring to the above quoted papers. Throughout this section we will be dealing with the following second order Hamilton-Jacobi equation in a separable Hilbert space X with norm | • \x and scalar product < •, • >x, omitting the subscript X when clear from the context.
=
Tr [Quxx}+ < Ax,ux > -H(t,x,ux),
t e]0,T], x € X
u(0, x) = (f(x), x € X. (14)
We will always assume the following Hypothesis 3.1 (i) X is a separable real Hilbert space and T is a fixed positive number.
(ii) The linear operator A : D(A) C X —>• X is the infinitesimal generator of a strongly continuous semigroup etA on X so there exist M >1 and u € IR such that
\\etA\\c(x) < Me"* (Hi)
We have Q € £+(X)
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and, for any t > 0,
Vt > 0
Hamilton-Jacob! Equations and Stochastic Optimal Control
IV [Qt] < +00,
263
(15)
where Qt = f*esAQesAtds. (iv) For every t > 0 we have etA(X) C Q\I2(X}.
(v) Setting, for t > 0, T(t) = <3t~1/2eiA, the map t 1-4 ||r(i)|| is locally integrable on [0,+00). Remark 3.2 The Hypothesis 3.1 above guarantees strong regularity properties of the Ornstein-Uhlenbeck semigroup associated to the linear operator
A
d
= \ Tr [QD2 . Zi
One can see e.g. [33, Appendix B], or [41] for more comments on Hypothesis 3.1 and its control theoretic interpretation. Sometimes, for simplicity of presentation, we will work with the stronger assumption that ||r(t) || < Ct~a for given C > 0 and a € (0,1) explaining how to work in the general case.
Regarding the data and H we could consider three possible groups of assumption: the first for the case of Lipschitz Hamiltonian, C1 solutions, the second for Lipschitz Hamiltonian, C2 solutions, the third for locally Lipschitz Hamiltonian. We will present here only the first case, the others can be found e.g. in [41, 42]. A delicate point in these assumptions (that do not appear in the quoted papers) is the fact that the Hamiltonian H can be unbounded with respect to p. As we said in the introduction, the domain of H(t,x, •) (i.e. with respect to p) is a space X\ C X. In the case of boundary control problems such unboudedness comes from the unboudedness of the operator B in the state equation (5) and in this case X\ = D(B*) C X. Such operator does not appear explicitely in the general form of the HJ equation (14). If we want study it in itself it is quite natural to set assumptions in the following way: first we assume that there exists a suitable closed operator B\ with suitable properties and then that the Hamiltonian (in p) is well defined in D(B\) C X and is Lipschitz continuous in such domain. More precisely we will assume first that Hypothesis 3.3 For a given dosed operator B\ : D(B\) C X i-4- X such
that D(A) C D(Bi), the operators T(t)Bi (and then also etABi = Ql/2r(t)) extend to a continuous operators on X and the function t -4- \T(t)Bi\ (and then also He^Bi}] < ||Qj/2||||r(t)||; is integrable on [0,T].
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Remark 3.4 Of course Hypothesis 3.3 above is always satisfied if the couple (A, Q) is null controllable and B = Identity. We will be interested in the case when A generates an analytic semigroup and B = A@ for some (3 € (0,1], the typical framework of boundary control problems like the one presented in §6 (see [33, Appendix B]. •
Then we assume that for such operator B we have Hypothesis 3.5 (Lipschitz Hamiltonian in p, C1 solutions)
(i) V£ (ii) H : [0,T] x X x -D(-BJ) —> IR is measurable and there exists a constant CH,O such that
\H(t,x,p)-H(t,x,q)\ < CH,o\B^(p-q)\ Vi € [0,T], x € X, p,q €
\H(t, x,p)\ < CH,o(l + \B*lP\) \/t € [0, T\,xeX,P Remark 3.6 As noted above we remark that here we will present a generalization of the results given in [12, 13, 41, 42], namely:
• we will take the map 11-4 ||r(t)|| simply integrable, and not dominated by Ct~a for some C > 0 and a
• the Hamiltonian H is written in a more general form, in particular it depends on t and it can be only Borel measurable with respect to x. So examples like H(t,x,p) =< F(t,x),p > +Ho(t,p) with F € Bb([0, T] X X) (that did not fit in the setting of [12, 13, 41, 42]) can be treated here. • the Hamiltonian can be defined in the narrow set D(B^) so that more general control problems (like the Neumann boundary control
described in Section 6) can be treated.
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We sketch the procedure used to solve problem (14). We first consider the linearized problem r\
-j
-^ = - Tr [Quxx}+ , dt 2
t > 0, x e X
^
u(0,x) = f ( x ) , x € X, which is the so-called backward Kolmogorov equation associated to the Ornstein-Uhlenbeck stochastic differential equation \ dZ(s) = AZ(s)d8 + ,/QdW(s), \ Z(0) = x, x€X.
s>0
As it is described in [33, Chapter 9] or in [41, 42], under Hypothesis 3.1 equation (16) has a unique classical solution (see again [33, Chapter 9] for
the concept of classical solution of (16)) represented by the formula u(t, x) = Rtf(x} = IE{f (Z(t, x ) ) } = I if>(y)J\f(etAx, Qt)(dy)
(18)
tJ j\
where {Rt, t > 0} is the so-called Ornstein-Uhlenbeck semigroup (see [33, Chapter 9]). By using the semigroup Rt we write problem (14) in the following integral form
u(t, x) = Rt
global solutions of (19) we need to prove suitable apriori estimates for the norm of the local solutions that requires some more assumptions, see on this [19, 42, 60]. The solution of (19) that we will find with the method described above will be called a mild solution of equation (14). To show the connection
between the concept of mild solution and the classical one (which is a key point for applications in control theory) we will show that the mild solution can be seen as the pointwise limit of classical (that we call strict, following
[33, Chapter 9]) solutions of suitable approximating problems, i.e. a strong solution, as defined in [21, 41]. We will make then use of three different concept of solutions for equation (14).
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• STRICT (see Definition 4.1) • STRONG (see Definition 4.3) • MILD (see Definition 3.7) From now on we always assume that Hypothesis 3.1 and 3.5 hold and we solve the mild equation (19) under Hypotheses 3.1 and 3.5 following the method introduced in [12] and developed in [41]. This method consists of applying to the equation (19) the contraction mapping principle in a suitable space of functions u : [0, T] X X -4- IR whose spatial derivative blows up in 0 at an integrable rate. The choice of the spaces in [12] presents some problems (see the discussion in [41, §1, §4]), so from now on we will mainly refer to the setting introduced in CPDE. To introduce it we recall the following definitions. For J C IR and X, Y Hubert spaces we denote by Bb(I x X, Y), C(I x X, Y) Cb(I x X, Y), UCb(I x X, Y) the set of all functions tp : / x X -4 Y which are, respectively, Borel measurable and bounded, continuous, continuous and bounded, uniformly continuous and bounded on IX X. The spaces Bb(I x X, Y), Cb(I x X, Y), UCb(I x X, Y) are Banach spaces with the norm |M|0=
sup
(t,x)£lxX
\(t,x)\Y.
We define
UCb(I x X;Y) d=f l € Cb(I xX;Y): sup \u(t, x) — u(t, y)\Y < p(\x — y\) for a suitable modulus p >. (20) It can be easily seen that UCb(I x H) is a closed subspace of Cb(I x H) because uniform continuity is stable with respect to the convergence in the norm || • ||Q. We now introduce suitable weighted spaces of continuous functions. Given T e (0, +00] and a weight 7 : [0, +00) M- [0, +00), -y(t) = ta (a € (0,1)) we Q£»t. set
([0, T} xX;Y)
= lv€ UCb([T, T] x X; Y) Vr e]0, T[, tav bounded j
Moreover, since we will be interested in space derivatives of functions belonging to the above spaces or even more regular we define, for a given closed
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operator E : D(E) C X M- X,
Dxv, Evx e £° ([0, \\v\\^,E
x
= \\v\\o +\\taDxv\\0 + \\taEDxv\\0
(22)
In all the definitions above we will omit Y when Y = M.
Definition 3.7 A function u : [0, T] x X i-v 1R is called a mild solution of problem (14) if u £ ^B ([0, T] x X] for some a E (0, 1) and u is a solution of the integral equation (19). We now give the main result of this subsection.
Theorem 3.8 Assume that Hypotheses 3.1 and 3.5 hold. Then the equation (14) has a unique solution u e £Q'BI ([0,T] x X) for every T > 0, for a suitable a € (0, 1). Remark 3.9 In fact the definitions of the weighted spaces above can be more general including more general type of weights (see on this [43]). This
would add some technical complexity that we want to avoid here.
4
•
Approximation of mild solutions
In this subsection we show that the mild solution obtained in the previous section can be characterized as the limit of classical solutions. As announced above at the end of Section 3 (see also [41] §4.3) we define
Definition 4.1 A function u : [0,T] xX —>• ]R is a strict solution of equation (14) if u has the following regularity properties
u(-,x)€Cl([Q,T}),
\/x£X
u(t) e D(Ao) Vt e [0,T]; supte[0)T] \\u(t)\\D(Ao) < +00
(23)
ut, AQu€UCb([Q,T}xX) and satisfies (14) in classical sense. To define the strong solution from now on we will use the notation
g(t,x) = —H(t,x,0) and Hi(t,x,p) = H(t,x,p) — H(t,x,0) so that we will substitute H(t,x,p) with Hi(t,x,p) — g(t,x). Moreover we recall the following.
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Definition 4.2
(i) A sequence (
lim sup \(pn(x] - f(x}\ = 0 for every compact set K C H In this case we will write
(25) lim
sup
\^Fn(t, x) — F(t, x)\ = 0
for any compact set K C H . In this case we will write, as before T = /C- lim J^n.
nj A subset Y ofUCb(H) is said to be fC -closed if for any sequence Y and if € UCb(H) such that
n)
C
fC-n-t+oo lim (fn —
we have that
A subset Y ofUCb(H) is said to be /C-dense if for any (f> € there exists a sequence (fn] C Y such that
A linear operator A : D(A) C UCb(H) —>• UCb(H) is said to be K,closed if, given a sequence (ifn) C D(A) such that
K.- lim tpn = (f and K,- lim Af>n = ib. n-»-+oo
we have € D(A) and A
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(v) Let A : D(A) C UCb(H] -)• UCb(H) and B : D(B) C UCb(H) -)• UCb(H) be two linear operators and assume that A C B and that B is K.-closed. We say that B is the K-closure of A, and we write B = A , if for every if € D(B) there exists a sequence (if>n) C D(A) such that K- lim (fn = n-H-oo
(26)
K,- lim A&n = B(p n-t+oo
Definition 4.3 Fixed B satisfying Hypothesis 3.3 we say that a function u : [0, T] X X —>• IR is a B-strong solution of the equation (14) if u € ^'B for suitable 77 € T\ and there^exist three sequences {
n
( wt(t, x) = Aow(t, x) + H]_(t, x, wx(t,x))) + Cw(t, x) + 9n(t, x) \ w(Q,x) = +00 n—»+oo
(fn = if
— n
i —y
Ifl
in
T°°
(\C\ T] v IT- Y\
<•'<' -L'Ar®/ir\l>'> J J x •"• i •"•)
and
{ K.- Um un = u ^—^-(-(X;,
mS?([0,T] x X) J.VL
/
J
/
Ac-Jim^ = «xSS([0,ri x X-X)
1C- lim 5*una; = B*uxY%([0,T\ x X;X) Now we apply the same arguments used in [41] to equation (14) to obtain the following. Theorem 4.4 Let T > 0 and B satisfying Hypothesis 3.3 and assume that Hypotheses 3.1, and 3. 5 hold true. Letu € S^ be the mild solution of equation
(i) The function u is the unique B-strong solution of the Cauchy problem (14)- Moreover the K.- convergence is also uniform on bounded subsets of X and, if if (E UCb(X) then we can choose the sequence (
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(ii) If the operator F(t)A can be extended to an operator in £(X) then Dxu(t,) € D(A*} and A*Dxu(t,) <E UCb(X) for every t <= (0,T].
We omit the proof because it is completely similar to the one of Theorem 5.1 in [41].
5
Application to stochastic control problems: verification theorems and optimal feedbacks
In this section we will consider stochastic control problems of the type described in §2 and we show • that the value function of the control problem coincides with the unique mild/strong solution of the associated HJB equation, • existence and uniqueness of optimal feedback controls • a feedback formulas in terms of the space derivative of the value function.
Let X, U be two separable Hilbert spaces (the state space and the control space respectively) and consider a stochastic controlled system governed by the state equation (5) (for results on existence and uniqueness of it see e.g. [33, Ch.7.1], or [12, 41]) where x £ X, A,Q satisfies Hypothesis 3.1, B satisfies the Hypothesis 3.3and F : X i-4 X, z : [0, T] x fl 1-4 U satisfy Hypothesis 5.1
(i) F is globally Lipschitz (ii) z^M^(t,T-X). Remark 5.2 We note also that one may consider more general situations, e.g. F mildly unbounded or time and control dependent (with suitable assumptions) but we avoid to do it here for simplicity. •
On the cost functional we will assume that Hypothesis 5.3 (i) h is convex and lower semicontinuous z and as
(ii) g and
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The value function of this problem is defined as
V(t, x) = inf { J(t, x; z) : z 6 M^(t, T; X)}
(28)
and a control z* € M^-(t, T; U) and V(t, x) = J(t, x; z*) is said to be optimal with respect to the initial state x. The corresponding Hamilton- JacobiBellman equation reads as follows -^ + \ Tr [Qvxx(t,x)} + < Ax + F(x),vx(t,x) > = H0(vx(t,x))+g(x),
te]Q,T\,x€X
( 29 )
. v(T,x) = (x), x£X, where the Hamiltonian HQ is given by
H0(p) = sup{-(Bz,P) - h(z}} = h*(-B*p)
(30)
We will denote by flb,cv(p; z) the function (sometimes called the current
value Hamiltonian) (Bz,p) — h(z). Remark 5.4 If h(z) - \z\2 then
The main aim of this section is to prove that, under the general assumption 3.1, 5.1 and 5.3, the value function V is the mild solution of the Hamilton-Jacobi-Bellman equation (29) and that, when H is smooth enough, there exists an optimal control z* (eventually unique) represented by a suitable feedback formula in terms of H and Vx. We want to emphasize here that this result is very interesting in terms of optimal control since it states that the value function is smooth (at least UCl(X) in x} for general costs J with h satisfying 5.3.
Theorem 5.5 Assume that Hypotheses 3.1, 5,1 and 5.3. Let HQ be as in (30). Let v € Z&B([0,T] x X) be the mild solution of (29). Then v = V on [0,T] xX. Then under a more restrictive assumption we have the following result that give existence of an optimal control in feedback form
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Theorem 5.6 Assume that Hypotheses 3.1, 5.1, 5.3, hold true. We also assume HQ and is Gateaux differentiable in p with continuous directional derivatives. Then (i) the unique mild solution v of (29) coincides with the value function V
given in (28); (ii) for any x € X, there exists an optimal control z* ; (Hi) z* is related to the corresponding optimal state y* by the feedback formula
z*(t] = argmaxH0>Cv (Vx(t, y*(t)); z)
W > 0;
(iv) If argmaxHotcv ( V x ( t , x ) ; z ) is always a singleton, then the optimal control is unique. Remark 5.7 Conditions that ensure the required regularity of the Hamiltonian can be found e.g. in [41].
6
Two examples
We present here the formulation of two examples
6.1
An example with distributed control, quadratic Hamiltonian
Let CN = [0, ir]N and X = jL2(Cjv), N < 3 and take the Laplace operator with Dirichlet conditions at the boundary defined as
D(A) = H*(CN) D H£(CN), Ax = Ax, for x e D(A) The operator A generates an analytic semigroup of compact operators. Moreover A is diagonal i.e. there is an orthonormal complete system (e^) A; € IN and a positive strictly increasing sequence aj. (such that limfc_>._|_00 a,^ = +00) in L 2 (Cjv) such that +00
Ax = ^T<x,ek > akek. k=l
It is enough to take, for (HI, ..,HN) € ESP^, 2\f and ani,..,n w (£) =»! + • • • + » & •
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(32)
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so that, by ordering the eigenvalues we obtain _2_
ctfc « kN as k —>• +00 Define now
where / € [/Cfc(IR), a, f3 € t7Ct(IR,). Consider the stochastic optimal control problem of Minimizing the Cost Functional
U
T
r
I h
JCN
I
(33) +
r
1
over all controls z progressively measurable satisfying fc \z(s,£)\2d£ < R2 almost surely for s € [t,T] where the state ?/(•,£) is the mild solution of the stochastic differential equation
dy(s, $) =
(34) driven by a White Noise W.
6.2 An example of boundary control problem A typical example where equations like (8) arise is the one of stochastic optimal control problem driven by parabolic SPDE with the control at the boundary. This is one of the motivating examples for the investigation of equation (1). We now present an example of problem where the operator A is the Laplacian with Neumann boundary conditions, reminding that a similar analysis hold true also for the case of Laplacian with Dirichlet boundary conditions (which is in some sense more difficlut to treat since it gives rise to a stronger unboudedness in the Bellman equation) . For further examples we refer to the
book of I. LASIECKA AND R. TRIGGIANI [54]. This book deals with deterministic boundary control problems. However it can be easily checked that our results apply to suitable additive stochastic perturbations (e.g. in the
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framework of Example 2 below) of examples belonging to the "first abstract class" treated in the book. Let fi C TRn be an open, connected and bounded set with smooth boundary. Consider the following stochastic controlled PDE •)
in (0, oo) x Q on ft
= a(t, £)
(35)
on (0, oo) x <9Q,
where WQ is a Wiener process with values in L2(Q) and with covariance operator Q, the initial datum is XQ G. L 2 (f£), and the control a : (0,oo) —>• L2(<9fi), is progressively measurable with respect to the Wiener process WQ. Moreover / : IR —>• IR is a Lipschitz continuous function, and Q need to satisfy some technical assumptions that are not discussed here for brevity. In this equation the control function a is applied only at the boundary and so it is called a "boundary" control. Using a standard procedure which can be found e.g. in [8] for the deterministic case (the stochastic case does not need any substantial change), problem (35) may be rewritten in an abstract form by finding an equation similar to the one treated in §2.
dy(s) y(i)
= \Ay(s) + F(y(s)) + A^?z(a)] ds + ^dW(s), = x,
t<s
x€X
(36) where X = L 2 (fi) is the state space, let U = L2(<9Q) is the spaces of control parameters, A is the Laplace operator with zero Dirichlet boundary conditions, and F : X x -4- X is defined as
F(x)($ = /(*(£), and finally C : U —> X is a continuous linear operator, the so called Neumann operator. Consider now the problem of minimizing the cost functional (for given horizon T > 0 and starting point t € [0, T)) a 'T g (y(t; XQ, a)) + l———dt (*)l 2 + (y(T; x , a)) 1\ , 0 z J
where A is a positive number, g : L2(Q) —> IR is a bounded measurable function and y(- ]XQ,Q.) is the solution of the Cauchy problem (35). The value function of this problem is defined as
V(t,x) = MJ(t,x;a)
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where the infimum is taken over all admissible controls a considered above. The associated HJB equation is -^ + - Tr [Qvxx}+ -H0(vx) + g(x) Cft
Z.
= 0,
xe[Q,T]
v(T,x) =
= sup {-{Bz,p} - M)} H/ supHcvfoz) z&J I
* )
(38)
z€U
where HCV is the so-called Current Value Hamiltonian. We only point out here that the presence of the boundary control term in the state equation (35) causes that the Hamiltonian H is only defined if Dv(x) € D(A@), where A is the Laplace operator with Neumann boundary conditions and (3 € (|, 1). Clearly, in the case of different boundary conditions, different values of (3 have to be considered, for example the case of Dirichlet boundary conditions gives /3 € (|, 1). This "bad behavior" appears
as a result of "transforming" the boundary control into a distributed one, as it is explained e.g. in A. BBNSOUSSAN, G. DA PRATO, M.C. DELPOUR AND S.K. MlTTER [8]. For other works on HJB related to boundary control problem in the deterministic case see e.g. [16, 17, 18] and in the stochastic case see e.g. [45, 35].
7
Some history
Problem (8) under different assumptions on the data has been treated by many authors. We try here to summarize the main results that can be found in the literature on Hamilton-Jacobi equations focusing on second
order infinite dimensional case.
The finite dimensional case Hamilton-Jacobi equations (in finite dimension) are a classical tool of analysis but it should be said that the study of them has been considerably intensified when their importance in optimal control theory (respectively: differential games theory) was discovered
by Bellman (respectively: Isaacs) (see [7, 49]). To be more precise we recall that the name Hamilton-Jacobi equations classically denoted first order equations (coming from classical calculus of variations applied to mechanics
theory) of the type u(x)
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= H(x,Du(x))
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for stationary case, and
ut(t, x) = H(t, x, Dxu(t, x)) for the evolution case, where the function H is usually taken as a supremum of functions that are linear in Du (see the introduction of the books [9, 72] for a history of these equations). Later on the words Hamilton-Jacobi equations have been also used to denote a wider class of PDE also of second order, more precisely u(x] = H(x, u(x), Du(x), D2u(x)) for stationary (elliptic) case
ut(t, x) = H(t, x, u(x), Dxu(t, x),DxU(t, x)) for the evolution (parabolic) case, where the function H is usually nondecreasing with respect to u and degenerate elliptic with respect to D2u. For equations of the above type related to control theory one generally uses the term Hamilton-Jacobi-Bellman (HJB) equations (characterized by the fact that H is found as a supremum of functions linear in Du) while for the more general class of Hamilton-Jacobi equations related to the theory of differential games (where H is found by a sup inf procedure) one generally uses the term Hamilton-Jacobi-Isaacs (HJI) equations. For these kind of equations one cannot expect in general existence and uniqueness of classical solutions even in simple cases (see [9] for a discussion on this point). To simplify the picture the research can then be developed in two directions: find special cases where one can hope for existence and uniqueness of classical solutions (which is very useful for studying the associated control problem) and/or find a general theory of existence and uniqueness of generalized solutions (in a suitable sense). Something in between is also possible, of course. The classical theory for the first order case (see [9] for an exposition of the classical theory) gave some results of existence and uniqueness of classical solutions and of existence of generalized solutions (in the sense of derivative almost everywhere) but was not able to state general existence and uniqueness results for generalized solutions. Concerning the more recent literature on the subject we recall that
• the problem of finding a general theory of existence and uniqueness of generalized solutions that covers cases where classical solution do not exist or are not unique has been solved for a wide class of HJB equations by Crandall and Lions by the introduction of viscosity solutions (that are in general non differentiable), see the survey paper [24] and the books [6, 4, 37] for a presentation of the argument;
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• the study of the cases when classical solutions of HJB (or HJI) equa-
tions in finite dimensions exists has become a part of the general regularity theory for second order PDE's and has been studied by many authors under various point of view, we recall e.g the books [58, 11].
The infinite dimensional case Infinite dimensional Hamilton-Jacobi equations (in Hilbert spaces to be more precise) were first studied by Barbu and Da Prato (see [I, 2, 3]), in the first and second order case, in connection with optimal control problems for distributed parameter systems (HJB equations). Then the study of them has been developed in many directions. • Regarding the first order case Barbu and Da Prato (see the book [3] and the references quoted therein) and then many others developed the approach of finding classical solutions and generalized solution (in the sense of almost everywhere derivatives) under convexity assumptions via suitable approximations of infinite dimensional convex functions. Moreover the viscosity solution approach was extended to a general class of equations by Crandall and Lions in the series of papers [26] and also by others (see e.g. [15, 16, 18, 14, 51, 61, 65, 66, 67]). • Regarding the second order case, which is the one we treat in this thesis, we give now a more detailed description of what is has been
done in the literature in this field concerning problems like (8). — Viscosity solutions The viscosity solution approach has been treated first in [57], and then in [50, 52, 62, 63, 53, 46, 45] In particular in [63] existence and uniqueness of viscosity solutions for a wide class of second order PDE's (including HJB equations) is stated. When Q is nuclear equation (8) fall into this class. When Q is not nuclear the approach of viscosity solution described in [63] does not work. This fact is due to the so called Alexandroff theorem (see [24, Appendix]) which is a key tool for the theory of viscosity solution in finite dimension that does not hold in infinite dimension. The infinite dimensional theory then is developed by approximating infinite dimensional problems with finite dimensional ones. But then strong approximation results are needed (namely convergence of projections) the holds only when Q is nuclear. A first attempt to extend the theory of viscosity solution to the case of non nuclear Q can be found in [45] for the case of HJB equations associated with stochastic boundary control problems (see [45] where we also prove that regular solution and viscosity solutions
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Gozzi
coincide, when they exists both). We remark that viscosity solution are not differentiable and so in this works differentiability of solutions (also in some weak sense) and existence of feedback controls for the associated stochastic optimal control problem has not been studied. — Regular solutions in the parabolic case The approach of finding regular solutions (i.e. at least differentiable in the space variable) or generalised solutions (in the sense of limits of classical solutions) of parabolic HJB equations has been treated first by Barbu and Da Prato in [3, Ch. 4]) that treated equation (8) in the case when Q is nuclear, F = 0, B = I and , g are convex, by getting existence and uniqueness of (71 solutions in the space variable. Then this approach has been developed in the case of nuclear Q and F = 0, B = I in [27] (avoiding convexity assumptions but without getting Cl regularity of the solution), in [48] (case A = 0). Moreover in the case when Q is not nuclear, F bounded and Lipschitz, B = 7, problem (8) was studied first in [12] for globally Lipschitz Hamiltonian H and for locally Lipschitz ones in [13]. In both paper it is supposed Q = I, A generator of an analytic semigroup of compact operators and existence and uniqueness of a global solution u of class (71 in the space variable x is proved. Then, in the papers [41, 42] the author of this thesis studied (8) for Q positive and selfadjoint, B = I, F bounded and Lipschitz continuous. In [41] the Hamiltonian H is globally Lipschitz while in [42] the case of locally Lipschitz H is studied. Other papers concerning the study of parabolic HJB like (8) are [40] that deals with the L2 approach treating stochastic delay control problems, [19, 20, 29, 30] where the authors study HJB equations related to optimal control of stochastic reaction-diffusion equations and of stochastic Burgers equations, respectively. — Regular solutions in the elliptic case Concerning the case of elliptic HJB (for the theory of regular solutions) we have the paper [44] where the case of stationary HJB equation (13) with A > 0, (related to infinite horizon discounted optimal control problems) is studied in the case of globally Lipschitz Hamiltonian H (the case of locally Lipschitz H is a work in preparation [60]). In the case when A > 0 the stationary HJB equation (13) is related to ergodic optimal control problems and it has been studied in [39] (see also [35] for results in this direction). Finally we recall [22] where existence and uniqueness of HJB equations like (13) are studied in the context of Gauss-
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Sobolev spaces, i.e. Sobolev spaces with respect of suitable Gaus-
sian measures in the underlying Hilbert space H. The results of this paper are obtained only in the case when A and Q commute and they do not cover the synthesis problem. A generalization of this approach that covers the non commutative case and concerns also the synthesis problem is in [40]. — The Linear Quadratic case An history apart deserves the HJB equations associated to the so called "linear quadratic" optimal control problems (i.e. when the state equation si linear and the cost functional is a quadratic form. It is well known that this case is easier to treat since the HJB equation can be reduced to the Riccati operator equation
P' = A*P where KI, K%, PQ depend on the data of the problem. There is a huge literature on this problem, which is the most common in applications. We recall, for the infinite dimensional first order case the book [8] and for the second order case the papers [36, 68, 69, 70, 31]
References [1] V. Barbu and G. Da Prato, A direct method for studying the dynamic programming equation for controlled diffusion processes in Hilbert spaces, Numer. Funct. Anal, and Optimiz. 4 (1), 23-43 (1981). [2] V. Barbu and G. Da Prato, Solution of Bellman equation associated with an infinite dimensional stochastic control problem and synthesis ofoprtimal control, SIAM J. Control and Optim. 21 (4), (1983), 531550. [3] V. Barbu and G. Da Prato, HAMILTON- JACOBI EQUATIONS IN HILBBRT SPACES, Research notes in Math., Pitman, Boston, 1983. [4] M. Bardi and I. Capuzzo-Dolcetta, OPTIMAL CONTROL AND VISCOSITY SOLUTIONS OF HAMILTON-jACOBI EQUATIONS, Birkhauser,
Basel 1997. [5] M. Bardi, P. Soravia, Time-Optimal control, Lie brackets, and Hamilton- Jacobi equations, preprint. [6] G.
Barles,
SOLUTIONS
DE
VISCOSITE
DBS
HAMILTON-JACOBI, Springer- Verlag, Paris 1994.
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EQUATIONS
DE
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[7] R. Bellman, DYNAMIC PROGRAMMING, Princeton University press, Princeton, 1957.
[8] A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, REPRESENTATION AND CONTROL OF INFINITE DIMENSIONAL SYSTEMS,
Birkhauser, Boston, 1992. [9] S.H. Benton, THE HAMILTON-JACOBI EQUATION: A GLOBAL AP-
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[12] P. Cannarsa and G. Da Prato, Second order Hamilton-Jacobi equations in infinite dimensions, SIAM J. Control Optim. 29, n.2 (1991) 474-492. [13] P. Cannarsa and G. Da Prato, Direct solution of a second order Hamilton-Jacobi equation in Hilbert spaces, Stochastic partial differential equations and applications, G. Da Prato & L. Tubaro Editors, Pitman Research Notes in Mathematics 268 (1992) 72-85.
[14] P. Cannarsa and G. Di Blasio, CHECK Second order HamiltonJacobi equations in infinite dimensions, SIAM J. Control Optim. 29, n.2 (1991) 474-492. [15]
P. Cannarsa, F. Gozzi and H.M. Soner, A boundary value problem for Hamilton-Jacobi equations in Hilbert spaces, Applied Mathematics and Optimization, 24, (1991), pp. 197-220.
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P. Cannarsa, F. Gozzi and H.M. Soner, A dynamic programming approach to nonlinear boundary control problems of parabolic type, J. Funct. Anal. 115 (1993) 25-61.
[17] P. Cannarsa and M.E. Tessitore, Cauchy problem for the dynamic programming equation of boundary control, Proceedings IFIP Workshop on Boundary Control and Boundary Variation, Marcel Dekker, 1993. [18] P. Cannarsa and M.E. Tessitore, Infinite dimensional HamiltonJacobi equations and Dirichlet boundary control problems of parabolic type, Siam J. Contr. Optim., 1996.
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[19] S. Cerrai Hamilton- Jacobi equation for reaction-diffusion control systems. To appear in SIAM J. Control Optimiz.. [20] S. Cerrai Stationary Hamilton-Jacobi equation for reaction-diffusion control systems. To appear in SIAM J. Control Optimiz..
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NITE DIMENSIONS, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge (UK), 1992.
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[38] M. Fuhrman and G. Tessitore (2000) A Malliavin Calculus approach to infinite dimensional HJB equations, draft. [39] B. Goldys and B. Maslowski (1998) Ergodic control of semilinear stochastic equations and Hamilton-Jacobi equations, J.M.A.A., 1999. [40] B. Goldys and F. Gozzi (1998) Second order Hamilton-JacobiBellman equations in Hilbert spaces and stochastic control: L2 approach, Preprint Dipartimento di Matematica, Universita di Pisa, n. 2.350.1211 - Ottobre 1999.
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[42] F. Gozzi, Global regular solutions of second order Hamilton- Jacobi equations in Hilbert spaces with locally Lipschitz nonlinearities,
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Approximations of Stochastic Partial Differential Equations ISTVAN GYONGY Department of Mathematics and Statistics, University of Edinburgh, King's Buildings, EH9 3JZ Edinburgh, United Kingdom, [email protected] 1
Abstract. We review some results on numerical approximations of parabolic
SPDEs, obtained with various methods, in particular, with the splitting up method.
1. INTRODUCTION Stochastic partial differential equations (SPDEs) play important roles in many applied fields of mathematics. They appear also in real world applica-
tions, for example in solving nonlinear filtering problems of signal processing. There are several theories developed during the last decades which provide existence and uniqueness theorems for the solutions. See, for example, [30], [31] [36], [39] for the L2-theory, [9] for the semigroup approach, [28] for the Lp-theory, [29] for the analytic approach, [32] for the characteristic method. In connection with these theories there are a great variety of methods of finding solutions numerically, for instance, finite difference methods, the splitting up method, Galerkin's approximation, finite element methods, the Wiener chaos decomposition. See, for instance, [5], [6], [13] [27] for the splitting up method, [42], [43], [44] [15], [1], [19], [20], [25], [37] for time discretization and finite difference methods, [14] for the finite element method, [33], [34] for the Wiener chaos decomposition, [8] for the particle method. In the present paper we mainly focus on some recent progress in the splitting up method and in the method of finite differences achieved in the framework of the L2-theory in [23] and in [43], respectively. In connection with applications in nonlinear filtering we discuss also Wong-Zakai type approximations for SPDEs.
The paper is organized as follows. In Section 2 we present an existence and uniqueness theorem which we use in other sections of the paper. This result J
This work was partially supported by OTKA T 032932 287
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is taken from [23]. Section 3 is devoted to Wong-Zakai type approximations. In Section 4 we present some recent results on the splitting up method from [23] , and in Section 5 we review some results of [42] and [43] on finite difference approximations. Section 6 is devoted to an application of the main result of Section 4 to nonlinear filtering. Moreover, applications of results from Section 2 are also discussed here. Like Section 4, this section is also based on [23] . Throughout the paper, d,do,d^ are fixed integers, K, T are fixed finite positive constants, p is a fixed number in (0, oo), and
Di = d/dx\
DH = d2/dxidxj .
We fix also a complete probability space (£), T, P) with an increasing rightcontinuous filtration (Ft)t>o of sub
2. EXISTENCE AND UNIQUENESS OF THE SOLUTION Let V = Vt be a continuous increasing process and let Y = (Y^, ...,Yd°) be a continuous martingale defined for t € [0, T] . Consider the following equation
du(t, x) = (Lu(t, x) + f ( t , x)) dVt + (Mku(t, x) + gk(t, x)) dYtk
(2.1)
for t € (0,T], x € Kd with initial condition u(0, x) = UQ(X), where the operators L and Mk are written as
L = aij(t, x)Dij + a^t, a:) A + «(*, x),
Mk = b{(t, x)Di + bk(t, x).
We will investigate the convergence of not only functions themselves but also of their derivatives in LI. Therefore, we need the spaces Hm of L% functions whose generalized derivatives up to the order m are also in L%. There are several ways to introduce the norm and the inner product in Hn. We choose the following
(u,v)m:= ^(Dau,Dav)0, |a|
where ( , )o is the inner product in L%, a = (ai, ..., ad) are multi-indices,
a:=ai + ... + ad, Da := We present now a result from [23] on the solvability of equation (2.1) in C([0, T]; Hm], which we use later. In order to formulate the assumptions we fix and integer m > 1 and recall that K > 0 and p €. (0, oo) are constants. Assumption 2.1 (smoothness of the coefficients). All the coefficients a^(t, x), a (i, x), a(t,x), b (t,x), bk(t,x) are predictable for any x € R , and their l
l
d
k
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derivatives up to order m and for a*-7 up to order 2 V m exist are continuous and by magnitude are bounded by K. Assumption 2.2. The function / is predictable £Tm-valued, gk are predictable .ffm+1-vamed, UQ is .ffm-valued and Jb measurable. Furthermore, for I < m and
Ki(t):=
Jo
where f ( s ) = f ( s , •), g(s) = g ( s , - ) and ||0(a)||?+1 := £ fc lls*(*)ll?+n we
have
Assumption 2.3. (stochastic parabolicity) For any u € fi, x, A € Md, we have
2aij(t, a)A'A> dt$ - 6|(t, a:)6f («, a)A'A' d 0 in the sense of measures on [0, T] . Definition 2.1. By a solution of (2.1) with initial data UQ we mean an L2-valued predictable function u(t) = u(t, •) defined on O x [0, T] such that
P(I
Jo
\\u(t)\\\dt«x>) = l
and, for any (f> € CQ°, the equation (u(t, .),^)o = («(0, • ) , <
+((o* - a^) A«(«) + a«(«) + f(8),)o] dVs£s
+ I\VkDiu(a) + bku(S) + 9k(s), e CQ°, the equation (n(t, -),(/))o = (u(t, -),0) holds for all i € [0,T] at once with probability 1. This is the reason why henceforth we only consider 1/2 continuous versions of solutions. Theorem 2.1. ([23]) Under Assumptions 2.1, 2.2, 2.3 equation (2.1) with initial condition UQ has a unique solution u = u(t). This solution is weakly continuous in Hm and strongly continuous in Hm~l . Moreover, if for I <m :=
Jo
and
for some p > 0, then for any integer I € [0, TO]
E sup ||«(t)||f
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where N is a constant depending only on d, do, K, m p and T. This theorem is a generalization of Theorem 3.1 of [31], and for p = 2 it can be proved in the same way. For its proof for p ^ 2 we refer to [23] . 3. WONG-ZAKAI TYPE APPROXIMATIONS
Let us first consider the following simple example of second order SPDE:
du(t, x) = ajlD.jiu(t, x) dt + VkDju(t, x) dWk,
(3.3)
where W is a cfo-dimensional Wiener process and a^1, bk are constants. Let u(t) denote the Fourier transform for each fixed t > 0 of the solution u(t) of this equation. Then it is easy to show that u(t, y) satisfies the stochastic differential equation
du(t, y) = -a'VV«(*, y) dt - iVkyju(t, y) dWk, for dy-almost all y = (y1,?/2, ...yd) € M.d. Hence u(t,y) = ^(y)e-<^^)»Vt-a^w?. This example shows, as it is explained in [30] , the significance of the assumption on stochastic parabolicity, Assumption 2.3. It can also be used to test some methods of approximations for SPDEs. Let us approximate equation (3.3) now by replacing the driving processes Vt '.= t and W with ^-adapted continuous increasing processes Vnt and ^-adapted continuous processes Wn of bounded variation, respectively. In the same line of reasoning as above we get un(t, y) = «0(|/)e-a'Wv,rt-fl^H& (3>4) for the Fourier transform un(t) of the solution un(t) of
dun(t, x) = ajlDjiUn(t, x) dVnt + VkDjUn(t, x) dW*. Letting n -> oo in (3.4) we get
un(t) -)- v(t, y) := uQ(y)e-aJl^lt-ib3^w"(a.S.)
(3.5)
in L\Rd),
under the condition that a^k is positive definite and UQ € L 2 (M d ). Notice that v(t) solves
dv(t, y) = -(a*1 + ^kblk)yjylv(t, y) dt - ttfa?v(tt y) dWtk. Hence by Parseval's identity we get that un(t) almost surely converges in L 2 (R d ) to v(t), which solves the equation
du(t, x) = (ajl + ^-b^Djiu^, x) dt + b{DjU(t, x) dWk,
(3.6)
instead of equation (3.3). Thus by approximating the Wiener process in equation (3.3) we created a new second order term ^bkbkDjiu(t,x) in the limit equation (3.6). This effect is a counterpart for SPDEs of a well-known
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phenomenon of Wong-Zakai type approximations for stochastic differential equations. In connection with the problem of robustness in nonlinear filtering this
type of approximations have extensively been studied. (See, e.g., [16], [17], [18] [22], [7], [40], [41] and the references therein.) An approximation result of this type for parabolic semilinear equations driven by space-time white noise is proved in [3]. Here we present a simple version of a result of [18], [22] for equation (2.1). Let Vnt and Ynt be an ^-adapted continuous increasing process and an J^-adapted continuous di-dimensional process of bounded variation, respectively, for every integer n > 1. Let us consider equation (2.1) with Vn and Yn in place of V and Y, respectively:
du (t, x) = (Lu (t, x) + f ( t , x)) dV + (M u (t, x) + g (t, x)) dY*. (3.7) n
n
nt
k
n
k
t
By Theorem 2.1 this equation admits a unique solution un, if Assumptions 2.1, 2.2 are satisfied, and a*-7' is nonnegative semidefinite. We are interested in the convergence of un if Vn —>• V and Yn —>• Y. We need the following condition on the driving processes. Assumption 3.1. Vnt -)• Vt
4 Yt,
A :=
Jo
; - Y*.)dY$. -> (Y\ Y*)t
*
(3.8)
in probability, uniformly in t e [0, T], and
i? \\(T) := [\YJ-Y*t\\dYi\ Jo
(3.9)
is bounded in probability for every i,j.
Remark 3.1. Many natural examples satisfy (3.8) and (3.9), the conditions on the convergence of Yn and of its 'area processes'. In particular, the smoothing /"* Ynt:=n Y(t-s)p(ns)ds n =1,2,3, ... Jo of Y with a nonnegative kernel p, supported on [0, 1] , such that J p(s) ds = 1, satisfies these requirements. In this section we use the following concept of convergence of the approximations un. Definition 3.1. Let un = un(t) be a sequence of stochastic processes taking
values in a Banach space B. We say that un —>• u in B in probability, uniformly in t € [0, T], if v is a /?-valued stochastic process and
lim P( sup \\un(t) -u(t)\\B >e) = ~
for every s > 0.
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Theorem 3.1. ([18], [22]) Let b\, bk and g be independent o f t . Let au be positive semidefinite and let Assumptions 2.1, 2.2 be satisfied with m + 3 in place ofm. Assume moreover that Assumption 3.1 holds. Then un(t) —>• u(t) in Hm in probability, uniformly in t € [0, T] , where u is the solution of the equation
du(t, x) = (Lu(t, x) + f)dVt + (MkMiu(t, x) + Mkgi) d(Yk, Yl}t +(Mku(t,x)+gk)dYk. Let C ([0,T] X K ) denote the space of bounded continuous functions / = f ( t , x ) on [0, T] x Rd which have bounded continuous derivatives in x € Rd up to the order n. The space C°n([0, T] x Rd) is a Banach space with the norm 0n
d
||/||c°»:= sup sup V \Daf(t,x)\.
Corollary 3.2. If I > 0 is an integer such that m > | + / then under the conditions of the above theorem un —> u in C°'([0,T] x R d ) in probability. The condition that the free terms gk and the coefficients of Mk are timeindependent is assumed for the sake of simple formulation of Theorem 3.1 For the case of SPDEs with time-dependent random (possibly unbounded) coefficients and free terms we refer to [18] and [22]. In [16], [17] and [22] a general approximation scheme is considered for equation (2.1), in which, together with the driving processes V and Y, the operators L, Mk are also approximated simultaneously. The results obtained in these papers can be applied to the situation when L and Mk are approximated with discretization operators in the space variable x. In particular, they can be applied to proving the convergence of the Galerkin approximations and of finite difference schemes. In these papers, however, there are no estimates on the rate of convergence. Let us now consider the special case when Vt = t and Y is a d\- dimensional Wiener process W. Then by virtue of Theorem 3.1 the equation du = (Lu + f ) d t + (Mku + gk) dWk
(3.10)
can be approximated by dun = (Lun + /) dVn + (MkUn + 9k) dYk
(3.11)
where L := L — ^MkMk, f := f — ^Mkgk. .[.From Theorem 3.1 and Corollary 3.2 we get the following result. Theorem 3.3. ([18], [22]) Let b\, bk and g be independent oft. Let the matrix au — \b\b3-k be nonnegative and let Assumptions 2.1, 2.2 be satisfied with m+3 in place ofm. Assume, moreover, that Assumption S.I holds with Vt := t and Y := W. Then un(t) -4- u(t) in Hm in probability, uniformly in t e [0,T], where u and u are the solutions of equations 3.10 and 3.11, n
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respectively. Moreover, if m > % + I for some integer I > 0, then this convergence holds also in C°([Q,T] x R ) in place of H . l
d
m
4. SPLITTING UP METHOD The splitting up method is widely used in solving partial differential equations numerically. A simple version of it can roughly be described in the language of semigroup theory as follows. Let A and B be the generators of some strongly continuous semigroups (Pt)t>o and (Qt)t>o, respectively, on a Banach space B. Then, under some conditions, for the solution u of the evolution equation
one has «(*)= Urn (QiPi)[ntl«o, n —^OO
n
n
called Trotter's formula. In other words, for large integers n > 1 we can approximate u(k/n) k > 1, with un(k/n), obtained (recursively on k) by computing the solution v of
at t = l/n, and then computing the solution of
The application of the splitting up method via adaptation's of Trotter's formula to stochastic equations and SPDEs was initiated by [2], [26] and [5], [6]. It was developed further in [13], [27] and [35]. Error estimates are given in [13] and [27] in the case of the filtering equations. The method of these papers are based on semigroup theory and does not seem to be extendible to the general case of filtering equations. In the situation of [13] the splitting up method is stated in the following way. Assume that we are given independent one-dimensional Wiener processes 10*, k = 1, ...,cfo, first order operators Mk, k = 1, ...,cfo> and a second order elliptic operator L acting on functions defined on M.d. Let the coefficients of L and Mj- be independent of time and suppose that we want to solve the equation
du(t,x) = Lu(t,x)dt + Mku(t,x)odw't,
x e Rd, t>Q
(4.12)
on [0, T] , with some initial data UQ = UQ(X) , where o stands for the Stratonovich differential. Let Tn := {tj = iT/n : i = 0, 1, 2, ..., n} be a partition of the interval [0, T\ for a fixed integer n > 1. Set 6 :— T/n and define the approximation un(t] for t € Tn, by n n (0) = UQ,
un(ti+i, •) = PsQtiti+1un(U, •)
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recursively, where {Pj : t > 0} and {Qst '. 0 < s < t} denote the solution operators corresponding to the equations dv(t,x) = Lv(t,x)dt,
v(0,x) = v(x)
(4.14)
and
dv(t,x) = Mkv(t, x) o dm* ,
v(s,x) = v(x),
(4-15)
respectively. In this way the approximation of equation (4.12) in each inter-
val [£j,ii+i] is split into two steps: solving the degenerate SPDE (4.15) and taking its solution at time ti-\-\ as the initial value at time tj while solving PDE (4.14) again on [ij,£j+i]. In [13] these steps are called correction and prediction steps, and it is proved that under appropriate conditions maxE\\u(t) - un(t)\\l < N/n2,
(4.16)
where || • ||o is the usual LI norm in Md. A different approach to proving the rate of convergence for splitting up method is used in [23]. It is based on stochastic calculus and not on semigroup theory, which allows the authors to improve the results of [5], [6] and [27] in various directions. In particular, estimate (4.16) is proved in [23] with the maximum inside of the expectation, with any exponent p in place of 2, and with Hm norms in place of the L2 norm. Then if m is large enough, the Sobolev embedding theorems provide estimates of the sup norm in x of u
n(t) ~u(t) afrd its derivatives. Thus, in particular, un — u is estimated uniformly in t (E Tn and x 6 Rd. Moreover, these estimates in [23] are obtained also for a multistage splitting up scheme which can be described briefly as
follows. Assume that L in (4.12) is the sum of finite number of elliptic operators, say L = LI + L%, where LI is a second order elliptic operator and L2 is a first order one. Define now the approximation un by
where v(t) :— P^v denotes the solution of (4.14) with Lj in place of L. Below we cite the main results of [23] on the multistage splitting up
method. An application of it to nonlinear filtering is presented in Section 6. In [23] the following equation is considered di
du(t, x) = \^(Lru(t, x) + fr(t,
x)) dt
r=\
+ (L0u(t, x) + f0(t, x)) dV? + (Mku(t, x) + gk(t, x)) dYtk, (4.17) for t 6 (0, T], x € ~R with initial condition u(Q,x) = UQ(X), where the d
operators Lr and Mk are of the form Lr = dlj (t, x) Dij + 4 (t, a;) A + ar (t, x),
Mk = b{ (t, x)Di + bk (t, x) .
(Here, as before, Y is a cfo-dimensional continuous ^i-martingale and V° is an .TVadapted continuous increasing process.)
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If the operators Lr and M& and the free terms /o, fr, 9k axe timeindependent then the multistage splitting up scheme is defined as follows:
i = 0,l,2,.., where Pj V
(4.18)
an
d Qsf0 := v(t) denote the solutions of the SPDEs
dv(t, x) = (L^v(t, x) + fi(x)} dt,
t>0
for fixed 7 = 1, 2, ..., di, and
dv(t, x) = (Lov(t, x) + f (x)) dV? + (M v(t, x) + g (x)) dY , k
0
k
k
t
t >s
with initial conditions v(0,x) = t^(x) and v(s,x) = i{>(x), respectively. The assumptions are the following (Recall that K is a fixed positive constant and p is a fixed number in (0, oo). ):
Assumption 4.1 (smoothness of the coefficients). All the coefficients cffft, x), alr(t, x), ar(t, x), bf.(t, x), bk(t, x) are predictable for any x € Rd and, for any (u>, t) G fJ x (0, oo), their derivatives up to the order m+3 exist are continuous and by magnitude are bounded by K. Assumption 4.2. For each u € fi, the functions fr(t) = fr(t, •) are weakly continuous as Hm+3 -valued functions, <%(£) = g^(t, •) are weakly continuous as fl"m+4-valued functions, and UQ €. I/2(^,-?"b,-ffm+3)' Furthermore, fr and gk are predictable and
sup \\f\\pm+3 + E sup where ||/||j^ = Er \\fr(t)\\2m+3 and ||5||^+4 = ^k \\9k(t)\\2m+,. Assumption 4.3. The process V^0 is predictable continuous increasing and starting at zero. We have V® +(Y}T < K. The matrices (($) are nonnegative and, for any uj e fi, x, A € Md, we have
2$ (t, x)Xi\j dVt° - b\(t, x)bi(t, x)*^^ d(Yk, Yr}t > 0 in the sense of measures on [0, T]. Let u denote the unique solution of (4.17) with initial condition n(0, x) = UQ(X), which exists owing to Theorem 2.1. Theorem 4.1. ([23]) Let ofi ,alr,ar,b'l!.,bk,b> fr and g^ be independent o f t . Then under Assumptions 4-1, 4-% and 4-3 there is a constant N depending only on d, do,di, K,p, m and T, such that
for all n > 1.
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Remark 4.1. The approximation u^ can be defined by splitting up in any order, i.e., by u^ by 2 1 (0 ....fp( )p( ) 7/ («)r/^ fg U (lt), s
5
in place of (4.18). Then one can easily see from its proof in [23] that Theorem 4.1 remains valid. In the time-dependent case the following additional assumption is used. Assumption 4.4. There exists a continuous Tt martingale
and for any x € Rd there exist bounded predictable functions
denned on O x (0, T] for r = 0, 1, ..., cfo, such that (i) d(Z}t < dVt,
ft ft I ho(s,x)dVs+ I hr(s,x)dZrs Jo Jo for all (jj and t, where, as usual, the summation in r is carried over all possible values, which in this case are 1,2, ...,^2- Furthermore, hr are continuously differentiable with respect to x up to the order m + 1 and \D@hr\ < K for 101 <m + l. Theorem 4.2. ([23]) Under Assumptions 4-1, 4-%> 4-3 and 4-4 there is a constant N depending only on d, do, di, d%, K,p, m and T, such that
£max||«(n)(i) - u(t)\\^ < Nn~p t£Tn
for all integers n > 1. Let Cl = Cl(S!td) denote the Banach space of functions / = /(#), x € R^, having continuous derivatives up to the order I, such that \\f\\ci '•= sup^gjjd ~Yj\B\ l + d/2 and nonnegative integer
I, then for some N = N(d, dQ,di,d%, K,p, m)
for all n > 1, where X := Cl, and \\\\x denotes the norm in X. The next corollary can be obtained easily by a standard application of the Borel-Cantelli lemma.
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Corollary 4.4. ([23]) Assume the conditions of the previous theorem. If Assumption 4-2 holds with p > K, for some n > 1, then there is a random variable £, such that almost surely max ||tr™'(£) — n(i)||x < £n for all n > 1, where X is H™', or we can also take X := Cl if m > I + d/2.
5. FINITE DIFFERENCE APPROXIMATIONS It is a natural idea to approximate partial differential equations by replacing the differential operators in the equations with computable discrete operators. In parabolic PDEs the time derivatives are often replaced with finite difference quotients and the differential operators in the space variable are approximated with various methods, with finite difference operators, Galerkin approximations, etc. There are several papers generalizing these methods to SPDEs. See, for example, [24], [25] [37] for time discretization, [1], [10], [19], [20] for finite difference schemes, [4], [15] for the Galerkin method, [14] for the finite element method. A general approximation scheme is considered in [43], [44] which covers various ways of discretizations in the space variable. By the help of an extension of the L2-theory of SPDEs to discrete evolution equations, general rate of convergence results are obtained, in particular, for finite difference schemes. Further results in this direction are obtained in [42] and [44] by an adaptation of the Lp-theory of SPDEs developed in [28] and [29]. Below we present some results from [43], [44] on finite difference schemes via the L2-theory. Consider
du(t, x) = (Di(aij Dju(t, x) + f(t, x)) dt + (Dibi(t, x) + gk(t, x)) dWtk «(0, x) = UQ(X)
(5.19)
x e Rd,
where W is a di-dimensional Wiener process and a*J', b\, /*, g^ are functions on fi x [0, T] x Rd. Let T, K and A > 0 be fixed positive numbers and let m > 0 be a fixed integer. Assume that the following conditions are satisfied. Assumption 5.1. The coefficients a '(t, x), b (t, x), are predictable for any x € Rd, and their derivatives up to order m are continuous and by magnitude are bounded by K. Furthermore, for I < m and tj
l
k
:= j
°AE ii^wii?+E
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Assumption 5.2. For all u € O, t > 0, x, z e Rd
(2aij(t, x) - bib{(t, x))zizj > A i
It is well-known from the L2-theory of SPDEs (see, e.g. [39]) that under Assumptions 5.1, 5.2 equation (5.19) has a unique solution it, and
E sup \\u(t)\\^ + E f 0
E(
JO
\\u(t)\\2m+l
£ \\f(t}\\l + £ \\gk(t}\\l] dt.} i k
J
o
with some constant N. Let 2d := {hz = (hz-[,hz2, ...,hz
5i~v(x) := h~l(v(x + h&i) — v(x}}, 5~v(x) := h-1(v(x) — v(x — hei)), where e\, e%, ...&d is the standard basis in Kd, and consider the equation duh(t, x) = 6~((aij(t, x)8fuh(t, x) + f(t, x)) dt t€[0,T],
x € 2%, (5.20)
The following theorem of [42], [43] shows that this equation can be solved in discrete Sobolev spaces H™, defined as the space of real valued functions / on 2d with norm
where /(y) := (2'K)~d^2z&Zde~lh^z>y^f(z) is the discrete Fourier transform of/. Theorem 5.1. ([42],[43]) Let Assumptions 5.1, 5.2 hold with m > d/2 + 1 and let m! > 0 be such that m > m' + d/2. Then equation (5.20) has a unique solution Uh, and
E sup \\uh(t)\\2n ,+E 0
H
™
>*
Jo
rT
JO
\\eh(t)\\2 H ,+1dt h
H
™
i
H
k
™
with a constant N. Assume moreover that m> m! + 2 + d/2. Then for the error e^ := u — u^ we have fT
E sup \\eh(t)\\* , +E \ 0
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h.
\\uh(t)\\2
JQ
"h
Approximations of SPDEs
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< Nh2{E\\u0\\2m + E A^\\f(t)\\l + £\\g k (t)\\ 2 m ) dt}, Jo i k with some constant N, depending only on X, d, K, m and T. Hence a discrete version of Sobolev's embedding of Hm into Cl yields the following corollary. For multi-indices a = (a 1,0:2, ...,a: 0 be an integer such that m > I + d + 2. Then there exists a constant N = N(X, d, K, T, m, I) such that
E sup sup \\6ae
rT +E J Jo
Let us now discretize equation (5.20) in time, along the partition Tn := {ti = iT/n : i = 0, 1, 2, ..., n} of [0, T] for a fixed integer n > 1. Let us use the notation At := T/n and Af(t) .- f(t + At) - f ( t ) . Then the Euler approximation applied to equation (5.20) gives
i, x) = 6-((av(t, x}6^ul(t, x) + f(t, x)) At t€Tn,
x € Z%, (5.21)
In order to estimate the error e% := Uh — u% one needs assumptions on the dependence of the coefficients au b\ and the free terms f1, gk on t. Assumption 5.3. For all uj e O, t € [0, T], x € Rd and for all multi-indices a <m ij
ik
m
m
Moreover, f /*(<) e H , ^(t) e H for all u, t, i,k and
ik
/« " ftt.~"^ ' ""°
< oo.
Theorem 5.3. ([42], [43]) Let Assumptions 5.1, 5.2 and 5.3 hold with m > m! + 2 + d/2 where m' is a nonnegative integer. Then there exists 0 < 9 = 0(X, K, m, d) such that if At/h2 < 0 then
n
2
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JO
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IK
where N is a constant depending only on X, d, T, K and m. Hence, as before, a discrete version of Sobolev's embedding Hm into Cl gives the following corollary. (Note also that A£ < Oh2 is assumed.) Corollary 5.4. f[42], [43],) Assume the conditions of the previous theorem and let I > 0 be an integer such that m > I + d + 2. Then there exists a constant N = N(X, d, K, T, m, /) such that Esup sup ||<5 Q e£(t,o;)|| 2 < JW»2s{j|tio||
i
ik
for all multi-indices \a\ < I.
^Prom Corollaries 5.2 and 5.2 one immediately gets the following estimate for the error u — ufc. Theorem 5.5. ([42], [43]) Let Assumptions 5.1, 5.2 and 5.3 hold with m > I + 2 + d where I is a nonnegative integer. Then there exists 0 < 6 = 9(\, K, m, d) such that if At/h2 < 6 then
E sup sup \5a(u(t, x) - ul(t, x ) ) \ 2 < 7V7i2£ j||w0||
ik
for all multi-indices \a\ < I, where N is a constant depending only on X, d, K, T, I and m.
6. APPLICATIONS TO NONLINEAR FILTERING Partially observable stochastic dynamical systems are often modelled by a pair Zt := (Xt, Yt) of multidimensional stochastic processes, satisfying some stochastic differential equations with given coefficients. Here Xt is a ddimensional process, called the unobservable component, or signal process, and It is a do-dimensional process, called the observation process. In a fairly general situation the evolution of these processes is governed by the equations
dXt = h(t, Xt, Yt) dt + a(t, Xt, Yt] dwt + p(t,Xt,Yt)dWt, X0 = £ dYt = H(t, Xt, Yt) dt + dWt, YQ = rj,
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(6.1)
Approximations of SPDEs
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where h(t,x,y) €_Rd, er(t,x,y) £ RdxJ, p(t,x,y) € R dx
and (wt, Wt) is a d + do-dimensional Wiener process, independent of the JF0measurable random vectors £, T?. The coefficients fo,
E((f>(Xt)\Y,,Q<
s
from the data P(0, dx),h, a, p, H and the observation {3^, s < t} for a given function I be an integer. Assume that (i) a^ have uniformly bounded derivatives in x up to the order m + 2; (ii) h and p have uniformly bounded derivatives inx up to the order m+1, andH have uniformly bounded derivatives in x up to the order m; (Hi) the conditional distribution of£ given rj has a density PQ (with respect to Lebesgue measure), which belongs to Hm. Then the conditional density 7it(a:) := P(t,dx)/dx exists and
*t(x) =p(t,x)/(p(t),l)0, where p = p(t, x) is the unique solution of the equation dp(t, x) = {D^tt, x, Yt)p(t, x)) + Di(h\tt x, Yt)p(t, x))} dt
+{Hk(t,x,Yt)P(t,x) + Di(pik(t,x,Yt)p(t,x))}dYtk (6.2) with initial condition PQ. Moreover, {p(t) : t € [0,T]} is a continuous Hm~lvalued stochastic process, and a weakly continuous Hm-valued stochastic process. This theorem describes the analytical properties of the conditional density
TTt, and presents a way of computing the estimate for ip(Xt), via equation (6.2), called the Zakai equation (or Duncan-Mortensen-Zakai equation) for the unnormalized conditional density pt.
In order to implement this result in practice one has to develop numerical methods to approximate the solution of equation (6.2) and needs also to control the error of the approximations. Therefore various methods of approximations have intensively been studied in the literature. See, for example, [5], [11], [12], [13] [27], [33], [34], [37] and the references therein. Notice first that in the signal and observation model (6.1) the observation process Y has infinite (first) variation on any finite time interval. In practice, however, the observations have finite variation. We can view the observations
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arising in practice as approximations Yn, obtained by measurements, for the 'theoretical' observation Y. Let pn denote the solution (6.2) with Yn in place of Y. Then the following important question arises: do pn and Trn(t) := pn/(pn(t), l)o converge to the unnormalized conditional density p and to the conditional density TT, respectively, as Yn —>• Y ? It is natural to assume that, due to the smoothing effect of the measuring instruments, these approximations Yn satisfy Assumption 3.1. (See Remark 3.1.) Thus we can apply Theorem 3.3 to answer the above question. We need the following assumption on the smoothness of the coefficients. Assumption 6.1. The coefficients OJQ = ( o)i i ( i ) continuous derivatives in x up the order m + 5, h = (h1} and p = (p*k) have continuous derivatives in x up the order m+4, and H — (Hlk] has continuous derivatives in x up the order m + 3. All these derivatives are bounded by a constant K. Consider the equation a
a
=
a
?
nave
dpn(t, x) = (L- \MkMk}pn(t, x) dt + Mkpn(t, x) dYtk pn(0,x)=po(x),
(6.3) (6.4)
with operators L, Mk defined by
Mk(x) = Hk(x)(x) + A(p*(aMaO). Theorem 6.2. ([16], [22]) Let p and H be independent oft and y and let Assumption 6.1 be satisfied. Assume that PQ G Hm+s. Assume, moreover, that Assumption 3.1 holds. Then the solution pn of problem (6. 3)- (6. 4) converges in Hm in probability, uniformly in t € [0,T]. Corollary 6.3. If I > 0 is an integer such that m > | + I then under the
conditions of the above theorem pn —±pin C°l([0,T] x Md) in probability. By Theorem 3.3 one can get the convergence of TT" to TT by using appropriate weighted Sobolev spaces (see [18] and [22]). The condition that p and H do not depend on t and y in the above results is assumed only for the sake of simplicity of presentation. We refer to [18] and [22] for the case of the general filtering model. We remark that the general approximation results in [22] can also be applied to prove convergence when the operators in equation (6.2) are discretized in the space variable. The results of [22], however, do not provide error estimates. Now we are going to discuss the application of the splitting up method to our filtering model. Notice that for equation (6.2) the condition of stochas-
tic parabolicity (Assumption 3.2) requires that the matrix 2ay — (pp*)^ = (o~o~*yi be nonnegative definite. Clearly, this is always satisfied. The degenerate case, a = 0, is of special interest. In this case the representation of the solution of equation (6.2) by the method of characteristics gives a relatively simple formula, which does not involve conditional expectation (see
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[31]). Using this representation one can obtain an approximation for the solution of equation (6.2) with au = 0$ , and the error can also be estimated (see [13]). This motivates the idea of splitting up equation (6.2) into the equations
du(t, x) = L0(t, Yt)u(t, x) dt + Mk(t, Yt)u(t, x) dYtk
(6.5)
and
dv(t, x) = Li(t, Yt)v(t, x) dt,
(6.6)
where
Lo(t, y)(x] := Dij(a^j(t, x, Li(t,y)(x) := DiAatffax^Mx
Mk(t, Yt)4>(x) := Hk(t, x, y}4>(x] + Di(pik(t, x, Let Pt(ti)if denote the solution, starting from (f>, of equation (6.6) with coefficients frozen at t = ti, Yj = Y^, where ti := Ti/n. Define the approximations Pn(ti), pn(ti) for U € Tn := {Ti/n :i:=Q, 1, 2, .., n} by pn(0) = pn(0) := Po, Pn(ti+l) := Ps(U+l)Qtiti+1Pn(ti),
Pn(U+l) •= Qtiti+iPS(ti)Pn(ti)
for i — 0, 1,2, ...,n — 1, where 8 = T/n and Qstf denotes the solution of equation (6.5) for t > s, with initial condition v(s) = if. We are going to apply Theorem 4.2 to these approximations. In addition to Assumption 6.1 we need also the following assumptions for a fixed integer m > 0 and real number p > 0. Assumption 6.2. The derivatives in x of a,\ and h up to the order m + 2 and m + 1, respectively, have continuous first derivative in t and continuous second order derivatives in y, which are bounded by the constant K.
Assumption 6.3. Almost surely po € H™- and £||p |lm+3 ^Theorem 6.4. ([23]) Under Assumptions 6.1, 6.2 and 6.3 there exists a constant N, depending only on d, do, d, K,p, m and T, such that +3
K
0
n(t)
-p(t)\\px < Nn~*
(6.7)
for all integers n > I, where || • \\x denotes the norm in X := Hm. One can prove this theorem by rewriting equation (6.2) in the form of equation (4.17) and then applying Theorem 4.2 (See [23]). Sobolev's embedding and the Borel-Cantelli lemma give the following corollary. Corollary 6.5. ([23]) // Assumptions 6.1, 6.2, 6.3 hold with m> d/2 + I, where I > 0 is an integer, then estimates (6.7) hold also with X := C rZ (K d ) in place of H"1 . If Assumptions 6.1, 6.2 hold, and E^PQ^^ < oo for some p > K, and K, > 1, then there is a finite random variable £, such that almost surely
max \\pn(t) -p(t)\\x < £n-1+1/«, *
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for all n > 1, with X := Hm, and if m> I + d/2, then also with X := Cl. We finish this section, and the whole paper, by the following remarks from [23]. Remark 6.1. In [13] a version of Theorem 6.4 is given in the time homogeneous situation, when the coefficients of (6.1) are independent of Yt, p = 2, m = 0, and with max's in (6.7) being outside of expectations. However, the number of derivatives required in [13] is smaller. This might be due to some confusion in [13], since in [13] the authors apply a theorem from [31], stated for the equations in the usual form, to equation (6.2) in conjugate form. Remark 6.2. One could easily consider the most general form of the signalobservation equations (6.1). In particular, one can put a uniformly nondegenerate smooth matrix-valued function G(t, Yt) in front of dWt. Then under natural assumptions on the smoothness of G one can get a result similar to Theorem 6.1. We have chosen not to deal with these generalizations just for simplicity of notation. Finally we remark that by using weighted Sobolev spaces in place of Hm one can extend our results to the case of SPDEs with unbounded coefficients. This kind of SPDEs are important from the point of view of applications, in particular, in nonlinear filtering (see for example [22], [38], [39] and the references therein). However, for the sake of simplicity of presentation we did not want to consider SPDEs with unbounded coefficients in this paper.
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[39] Rozovskii, B. (1990). Stochastic evolution systems. Linear theory and applications to nonlinear filtering. Kluwer, Dordrecht. [40] Twardowska, K. (1993). Approximation theorems of Wong-Zakai type for stochastic differential equations in infinite dimensions. Disserta-
tiones Math. (Rozprawy Mat.) 325. [41] Twardowska, K. (1995). An approximation theorem of Wong-Zakai type for nonlinear stochastic partial differential equations. Stochastic Anal. Appl. 13 No 5, 601-626. [42] Yoo, H. (1998). An analytic approach to stochastic differential equations and its applications. Thesis, University of Minnesota. [43] Yoo, H. (1988) On .Z^-theory of discrete stochastic evolution equations and its application to finite difference approximations of stochastic PDEs. Preprint [44] Yoo, H. (2000). Semi-discretization of stochastic partial differential equations on K1. Mathematics of Computation 69 653-666.
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Regularity and Continuity of Solutions to Stochastic Evolution Equations ANNA KARCZEWSKA Institute of Mathematics, Technical University of Zielona Gora, ul. Podgorna 50, PL-65246 Zielona Gora, Poland, e-mail: [email protected] *
Abstract. This paper is a review note concerning regularity and continuity of solutions to stochastic heat, wave and Volterra equations. In the paper recent results obtained by several authors are discussed.
1
Introduction
Starting from Walsh [Wa], one may observe an increasing interest in studying stochastic differential equations. Particularly, during some recent years there have been written several papers devoted to existence, regularity and continuity of solutions to stochastic evolution equations. Authors have studied equations driven by more or less general noises and obtained corresponding regularity results which in some cases depend on the spatial dimensions. In this situation it seems be profitable to collect and compare some of those
results. The main purpose of this note is to give an updated and an as complete as possible on twelve pages presentation of regularity results. We try to show what has been achieved so far and what the aims are,
rather than to present
detailed proofs. In this note we consider stochastic heat, wave and Volterra equations, linear and nonlinear, as well. In order to do this, first we should recall some notation, definitions and auxiliary results. But because of the
lack of place in such a short note, we refer the reader to proper references. A word of caution is necessary concerning the list of references. We have not tried to present a complete bibliography for the subject of this note. We
wanted to have a representative one,
restricting it to the more important
Research partially supported by KBN Grant No. 2 P03A 017 17 and Technical University of Zielona Gora.
309
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Karczewska
contributions. Since it depends very much on personal views and author's taste, the selection of the bibligraphy is not impartial. Finally, it is a great pleasure to thank professors G. DaPrato and L. Tubaro for inviting me to prepare a review article.
2
Stochastic heat and wave equations
This section is devoted to the stochastic heat and wave equations with spa-
tially correlated noise. By the occasion, we mention results concerning more general kind of stochastic partial differential equations, as well. First, let us consider the stochastic linear heat and wave equations of the form:
u(0,0) = 0,
G e Rd
and
,6»),
t>0,
where W is an appropriate Wiener process with the space correlation F.
When W is a spatially homogeneous Wiener process (for definition and investigations, see e.g. [DaSa], [DaPrZa2], [HoSt], [KaZaS], [No] or [PeZal]), the correlation F may be any positive definite distribution. Then the correlation F defines the covariance operator Q of the Wiener process W by the formula: Q 1. First, this problem has been investigated for stochastic wave equation by Dalang and Frangos [DaFr] and also Mueller [Mu] in dimension d = 2. Especially Dalang and Frangos' paper was important and stimulating the develope of interest of many people in this problem. Actually, in order to obtain results in higher dimensions, even for nonlinear stochastic partial differential equations, Dalang and Frangos have proposed a new approach. They considered noise with a spatial correlation and suggested to find the weakest posible conditions on the correlation that provides solutions of equations, linear and nonlinear, in the space of real-valued processes. Particularly, they considered Wiener process W(t,0), t > 0, 0 € R2, with the
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space correlation function F:
E[W(t, 0)W(s, 77)] = t A s T(0 - ri) ,
6, rj e M2 ,
where F(0) = f ( \ 6 \ ) , 0 G M2, and / is non-negative function, continuous outside zero. For such a noise Dalang and Frangos have proved (see [DaFr] , Theorem 1) that the stochastic wave equation (2) has a function-valued solution if and only if
/
J\e\
/(|0|)ln-U?<+oo.
W\
(3)
Because the proof in [DaFr] was based on explicit representation of the fundamental solution of the deterministic wave equation in dimension d = 2, it couldn't be extended, using that method, to higher dimensions. It was evident that more powerful tools for obtaining stronger results were needed. Strong enough tools have been provided by Peszat and Zabczyk [PeZal] in 1997. They introduced, among others, stochastic integral with respect to spatially homogeneous Wiener process and provided some auxiliary results needed for generalization of reguliarity theorems. The general case of arbitrary dimension d and of arbitrary spatially homogeneous noise for both stochastic linear heat and wave equations has been studied first by Karczewska and Zabczyk in [KaZal] and [KaZa2] . These authors used results from [PeZal] and obtained the following main results.
Theorem 1 ([KaZa2], Theorem 1) Let T be a positive definite, tempered distribution on Kd, with the spectral measure ft. Then the equations (1) and (2) have function-valued solutions if and only if
Theorem 2 ([KaZa2], Theorem 2) Assume that F is not only a positive definite distribution but also a non-negative measure. The equations (1) and (2) have function-valued solutions: i) for allT ifd= I ; ii) for exactly those F for which Li^ In \Q\ T(dd) < +00 if d = 2 ; Hi) for exactly those F for which J r(d0)/\d\d-2 ifd>3. Comment: 1. Note that condition (3) is a special case of ii). 2. This is worth to emphasize that for both equations, (1) and (2) on Rd and on Td (c?-dimensional torus), the necessary and sufficient conditions are exactly the same.
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Karczewska
These above results have been followed by several next results concerning stochastic evolution equations, linear and nonlinear, as well. In the study of the wave equation in dimensions d > 3 the following difficulty appears. The Green's function or fundamental solution of the wave equation is in fact not a function but a distribution. This does not accur for the heat equation, whose kernel is very smooth in all dimensions. This difference is a reason why the papers [DaFr], [MiSaSol], [MiSaSo2], [Mu] and [ObRu], which are based on explicit representation of the fundamental solution of the wave equation, have considered only the cases d = 1,2. As we have already written, Dalang and Frangos in their important paper [DaFr] presented approach useful for studying nonlinear wave equation. They have proved ([DaFr], Th. 2) that if the condition (3) on the covariance function of the noise is satisfied, not only the linear form of the equation (2), but also nonlinear versions, have real-valued process solutions. Additionally, these authors gave ([DaFr], Th. 3) conditions on the spatial covariance for the solution of the linear equation to be continuous. As we have already mentioned, in the previous papers with the exception of [KaZal] and [KaZa2], stochastic wave equation in the spatial dimensions d = 1, 2 was studied. In his important paper [Da], Dalang extended the definition of Walsh's stochastic integral with respect to martingale measure. As the consequence, it was possible to solve stochastic partial differential equations which fundamental solutions are not functions but Schwartz distributions. This possibility comes from the "nice" feature of the new stochastic integral. Namely, even when the integrand is a distribution, the value of the stochastic integral process is a real-valued martingale. Using this extended integral, Dalang proved the following important results. Theorem 3 ([Da], Theorem 11) Suppose that the fundamental solution U of Lu = 0 is such that (s,£) i-4 FU(s, •)(£) is a jointly measurable function and for each £, s i->- FU(s, •)(£) is locally Lebesgue-integrable. Let u be the distribution-valued solution to the linear s.p.d.e. Lu = F given by special formula (see formula (44) in [Da]). If there exists a jointly measurable locally mean-square bounded process X : (t,x,u) i-4 X(t,x, u) such that a.s., for all
(u,)=
I
I X(t,x)
*/lK,-f-
then for all T > 0 i
I JR
2 d
<+oo.
In Theorem 3 linear stochastic equations, such as the wave and heat equations, are considered. Dalang has determined the necessary and sufficient
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Regularity and Continuity for Stochastic Evolution Equations
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condition under which such an equation has a real-valued solution. This way the author recovered results of papers [DaPr] and [KaZa2]. Under some additional assumptions on fundamental solution U (see Hypothesis B, page 20 in [Da]) Dalang has obtained the following stronger result.
Theorem 4 ([Da], Theorem 13) If Hypothesis B from [Da] is satisfied and a(-) and /?(•) are Lipschitz functions, then the equation Lu = a(u)F(t,x) + (3(u) with zero initial conditions has a unique solution u(t,x). Moreover, this solution is L? -continuous and for any T > 0 and p > 1,
sup
sup E(\u(t, x)\p) < oo .
0
Dalang's result, mentioned in Theorem 4, shows that the nonlinear form of the three dimensional wave equation and the heat equation in any dimension also have a global solution that is I^-bounded and L2-continuous. This result recovers the long term existence result due to Peszat and Zabczyk ([PeZa2], Theorem 0.1) for the nonlinear wave equation in three dimensions. In fact, Dalang's and Peszat and Zabczyk's results have been obtained nearly simultaneously. In [PeZa2] the nonlinear wave equation in three spatial dimensions was considered and (like in [KaZa2]) an abstract approach via stochastic equations in infinite dimensions was used. Recently, Peszat [Pe] extended results obtained in paper [PeZa2] to nonlinear stochastic wave equation for an arbitrary space dimension d. The main result of his paper is the following (for details see [Pe] ) .
Theorem 5 ([Pe], Theorem 1.1) Let Q € M. Assume that there is a q > 2 such that nonlinear functions appearing in the considered stachastic wave equation belong to the class Lip(g, q) of Lipschitz mappings. Additionally, let the spectral measure fj,ofW satisfy the following inegrability condition sup
Then for any X(Q) = (UQ,VQ)T there exists a unique solution X = (u,v) such that u^I?1 and v € H~* .
Very recently, in May 2000, three authors: Marquez-Carreras, Mellouk and Sarra provided, in the paper [MaCaMeSa], smoothness of the law of the solution to the general kind of stochastic partial differential equations. They dealt with the following equation
L u(t, 0) = a(u(t, 0))F(t, 6} + /3(u(t, 6)), tt(0,0) = 0, 6£Rd
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t>Q,0€Rd (4)
314
Karczewska
where L is a second order partial differential operator, a,/3 : K M- M, and F is a Gaussian noise, white in time and correlated in space. First, using Malliavin Calculus, they proved that the solution u(t, 0) possesses a smooth density p^e for every t > 0, 9 £ R^. Next, they applied the obtained result to two particular cases: the d-dimensional heat equation with d > 1, and the wave equation for d = 1,2. Let us notice, that under conditions from [MaCaMeSa], assuming a, (3 are Lipschitz functions, one may arrive at Dalang's [Da] theorem about the existence of a unique solution to (4). In their study, the authors came back to ddimensional spatial stochastic wave equation with d= 1,2 and (/-dimensional spatial heat equation, d > 1. Although these equations have been widely studied by many authors (see e.g. [Da], [DaFr], [KaZal], [KaZa2], [MiMo], [MiSaSol], [PeZal], [PeZa2]), Marquez-Carreras, Mellouk and Sarra were able to generalize some results obtained earlier. These authors introduced the following condition, well-known from other papers (see e.g. [KaZa2], [KaZa3]) 1
'"*<+oo
(5)
for some ij € (0,1]. If condition (5) is assumed for some rj € (0,1), the Holder continuity in t and 0 has been provided for a class of wave and heat equations (see [MiMo], [MiSaSol], [SaSoSal] or [SaSoSa2]). In papers [MiMo] and [MiSaSol] the two-dimensional spatial stochastic wave equation was studied. The main assumption was that the correlation measure was absolutely continuous with respect to the Lebesgue measure. Among others, the authors proved the existence of a smooth density. This fact required assumption that density of the measure is integrable in some sense. Moreover, proof of smoothness was difficult and technical. In paper [MaCaMeSa] the authors proved the regularity of the density for the wave equation under condition (5) for some 77 € (0,3/4). For the heat equation, condition (5) for some t] €. (0,1/2) is needed. (Let us recall, that necessary and sufficient conditions obtained in papers [KaZal] and [KaZa2] for regularity of solutions to both equations, heat (1) and wave (2), on Md were exactly the same.) In [MaCaMeSa] the authors have eliminated integrability conditions used in the previous papers [MiMo] and [MiSaSol] and, this way, their result could be proved easierly. This might be reached by using the techniques introduced by Dalang in [Da].
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Regularity and Continuity for Stochastic Evolution Equations
3
315
Stochastic Volterra equation
In this part of the paper we consider the following stochastic Volterra equation
X(t, 6}= t v(t - r}AX(r, G}dr + XQ(8) + W(t, 0) ,
JQ
(6)
where t G R+, 9
Analogously, taking v(t) = t,t € [0, oo), after twice differentiating (6) we obtain
We have to notice that the noise term in the above wave equation differs from the one considered by many authors, see e.g. [DaFr], [KaZal], [KaZa2], [MiMo], [MiSaSol], [Mu], [PeZa2] and [Wa]. Let us rewrite the equation (6) in the simpler form
X(t)
=
ft
Jo
v(t- T)AX(r)dT + X0 + W(t) .
(7)
As in deterministic case, the solution to the stochastic Volterra equation (7) is of the form: ft
X(t] = n(t}XQ +
Jo
U(t- r)dW(r) , t > 0 ,
(8)
where 72.(i), t > 0, is the resolvent family of the equation (7) determined by the operator A and the function v. In order to study equation (8) it is enough to consider the stochastic convolution ft WR(t) = U(t- r)dW(r) , t > 0 , (9)
Jo
where the stochastic integral is defined like in [Ito2], [DaPrZal], [PeZal] or [KaZaS], according to particular case under consideration.
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Karczewska
Our aim is to recall and compare conditions Tinder which solutions to the stochastic Volterra equation (6) are function-valued and even continuous. The stochastic Volterra equations have been treated by many authors, see e.g. [ClDaPrl], [ClDaPr2], [CIDaPrPr], [ClDaPr3], [KaZa3] or [RoSaSol], [RoSaSo2]. In the first three papers stochastic Volterra equations are studied in connection with viscoelasticity and heat conduction in materials with memory. The paper due to Clement and DaPrato [ClDaPrl] is very significant because the authors have extended the well-known semigroup approach, applied to stochastic differential equations, to the equation (7). The second and the third reasons why this paper is very important are: the inspiration for other people for studying stochastic Volterra equations and application of results obtained in [ClDaPrl] to the study of integro-differential equations arising in the theory of viscoelasticity forced by white noise, see e.g. [ClDaPr2] or [CIDaPrPr]. Clement and DaPrato studied Volterra equation (7), where A was selfadjoint, negative operator in a separable Hilbert space H, and diagonal with respect to the basis {e^} C H:
Aek = -Vkek,
Wfc > 0,
k €N.
In their paper, Clement and DaPrato considered stochastic Volterra equation (7) driven by the noise term W of the form +00
(W(t), h}H = £>, ek)H 0k(t), fc=i
h£H,
(10)
where {/?&} w&s a sequence of real-valued, independent Wiener processes. In that paper the authors assumed that the problem (7) is well-posed (for notions and definitions, see the monograph [Pr]). Moreover, they assumed that the kernel function v is completely positive. Both these assumptions naturally arise in the applications, see [Pr] . The consequence of completely positiveness of the function v is that the solution s(-,7), 7 > 0, to the integral equation
r*
s(t) + 7 / v(t-
Jo
T}s(r)dT = 1, t>0,
(11)
is nonnegative and nonincreasing for any 7 > 0. (Let us notice that s(t) £ [0, 1].) Clement and DaPrato [ClDaPrl] studied regularity of stochastic convolutions (9) corresponding to the Volterra equation (7) perturbed by the white noise of the form (10). Additionally, under suitable assumptions, they proved holderianity of the corresponding trajectories. Their main assumptions were the following.
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Regularity and Continuity for Stochastic Evolution Equations
Hypothesis 1
317
(i) A is a self-adjoint negative operator and Ae = —/ k
(ii) v is completely positive. (Hi)
-Tr^-1) = EfctiCVwO < +00.
Hypothesis 2 There exists 6 € (0,1) and Ce > 0 such that, for all 0 < T < t we have
I s2(fj,,o-)d(T
,
JT fT
JQ
S
V,T
and k=l
Hypothesis 3 There exists M > 0 such that { \ek(e)\<M,
k€N, 6<
where O is a bounded open subset ofR . d
Clement and DaPrato proved the following main results.
Theorem 6 ([ClDaPrl], Theorem 2.2) Assume that Hypothesis 1 holds. Then for any t > 0 the series:
E
ft I s(fj,k,t-T)ekdpk(T),
fc=i Jo
is convergent in L 2 (fi) to a Gaussian random variable Wn(t) with mean 0 and covariance operator Qt determined by rt
Qt&k= I s2(fjlk,T)dTek, Jo Theorem 7 ([ClDaPrlJ, Proposition 3.3) Under Hypotheses 1 and 2, for
every positive number a < 0/2, the trajectories of WR are almost surely a-Holder continuous. Theorem 8 ([ClDaPrl], Theorem 4-1) Under Hypotheses 1, 2 and 3, the trajectories of WR are almost surely a-Holder continuous in (t, 6) for any a €(0,1/4).
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Karczewska
In their second paper [ClDaPr2] , Clement and DaPrato studied white noise perturbation of an integro-differential equation arising in the study of evolution of material with memory. In the next paper [CIDaPrPr], the authors considered evolutionary integral equations as appearing in the theory of lin-
ear parabolic viscoelasticity forced by white noise. As earlier, they studied the stochastic convolution that provides regular solutions. Additionally, un-
der suitable assumptions the authors proved that the samples are Holdercontinuous. In the remaining part of the paper [CIDaPrPr], the results obtained of that paper were put in a wider perspective by consideration of equations with fractional derivatives.
In their third paper [CIDaPrS], Clement and DaPrato consider general stochastic convolutions of the form (9) , too. In the paper, the authors first prove that Wp,(t) is a Gaussian random variable for any t > 0. Next, the transition function Pt, t > 0, associated with Tl(t} is considered. When 1Z(t) is the resolvent operator of a stochastic Volterra equation, the convolution WR(t), t > 0, is not a Markov process. This fact has the consequence that Pt) t > 0, is not a semigroup and then it is not possible to associate to Pt a Kolmogorov equation. However, the authors are able to characterize those transition functions such that Pt f is differentiable for any uniformly continuous and bounded function
some results in the case of Markov process may be generelized in this new situation. In the remaining part of this section we shall follow the paper [KaZaS]
and consider regularity of stochastic convolution (9) corresponding to the Volterra equation (7) on W1. In this case the equation (7) is driven by a correlated, spatially homogeneous Wiener process W, which takes values in the space of real, tempered distributions S'(M.d). We denote by F the covariance
of W(l) and the associated spectral measure by fj,. To underline the fact that the distributions of W are determined by F we will write WY • We consider
existence of the solutions to (7) in S'($&d) and derive also conditions under which the solutions to (7) are function-valued and continuous. In our case, in the equation (7) the initial value XQ € S'(K. ), v is a locally integrable d
function and A is an operator given in the Fourier transform form:
(12) We shall assume the following hypothesis. Hypothesis 4 1. For any 7 > 0, the equation (10) has exactly one solution s(-,7) locally integrable and measurable with respect to both variables 7 > 0 and t > 0. 2. Moreover, for any T > 0,
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supt€[0,T]
SU
P-7>0 l s (^)7)l < +00.
Regularity and Continuity for Stochastic Evolution Equations
319
Let us notice that for some special cases the function s(t; 7) may be found explicitly. For instance, we have (for more information, see [Pr]): for v(t) = l, s(t;^ = e-^, t > 0, 7 > 0; for v(t) = t, s(t;i) = cos(^/jt), t > 0, 7 > 0; (13) for w(t) = e-*, s(*;7) = (l + 7)~1[l+7e-(1+7)t], t > 0, 7 > 0. In our case, the resolvent family 7£(-) determined by the operator A and the function v is given by the formula (12) and has the form
where
t,a>,
t>0.
We have obtained the following results on stochastic convolution.
Theorem 9 ([KaZaS], Theorem 2) Let Wp be a spatially homogeneous Wiener process andH(t),t > 0, the resolvent for the equation (7). If Hypothesis 4 holds then the stochastic equation ft
= \ Jo
t>0
is a well-defined S' (R )-valued process. For each t > 0 the random variable H * Wr(i) is generalized, stationary random field on Rd with the spectral measure it:
Theorem 10 ([KaZaS], Theorem 3) Assume that Hypothesis 4 holds. Then the process 7£* WT(£) is function-valued for allt>0 if and only if
JR'
ft \ I (s(a, a(A))) 2 cfcr} /x(dA) < +00, t > 0. o /
If for some e > 0 and all t > 0, /** I
JO JRRd
then, for each t > 0, 72.* Wr(t) is a sample continuous random field. Comment: The above results are consequences of previous results (see [KaZaS], Theorem 1) on stochastic integration. For more information on stochastic integral with values in the Schwartz space of tempered distribution S'(Rd) we refer to Ito ([Itol], [Ito2]), Bojdecki and Jakubowski ([BoJal],
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Karczewska
320
[BoJa2]), Bojdecki and Gorostiza ([BoGo]) and Peszat and Zabczyk ([PeZal], [PeZa2]). This is possible to provide sufficient conditions for continuity of the solutions in terms of the covariance kernel Fof the Wiener process Wr rather than in terms of the spectral measure.
Theorem 11 ([KaZaSj, Theorem 4) Assume that d > 2, F is a non-negative measure and A — — (— A) Q / 2 , a e (0, 2]. If for some 5 > 0,
then for the cases (13) the solution of the stochastic Volterra equation (7) has continuous version. Comment: Analogical conditions for existence of function- valued solutions can be derived in a similar way.
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Bojdecki, T. & Jakubowski, J., Stochastic integral for inhomogeneous Wiener process in the dual of a nuclear space, Journal of Multivariate Analysis 34 (1990), 185-210.
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Clement, Ph. fe DaPrato, G., Some results on stochastic convolutions arising in Volterra equations perturbed by noise, Rend. Math. Ace. Lincei s.9, 7 (1996), 147-153.
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Clement, Ph. &; DaPrato, G., White noise perturbation of the heat equation in materials with memory, Dynamic Systems and Applications 6 (1997), 441-460.
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Clement, Ph. & DaPrato, G., Stochastic convolutions with kernels arising in some Volterra equations, in: Volterra equations and applications (Arlington, TX, 1996) 55-65, Stability Control Theory Methods Appl. 10, Gordon and Breach, Amsterdam, 2000.
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[CIDaPrPr] Clement, Ph., DaPrato, G. & Priiss, J., White noise perturbation of the equations of linear parabolic viscoelasticity, Rendiconti Trieste, 1997. [Da]
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Dalang, R. & Frangos, N., The stochastic wave equation in two spatial dimensions, The Annals of Probability 26-1 (1998), 187212.
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Holley, R. & Stroock, D., Generalized Ornstein-Uhlenbeck processes and infinite particle branching Brownian motion, Publ. RIMS, Kyoto Univ. 14 (1978), 741-788.
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ltd, K., Distribution valued processes arising from independent Brownian motions, Mathematische Zeitschrift 182 (1983), 1733.
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Karczewska, A. fe Zabczyk, J., Stochastic PDE's with functionvalued solutions, Preprint 33, Scuola Normale Superiore di Pisa, Pisa, 1997, and in : Infinite Dimensional Stochastic Analysis, P Clement, F. den Hollander, J. van Neerven and B. de Pagter, eds., Proceedings ofhe Colloquium of the Royal Netherlands Academy of Arts and Sciences 1999, North Holland, Amsterdam, 2000.
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Karczewska, A. & Zabczyk, J., Regularity of solutions to stochastic Volterra equations, Preprint 17, Scuola Normale Superiore di Pisa, Pisa, 1999.
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Stochastic Processes and their Applications 86-1 (2000), 141162. [MiSaSol]
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Millet, A. & Sanz-Sole M., Approximation and support theorem for a two space-dimensional wave equation (preprint 1997), Bernoulli 6(5) (2000), 887-915.
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Muller, C., Long time existence for the wave equation with a noise term, The Annals of Probability 25-1 (1997), 133-151.
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Nobel, J., Evolution equation with Gaussian potential, Nonlinear Analysis: Theory, Methods and Applications 28 (1997), 103-135.
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Oberguggenberger, M. & Russo F., The nonlinear stochastic wave equation, Integral transforms and special functions 6 (1997), 58-70.
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Peszat, S., The Cauchy problem for a nonlinear stochastic wave equation in any dimension, submitted.
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Peszat, S. &: Zabczyk, J., Stochastic evolution equations with a spatially homogeneous Wiener process, Stochastic Processes and their Applications 72 (1997), 187-204.
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Priiss, J., Evolutionary integral equations and applications, Birkhauser, Basel, 1993.
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[RoSaSol]
Rovira, C. & Sanz-Sole, M., Stochastic Volterra equations in the plane: smoothness of the law, Preprint 226, Universitat de Barcelona, Barcelona, 1997.
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Rovira, C. & Sanz-Sole, M., Large deviations for stochastic Volterra equations in the plane, Preprint 223, Universitat de Barcelona, Barcelona, 1997, to appear in Potential Analysis.
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Sanz-Sole, M. & Sarra, M., Holder continuity for the stochastic heat equation with spatially correlated noise. Preprint 273, Universitat de Barcelona, Barcelona, 1999.
[Wa]
Walsh, J., An introduction to stochastic partial differential equations, Ecole d'Ete de Probabilites de Saint Flour XIV-1984, Lecture Notes in Mathematics, Springer-Verlag, New York-Berlin
1180 (1986), 265-439.
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Some New Results in the Theory of SPDEs in Sobolev Spaces N.V. KRYLOV University of Minnesota, 127 Vincent Hall, Minneapolis, MN, 55455, USA, email: [email protected] *
Abstract. A description of the modern L-theory of SPDEs is presented. The emphasis is on equations in domains, Sobolev spaces with weights and different power of summability in time and space variables. p
1. INTRODUCTION Evolutional stochastic partial differential equations (SPDEs) arise in many applications of probability theory and have been treated since long ago. Usually they have the following form
du = (aijuxixj + Vu^ +cu + f ) d t
+ (crikuxi + vku + gk) dwk, t > 0, (1.1)
where u = u(t,x) = u(u,t,x), a^ = a^(t,x) = a^(u!,t^x}r.. are random variables looked for or defined for any t > 0 and x € Rd, wk are independent one-dimensional Wiener processes, the summation convention over the repeated indices i,j,k is assumed as usual, i and j go from 1 to d, and k may run through 1,2, .... Underlying is a complete probability space (fl, T, P) and a nitration (^t)t>o of sigma fields Ft C J- '. One of the simplest situations when such equations appear is as follows. Example 1.1. Take a d\ -dimensional Wiener process wt, a constant d x d\ matrix cr, and a smooth function 4>(x) on Kd and define u(t, x) = <j)(x + crwt) . Ito's formula reads d(f>(x + awt) = xi(x + awt) d(awt}1 + (l/2)(pxixj(x
which is rewritten as
du = aijuxixj dt + &tkuxi dwk. with 2a = 2(aij) = aa*. lr
The work was partially supported by NSF Grant DMS-9876586
325
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(1.2)
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This example shows many interesting features of equations like (1.1). In
the general theory it is always required that
[2aij - <7*Vfc]A*AJ' > <5|A|2 VA € Rd, where 5 > 0 is a constant. In the case of equation (1.2) we have 2a — a a* =
0. What happens if, say d = 1, a,<7 are constant and in (1.2) we have 2a - era* = -2? It turns out that the reverse change of variables Xx dt, which is a (deterministic) backward heat equation. One knows that even if the given values of u or <$> at t — 0 are smooth this equation may not have any solution for t > 0. We also learn that 6 = 0 is a degenerate case: in that case the solution (f>(x + o"Wt) is not smoother in x than its initial value. However, if 8 > 0, the solution can be much smoother. Indeed, again let d = 1 and let d\ = 2,
so that a = (a,0). Assume j3 ^ 0. By taking conditional expectation in equation (1.2) given w],t € [0, oo) and denoting u(t, x) = E{u(t,x)\w\,s € [0, oo)}, we easily convince ourselves that
du = auxx dt + aux dw\ . 2
2
Now 2a — a = /3 > 0 and, for t > 0, u(t, x) is infinitely differentiate in x for any bounded
(aw] + y}e-*-y
dy.
This discussion is based on the fact that one can write solutions of (1.2) explicitly. This can be done for most general equations (1.1) as well (see [6], [21]). In the above example we have seen that taking conditional expectations converts a degenerate SPDE into a nondegenerate one. Considering conditional expectations is part of so called filtering problems.
Example 1.2. The filtering problem (estimation of a "signal" by observing its mixture with a "noise") is one of classical problems in the statistics of
random processes. In a fairly general situation we are given a couple of multidimensional processes governed by the system
dxt = h(t, xt, yt) dt + p(t, xt, yt) dWt + a(t, xt, yt) dwt, dyt = H(t, xt,yt)dt + dWt, where h(t,x,y) € E. ,
and (wt, Wt) is a d + do-dimensional Wiener process, independent of the
.Fo-rneasurable random vectors XQ, yo. The filtering problem is to compute at time t the best mean square estimate for
E(
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from the data h, s < t}. In 1972 Fujisaki, Kallianpur, and Kunita derived an equation for P which turns out to be a (nonlinear) SPDE. This first derivation is based on Clark 's representation theorem. Later on other ways were found and smoothness properties of filtering densities were investigated on the basis of SPDEs (see [24]). As of now the shortest way to derive the equation and at the same time get that the solutions are smooth is given in [22]. Example 1.3. Branching measure-valued processes (or super diffusions) which arise in many issues related to population dynamics have attracted a great deal of effort of many investigators (see [3]). They turn out to be solutions of SPDEs. In [5] it is proved that the one-dimensional super-Brownian process at any time t > 0 has a density u = u(t, x) which satisfies the following stochastic partial differential equation oo
du = uxx dt + \/u \~]
where {fi} is an orthonormal basis in LZ(&) and wl are independent Wiener processes. Actually, the equation in [5] is written differently in terms of socalled space time white noise. The advantage of the above form is that it admits a natural multidimensional generalization. It turns out (see [10]) that the super-Brownian process ^ in Kd can be viewed as a solution of the equation
dj,t = A t dt where {y«(/u, x)} is a so-called frame function defined for any finite measure fj,. We see that considering infinitely many Wiener processes is quite essential in this application. This point of view includes super diffusions into SPDEs and allows to construct them on the basis of solving equations like (1.3) (see [15]). Also considering the equations driven by space time white noise as usual SPDEs allows for deriving some difficult results easier (nonexplosion [12], compact support property [9]).
2. THE CAUCHY PROBLEM In [12] a solvability theory of the Cauchy problem for SPDEs like (1.1) is developed in spaces of summable functions with exponent of summability p > 2. If p = 2, so that we are concerned with solutions belonging to the Sobolev spaces W^(^d), such a theory was developed long before (see, for instance, [24]). Recall that W^W1} is the set of all generalized functions on Md whose derivatives up to and including the rath order belong to Lp = Lp(Rd). Roughly speaking, the main tool in W^-theory is integration by parts. There are also approaches based on semigroup methods [2] , [4] , which
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work well for the equations with nonrandom leading coefficients au and again in the Hilbert-space framework. The necessity of the Lp-theory arises, for instance, when one wants to find the solutions numerically. The convergence rate and the way the finite difference should be chosen depend on smoothness properties of solutions. Here by the smoothness properties we mean continuity, Holder continuity, differentiability, continuity of derivatives, and so on. One of inconveniences of W^-theory is that W$(Rd) C Cn~d^(Rd) only if 2n > d, and one can prove that the solutions belong to W^(M.d) only if the coefficients are n — 2 times continuously differentiate. Therefore, if we want to get the solutions m times continuously differentiate with respect to x e Rd, we have to suppose that the coefficients of the equation are more than m + d/2 — 2 times continuously differentiate even if the free terms are of class Cg°(Rd). At the same time, W£(Rd) C Cn~dlp(^d} if pn > d, and, by taking p sufficiently large, we see that the solutions have almost as many usual derivatives as generalized ones. Actually, exactly for this purpose the spaces Wp(Rd) with p > 2 have already been used in SPDB theory (see, for instance, [24]), but the corresponding results, obtained again by integration by parts, were not sharp. Another advantage of the W£ setting with p > 2 can be seen in the case of equations like (1.3). Although these equations are covered by the general WT-theory for any p > 2, for p = 2 we get only the solutions summable to any degree, and the solutions become continuous only for p > 2. By the way, while considering (1.3) it is extremely convenient to allow n to take any values in R. For general n we are working in the spaces of Bessel potentials Hp = Hp(Rd) rather than W^(R d ), and, in the case of equations with cylindrical white noise, we take 7 slightly less than (—3/2). Here we present generalizations of some basic results from [12]. We fix some numbers satisfying q > p > 2 and follow [16]. In [16] only equations
du = (aijuxixj + f ) d t + (o-ikuxi + gk) dw^
(2.1)
with coefficients independent of x are considered. Usual techniques developed for instance in [12] can be easily applied for full equation with variable coefficients. We assume that a u , alk are predictable functions of (u,t) (independent of x) and, for a constant S € (0,1), they satisfy amn = anm and
S"1^!2 > a^A'A^ > <5|A| 2 ,
(1 - S)ai:iXiXj > aijXiX:i,
(2.2)
for all A e Rd and n, m, u, t, where aij = (l/2)aikcrjk. The notation Hp is also used for l^-v&lued functions g, where 1% is the set of all real-valued sequences g = {gk; k = 1,2,...} with the norm defined by \9\l-=Ek\9k\2- Specifically, \\9\\p :=\\\9\l,\\p,
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where || • \\p is the norm in Lp. For stopping times T, we denote (0, r| =
where P is the sigma field of predictable sets. The norms in these spaces are defined in an obvious way. Definition 2.1. For a function u € P|r>0Hp'9(r A T) with values in the space of distributions on Rd, we write u € U^(T) ifuxx € H$~2'q(r), «(0, •) € Lq(£l,F0,H2~2/q), and there exists / e H^~ 2>9 (r) and g 6 H^~1>9(r) such that, for any (f> € CQ°, the equality
/(*, '),) ds + f" f (gk(s,-),4>)dwks
S^0
(2.3)
holds for all £ < r with probability 1. As a shorthand for (2.3) we write du = f dt + gk dwk, Du = /, and Sn = g and always understand equations like (1.1) in the above sense. We also define
The main result of [16] for the Cauchy problem is as follows. Theorem 2.2. Let T G (0, oo), r < T, 7 € R, q > p > 2, / € B$>9(r), 0€HZ +1 '*(T)- Then, (i) in "Hp+ >9 (r), equation (2.1) with zero initial condition has a unique solution. For this solution we have where N = N(d, S,p, q);
(ii) if in addition we are given a function UQ € Lq(ti, FQ, H^+ ~ ), where e > 0, then in ~Hp >9 (r), equation (2.1) with initial condition UQ has a unique solution. For this solution we have
where N = N(d, 6, p, q, s, T) . Moreover if q = p, one can take e = 0. The basis of the results in [12] is the stochastic version of the HardyLittlewood inequality proved in [8]. Theorem 2.2 is based on a further generalization of [8] . Interestingly enough even if there is no stochastic terms in (2.1) the result of Theorem 2.2 is new and in this situation one can take a n y p , g € (l,oo) (see [17]). In the corresponding counterpart of Theorem 2.2 in [12] we assume q = p. Therefore it is worth discussing what we gain allowing q > p. The point is that in many applications, say to filtering equations / = g = 0, so that they belong to any Hp' 9 (r), and at the same time the properties of u(t, •) for each
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individual t are much better when q is large. Here is one of the results from [18]. Theorem 2.3. Let 7 € R, q > p > 2, T € (0, oo), r < T. Assume that we are given some numbers a, a, (3, and J3 satisfying
, Tften /or any function u € Hp' (r), we have /
q
)
(2.4)
fa.s.,) and for any constant a > 0, stopping time f]
E
IHIcflOTiir-'> C([0,T\,H , , qq
)
t l
q
where and the constants N are independent of a, 7, T, T, and u. If we apply Theorem 2.3 to the situation in Theorem 2.2 assuming that / and g do not depend on t and deterministic UQ € Hp , then u € Up '^(T) for any q > p, one can take a, a as close to 0 and /3 as close to d/p as one wishes and from (2.4) one gets that u(t) is a continuous Hq ~ 'p~s function for any large q and small e. By embedding theorems it follows that u(t] is a continuous C^+<1~d/p~e([Q, T}) function for any e 6 (0, 7 4- 2 - d/p]. In the same situation one can take a as close to 1 as one wishes and one gets that u(t) is a 1/2 — K Holder continuous function with values in <77+i-
3. EQUATIONS IN DOMAINS The theory of SPDEs in domains is much more sophisticated. The reason for that can be seen from the fact that even if we consider the simplest one-dimensional equation
du = uxx dt + g dwt for x € (0,1), t > 0 with nonrandom function g and with zero boundary data, then the fact that uxx is square integrable in (t, x) implies that g(t, 0) = g(t, 1) = 0 for all t > 0. Therefore, either we should impose some strange compatibility conditions for all t > 0, or we should look for solutions with different properties of derivatives allowing them to blow up near the boundary. Another example with interesting features is the following.
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Example 3.1. For d — 1, consider the equation du = uxx dt + crux dwt
with nonrandom initial data UQ € Co°(R+), UQ > 0, UQ ^ 0. Here u is a nonnegative number satisfying
du(t, x) = (1 — <72 /2)uxx(t, x) dt in the random domain t > 0, x > 0, x' e R^-1}. Definition 3.2. Take and fix a nonnegative function £ € CQ°(]R+) such that e*~n > 1 Vz € R.
For 7, 6 € R, and p € (l,oo) let H^g be the set of all distributions on such that :
= E en'He
Denote by Ma the operator of multiplying by (xl)a and Dj — d/dx? . It turns out that (see [14]) that H2e spaces are generated by fractional powers of the operator L := M2A - c acting on H^>g = LP(E.^_, (xl)6-ddx). Here c is a large constant. It is also worth noting that H?Q spaces are complex interpolation spaces with natural duality and with CnH^n6 = H®e if n is an integer > 1. Denote
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Definition 3.3. For a function u — u(t) with values in the space of distributions on R£, we writer e Spf$(r) if and only if M2/«~XO) 6 L9(Q, JF0, #JJ2/< M-1^ € H^f (T) and there exist / € H^02'9(r) and 5 € ^p~e"q(T} in the sense of Definition 2.1 we have
sucl1 that
>
In this case we write M~lf = Bn, g = Sw and define
It is important to notice that, if u e ^p'9(r), then n e Hp'9(r), D« 6 IOjr2'9(T)i and the 22-valued function Sn € H^"1'9^), whereas if u e 5j|(r), then M~lu € Hj|(r), MB« € H^'V), and 5 € BjJll9(T). Before stating the main results of of [16] for the initial boundary value problem in R^ we point out that the conditions (3.1) and (3.2) below play an important role. Therefore, it is worth noting that, if g = np with n = 1, 2, ..., they become
d-l+p>e>d-l+p-l/n-x, ij
(1 - <5)a »AiAj > a'iMXX* + au(t)(\l)2(q -p)/(p - 1) respectively. Notice that these conditions are much more restrictive than d— 1+p > 0 > d — 1, which we have in the deterministic case (see [17]). Theorem 3.4. Let q > p > 2, e <E (0, 2/q), T € (0, oo), T < T, 7 e R. Let Mf € ^ f (r], g e tf£ ' (r), M^-^UQ € ^(Q,^, ^ ) and one o/ the following conditions hold q
l
9
2/9+£
p
d-l+p>8>d-l+p-x-l/lq/P\, where x = x(d,P, q,S)>Qisa (small) constant,
{
d-l+p>6> d-2 + p and (1 - <J)a«(*)A*A^' > Q«(t)VA^ + an(t)(Xl)*(\q/p]
(3.1)
- l)p/(p - 1) (3.2)
for all A € Rd and t > 0, w/iere [r] is #ie smallest integer > r. Then in ^p'^(r) there is a unique solution of equation (2.1) with initial condition u(0) = UQ. For this solution, we have
, P,9
'
(3.3)
p,0
where the constant N = N(^, d,p, q, 6, £,T). In addition, if UQ = 0, then estimate (3. 3) holds with N = Nfr, d, p, q, 6) and if q = p, one can take £ = 0 and N = N(y,d,p,q,6).
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Theorem 3.5. Let q > p > 2, e € (0, 2/q), T € (0, oo), r < T, 7 e K,
d-l+p>e>d-x,
Mf e D ^ C T ) , 5 6 H ^ C r ) , M2/*-i-%0 e L q ( ^ ^ / q + £ } . Then all the assertions of Theorem 3-4 hold true again. Remark 3.6. If q = p, (3.1) becomes d-l+p>e>d-2 + p~x- We know from [20] that, if p = q, then the assertions of Theorems 3.4 and 3.5 hold with e = 0 and N independent of T under somewhat weaker but very similar assumptions on Q:
where <5i is any constant satisfying (1 — 8\}a > a. Remark 3.7. Condition (3.4) and, for q = p, even its slightly stronger version (3.1) look quite reasonable. To see this in Example 3.1, take m = 1,2, ..., q = mp, and Q — p — l/m — %. Then condition (3.1) is satisfied and by Theorem 3.4 we get that u € -5- 0"P(1) for any 7. By embedding theorems (see [13])
E [ supx-i-mx\u(t,x)\m*>dt
(3.5)
JO x>0
It turns out that, for any n € {2,3, ...} and e > 0, one can find
Remark 3.8. It turns out that Theorem 3.4 yields better properties of ttraces of u than the ones following from [20]. For instance, in Example 3.1 let p > 2, n € {1, 2, ...}, q = np, and 6 = p — l/n — %. Then condition (3.1) is satisfied and by Theorem 3.4 we get that u € ^2'gp(l) for any 7. By an embedding theorem similar to Theorem 2.3 and proved in [18] this implies
E sup ( / t
1
1 u(t, x) \p dx) n < oo
JO
for any p > 2 and n > 1. Observe that the bigger n the better information about the behavior of u at x = 0 we get.
Finally we state a trace theorem from [18] for equation in half spaces and for traces in Holder spaces. Theorem 3.9. Let 7, 6 € M, q > p, T € (0, oo), r < T. Assume
2/q
f-/3
where k € {0, 1, 2, ...} and e € (0, 1]. Denote
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Then for any u € &I'Q(T) and any multi-indices i and j such that \i\ < k and \j\ = k, we have that (i) on the set of probability one, the functions Du(t, x) are well defined and continuous for (t, x) € [0, T] x Rd and l
(ii) for any constant a > 0,
\t - sf
,,
_2
,0v '
9
+ i ^ '
.,
), ^
'
where N is independent of u, r, and T.
REFERENCES [1] Z. Brzezniak, Stochastic partial differential equations in M-type 2 Banach spaces, Potential Anal. 4 (1995), no. 1, 1-45. [2] G. Da Prato and J. Zabczyk: "Stochastic equations in infinite dimensions", Cambridge University Press, New York, NY, 1992. [3] E.B. Dynkin, "An introduction to branching measure-valued processes", CRM monograph series, Centre de recherches Math. Univ. de Montreal, Vol. 6, AMS, Providence, RI, 1994. [4] F. Flandoli, Dirichlet boundary value problem for stochastic parabolic equations : compatibility relations and regularity of solutions. Stochastics and Stochastics Reports, 29 (1990), 331-357. [5] N. Konno, T. Shiga, Stochastic partial differential equations for some measure-valued diffusions, Probab. Theory Relat. Fields, 79 (1988), 201-225. [6] N.V. Krylov, On explicit formulas for solutions of evolutionary SPDE's (a kind of introduction to the theory), Lect. Notes in Control and In-
form. Sci., 176 (1992) 153-164. [7] N.V. Krylov, A W% -theory of the Dirichlet problem for SPDE in general smooth domains, Probab. Theory Relat. Fields, 98 (1994) 389-421. [8] N.V. Krylov, A generalization of the Littlewood-Paley inequality and some other results related to stochastic partial differential equations, Ulam Quarterly, 2 (1994) 16-26, http://www.ulam.usm.edU/VIEW2.4/krylov.ps [9] N.V. Krylov, On a result of C. Mueller and E. Perkins, Probab. Theory Relat. Fields, 108 (1997), 543-557. [10] N.V. Krylov, On SPDEs and superdiffusions, Annals of Probability, 25
(1997), 1789-1809.
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[11] N.V. Krylov, One-dimensional SPDEs with constant coefficients on the positive half axis, in Proceedings of Steklov Mathematical Institute Seminar, Statistics and Control of Stochastic Processes, The Liptser Festschrift, Kabanov, Rozovskii, Shiryaev eds., World Scientific, Singapore-New Jersey-London-HongKong, 1997, pp. 243-251. [12] N.V. Krylov, An analytic approach to SPDEs, pp. 185-242 in Stochastic Partial Differential Equations: Six Perspectives, Mathematical Surveys and Monographs, Vol. 64, AMS, Providence, RI, 1999. [13] N.V. Krylov, Weighted Sobolev spaces and Laplace's equation and the heat equations in a half space, Comm in PDE, Vol. 24, No. 9-10 (1999), 1611-1653. [14] N.V. Krylov, Some properties of weighted Sobolev spaces in R^, Annali Scuola Normale Superiore di Pisa, Sci. Fis. Mat. Serie 4, Vol. 28 (1999), Fasc. 4, 675-693. [15] N.V. Krylov, Constructing the super-Brownian process by using SPDEs and Skorohod's method, in Proceedings of the Institute of Math, of the Nat. Acad, of Sci. of Ukraine, V. Korolyuk, N. Prortenko, H.Syta eds, Vol. 32, 232-240, Institute of Math., Kyiv, 2000. [16] N.V. Krylov, SPDEs in Lq((Q,T\,Lp} spaces, Electronic Journal of Probability, Vol. 5 (2000), paper 13, 1-29 http://www.math.washington.edu/ ejpecp [17] N.V. Krylov, The heat equation in Lq((0,T),Lp)-spaces with weights, to appear in SIAM J. on Math. Anal. [18] N.V. Krylov, Some properties of traces for stochastic and deterministic parabolic weighted Sobolev spaces, submitted to Journal of Functional Analysis. (Preprint 99-091, SFB-343, University of Bielefeld, http://www.uni-bielefeld.de/homedir/index.html) [19] N.V. Krylov, S.V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients on a half line, SIAM J. Math. Anal., 30 (1999), 298 - 325. [20] N.V. Krylov, S.V. Lototsky, A Sobolev space theory of SPDEs with constant coefficients in a half space, SIAM J. on Math. Anal., 31 (1999), 19-33 [21] N. V. Krylov, B. L. Rozovskii, On the characteristics of degenerate second order parabolic ltd equations, Trudy seminara imeni Petrovskogo, 8 (1982) 153-168 in Russian; English translation: J. Soviet Math., Vol. 32
(1986), 336-348. [22] N. V. Krylov, A. Zatezalo, A direct approach to deriving filtering equations for diffusion processes, to appear in Applied Mathematics and Optimization. [23] S.V. Lototsky, Dirichlet problem for stochastic parabolic equations in smooth domains, Stochastics Stochastics Rep., 68 (1999), no. 1-2, 145175.
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[24]
Krylov
B. L. Rozovskii, "Stochastic evolution systems", Kluwer, Dordrecht, 1990.
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Lyapunov Function Approaches and Asymptotic Stability of Stochastic Evolution Equations in Hilbert Spaces — A Survey of Recent Developments KAI LIU and AUBREY TRUMAN Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, UK, e-mail: [email protected]
1. INTRODUCTION In this paper we shall give an expository survey on the topic of asymptotic stability of stochastic evolution equations in Hilbert spaces. The study of these kinds of equations is motivated by the internal developments of analysis and the theory of stochastic processes such as stochastic partial differential
equations and stochastic delay differential equations on one side, and by a need to describe random phenomena studied in natural sciences like chemistry, biology and in control theory, on the other. In particular, we shall content ourselves with the presentation of some recent progress made by various probabilists and authors, including ourselves over the last several years. In a wide variety of applications where dynamical system models are used to describe the behavior of real world systems, stochastic components and random noises are included in the models. The stochastic aspects of the models are used to capture the uncertainty about the environment in which the system is operating and the structure and parameters of the models of physical processes being studied. The analysis and control of such systems then involve evaluating the stability properties of them. Stability is a qualitative property and is often regarded as the first characteristic of dynamical systems (or models) studied. In the early stages (1940s to 1960s) of the development of the theory and methods of stochastic stability (mainly in finite dimensional spaces), some confusion about the stability concepts, their usefulness in applications, and the relationship among the different concepts of stability existed. Kozin's readable survey [36] clarified some of the 337
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confusion and provided a good foundation for further work. Subsequently, Has'minskii's celebrated monograph [24] and the references cited therein provided a comprehensive statement of stability theory of stochastic processes, mainly, diffusion processes interpreted as the solution processes of finite dimensional stochastic systems governed by standard Ito stochastic differential equations. In the history of the study of asymptotic properties, Lyapunov's method is probably the most effective tool to handle stochastic stability of systems. One of the most important advances in these aspects is the development of the Lyapunov exponent approach for finite dimensional stochastic systems. This is the stochastic counterpart in some sense of the notion of characteristic exponents introduced in Lyapunov's classic work on asymptotic (exponential) stability of deterministic systems. Under some circumstances, this approach provides necessary and sufficient conditions for asymptotic (exponential) stability, but significant computational problems must be solved. For infinite dimensional cases, the Lyapunov exponent method, especially for nonlinear stochastic systems, needs to use sophisticated tools from stochastic process theory and other related branches of mathematics, and it is not relatively a mature subject area.
Another of the most fruitful advances in the stochastic stability theory is the Lyapunov function approach. This method, i.e., Lyapunov's second (direct) method, provides a powerful tool for the study of stability properties of infinite dimensional dynamical systems because the technique does not require ones to solve the system equation explicitly. In the remainder of this work, we shall mainly focus attention on the Lyapunov function approach for analyzing the stability of infinite dimensional stochastic evolution systems. (A). Strong solutions and mild solutions
Generally speaking, there are two main ways of giving a rigorous meaning to solutions of stochastic evolution equations in infinite dimensional spaces, that is, the variational one (for example, see [67] [71]) and the semigroup one (for example, see [17]). As in the case of (deterministic) partial differential equations, we have two notions of strong and mild solutions. Consider the following nonlinear stochastic evolution equation ft
ft
Xt= I A(s,Xs)ds + I B(s,Xs)dW(s),
H
Xto = XQ,
where A(t, •) and B(t, •) are in general nonlinear mappings from a certain
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Banach space V (<->• H) into V*, the dual of V, and Hilbert space H into £(K, H), the family of all bounded linear operators from Hilbert spaces K into H, respectively. Here "<—>•" denotes the injection which is supposed to be continuous and dense. W(t) is a given .Revalued Wiener process with finite trace covariance operator Q (cf. see [17] or [71]). Definition 1.1.
Let (fi, JT,{jTt}, P) be some basic probability space and
p > 1. Suppose that XQ is an H-valued random variable such that .E||o;o||# < oo. A stochastic process Xt is said to be a strong solution on (fl, T, {^t}, P) of the SDE (1.1) for t € [to, T] if the following conditions are satisfied: (a) Xt is a F-valued Jt-measurable random variable;
(b) Xt € MP(t0,T; V) nL 2 (fi;C(t 0 ,T;H)), T>t0>0, where Mp(to,T;V) denotes the space of all F-valued processes (Xt)te[t0,T], measurable (from [to, T] x Q into V) and satisfying rrt
E I \\Xt\\^dt < oo. Here C(to, T; H) denotes the space of all continuous functions from [to, T] to
H; (c) The equation (1.1) in V* is satisfied for every t € [to,T] with probability one.
If T is replaced by oo, Xt is called a global strong solution of (1.1).
Generally, the strong solution concept is too strong exactly as its name shows for some practical purposes and the weaker one, mild solution, turns to be more appropriate at some times. Consider the following semilinear stochastic evolution equation on I = [to,T], VT > to > 0,
f dXt = (AXt + f ( t , Xt))dt + g(t, Xt)dW(t), I Xto = XQ,
where A is the infinitesimal generator of some Co-semigroup T(t), t > to, over H. f ( t , •) and g(t, •) are in general measurable, nonlinear mappings from H to H and H to C(K, H), respectively. Definition 1.2. A stochastic process X, t £ I, defined on (O,F,{F }>P} is a mild solution of (1.2) if t
t
(i) Xt is adapted to ft; rrt
(ii) Xt is measurable and Jto ||Xt||^dt < oo almost surely and
X = T(t - t )x + f T(t- s)f(s, X )ds + t T(t - s)g(s, X )dW(s) t
0
0
s
Jt0
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Jto
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for all t € I with probability one.
There exists an extensive literature on the study of existence and uniqueness of solutions (strong or mild) and associated properties of the above equations. For instance, under some conditions such as coercivity and monotonicity, Pardoux [67] initially established the existence and uniqueness of the strong solution of (1.1), which subsequently became the standard and fundamental setting for a great deal of further work on the theory of stochastic partial differential equations. The reader is referred to the standard monographs such as Da Prato and Zabczyk [17] or Rozovskii [71] for a detailed statement on these aspects. Among some recent developments on this topic of other types of equations closely related to stability analysis below, we would only like to mention that, for the time delay case, Real [70] initially investigated the existence and uniqueness of strong solutions of linear stochastic evolution systems. By carrying out a Caratheodory approximation program, Liu [46] investigated the same problems for a class of semilinear stochastic delay evolution equations. Under some reasonable conditions as in [67], Caraballo, Liu and Truman [7] established the existence and uniqueness of strong solutions of a class of non-autonomous Hilbert space-valued stochastic functional differential equations with bounded time delays. Since the existence and uniqueness of solutions are beyond our main concerns in this paper, we shall not discuss them in more detail. (B). Some stability definitions
By analogy with the finite dimensional case, there are at least three times as many definitions for the stability of stochastic evolution systems as there are for deterministic ones. This is certainly because in a stochastic setting there are three basic types of convergence: convergence in probability, convergence in mean and convergence in an almost sure (sample path, probability one) sense. The reader should be cautioned to examine carefully the definitions of stochastic stability used when interpreting any stochastic stability results. The following is a brief list of some definitions used quite often in applications. Definition 1.3. (Stability in Probability) The solution X € H, t > 0, of (1.1) or (1.2) is said to be stable in probability if for arbitrarily given e, e' > 0, there exists £(e,e') > 0 such that for all ||#o||.ff < ^> t
t>0
Here, P denotes the probability on the probability space (fi,.F, {.Ft},P) in which the solution process Xt € H, t > 0, is formulated.
Definition 1.4. (Stability in p-th Moment) The solution X e H, t > 0, of (1.1) or (1.2) is said to be stable in p-th moment, p > 0, if for arbitrarily t
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given s > 0, there exists 6(s) > 0 such that ||JCO||H < 8 guarantees that
E\sup\\Xt(x0)\\pH}<£. *• t>o ) Definition 1.5. (Almost Sure Stability) The solution Xt € H, t > 0, of (1.1) or (1.2) is said to be almost surely stable if for each E > 0, there exists a 8 > 0 such that ||a;o||if < 8 guarantees that p{sap\\Xt(x0)\\H<£} = l. *- t>0
'
Note that almost sure stability is equivalent to saying that with probability one, all sample solutions are stable. Similar statements can be made for asymptotic stability and exponential stability. Definition 1.6. (Asymptotic Stability in Probability) The solution Xt € H, t > 0, of (1.1) or (1.2) is said to be asymptotically stable in probability if it is stable in probability and for each e > 0 and XQ € H,
•
lim P{ sup \\Xt(x0)\\H > s\ = 0. ¥0 °
*• t>T
'
Definition 1.7. (Asymptotic Stability in p-th Moment) The solution Xt € H, t > 0, of (1.1) or (1.2) is said to be asymptotically stable inp-th moment, p > 0, if it is stable in p-th moment and for any XQ € H,
lim .
Definition 1.8. (Asymptotic Almost Sure Stability) The solution Xt € H, t > to, of (1.1) or (1.2) is said to be asymptotically almost surely stable if it is stable in Probability and for any XQ € H
P{ lim Definition 1.9. (p-th Moment Exponential Stability) The solution Xt € H, t > 0, of (1.1) or (1.2) is said to be p-th moment exponentially stable if there exist positive constants a > 0 and P(XQ) > 0 such that
E\\Xt(xo)\\pH <
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*-at
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Definition 1.10. (Almost Sure Exponential Stability) The solution Xt € H,
t > 0, of (1.1) or (1.2) is said to be almost sure exponentially stable if there exist a > 0, /3(zo) > 0 and random time T(a>) > 0 such that t > T(u] guarantees almost surely
\\Xt(x0)\\HQ,is given
by
Xt = exp {b/3t + (a - b2/2)t}x0, t>0. (1.4) By (1.4) and the properties of Brownian motion, it is easy to see that for no positive constant e > 0 does there exist a number 6 > 0 such that almost all the sample trajectories of the solutions originating at XQ ^ 0, \XQ\ < <5 remain in an e-neighborhood of x = 0 even if the unperturbed term is very stable (i.e., a < 0) and \b\ is very small.
In the remainder of the paper, we shall present an up-to-date survey on stability properties, especially those in association with our recent work on the stability of non-linear stochastic evolution equations. The organization of this survey is as follows. In Section 2, we begin with a brief statement of classic work on the stability of stochastic linear evolution equations, but concentrating on those results which will motivate future work in connection with the nonlinear case. Section 3 is devoted to the investigation of general nonlinear stochastic evolution equations and Section 4 to a statement of certain more specific types of topics such as the stability of stochastic delay evolution equations and ergodicity for infinite dimensional stochastic systems.
2. STABILITY OF LINEAR SYSTEMS (A). Linear deterministic equations
We start our statements from the following deterministic linear Cauchy problem
= AXt(x0),
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X0(x0) = x0£S,
(2.1)
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where A is the infinitesimal generator of a Co-semigroup T(t), t > 0, on a Banach space S. If x0 € T>(A), then T(t)x0 € T>(A), and
^-(T(t)x0) = AT(t)x0 = T(t)Ax0, cit
t > 0.
Hence Xt = T(t)xo is the solution of the linear system (2.1). It is immediate to see that the solution is exponentially stable if and only if for some positive constants M > 1, // > 0, and all t > 0
\\T(t)\\ <M-e-^.
(2.2)
If the state space S is finite dimensional, there are many equivalent conditions for the solution to be asymptotically or exponentially stable. They are based either on the properties of the spectrum of the matrix A or on the existence of an appropriate Lyapunov function. For instance, the following proposition holds.
Proposition 2.1. Let S be finite dimensional, for instance, R , n>l. The solution is exponentially stable if and only if one of the following conditions holds: n
(i) all eigenvalues of the matrix A have negative real parts, max{EeA : det(A7 - A) = 0} < 0;
(2.3)
(ii) there exists a nonnegative definite matrix P > 0 such that
PA + A*P = -I.
(2.4)
In the latter case, the function V(x) = < Px, x > is the Lyapunov function for (2.1) in the sense that for every Xt, t >Q, of (2.1)
where < •, • > denotes the standard inner product in S. If the Banach space S is infinite dimensional, the above Proposition 2. 1 is only partially true. For instance, the condition (2.3) does not imply stability of the Cauchy problem unless some extra restrictions on A are made and on this occasion, (2.3) is, of course, replaced by
sup{ReA : A € a(A}} < 0
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where o(A] is the spectrum of the linear operator A. This behavior is certainly a consequence of the fact that linear operators in finite dimensional Banach spaces have only point spectrum. Since this is not the case in general infinite dimensional spaces one does not expect this result to remain true. In particular, in this general framework, (i) of Proposition 2.1 can be formulated as follows:
Proposition 2.2. Let us define for every generator A the lower and upper stability indices: l(A) = sup $.Re\ : X e
r(A)
= inf -{V : \\T(t)\\ < Me^, for some M>land all t > 0 j.
Then
(2.6) and therefore if the system (2.1)
is exponentially stable,
< 0.
(2.7)
Moreover, if (i) the semigroup T(t)
is differentiate
on t € [0, +00),
or
(ii) for some to > Q, T(to) is a compact operator, i.e., T(to) bounded sets in S into relatively compact ones, then
takes any
l(A) = r(A),
(2.8)
and consequently (2. 7) implies exponential stability. In particular, if S is finite dimensional, the equality (2.8) holds. In the case where S is infinite dimensional, although it is not generally true for (2.5) to imply the stability of the semigroup, the Lyapunov condition (2.4) has its complete infinite dimensional counterpart. Precisely, we have the following result which is due to Datko [19].
Theorem 2.1. (Datko [19]) The following statements are equivalent:
(i) T(t] is stable, i.e., (2.2) is true; (ii) /0°° \\T(t)x\\2Hdt < oo for each x € H; (Hi) there exists a self-adjoint nonnegative operator P in C(H] such that
2 < Ax, Px >H= — < x,x >H for each x 6
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345
Linear stochastic evolution equations
The fundamental idea for our purposes is to hopefully establish a stochastic version of the Datko's result above for linear stochastic evolution equations. Consider the following linear stochastic evolution equation
f dXt = AXtdt + B(Xt)dW(t), 1 XQ = XQ € H
where A is the infinitesimal generator of a strongly continuous semigroup T(t)
over H, B(-) e £(H, £(K, H)) and W(-) is a given Wiener process with
covariance operator Q. Definition 2.1. The mild solution of (2.9) is exponentially stable (or stable) if the solution Xt satisfies
/'
Jo
< oo for each XQ e H.
The following powerful result which is due to Zabczyk [79] and Ichikawa [26] (also see Haussmann [25]) states that stochastic stability of the mild solution of (2.9) is equivalent to the existence of a solution of a certain Lyapunov equation as well as to exponential stability of the mild solution in mean square sense.
Theorem 2.2. (Zabczyk [79] and Ichikawa [26]) The following statements are equivalent: (i) the mild solution of (2.9) is stable; (ii) there exists a non-negative operator 0 < P < £(H)
such that
2 < Ax, Px >H +(A(P)a;, x) = - <x,x>H for all x € V(A);
(2.10)
(Hi) there exist positive numbers a > 1, a > 0 such that
E\\Xt(x0)\\2H
x, y e H, P € L(H).
Generally, Theorem 2.2 is somewhat inconvenient to use directly (also see Da Prato and Zabczyk [17] for an example whose stability can be completely characterized by Theorem 2.2). It is therefore expected to find some more effective sufficient conditions by means of Theorem 2.2 to secure the stability properties. One of the conditions which is due to Haussmann [25] states that if
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(HI)
A generates an exponentially stable semigroup;
(H2)/ 0 ~r(*)*A(W)# <1> then the mild solution of (2.9) is exponentially stable in mean square. Combining with an approximation argument, Curtain [13] also applied this sufficient condition to some stochastic second order partial differential equations to derive stochastic stability properties.
In practice, moment stability criteria are often too conservative to be useful in applications. This is not surprising because the sample paths rather than moments or probabilities associated with trajectories are observed in real systems and the stability properties of sample paths can be most closely related to deterministic counterparts. During the past twenty years, pathwise stability with probability one studies of infinite dimensional stochastic systems have attracted increasing attention of researchers. For the stochastic linear equation (2.9), Haussmann [25] presented a condition to handle the pathwise stability provided the mild solution of (2.9) is mean square stable. Precisely, assume the following conditions:
(H) (i) {T(t)} is an analytic semigroup; (ii) there exists a function / > 0 such that for all to < t < oo /"* / f ( s ) 2 d s < oo Jto
and for all i, t > to > 0, x € H,
\\AT(t)B(x)i\\H < f(t)\\x\\H where {fa}, i > 1, is a complete set of orthonormal eigenvectors of the covariance operator Q.
Theorem 2.3. (Haussmann [25]) Assume the condition (H) holds. If Xt is a mild solution of the equation (2. 9) satisfying the conditions (HI) and (H2), then there exist a, (5 > 0, T(UJ) < oo, such that for t > T(UJ)
Xtxo2
a.s.
Generally, Condition (H) is rather restrictive so that it is difficult to apply it to many practical situations. Another method of attacking almost sure stability of Equation (2.9) is to consider its strong solution. Let V be a dense Banach subspace of the real Hilbert space H with norms || • ||y and || • \\H, respectively. Let V* be its dual with norm denoted by || • ||*.
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Then V <—>• H <-> V * , and we assume that the injections are continuous, i.e., II • \\H < /3|| • || v for some (3 > 0. We write < x, y > for the dual between x € V* and y € V. Consider the following linear stochastic system
= AXt+B(Xt)dWt, 1 XQ = XQ € H
where A : V —> V* is a bounded mapping which is coercive, i.e., there exist a > 0, A € R1 such that for any x € V,
<-a\\x\$ + X\\x\\2H and B is an element of £(H,£(K,H)). following result.
(2.12)
As a consequence, we have the
Theorem 2.4. Assume the coercive condition (2.12) holds. If Xt is a strong solution of (2. 9) which is exponentially stable in mean square, then there exist a, /3 > 0, T(u) < oo such that for t > T(u],
Xt2<^-x02e-^
a.s.
It is possible to relax the conditions on B, i.e., B will only be required to lie in £(V, £(K, H ) ) , but for our stability purposes the coercivity (2.12) is strengthened to
2 + < &(I)x,x><-a\\x\\l- + X\\x\\2H,
Vx € V.
(2.13)
On this occasion, for P e £(H, H), A(P) € £(V, V*) is defined by
)z, y >= tr[B(x)*PB(y)Q],
x,y<=V.
In particular, we can have the following stability result similarly to Theorems 2.3 and 2.4.
Theorem 2.5. (Haussmann [25]) Assume B €. £(V,£(K,H)) satisfying (2.13). If (HI) and (H2) hold, then there exist positive constants M, /J, > 0 such that for any strong solution Xt of (2.9) 2
2 H<M-\\x0\\ He-^,
t>Q.
Moreover, under the same conditions the solution is also pathwise exponentially stable with probability one, i.e., there exist a, (3 > 0, T(tu) < oo such that for all t>T(u], Xt2
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a.e.
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3. STABILITY OF NONLINEAR EVOLUTION EQUATIONS (A) . Equivalence of p-th moment integrability and exponential stability
Let T(t), t > 0, be a strongly continuous semigroup of bounded linear operators on the Hilbert space H with norm || • ||#, inner product < •, • >#, and its infinitesimal generator A. Then T(t)xo, XQ € 'D(A), corresponds to the solution of the differential equation
(3.1) It is shown in Theorem 2.1 that two statements below are equivalent:
(a) | T(t)\\ < Me-v*, t > 0, for some M > 1 and // > 0; (b) J0°° \\T(t)x\\2Hdt < K\\x\\*H, x e H, for some K > 0. Therefore, it is natural to ask whether the equivalence of (a) and (b) above still holds for certain nonlinear semigroups such as perturbed systems, or more generally, for general semilinear stochastic evolution equations. To put this more precisely, consider the following autonomous semilinear stochastic evolution equation:
dXt = [AXt + F(Xt)]dt + G(Xt)dW(t),
X0 = x0€H
(3.2)
where F : H —» H and G : H —> C(K, H) are nonlinear and satisfy the usual Lipschitz conditions together with F(0) = 0 and G(0) = 0. The question is whether or not two statements below are equivalent for each p > 1:
(a'). E\\Xt(xQ)\\pH < Me-^\\XQ\\pH, XQ € H, for some M > 1 and /x > 0; (b'). /0°° E\\Xt(xQ)\\pHdt < K\\xQ\\pH, XQ € H, for some K > 0. It is worth pointing out that the question above is also important, as one may expect, from the viewpoint of Lyapunov's function methods even for deterministic case (i.e., (?(•) = 0 in (3.2)). The point is that there exist some situations where one may easily find "Lyapunov functions" which are not strictly positive-definite (i.e., ^ c||a;||^, c > 0) but assure p-th moment integrability, i.e., (b) or (b'). To be more specific, let P > 0 satisfy (2.10) and suppose further that
2 < PAx, Ax + F(x) >H < -A||x||^,
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x € T>(A)
for some A > 0.
(3.3)
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Then applying the usual Lyapunov arguments to (3.2) with G(-) = 0 and < Px, x >H, it is possible to obtain
< PXt(xo), Xt(xo) >H < Me ^Uxolljj f°r where Xt(xo) is the mild solution of (3.2)
some
M > 0 and fj, > 0,
with (?(•) = 0. However, we cannot
in general conclude that ||^t(£o)||tf < -/Ve-I/t||a;o||H for some N >1 and v > 0. The reason is that for some A, for instance, A is the generator of a stable
analytic semigroup, it can be proved that < Px, x >H cannot be equivalent In spite of the remarks mentioned above, fortunately, in some situations we can deduce from (3.3) the mean square integrability, i.e., oo
/ .
\\Xt(x0)\\2Hdt < K\ x0\\2H, x0€H for some K > 0.
In Ichikawa [30], it is judged that under some rather weak conditions on Equation (3.2), the condition above or (b') actually implies exponential stability in square or mean square sense. Another of the methods handling stability of mild solutions is to carry
out a comparison procedure which was introduced in Ichikawa [28] and states that mean square stability of a class of nonlinear stochastic evolution equations is equivalent to the same property of a certain linear stochastic evolution equation provided noise terms in the former are dominated by those of the
latter in a suitable sense. This consequence certainly generalized Morozan's well-known stability results concerning the stochastic differential equation of Lur'e type in the finite dimensional case to a much wider class of stochastic evolution equations in Hilbert spaces. (B).
Stability criteria based on a coercivity condition
The investigation of stability of non-autonomous stochastic differential equations in Hilbert spaces has received increasing amount of attention over the last several years. Following the classic work of deterministic partial differential equations from Lions and Magenes [42], Pardoux [67] initially established
the existence and uniqueness of the solution of (1.1) in a strong sense in which a coercive condition plays a fundamental role. Chow [11] pointed out that if ones take the Lyapunov functional to be quadratic, under some reasonable assumptions the coercive condition then turns out to play the role of a stability
criterion. In the same paper, based on a Lyapunov functional argument the
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results of almost surely asymptotic stability and boundedness were also investigated. The first attempts to generalize Chow's results to non-autonomous cases were made by Caraballo and Real [9]. As a matter of fact, Caraballo and Real obtained that under a slightly weaker hypotheses than those in [11], the strong solution of the equation (1.1) is exponentially stable not only in mean square but in a pathwise with probability one sense. Caraballo and Liu [4] improved and generalized their results to the non-autonomous case in a systematic way. Precisely, the following coercive condition playing a role of the stability criterion was presented in [4] . Consider the stochastic differential equation (1.1) satisfying the standard conditions of existence and uniqueness of strong solutions from [67] . Suppose there exist constants a > 0, fj, > 0, A € R1, and a nonnegative continuous
function 7(£), t € R+, such that
'«,
v£V, (3.4)
where p > I and, for arbitrary 6 > 0, 7(i) satisfies ^(t) = o(est), as t —¥ oo, i.e., lim^oo ^(t)/est = 0. Here < •, • > denotes the duality between V* and V (< x € V*,y € V >) and \\B(t,v}\\% = tr(B(t,v)QB(t,v)*).
Theorem 3.1. (Caraballo and Liu [4]) Assume the coercive condition (3-4) holds, then there exist constants T > 0, C(XQ) > 0 such that for the strong solution Xt(xo) € V of (1.1),
E\\Xt(x0)\\2H < C(x0) • e~Tt,
V* > 0,
(3.5)
if either one of the following hypotheses holds
(ii) A / ? 2 - a < 0 (p = 2), where \\ • \\H < /3\\ • ||y. Furthermore, under the same conditions the strong solution is also exponential almost surely stable. The exponential decay term appearing on the right hand side of (3.4) is essential for the stability result. In fact, to interpret this, it suffices to consider the following one-dimensional linear ltd equation (note that on this occasion V = H): Example 3.1. Assume Xt satisfies the following stochastic differential equation
dXt = -pXtdt + (1 + t)-*dpt,
t > 0,
with initial datum XQ = 0, where p, q > 0 are two positive constants and f3t is a one-dimensional standard Brownian motion. Clearly, the corresponding coercive condition (3.4) now turns out to be
2 < -pv, « > + [ ( ! + t)~q] = -2pv2 + (1 + t)~2q,
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where < •, • > denotes the standard inner product in R1. However, on this occasion the solution is exponentially unstable. Indeed, it is easy to obtain the explicit solution I Jo
t>0,
which, by a direct computation, immediately yields that for arbitrarily given p > 0, q > 0, the Lyapunov exponent
t —J-00
t
Similarly, by using the law of iterated logarithm
,.hmanp ____ Mt _____ = 1 V2 < Mt > loglog < Aft >
-,
a.S..
we can easily deduce the almost sure Lyapunov exponent limsup ——— — = 0
a.s.
In other words, despite the typical stability of ordinary differential equation
dXt = -pXtdt, the polynomial decay types of noise terms are not sufficient to guarantee the exponential stability of its stochastically perturbed system.
Under some Lipschitz and linear growth types of conditions on the coefficients of Equation (1.2), Ichikawa [27] and Liu and Truman [52] presented a Lyapunov function programme to investigate the exponential stability for mild solutions of a class of semilinear non-autonomous stochastic evolution equations, (cf. also see Theorem 4.3 in Section 4). When one extends finitedimensional results based on Lyapunov functions to Hilbert spaces, a difficulty encountered is that one needs strong solutions in order to use the classic tools such as Ito's formula, Burkholder's inequality, etc. An effective technique to solve this problem is introducing suitable approximation systems with strong solutions and carrying out a limiting procedure for mild solutions, a method which is quite useful even in the stability study of stochastic functional partial differential equations. However, it is also worth pointing out that in order to keep this programme going through, some suitable restrictions to Lyapunov functions must be made. A direct method by employing some properties of stochastic convolution and semigroups instead of using the approximation procedure mentioned above is possible. In particular, by
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this method, in Taniguchi [74] some stability criteria of mild solutions are established for a class of semilinear stochastic evolution equations.
(C). Lyapunov functions and stability
In Theorem 2.2, we extend a celebrated result of Lyapunov concerning Hurwitzian matrices (n x n matrices with eigenvalues in the half-plane Re z < 0) to a class of linear stochastic evolution equations. It is well-known that the usefulness of Lyapunov's theorem in ordinary differential equations is that it allows for an explicit representation of a Lyapunov function as a positive definite quadratic form. Using this representation one may then, for instance, study the effects of perturbations on asymptotically stable, linear constant coefficient systems of ordinary differential equations. To make this more specific, consider the finite dimensional ordinary perturbed system
Let A have its eigenvalues in the left half-plane Rez < 0. By Lyapunov's theorem we then have a positive definite Hermitian matrix B such that
±(BXt,Xt) = -\Xt\2+2Re(BXt,f(t,Xt)). Hence, one may obtain conditions on the nonlinear perturbation /(•, •) to insure that d(BXt,Xt)/dt is negative definite (negative semidefinite) thus insuring asymptotic stability (stability) of the zero equilibrium. It is essential, however, that for (BXt, Xt) to be a Lyapunov function, we have an estimate of the form (BXt,Xt) > c\Xt\2 for some c > 0. In other words, the Hermitian form B has to define an equivalent norm on H. If H is finite dimensional, this is always the case. However, if H is infinite dimensional, this could be false. Consider the partial differential equation
where a, b and c are real numbers, with the boundary conditions XQ(X) € L? (— OO, +Oo) fl L 1 (— OO, +OO).
XQ(X) — X Q ( X ) ,
By a direct computation, it can deduce that +00
|zo(A)| 2 exp{(-2a2A2 + 2c)t}d\. /
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•00
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Here | • \H denotes the norm of Hilbert space H = L 2 (—oo,+00). Assume c < 0, then If we further consider as in the finite dimensional case, /•+00
/>00
\X?°(-)\'2Hdt =
A(x 0 ) := / Jo
/-+00
z 0 (A)| 2 exp{(-2a 2 A 2 +2c)t}dAdt
I Jo
J—oo
+00
/
|£ 0 (A)| 2 /(2a 2 A 2 -2c)dA, •OO
then A(#o) does not satisfy the condition A(#o) > ^l^olf^ f°r
some a
> 0-
The Lyapunov function characterization method goes back at least to Has'minskii [24] for the study of stability of finite dimensional Ito stochastic equations and to Datko [19] for infinite dimensional deterministic evolution equations. For infinite dimensional stochastic evolution equations, an appropriate description of Lyapunov function characterization remains open for a long time until recently Khasminskii and Mandrekar [35] firstly took an essential step on this aspect in the linear case. Precisely, suppose A : V —t V* and B : V -» £(K, H) satisfy
||Ac||* < ai||o;||v
and
||-B«||/;(K,H) < MNIv
for all x e V
(3.7)
for some a\ > 0, 61 > 0. Furthermore assume the following coercive condition holds: there exist a > 0 and A € R1 such that:
2
^ Av
"^ )
v ~^
I II RT||2
^ ~~>
<^
C*CK H} — —
/->,ll/v.||2
IV
''
i 111/1.112
If'
WT c. T/
VO/*CK.
C\ S\
10.01
Theorem 3.2. (Khasminskii and Mandrekar [35]) Assume the assumptions (3.7) and (3.8) hold. Suppose Xt, t > 0, is the strong solution of (2.9). //
there exists a function A : H —> R1 which satisfies the following: (i) A(-) satisfies all the conditions of using Ito's formula; (ii) ci^x\(jj < A(o:) < C2||a;|||f, Va; € V; (Hi) (LA)(x) < -c3k(x), Vx € V, where L is the infinitesimal generator of the Markov process Xt, that is, (LA)(s) := < Ax, A'(z) > +1/2 • tr(k"(x)(Bx)Q(Bx)*), x € V, and a, i = 1,2,3 are positive constants, then X, t > 0, is exponentially mean square stable, i.e., there exist positive constants c > 0, (3 > 0 such that t
xQ)\^H
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for all x0 6 H,
t>0.
(3.9)
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Conversely, suppose (3.9) holds and let />oo
K(xQ}=
Jo
E\\X3(xo)fvds
for all x0 € H,
then there exist constants c, > 0, i = I, 1, 3 such that Conditions (i), (ii) and (Hi) above hold. If, in addition, we assume t -> £'||Jft(3;)||y is continuous at zero for x
Theorem 3.2 provides a very effective tool in the stability study by the so-called first order approximation argument (cf. see Theorem 3.4 below). However, it is worth mentioning that although some appropriate nonlinear
extensions are possibly made as argued in [35], this result is somewhat restrictive to be applied to even linear stochastic evolution systems of nonautonomous type such as Example 3.2 below. Example 3.2. Consider the following linear stochastic partial differential equation:
' dYt(x) =
t>Q, Y0(x)
= yo(x),
xe(o,i),
'
Yt(Q) = Yi(0) = 0,
where (3(t) is a standard one-dimensional Brownian motion and p, is a positive
number. By a direct computation, it shows that on this occasion Theorem 3.2 cannot be applied to this case since Conditions (3.7) and (3.8) are not satisfied. (By employing the following Theorem 3.3, it can be proved the solution of (3.10) is exponentially stable in mean square.) In [49] , Liu extended and improved Theorem 3.2 to obtain a non-autonomous, non-linear version which also contains as a special case some results such as ultimate boundedness characterization (cf. Theorem 4.4 in Section 4) in [57]. Precisely, assume there exist constants a > 0, 7 > 0, // > 0 and A € R1 such that:
2 < A(t,v),v> +\\B(t,v)\\2c(K m < -alHIv+AlHllf+T-e-^, W € V, t > 0 (3.11) Theorem 3.3. (Liu [49]) Suppose Xt, t > 0, is the strong solution of (1.1) and (3.11) holds. If there exists a function A.: H x R+ —>• R1 which satisfies the following: (i) A(-,-) satisfies all the assumptions of using ltd 's formula; (ii) ci H^ll^ - he-*** < A(x, t) < c2\\x\\2H + k2e-^, Vz e V, t> 0; (in) (LA)(x, t) < -c3A(z, t) + kAe-»3t, Vx € V, t > 0, where Cj > 0, A^ > 0, //, > 0, i = I, 2, 3, and L is the corresponding infinitesimal generator of Markov process Xt, i.e., (LA)(£, x) := A.'t(t,x)+ <
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A(t,x), A.x(t,x) > +l/2-tr(A'xx(t,x)B(t,x)QB(t,x)*), exponentially mean square stable.
then Xt, t > 0, is
Conversely, suppose Xt, t > 0, is exponentially mean square stable and
let
fT+t0
A(z0,to)=/ Jto
ru
eet°( \ Jto
e-9'E\\Xf°\$da)du
(3.12)
'
where Xt0 = XQ and T, 6 are two proper positive constants and assume A(-, •) satisfies all the conditions of using ltd 's formula, then there exist constants Ci > 0, ki > 0, fii > 0, i = 1, 2, 3 such that the conditions (ii) and (Hi) above hold. A typical procedure employing Theorem 3.2 or 3.3 to treat the stability of certain nonlinear stochastic evolution equations can be carried out by virtue of the so-called first order approximation method. For instance, we actually have the following result. Theorem 3.4. Consider the equation (2.9) satisfying the conditions (3.7), (3.8) and let {X(xo),t > 0} be its strong solution with t —>• £/||X (a;o)||y continuous. Assume that this solution is exponentially stable in mean square. Let {Xt(xo),t > 0} be the strong solution of nonlinear equation t
t
Xt = x0 + f A(Xs}ds + I B(Xs)dW(s), Jo Jo
(3.13)
where XQ € H, A : V -v V* with \\A(v)\\* < /3\\v\\v, P > 0, B(v) € C(K,H) for all v £ V. If, further, \\A(v) - Av\\l + \tr[B(v)QB(v)* - (Bv)Q(Bv)*
cv
v
for c > 0 small enough in a sufficiently small neighborhood ofv = 0 in \\ • ||y. Then the strong solution of (3.13) is stable in probability in || • \\jj.
(D). Stabilization of stochastic (deterministic) evolution equations
Based on the viewpoint of perturbations of infinite dimensional dynamical systems, one of the most important problems in stability theory is the socalled stabilization by white noise sources of deterministic dynamical systems. The first work on this aspect in finite dimension goes at least back to
Has'minskii [24], who stabilized a system by using two white noise sources. Mao [60] also presented a theory on the stabilization and destabilization for finite dimensional nonlinear systems by Brownian motion. By employing some results from Liu [44] and Liu and Mao [51], Caraballo, Liu and Mao [8] reformulated some of them as criteria for stabilizing (deterministic and
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stochastic) partial differential equations. As a consequence, these results extend to infinite dimensional systems the corresponding ones from Mao [60] and moreover improve the stability criteria in Caraballo and Liu [4] and Chow [11].
For the motivation, let us consider a one-dimensional rod of length TT whose ends are maintained at temperature zero and whose sides are insulated. Assume that there is an exothermic reaction taking place inside the rod with heat being produced proportionally to the temperature. The temperature in the rod may be modeled to satisfy
X0(x) = where r depends on the rate of reaction. If we assume r = TO, a constant, then we can get the explicit solution as follows Xt(x) = n=l
where XQ(X] — Y^Li an sinra. Hence we obtain exponential stability if n2 > TO for all n € N, or equivalently, TQ < 1. Observe that, in general, for TQ > 1 the trivial solution is not stable. Suppose now that r is random, and assume it is modeled as TO + so that the equation (3.14) becomes
dXt(x) =
j + roxt(x) dt +riXt(x)d(3t, t>Q, 0 < x < TT,
where 0t is a one-dimensional Wiener process. Haussmann proved in [25] that when ro < 1 (i.e., the unperturbed system (3.14) is very stable), the perturbed system (3.15) remains pathwise exponentially stable if r\ < 2(1 — TO), i.e., if the perturbation is sufficiently small so that this relation is satisfied. On the other hand, since a new noise term is included in the perturbed system (3.15), a natural problem now arises: as TO > 1 (a case to which Haussmann's results fail to be applied), is it possible to deduce any exponential stability results for the perturbed system (3.15)? In other words, is it possible to stabilize the unperturbed (3. 14) by a suitable noise source? In [8] , a general result which can be specially applied to answer this question was presented
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to stabilize a class of deterministic nonlinear partial differential equations. Precisely,
Theorem 3.5. (Caraballo, Liu and Mao [8]) Assume the strong solution
of (1.1) with W(t) = /3t, t > 0, satisfies that \\Xt(x0)\\H ^ 0 for all t > 0 provided \\XO\\H 7^ 0 almost surely. Suppose further that
(i)
< B(t,x),x >2H> p(t)\\x\\*H,
VzeF; (Hi) v(t}, \(t) and p(t] are three continuous functions such that 1 /"*
limsup- / v(s}ds < VQ € R1, t-*00
t JQ
1 f* limsup- / \(s)ds < \Q € R+, t-XX>
t JQ
i r*
limsup - / p(s)ds > po € R+. t-^oo
t JQ
Then, the strong solution of (1.1) satisfies limsup 10g|l*fo)l1* < -(Po - ,0 - Ao/2),
a.,.
t— >-oo
for any XQ € H. In particular, if2po > IVQ + AQ the equation (1.1) is almost surely exponentially stable.
Another interesting fact is that noise sources can also be used to stabilize some stochastic partial differential equations. For instance, returning once again to the previous equation (3.15), i.e., if T-Q > 1 and r\ € R1 satisfies PI < 2(ro — 1), we actually do not know whether the trivial solution is stable or not. But, if the system is perturbed by another multiplicative noise of the same type, say r^X^\^ where $ is another one-dimensional Wiener process independent of /?*, as a special case of another result (i.e., Theorem 5 from Caraballo, Liu and Mao [8]), it can be specifically deduced that the system
dXt(x) -
^ + roxt(x) dt + nXt(x t>0,
0 < X <7T,
becomes pathwise exponentially stable once again if r^ is chosen largely enough.
4. STABILITY AND SOME RELATED TOPICS
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(A).
Liu and Truman
Stability with a certain decay rate
The stochastic evolution system (1.1) or (1.2) could be commonly illustrated as a perturbed stochastic system by space-time white noise sources of a deterministic dynamical system. For instance, consider the system (1.1) which could be regarded as the perturbed stochastic system of the corresponding deterministic one
= A(8,X8),
Xto=x0£H.
(4.1)
As we mentioned in the previous sections, even if the solution of (4.1) is exponentially stable, for instance, suppose A(t,Xt) = AXt and A generates a stable Co-semigroup T(t) on H, it cannot be generally deduced that its perturbed stochastic system (1.1) remains exponentially stable. However, under certain circumstances, it is found that although its perturbed system fails to be exponentially stable, it indeed remains stable instead with a slower decay, a case which is very important for a variety of stochastic control problems. Particularly, in many situations, the estimate of suitable decay rates is always involved in the interested models. In fact, to put our motivation more precisely, let us turn back to Example 3.1 once again.
Recall that Xt satisfies the following one-dimensional system
dXt = -pXtdt + (1 + t)-qd(3t,
t > 0,
with initial datum XQ = 0, where p, q > 0 are two positive constants and (3t is a one-dimensional standard Brownian motion. It is shown there that for arbitrarily given p > 0, q > 0, the Lyapunov exponents
logE\Xt\2
hm — — •——— = 0
t—s-oo
t
and
log\Xt\
hm ——!——- = 0
t-*co
t
a.s.
However, by a direct computation, we can actually deduce rhm — — '——— = —o2g
t-*oo
log I
^ and
r hm
t-too
'
lOgt
' = —q
a.s.
That is, the solution remains stable but subject to a polynomial decay rate. Based on the viewpoint of a general decay rate which contains as a special case the exponential decay, Liu and Mao [51] initially presented some effective sufficient conditions to guarantee almost sure stability subject to the given decay rates of solutions of the usual Ito stochastic equations in the
finite dimensional space. In order to consider the almost sure decay rate, a difficulty encountered here is that the classic method, i.e., investigating the
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stability in mean, and then as a consequence deducing pathwise stability,
does not work any more. This is because in the usual study of exponential stability, a Borel-Cantelli lemma type of argument is carried out. This is usually involved with the investigation of convergence of a certain series with suitable terms. For exponential stability, this series consisting of exponential decay terms is always convergent. But, for general decay rates, this series could be divergent. In [51], a new technique was introduced to overcome this difficulty. For infinite dimensional cases, by carrying out an approximation solution procedure, the analogous results were generalized to mild solutions for a class of semilinear stochastic evolution equations in Liu [44]. Precisely, consider the stochastic evolution equation (1.2) with to = 0. Assume V(x, t) : H x R+ —> R+ is a C*2'1-positive function. We define operators L and Q as follows: for any ( x , t ) € (D(A),R + ) with Vj.(x,t) € T>(A),
(LV)(x,t) := Vt'(x,t)+ < V^(x,t), Ax + A(t,x) > +±tr\Vt?x(x,t)B(t,x)
QB(t, x)4 (4.2) and
(QV)(x, t) := tr l£ ® Vx(x, t)B(t, x)QB(t, a)* .
(4.3)
Theorem 4.1. (Liu [44]) Assume A(£) f +00 is a positive continuous function defined for all sufficiently large t > 0 and satisfies (i). log A(i) is uniformly continuous over t > to for some to > 0; (ii). There exists a nonnegative constant T > 0 such that
loglogi
km SUp-——r-r-7-
Let V(x,t) € C ' ^ x R ;R ) and^i(t}, ^(t] be two non-negative continuous functions. Assume that there exist positive constants p > 0, m > 0, constants /j,, v, G € R1 and £(t) > 0, a non-increasing continuous function such that 2
1
+
(1) (2)
+
\\x\\pH\(t)m < V(x, t), (x, t)€Hx R+; (LV}(x,t)+£(t)(QV)(x,t)<^(t) + ^(t)V(X,t),
logA(t)
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x€V(A),
t € R+;
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Liu and Truman
Then the mild solution of equation (1.2) satisfies _
logA(t) p In other words, as m > O + T + V\/ n, the mild solution is almost surely stable with the decay rate X(t). In a similar manner, Theorem 4.1 was generalized to deal with mild solutions of a class of stochastic delay evolution equations in Liu [47]. In [54], Liu and Zou presented a robustness analysis with a general decay for a class of perturbed stochastic evolution equations in infinite dimensional spaces.
(B). Stability of stochastic delay evolution equations
The Lyapunov's second (direct) method provides a powerful tool in the study of stability properties of stochastic dynamical systems because the technique does not require solving the system equations explicitly. It is particularly worth pointing out that this interesting and fruitful technique has gained increasing significance and has given decisive impetus for the modern development of stability theory of stochastic evolution equations with variable delays, generally, stochastic functional evolution equations. By analogy with the approaches from [25], Caraballo [3] initially presented an extension of stability properties of stochastic linear evolution equations considered in [25] to a class of stochastic linear delay evolution systems. Subsequently, much effort is devoted to the investigation of stability of nonlinear stochastic dynamical systems in infinite dimensional spaces. Caraballo, Liu and Truman [7] studied the stability for strong solutions of a class of stochastic partial differential equations with variable delays, in which a version of coercive condition once more plays the role of an exponential stability criterion similarly to Theorem 3.1. By using some properties of stochastic convolution, Caraballo and Liu [5] considered the exponential stability of mild solutions in a straightforward way for a class of autonomous stochastic evolution equations and Taniguchi [75] studied the same problems for mild solutions of stochastic partial functional differential equations. Removing the coercive condition and carrying out instead a Lyapunov function argument, Liu and Truman [52] presented an investigation of stability properties for a class of semilinear stochastic evolution equations with bounded time delays. More precisely, consider the following stochastic delay evolution equations over the Hubert space H:
Vt € [0,+oo), € [-h, 0], (4.4)
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where A is the infinitesimal generator of a certain Co-semigroup T(t), t > 0, over H and A(t, •, •) and B(t, •, •) are in general nonlinear mappings, satisfying certain proper Lipschitz continuous conditions and linear growth conditions,
from H x H to H and H x H to £(K, H), respectively. Here (f>(t) : [-h, 0] x fi —>• H, h > 0, is a given initial datum such that <j)(t] is TQ -measurable and sup_h(r)\\2H < oo. r : [0, +00) -> [-h, +00), /i > 0, is some given differentiate function with T'(t) > 1, which will play the role of variable delays. Theorem 4.2. (Liu and Truman [52]) Let v(x] : H -4 R+ satisfy
(i) v(x) > ci • \\x^2H for some c\ > 0; (ii) v(x) is twice Frechet differentiate over H and v'(x), v"(x) are continuous in H and £(H) with v'(x) e ~D(A) as x €. 'D(A), respectively, and v(x)\ + \\x\\ \v'(x)\\ + \\x\\ \\v"(x)\\ ) < c \\x\\ 2
H
H
2
H
c(HiH
2
H
jar some c > 0; 2
(Hi) there exist constants a > 0, A € R1 and a nonnegative function 7(t), t € R+, such that
where (Lv)(t,x,y) := < Ax + A(t, x,y), v'(x) >H +1/2 • tr(v"(x)B(t, x,y) QB*(t,x,y)), x e 'D(A), y € 'D(A), t > 0 and -f(t) satisfies that there exists [i > 0 such that e/i*7(t) is an integrable function on [0, +00) . Assume furthermore the condition a > A holds, then there exist constants T > 0, C(4>] > 0 such that for the mild solution Xf of (4-4) >
E\\xt\\2H)-e-^,
Vi>0.
Furthermore, under the same conditions the mild solution is also exponential almost surely stable. As a direct consequence, it is interesting to deduce the following result which generalizes those in Ichikawa [27] (i.e., Theorem 3.1 and Theorem 5.1 there) to a class of non-autonomous stochastic evolution equations.
Theorem 4.3. (Ichikawa [27] and Liu and Truman [52]) Suppose r(t) = t for any t > 0, in Theorem 4-2 and the corresponding Lipschitz and linear growth types of conditions hold. Let v(x] : H -4- R+ satisfy
(i) v(x] > ci • ||:c||# for some c\ > 0; (ii) v(x) is twice Frechet differentiate over H and v'(x), v"(x) are continuous in H and C-(H) with v'(x] € 'D(A) as x € 'D(A), respectively, and \v(x)\ + ||z||H||*/(a;)||fl- + INll-llv"^)!!^,.?/) < c2||z||^ for some
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Liu and Truman
(Hi) there exists constant a > 0 and a nonnegative function ~i(t), t € R+, such that
(Lv)(t,x) < -av(x) + 7(4),
x € V(A),
where (Lv)(t, x) :=H +1/2 • tr(v"(x)g(t, x)Qg(t, x)*), x € 'D(A), t > 0 and ~f(t) satisfies that there exists fj, > 0 such that e^^it) is an integrable function on [0, +00). Then there exist constants r > 0, C(XQ) solution Xt(xo) of (1.2), E\\Xt(x0)fH
> 0 such that for the mild V*>0.
(4.6)
That is, the mild solution is mean square exponentially stable. Furthermore,
under the same conditions the solution is also exponential almost surely stable. (C). Ultimate boundedness and invariant measures Generally, exponential stability of stochastic evolution equations is a considerably strong concept for the investigation of existence or uniqueness of invariant measures in association with given stochastic systems. In some
cases such as linear stochastic systems, this usually leads to the trivial invariant measure ^ = 5$. A weaker concept in studying invariant measures is the ultimate boundedness which was originally considered by Zakai [80]
and Miyahara [64],[65] to try to study invariant measures and weak (positive) recurrence of solutions of finite dimensional Ito stochastic differential equations. For the infinite dimensional situation, Zabczyk [79] developed a
study of invariant measures and recurrence properties for linear stochastic evolution equations with additive noises. For nonlinear situations, we have the following result which was firstly obtained by Liu and Mandrekar [57] and subsequently, as a by-product of Theorem 3.3 in Section 3, by Liu [49] in the study of Lyapunov function characterization of non-linear stochastic evolution equations. More precisely, we have
Theorem 4.4. (Liu and Mandrekar [57] and Liu [49]) Assume the condition (3.11) holds with n = 0. Suppose Xt, t > 0, is the strong solution of (1.1). If there exists a function A : H —>• R+ which satisfies the following: (a')
A(-) satisfies all the conditions of using Ito's formula;
(b>) Cl\\x\ 2H - fci < A(x)
< c2\\x\\2H + fc 2>
Vz e V;
(c') (LA)(a?) < -c3A(a:) + kz, \/x€V, where GJ, ki, i = 1,2,3 are positive constants and L is the infinitesimal generator of Markov process Xt, then Xt, t > 0, is exponentially ultimately
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bounded in mean square. In other words, there exist positive constants c, v and M such that
E\\Xt(x0)\\2H < c • e-vt\\x0\\jj + M
for all x0 € H.
Conversely, suppose to = 0, 6 = 0 in (3.12) and assume the corresponding A(-) =: A(-,0) satisfies all the conditions of using Ito's formula, then there exist constants Ci, ki, i = 1, 2, 3 such that the conditions (b'J and (c') above hold. The following results obtained by Chow and Khasminskii [12] (Theorem 4.5) and Liu and Mandrekar [57] (Theorem 4.6) give effective criteria to treat the existence of invariant measures, which is a direct generalization in infinite dimensions of the results from Miyahara [64] [65]. Theorem 4.5. Assume the embedding V -V H is compact. Suppose that £
the strong solution Xt(xo) of the equation (1.1) with XQ = XQ satisfies the condition: for some XQ € H, there exists a sequence Tn f oo such that I fT" ( i — / P\ \\Xt(xo)\\v > R\dt ->•() uniformly in n as R-+OO. (4.7) Tn Jo <• J Then Equation (1.1) has an invariant measure on (H,B(H)).
Theorem 4.6. (Liu and Mandrekar [57]) Suppose V <—>• H is compact and the strong solution {Xt(xo),t > 0} of (1.1) under coercive condition (3.11) with [i = 0 is ultimately bounded, then there exists an invariant measure v for {X(xo),t > 0}. Moreover, any invariant measure v of {X(xo),t > 0} satisfies f t
t
I
\\x\\yv(dx) < co.
As a typical example, Theorem 4.5 can be applied to study the invariant measures of a class of randomly perturbed Navier-Stokes equations in two dimensional spaces, a model in turbulent flow of Vishik and Fursikov [78]. Example 4.1. (Stochastic Navier-Stokes Equation) Let D C R be a 2
bounded domain with a regular boundary dD. Let u(t, x) = (vi,V2)(t,x) be the velocity field and p(t, x) be the pressure field in an incompressible fluid. Then, under a random perturbation by a Gaussian white noise, the fluid flow is governed by the stochastic Navier-Stokes equations:
dvi(t,x)
dt 2
±^L _—— n \Jtr n ^ IJ.7 -& —— 1J.. 2,£1. ~" SJSY* •
j=l
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Liu and Truman
where p is the fluid density, v the kinematic viscosity, cr^'s the variance parameter, and Wt — (Wt, W2) is a random force. In a vectorial notation, the above equations take a simpler form:
du(t x} 1 —^—L + (u- V ) u = --Vp + vAu + aWt(x), c/c p V • u = 0,
t>0,
x e D,
where (V-), V and A are the conventional notations for the gradient, the
divergence and the Laplacian operator, respectively, and a = I * The equations are subject to the initial-boundary conditions:
where £(x) is the initial random velocity field. Let Cg° = {v € [C£°(D)]2 : V • v = 0} and H the closure of C$> in [L2(D)}2, V = {v € [H&(D)]2 : V-t> = 0}. Let y* denotes the dual of V so that V <-> H «—>• F* and the embedding are compact. It is known from Vishik and Fursikov [78] the space [L2(D)]2 has the direct sum decomposition:
where HL is the orthogonal complement to H characterized by HL = {v = Vp, for some p € Hl(D)}.
Let II be the orthogonal projection from [L2(D)]2 to H1- and define for
g(v) = i/HAv - IL[(v • V)w]. Then g(-) can be extended as a continuous operator from V to V* . The equation can be recast as a stochastic evolution equation in the form:
J du(t) = g(u(t))dt + crdWt,
\ «(o) = e, £ e v,
where Wt is a Q-Wiener process in H, instead of £?(D). Here, again, we denote the norms of H and V by | • | and || • ||, respectively. It can be also deduced from Vishik and Fursikov [78] that the above equation has an unique strong solution {?/(£); t > 0} satisfying: VE O
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=
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In particular, by using Theorem 4.6, it can be deduced that the equation has an invariant measure on (H,B(H)). A Lyapunov function method is carried out by Chow and Khasminskii [12] to investigate the uniqueness of invariant measures for strong solutions. Their results can be applied to the above example to deduce that under some conditions, the invariant measure in the problem is unique.
For mild solutions of (1.2), the condition (4.7), for instance, makes no sense. There have existed several approaches to handle this difficulty. For instance, Maslowski [61] proved a similar result to Theorem 4.5 with the F-norm in (4.7) replaced by a H-norm plus some additional assumptions on the associated semigroup generated by A. A proper version of Theorem 4.4 was established in Liu and Mandrekar [58] and applied to the study of invariant measures for mild solutions of the equation (1.2). We shall not go into the discussion on this aspect and refer the reader for further information to Da Prato and Zabczyk [17] [18] in which a systematic study is presented, particularly, some special techniques are used to handle the uniqueness and existence of mild solutions for various specific stochastic models. (D). Some related topics
In a wide variety of situations, we find the following problem is important: whenever the stability is invalid for a given stochastic evolution system, under what conditions does the solution of the system have boundedness or certain possible asymptotic growth properties? Based on a Lyapunov function type of argument, Ichikawa [29], Chow [11] presented an investigation of boundedness for mild and strong solutions. Liu [45] [48] carried out a similar investigation of the supreme growth upper bounds under some suitable sufficient hypotheses for a class of nonlinear stochastic evolution equations. There has been serious effort to investigate the stability of stochastic functional evolution equations. Based on Lyapunov's second (direct) method, some stability criteria are obtained in [5] [6] [47] [77] to tackle strong or mild solutions for various stochastic evolution equations with finite time delays. To avoid the difficulty for the constructions of the Lyapunov functionals, Taniguchi [76] established a suitable stochastic version to the asymptotic results of solutions to the classic finite dimensional functional differential equations by the well-known Lyapunov-Razumikhin method. The stability of a class of semilinear stochastic evolution equations which describe a range of problems such as stochastic nonlinear boundary value parabolic problems with boundary or pointwise noise was studied in [31] [32] [62]. Usually, to this end, more delicate analysis is necessarily carried out to treat this kind of models. There is a method by using Lyapunov function programme to study weak (positive) recurrence of solutions which goes back
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at least to [64] [65] in the finite dimensional case. For infinite dimensional spaces, some generalizations have been made in [29] [56].
There is a long history of investigating optimal control for linear stochastic evolution equations with state and control dependent on noise, for instance, Ichikawa [26]. For nonlinear stochastic evolution systems, we mention that the problem of stabilization of systems governed by stochastic evolution
equations on Hilbert spaces containing randomly perturbed operators was considered recently by Li and Ahmed [40] [41]. In particular, it is shown that under some circumstances a state feedback control law can be found to make the system exponentially stable not only in the mean square but in sample paths with probability one sense even in the presence of unbounded
perturbations of the generators.
ACKNOWLEDGEMENTS The authors would like to thank Professor T. Caraballo for his careful and constructive comments on the first drafts.
REFERENCES
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Strong Feller Infinite-Dimensional Diffusions
1
BOHDAN MASLOWSKI and JAN SEIDLER Mathematical Institute, Academy of Sciences, Zitna 25, 11567 Praha 1, Czech Republic e-mail: {maslow, seidler}@math. cas. cz
Abstract. Some recent results on the strong Feller property of infinitedimensional diffusions are reviewed.
0. Introduction The aim of this paper is to review some results on the strong Feller property of Markov semigroups defined by stochastic equations in infinitedimensional spaces, that have been achieved in recent years. The strong Feller property of infinite-dimensional diffusions has been studied in numerous papers and we cannot suppose that the attached bibliography, though rather extensive, is complete. The strong Feller property may hold for deterministic systems only in very special cases, which easily follows from its definition; therefore it indicates that a stochastic system is sufficiently nondegenerate. The strong Feller property has been studied for two main reasons. First, it is a very useful tool in ergodic theory of Markov processes as will be explained in some detail in Section 1. Second, the transition semigroup of a diffusion process in many cases coincides with the solution semigroup of the associated backward Kolmogorov equation, and the strong Feller property may be viewed as a smoothing property of the Kolmogorov equation; this relation is useful in both directions. However, the strong Feller property may be useful in many other problems; let us mention at least the problem of continuous dependence of an invariant measure on a parameter, which arises in stochastic control lr
This research was supported by the GA CR Grant 201/98/1454.
373
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theory (cf. [20]). The paper is divided into four sections, including this Introduction. In Section 1, basic definitions are recalled, a general ergodic theorem for strong Feller processes on Polish spaces is stated and the recurrence/transience dichotomy is explained. In Section 2, an if and only if condition for 6u;-ultraFellerness of Ornstein-Uhlenbeck processes on Hilbert spaces is stated, and then analytic methods for proving the strong Feller property are reviewed and basic references are given. The two main methods discussed there are the one relying on smoothing properties of Kolmogorov equations in infinitely many variables and the procedure based on the so-called Bismut-Elworthy formula for directional derivatives of a transition semigroup. Section 3 is devoted to what we call "probabilistic approach". The idea is to show that the strong Feller property is preserved under the Girsanov transform. Since the results contained in this section are rather recent, they are amended with a few examples indicating usefulness of this approach. Also, the concept of a 6tostrong Feller transition function is discussed here to complete the recurrence/transience result. We close this section with introducing some notation. If T is a topological space then 38(T) (or simply 38, if there is no danger of confusion) denotes the Borel a-algebra over T, b^1 the space of all bounded Borel functions T —> R equipped with the sup-norm || • H^, ^&(T) its subspace of all bounded continuous functions, ^K(T] the space of all bounded (signed) Borel measures
on T and ||| • ||| the variation norm on ^C(T). If X, Y are Banach spaces, we denote by =£?(X, Y) the space of all bounded linear operators between them, by I € -^(X) the identity operator and, provided that X and Y are Hilbert, by ||A||Hs the Hilbert-Schmidt norm of an operator A e 3?(X,Y). We use ^(X;Y) to denote the space of all continuously differentiable Yvalued functions on X having bounded continuous derivatives up to the k-th order, tfbk(X} = ^bk(X;R).
Hypothesis 0.1. Throughout the paper we real separable Hilbert spaces (the norm and denoted by • and (-,-}, respectively), W process on T, and A : Dom(A) —> H is Co-semigroup (eiA) on H.
shall suppose that H and T are the inner product in both being is a standard cylindrical Wiener an infinitesimal generator of a
Acknowledgement. Thanks are due to B. Goldys who offered valuable comments. 1. Basic definitions and general results In this section, a transition function P = (Pt) on a Polish space (E, d) is considered. That is, for any t > 0, Pt : E x 88 —> [0,1] is a Markov kernel (a Borel measurable function in the first variable and a probability measure
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in the second one) and the Chapman-Kolmogorov identity
pt+t(;r)=
JE
pt(x,r)ps(-,dx),
M>O, re^
is satisfied. We shall denote the transition semigroup on \>3$ by the same symbol (P t ),
Pt 0, 7s
and ( P t ) will stand for the adjoint semigroup, which acts on the space by the formula (1.1)
P>=
P t (a;,-)di/(x),
JE
p€J?(E), t > 0.
Using (1.1), we may obviously define Pt*i/ for any nonnegative Borel measure v on jB. A transition function P such that the mapping Pt : E — > ^f(E), x \— > Pt(x, •) is continuous for every t > 0 if ^(E) is equipped with the topology of setwise convergence is called strong Feller. In other words, P is strong Feller provided Pt(-,F) is a continuous function on E for each F 6 38 and t > 0, (equivalently: Pt(b&) C 0). This property was introduced by I. Girsanov [32] in connection with probabilistic potential theory. It turned out immediately that strong Feller transition semigroups have numerous applications in ergodic theory of Markov processes. First, if P is also (topologically) irreducible, Pt(-,U) > 0 on H for every open set U ^ 0 and all t > 0, then
(1.2)
all measures Pt(x, •), t > 0, x E E, are equivalent.
The property (1.2) implies uniqueness of a finite invariant measure, and even more: Assume that a Markov process on E has right-continuous paths, its transition function P satisfies (1.2) and there exists an invariant probability measure ^ for P, Pt* JJL = /u, t > 0, then (1.3)
lim \\\P? 8 — p\\\ = 0
t—s-oo
for any probability measure Q on £$.
This result is due to J. Doob, who proved setwise convergence to /j,, the strengthened form above comes from [48], [47]. In many papers on uniqueness of invariant measures, the strong Feller property is invoked only as a tool for obtaining (1.2), however, it may yield much more complete description of the long-time behaviour of a Markov process and help to establish a recurrence/transience dichotomy. Pioneering results in this direction appeared
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Maslowski and Seidler
in the papers [36], [37] by G. Maruyama and H. Tanaka, who considered processes in W1 (and used the strong Feller property only implicitly). Basic theorems on ergodic behaviour of strong Feller processes on cr-compact complete metric state spaces were obtained by R. Khas'miskii [34]. The method of embedded chains as employed by these authors may be extended to Polish spaces yielding the following result ([41], [47]): Let (X, Px) be a homogeneous Markov process on E with continuous trajectories, whose transition function P is strong Feller and satisfies (1.2). Suppose that (1.4) lim sup Py{ sup d ( X s , y ] > 77} = 0 for all compacts K C E and all r\ > 0. -
0<S
For any Borel set B, denote by
TB = inf{t > 0; Xt € B},
LB = sup{t > 0; Xt e B}
the first hitting time of B and the last exit time from B, respectively. The set B is called recurrent, if TB < oo Px-almost surely for all x € E, and transient, if LB < oo Px-almost surely for all x E E. Theorem 1.1. (i) Let there exist a recurrent compact subset of E. Then there exists a a -finite invariant measure /i for P. Up to a multiplicative constant, JJL is the unique a-finite invariant measure and it is a locally finite Radon measure. The process X is Harris recurrent, oo
/.
Tir(Xs) ds = +00
Px-almost surely
for all x € E and F G 38 with f i ( r ) > 0. The ratio ergodic theorem holds: for all /, g 6 Ll(fj,), g > 0, fH g d/x > 0 we have ft
aI
-y \
-,
f
-f A
— ———-—— ——— • - — —
Px-almost surely for every x G E
for ^-almost all x e E. For all Borel probability measures g, v on E, (1.5)
holds.
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(ii) If there is no recurrent compact in E, then every compact subset K C E is transient and satisfies />oo
(1.6)
sup/
Pt(x,K)dt < oo.
x£E Jo
Remarks, (i) Denote by (Ot) the shift operators. The proof of Theorem 1.1 yields that if the process is positive recurrent, sup y e j Ey9&Tj < oo for some 6 > 0 and a compact set J C E, then (J,(E) < oo. (Let us recall that ^ + dgTj is the first hitting time of J after <5.) (ii) If the invariant measure p. is a probability measure then (1.5) implies (1.3). In the opposite case /i(E) = +00 one gets Pt(x,D) —>• 0 as t —t oo for all x G E and every Borel set D of finite measure. (iii) It follows easily that there is no finite invariant measure if (1.6) holds for all compacts. (iv) The assumption (1.4) is stronger than its counterpart in the locally compact case, however, it may be checked easily for many stochastic PDEs by means of maximal inequalities for stochastic convolutions. Further, the hypothesis (1.2) follows from the strong Feller property and irreducibility, as we have already mentioned. For linear stochastic PDEs, irreducibility is a consequence of the strong Feller property; for nonlinear ones, it is often much easier to establish than the latter. The strong Feller property depends on the topology of the state space. Strengthening the topology may make the proof of the strong Feller property easier (this procedure is used e.g. in the papers devoted to the twodimensional stochastic Navier-Stokes equation), on the other hand, considering weaker topologies we may get more interesting results. An example of this kind is touched upon at the end of Section 3, here we only introduce the necessary notions. Define the bounded weak topology few on H as the strongest topology that agrees with the weak topology on any (norm) bounded subset of H. Note that a function / : H —> R is few-continuous if and only if it is sequentially weakly continuous, and few-compact sets in H coincide with weakly compact sets. Naturally, we call a transition function P on H few-strong Feller if Pt(-,F) is few-continuous for alii > 0 and every Borel set F in H. Finally, let us note that the strong Feller property of the transition function P is equivalent to the (seemingly much stronger) ultra-Feller property: for any t > 0, Pt : H —>• ^(E), x i—> Pt(x, •) is continuous if ^(E) is
endowed with the norm ||| • |||. (A proof for separable metric state spaces may be found e.g. in [19]; the same argument applies also if processes on (H, few) are considered.)
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2. Analytic approach At first we will formulate a basic necessary and sufficient condition for the strong Feller property of an Ornstein-Uhlenbeck process. Consider the equation
dZ --
(2.1)
in H. We assume that S <E 3?(T, H) and T
\\eiAS\\ /
dt < oo for some T > 0,
_
which implies existence of a (unique) mild solution Zx to (2.1) satisfying an initial condition Zx(0) = x for every x 6 H. The equation (2.1) then defines a Markov process on H whose transition function is denoted by R,
Rt(x,r) = p{zx(t) e r}, t > o, x e H, r e %. Set (2.3)
Qt = I esASS*esA' ds,
t > 0.
We have the following result. Theorem 2.1. Assume (2.2). Then the following statements are equivalent: (i) The transition function R is strong Feller. (ii) The transition function R is bw-strong Feller. (Hi) The transition function R is bw-ultra-Feller, that is,
\\\Rt(yn,-)-Rt(yo,-)\\\——>0 n—>oo whenever t > 0 and yn —> yo weakly in H. (iv) For any t > 0, (2.4)
Rnge
holds. The equivalence (i) & (iv) is well known (see [39] or [17]), the equivalence (iii) 44- (iv) is proven in [44]. Note that the condition (2.4) may be verified by means of deterministic control theory; in fact, (2.4) is equivalent to exact null controllability of the equation with control y = Ay+Su, u e L 2 (0, t; H), over each time interval [0,<] (see [50]). In the rest of the section we shall consider a semilinear equation of the form (2.5)
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dX = (AX -
Strong Feller Infinite-Dimensional Diffusions
379
where Q G ^ f ( T } is a symmetric nonnegative operator, and / : Dom(/) C H — > H, a : Dom(cr) C H — > ^(RngQ 1 / 2 ,^) are Borel mappings. The coefficients /, a need not be defined on the whole space H, which is important in applications, for instance, to stochastic reaction diffusion equations, equations of stochastic hydrodynamics, and others. The first simple results on the strong Feller property of a transition semigroup defined by (2.5) were proved in [38] and [35] using finite-dimensional approximations; the argument works only under rather restrictive hypotheses. More efficient analytical methods are based on a common basic idea, which we shall try to indicate now. Assuming the solution of (2.5) to exist and define a Markov process ( X , P ) , let us denote by (P) the corresponding transition semigroup, i.e. P(p(x) = E ip(X) for x G H , t > 0, (p G b^. If one shows that the mapping x >->• Pt(p(x) is continuously Gateaux differentiable and its Gateaux derivative DxPt
t
t
(2.6)
x
t
(({A^^O^L^tNU/*!
for a constant Kt < oo and all (p G b& and h G H, then by the mean value theorem we have
(2.7)
P
t
4
x,y&H,
hence |||Pt(o;) •) — P t (y, -)||| < Kt\x — y\, which yields the ultra-Feller property. It is worth noting that (2.7) holds for all ip G b^ if and only if it holds for all (p G ^(H). The function u, u(t, x) = Ptip(x), t > 0, x G H , may be expected to solve the Kolmogorov equation of the form (2.8) r\
-i
-£(t,x) = -Tr[Dxxu(t,x)a(x)]
u(0, x) =
+ (Ax + f(x),Dxii(t,x)},
t > 0, x G Dom(A),
with a(x) = cr(x}Ql^(<j(x)Qll2Y , x G H, as an analogy with the finitedimensional case shows. In this case (dimff < oo, dimT < oo, A = 0) it may be proven under suitable hypotheses that each measure Pt(x,-) has a density p(t,x, •) with respect to the Lebesgue measure, p being the fundamental solution of (2.8), and the strong Feller property follows easily (see e.g. [28]; more refined results for stochastic differential equations may be found e.g. in [49], [33], [46], [22]). For the infinite-dimensional equation (2.5) with an additive noise (a = I), smooth (mild) solutions to (2.8) with ip G b^? were obtained in [16], [17], [7] by using properties of the corresponding Ornstein-Uhlenbeck semigroup and applying a fixed point theorem or a
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Maslowski and Seidler
perturbation argument and the solution semigroup of (2.8) was identified with the transition semigroup of (2.5). These results suffice (in combination with some other procedures, like truncations or the Girsanov theorem) for verifying the strong Feller property, as it was shown in [40] (based on the work [16]) and in [7]. As an example we present the following statement from [7]. The operators Qt are defined again by (2.3), with £ = Q 1 / 2 .
Theorem 2.2. Let H = T, a = I and f € tfb(H). sup t>0 Tr Qt < oo and
Assume (2.4),
rT
/ \\Qtl/2etA\\ 0. Jo
(2.9)
Then Pt
on H for each (f (E b^ and
for a locally integrable function if) and all (f 6 A rather powerful tool for finding an estimate of the form (2.6) is the so-called Bismut-Elworthy formula giving a probabilistic representation of directional derivatives of Pt
and a £ <^ 2 (ff ;«£?(#)). Let a(y) be
sup \\(T~l(y)\\jz>(H) yen Assume that
< oo.
rT A
||H S di < oo for some T > 0. o Then for each ip 6 ^^(H) the directional derivatives of Pttp are given by (2.10)
(DxPtil>(x),h) = -E(^(X t(x}} 1 (.
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f (a-1(Xs( Jo
Strong Feller Infinite-Dimensional Diffusions
381
where DX^ is the mean-square derivative of the mapping x i— > Xt(x) in the direction h 6 H .
The formula (2.10) yields an estimate of the form (2.6) and the assumptions on smoothness of / and 0, and the measures P(x ,-), k > 1, are uniformly countably additive, that is t
b
(3.1)
b
t
k
lim supPt(xk, An) — 0 for all Borel sets An, An \. 0,
whenever t > 0 and {xk} is a Cauchy sequence in H. (Sufficiency of this condition was noted first by L. Stettner, cf. [21], a different proof yielding also necessity appeared in [43].) The reformulation (3.1) of the strong Feller property proves itself useful in connection with the following procedure: Assume that we have two stochastic differential equations linked together via
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the Girsanov theorem. If one of them defines a strong Feller process, i.e. its transition function satisfies (3.1), then it is easy to find hypotheses on the Girsanov exponentials implying that the transition function of the other equation also obeys (3.1). More precisely, let us consider a pair of equations
(3.2)
dX = (AX + f(X)) dt + a(X)Q^2 dW,
(3.3)
dZ = (AZ + g(Z}} dt + a(Z)Ql/2 dW,
in H, supposing that Q G =£?(T) is a symmetric nonnegative operator, and f,g : H — > H, a : H — > ^(RngQ1/2,!!) are Borel mappings. Assume further that for every deterministic initial condition x € H there exists a unique in law mild solution Zx to (3.3), defined on a fixed probability space (J7, ^", P), and a unique in law martingale solution to (3.2), both with continuous trajectories. Denote by Pt, Rt the transition functions defined by (3.2), (3.3), respectively. Let us quote a version of the main result from [43]:
Theorem 3.1. Let there exist a Borel function u : H — > T satisfying f = g + oQ^/2u. Set
U(y, t) = exp (/ (u(ZV(S)}, •> dW(a) - ~ /* «(Z»(a))| 2 ds \Jo * Jo Assume that EU(y, t} = I for all t > 0, y £ H,
(3.4)
and
(3.5)
U(yn,t) — £-> U(y,t), n—>oo
n—foo
whenever t > 0 and {yn} is a convergent sequence in H, yn —>• y. If the transition function Rt is strong Feller then Pt is also strong Feller. Note that the hypothesis (3.4) yields that the Girsanov theorem is applicable, so we may get martingale solutions to (3.2) in this way, and sometimes their uniqueness in law as well. The first of the hypotheses (3.5) looks rather stringent, but it follows from
ft lim E / \u(Zy«(s))-u(Zy(s))\2ds
n^oo
JQ
I
= 0 for all t > 0, yn € H, yn -> y,
thus it may be checked in a straightforward manner under suitable assumptions on / , a and Q. The proof of Theorem 3.1 is quite simple. The theorem
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is of a "perturbative" nature: we have to start with a strong Feller process to obtain another strong Feller process, hence there are two basic types of applications. Either we start with a linear problem (g = 0, a — const.) and arrive at an easy self-contained proof of the strong Feller property for a semilinear equation with a rather general drift and an additive noise. Or we may consider diffusions whose strong Feller property was established by analytical methods and use the probabilistic procedure to relax assumptions upon the drift. The necessity to employ the Girsanov theorem imposes unpleasant restrictions on the (range of the) drift, on the other hand it is possible to treat equations with rather degenerate diffusion coefficients, provided drifts degenerate in an analogous way, covering problems that cannot be dealt with by the available analytic methods. Such examples arise already in finite dimensions. From Theorem 3.1 it follows that a second order stochastic differential equation
dx + F(x, x) dt = SdW in En defines a strong Feller process provided S is an invertible matrix and F is either a bounded continuous function or F(x, y) = VG(x) for a potential G € *if2(]Rn) bounded from below. Infinite-dimensional examples include e.g. stochastic delay equations
dx(t)
=
/ /.o / x(t + s) dn(s) + F ( x ( t ) )
dt + SdW(t)
\J — r
or a heat equation with a white noise boundary condition
du _ 82u <~\ r\ zo 1 ai ax t
fdu I r-v \ox
\
the reader is referred to [43] for a detailed discussion of these and some other examples. The idea behind the proof of Theorem 3.1 makes it possible to investigate also the friw-strong Feller property. Strong Feller transition functions are automatically 6w-strong Feller, if they satisfy the following modified (ordinary) Feller property:
(3.6)
Pi maps ^(H] into bounded bw-continuous functions for all t > 0.
It is shown in [42] that (3.6) holds for the transition semigroup (Pt) defined by the equation (3.2) if / and a are globally Lipschitz and the semigroup (etA) is compact. Relaxing the Lipschitz continuity assumption on the drift / may be based on a straightforward modification of Theorem 3.1, see [44].
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The foii-strong Feller property being established, we may complete results on the recurrence/transience dichotomy.
Corollary 3.2. Assume, in addition to hypotheses of Theorem 1.1, that the transition function P is bw-strong Feller. (i) If there exists a recurrent ball in H, then there exists also a recurrent compact. (ii) If no (norm) compact subset of H is recurrent, then all weakly compact subsets are transient. Let us note that Lyapunov functions techniques may be used to look for recurrent balls almost in the same way as in the finite dimensional case, while to find a direct proof that some compact is recurrent usually seems to be rather difficult. Finally, the transience of all closed balls (which are, of course, weakly compact) yields that Xt —> oo as t —> oo Px-almost surely for each x 6 H, which accords with the intuitive notion of transience. References [1] S. Bonaccorsi, M. Fuhrman: Regularity results for infinite dimensional diffusions. A Malliavin calculus approach, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 10(1999), 35-45 [2] E. Cepa, S. Jacquot: Ergodicite d'inegalites variationnelles stochastiques, Stochastics Stochastics Rep. £3(1998), 41-64 [3] S. Cerrai: Elliptic and parabolic equations in R™ with coefficients having polynomial growth, Comm. Partial Differential Equations 21(1996), 281317 [4] S. Cerrai: Smoothing properties of transition semigroups relative to SDEs with values in Banach spaces, Probab. Theory Related Fields 113(1999), 85-114 [5] S. Cerrai: Ergodicity for stochastic reaction-diffusion systems with polynomial coefficients, Stochastics Stochastics Rep. £7(1999), 17-51 [6] S. Cerrai: Differentiability of Markov semigroups for stochastic reactiondiffusion equations and applications to control, Stochastic Process. Appl.
53(1999), 15-37 [7] A. Chojnowska-Michalik, B. Goldys: Existence, uniqueness and invariant measures for stochastic semilinear equations in Hilbert spaces, Probab. Theory Related Fields 102(1995), 331-356 [8] G. Da Prato: Dirichlet operators for dissipative gradient systems, Scuola Normale Superiore Pisa, Preprints di Matematica n. 4, 1999 [9] G. Da Prato: Monotone gradient systems in L2 spaces, Scuola Normale Superiore Pisa, Preprints di Matematica n. 2, 2000 [10] G. Da Prato: Some properties of monotone gradient systems, Dynam. Contin. Discrete Impuls. Systems, to appear
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[11] G. Da Prato: Elliptic operators with unbounded coefficients: construction of a maximal dissipative extension, J. Evolution Equations .7(2001), 1-18 [12] G. Da Prato, A. Debussche: Stochastic Cahn-Hilliard equation, Nonlinear Anal. 26(1996), 241-263 [13] G. Da Prato, K. D. Elworthy, J. Zabczyk: Strong Feller property for stochastic semilinear equations, Stochastic Anal. Appl. ^5(1995), 35-45 [14] G. Da Prato, D. Gatarek: Stochastic Burgers equation with correlated noise, Stochastics Stochastics Rep. 52(1995), 29-41 [15] G. Da Prato, D. Nualart, J. Zabczyk: Strong Feller property for infinitedimensional stochastic equations, Scuola Normale Superiore Pisa, Preprints di Matematica n. 33, 1994 [16] G. Da Prato, J. Zabczyk: Smoothing properties of transition semigroups in Hilbert spaces, Stochastics Stochastics Rep. 55(1991), 63-77 [17] G. Da Prato, J. Zabczyk: Stochastic equations in infinite dimensions, Cambridge University Press, Cambridge 1992 [18] G. Da Prato, J. Zabczyk: Ergodicity for infinite dimensional systems, Cambridge University Press, Cambridge 1996 [19] C. Dellacherie, P.-A. Meyer: Probabilites et potentiel. Chapitres IX a XI: Theorie discrete du potentiel, Hermann, Paris 1983 [20] T. Duncan, B. Maslowski, B. Pasik-Duncan: Ergodic boundary/point control of stochastic semilinear systems, SIAM J. Control Optim. 56(1998), 1020-1047 [21] T. Duncan, B.Pasik-Duncan, L. Stettner: On ergodic control of stochastic evolution equations, Stochastic Anal. Appl. .75(1997), 723-750 [22] J.-P. Eckmann, M. Hairer: Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators, Comm. Math. Phys. 2.72(2000), 105-164 [23] J.-P. Eckmann, M. Hairer: Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise, archived in: www.ma.utexas.edu/mp^arc #00-361 [24] K. D. Elworthy, X.-M. Li: Formulae for the derivatives of heat semigroups, J. Fund. Anal. .725(1994), 252-286 [25] B. Ferrario: Ergodic results for stochastic Navier-Stokes equation, Stochastics Stochastics Rep. 60(1997), 271-288 [26] B. Ferrario: Stochastic Navier-Stokes equations: Analysis of the noise to have a unique invariant measure, Ann. Mat. Pura Appl. .777(1999), 331-347 [27] F. Flandoli, B. Maslowski: Ergodicity of the 2-D Navier-Stokes equation under random perturbations, Comm. Math. Phys. 171(1995), 119-141 [28] A. Friedman: Stochastic differential equations and applications, Vol. 1, Academic Press, New York 1975 [29] M. Fuhrman: Smoothing properties of nonlinear stochastic equations in Hilbert spaces, NODE A Nonlinear Differential Equations Appl. 5(1996),
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445-464 [30] M. Fuhrman: On a class of stochastic equations in Hilbert spaces: solvability and smoothing properties, Stochastic Anal. Appl. 17(1999), 43-69 [31] D. Gatarek, B. Goldys: On invariant measures for diffusions on Banach spaces, Potential Anal. 7(1997), 539-553 [32] H. B. FHpcanoB, Cujibno-fiejiJiepoecKue npou,eccw I. O6w,ue ceoucmea,
Teop. BepoflTHOCT. H IIpHMeHeH. 5 (1960), 7-28 [33] K. Ichihara, H. Kunita: A classification of the second order degenerate elliptic operators and its probabilistic characterization, Z. Wahrscheinlichkeitstheorie verw. Gebiete 50(1974), 235-254, 55(1977), 81-84 [34] P. 3. XactMHHCKHH: Bpro^HHecKHe CBoMcTBa BOSspaTHbix 3HOHHBIX npon;eccoB H CTa6HJiH3aii;Hfl penieHnii sa^a^ia Koran napa6ojinHecKHX ypaBHemiH, Teop. Beposrmnocm. u UpuMeneH. 5 (1960), 196-214 [35] R. Manthey, B. Maslowski: Qualitative behaviour of solutions of stochastic reaction-diffusion equations, Stochastic Process. Appl. ^5(1992), 265-289 [36] G. Maruyama, H. Tanaka: Some properties of one-dimensional diffusion processes, Mem. Fac. Sci. Kyusyu Univ. Ser. A 11(1957), 117-141 [37] G. Maruyama, H. Tanaka: Ergodic property of JV-dimensional recurrent Markov processes, Mem. Fac. Sci. Kyushu Univ. Ser. A .75(1959), 157-172 [38] B. Maslowski: Strong Feller property for semilinear stochastic evolution equations and applications, Stochastic systems and optimization (Warsaw, 1988), 210-224, Lecture Notes in Control Inform. Sci. 136, SpringerVerlag, Berlin 1989 [39] B. Maslowski: On ergodic behaviour of solutions to systems of stochastic reaction-diffusion equations with correlated noise, Stochastic processes and related topics (Georgenthal, 1990), 93-102, Akademie-Verlag, Berlin 1991 [40] B. Maslowski: On probability distributions of solutions of semilinear stochastic evolution equations, Stochastics Stochastics Rep. ^5(1993), 17-44 [41] B. Maslowski, J. Seidler: Ergodic properties of recurrent solutions of stochastic evolution equations, Osaka J. Math. 5.7(1994), 965-1003 [42] B. Maslowski, J. Seidler: On sequentially weakly Feller solutions to SPDE's, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat.
Appl. 10(1999), 69-78 [43] B. Maslowski, J. Seidler: Probabilistic approach to the strong Feller property, Probab. Theory Related Fields 118(2000), 187-210 [44] B. Maslowski, J. Seidler: Strong Feller solutions to SPDE's are strong Feller in the weak topology, Studia Math., to appear
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[45] S. Peszat, J. Zabczyk: Strong Feller property and irreducibility for diffusions on Hilbert spaces, Ann. Probab. 55(1995), 157-172 [46] M. Rockner, T. S. Zhang: On the strong Feller property of the semigroups generated by non-divergence operators with Lp-drift, Stochastic analysis and related topics VI (Geilo, 1996), 401-408, Birkhauser, Boston 1998 [47] J. Seidler: Ergodic behaviour of stochastic parabolic equations, Czechoslo-
vak Math. J. 47(122)(1W7), 277-316 [48] L. Stettner: Remarks on ergodic conditions for Markov processes on
Polish spaces, Bull. Polish Acad. Sci. Math. ^2(1994), 103-114 [49] D. W. Stroock, S. R. S. Varadhan: Multidimensional diffusion processes, Springer-Verlag, Berlin 1979 [50] J. Zabczyk: Structural properties and limit behaviour of linear stochastic systems in Hilbert spaces, Mathematical control theory, 591-609, Banach Center Publications Vol. 14, PWN, Warszawa 1985
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Optimal Stopping Time and Impulse Control Problems for the Stochastic Navier-Stokes Equations J.L. MENALDI Wayne State University, Department of Mathematics, Detroit, Michigan 48202, USA, e-mail [email protected])
S.S. SRITHARAN US Navy, SPAWAR SSD - Code D73H, San Diego, CA 92152-5001, USA,
e-mail [email protected])
Abstract. In this paper we will review certain recent developments in the optimal stopping and impulse control problems for the stochastic NavierStokes equation. One of the main ingredients of this work is a new existence and uniqueness theorem for strong solutions in two dimensions. This result is obtained by utilizing a local monotonicity property of the sum of the Stokes operator and the nonlinearity. This gives a realization of the Markov-Feller process associated with the stochastic Navier-Stokes equation. The dynamic programming equations for the optimal stopping and impulse control problems arise as variational and quasi-variational inequalities respectively in infinite dimensions. These problems are then solved in a weak sense using the semigroup approach.
1
Introduction
Optimal control of fluid mechanics has numerous applications in engineering sciences. During the past decade several fundamental advances have been made by a number of researchers as documented in Sritharan [23]. In this paper we describe a new direction to this field. Namely we study impulse and stopping time problems for turbulence. In optimal weather prediction the task of updating the initial data optimally at strategic times can be reformulated precisely as an impulse control problem for the primitive cloud equations (which consist of the Navier-Stokes equation coupled with temperature and species evolution equations, cf. Dymnikov and Filatov [12]). 389
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Optimal stopping and impulse control problems have an extensive literature in particular for diffusion processes (e.g. Bensoussan and Lions [3, 5]), for degenerate diffusion with jumps (e.g., Menaldi [16]) and for general Markov process (e.g., Robin [20], Shiryayev [21], Stettner [22]). The main technical challenge is to give a characterization of the value function (or optimal cost) and to exhibit an optimal control. The general developments in the literature however do not apply to control problems in fluid mechanics due to the restrictions imposed on the data. Variational technique to treat these control problems has been adapted to Gauss-Sobolev spaces (e.g., Chow and Menaldi [10], Zabczyk [28]) with partial results. However, because of the technical difficulties associated with the domain of the generator we prefer to follow the semigroup approach. Most of the effort is dedicated to give a suitable sense to the stochastic Navier-Stokes equation in a two-dimensional domain in order to produce a Markov-Feller process in a Hilbert space (which is not locally compact) with a weakly continuous semigroup. Some related results in this context can be found in Bensoussan [1] and Zabczyk [27] but they are not directly applicable to our model. The dynamic programming approach is used to discuss a simple optimal stopping time problem for the Navier-Stokes equation. We are forced to use sufficiently weak conditions on the data because our final objective is the optimal impulse control problems. In order to facilitate the use of the semigroup technique we first consider the 2-D Navier-Stokes equation with random (Gaussian) forcing field. Several approaches have been proposed in the literature starting from the classic paper by Bensoussan and Temam [4] to some more recent results, e.g., Bensoussan [2], Capinski and Cutland [7], Flandoli and Gatarek [13], Flandoli and Maslowski [14] and Sritharan [24]. The reader is referred to the books by Vishik and Fursikov [26] and Capinski and Cutland [8] for a comprehensive treatment. The local monotonicity method described in this paper seems to be simpler and more direct than the other methods. It is essentially based on the monotonicity (on bounded L4-balls) of the nonlinear Navier-Stokes operator and it allows us to deduce the existence (and uniqueness) of a strong (variational) solution in 2-D. This argument can be used in lieu of the classic compactness method to show existence of solution (even in the deterministic case with an unbounded 2-D domain). In this way a Markov-Feller process is constructed. We then proceed to treat the infinite dimensional variational and quasi-variational inequalities to deal with the optimal stopping and impulse control problems in a weak sense. Complete proofs of the results stated in this paper can be found elsewhere (cf. [17] and [18]), here only the main ideas are given.
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Optimal Stopping Time for Stochastic Navier-Stokes Equations
2
391
Stochastic 2-D Navier-Stokes Equation
Let O C K2 be a bounded domain (for the sake of simplicity) with smooth boundary and u the velocity field. The Navier-Stokes problem (with Newtonian constitutive relationship) can be written in the abstract form as follows fyu + Au + B(u) = f
inL 2 (0,T;V),
(2.1)
with the initial condition u(0) = u0 in H,
(2.2)
where UQ belong to H and the (force) field f is in L2(0,T;]HI). Let us begin by defining some standard function spaces, V = {v e Hj(O, M 2 ); V • v = 0 a.e. in O},
with the norm
a a
(2.3)
v 1/2
VvM ? / 2 and H is the closure of V in the L -norm
= ||v||,
. \1/2 Ivfdz =|v|. > /
(2.4)
(2.5)
We will also define the following linear operators PK : l?(O, M2) — >• H 2
2
A : H (O, R ) n V — y H,
is the Helmhotz-Hodge orthogonal projection and
Au = -i/PH Au, v > 0, is the Stokes operator, (2.6)
where v is the coefficient of kinematics viscosity. The inertia term is represented by the nonlinear operator
B : VB C H x V —->• H,
B(u, v) = PH(u • Vv),
(2.7)
with the notation -B(u) = B(u, u). The domain of B requires that (u • Vv) belongs to the Lebesgue space L 2 (0,R 2 ). Using the Gelfand triple V C H = H' C V we may consider A as mapping V into its dual V'. The inner product in the Hilbert space H (i.e., L2-scalar product) is denoted by (•, •) and the induced duality by (•, •}. Let us consider the Navier-Stokes equation subject to a random (Gaussian) term i.e., the forcing field f has a mean value still denoted by f and a noise denoted by G. We can write (to simplify notation we use timeinvariant forces) f(t) = f ( x , t ) and the noise process G(t) = G(x,t) as a series dGk = Y^k Sk(x, t}dwk(t), where g = (ga, g2, • • • ) and w = (wi,W2, . . .)
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are regarded as I2 -valued functions. The stochastic noise process represented by g(t)dw(t) — J^fc gk(x, t)dwk(t, w) is normal distributed in H with a traceclass co- variance operator denoted by g2 = g2(i) and given by (g 2 (i)u,v) = £( a (t),u) (»(*), v) k
(2.8)
k
We interpret the stochastic Navier- Stokes equation as an Ito stochastic equation in variational form
d(u(t), v) + (Au(t) + B(u(t)), v) dt = (f , v) dt + Y^(Sk, v) dwk(t),
(2.9)
k
in (0,T), with the initial condition (u(0),v) = (uo,v),
(2.10)
for any v in the space V. A finite-dimensional (Galerkin) approximation of the stochastic NavierStokes equation can be defined as follows. Let {ei,62, . . .} be a complete orthonormal system (i.e., a basis) in the Hilbert space H belonging to the space V (and L4). Denote by Hn the n-dimensional subspace of H and V of all linear combinations of the first n elements (ei, 62, . . . , en}. Consider the following stochastic ODE in W1
k in (0,T), with the initial condition (u(0),v) = (uo,v),
(2.12)
for any v in the space Hn . The coefficients involved are locally Lipschitz and we need some a priori estimate to show global existence of a solution un(t)
as an adapted process in the space C°(0,T, Hn). Proposition 2.1 (energy estimate). Under the above mathematical setting let
f eL 2 (0,T;e),g€L 2 (0,7V 2 (H)) and u0 € H. n
(2.13)
Let u (t) be an adapted process in C°(0, T,Mn) which solves the stochastic ODE (2.11). Then we have the energy equality d\un(t)\2 + 2u \Vun(t)\2dt = [2 (f(t),un(t))
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+ Tr(g2(0)] dt+
Optimal Stopping Time for Stochastic Navier-Stokes Equations
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which yields the following estimate for any e > 0 rT v I E{\Vun(t)\2}e-etdt< J ° T L
(2.15)
£
for any 0 < t < T. Moreover, if we suppose
feLP(0,T;H),
ge ^(0,7^2(11))
(2.16)
then we also have
E{ sup \un(t}\pe-et+pv f 0
|Vun(t)|2|un(t)|p-2e-etdi} <
JO
< |u(0)|* + Ce,p,r /
Jo
[\f(t)\P + Tr(S\t))^]e-stdt,
(2.17) for some constant C£jptT depending only on e > 0, l < p < o o and T > 0. Now we deal with the uniqueness of the SPDE and its finite-dimensional approximation
Proposition 2.2 (uniqueness). Let u be a solution of the stochastic NavierStokes equation (SPDE) with the regularity ueL 2 (fi;C 0 (0,r;M)nL 2 (0,r;V)),
u € L4(O x (0,T))
(2.18)
and let the data f, g and UQ satisfy the condition f eL 2 (0,T;V),
g eL
2
(0,T;4(EQ),
u0 € H.
(2.19)
//v in L 2 (fi;C r °(0,r,H) nL 2 (0,r,V)) is anotfier soZufton then 2 4 \u(t) - v(t)| exp [7 /-* J ^| / ||u(s)|| i\/ v/i / Lice'1 4
t/0
ds] < |u(0) - v(0)|2, —
(2.20)
u;z^ probability 1 for any 0
dtvf + Aw = £(u) - £(v) inL 2 (0,T;V), and setting r(«) = f| /„* ||u(s)||J4(o)ds we have
- B(u(<)), w(t)) dt < 0,
and integrating in t, we conclude.
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D
394
Menaldi and Sritharan
Each solution u in the space L2(Q;L°°(0,T;H) n L 2 (0,T;V)) of the stochastic Navier-Stokes equation actually belongs to L2(Q; C°(0, T;H) n L4(O X (0,T))) in 2-D, O C M2. Thus in 2-D, the uniqueness holds in the space L 2 (ft;L 2 (0,T;V)). If a given adapted process u in L2(fi; L°°(0, T; H) n L2(0, T; V)) satisfies
d(u(t), v) = (f(t),v) dt + (g(t), v) dw(t),
(2.21)
for any function v in V and some f in L2(0,T; V) and g in L2(Q,T;£2(E.)), then we can find a version of u (still denoted by u) in L2(Q; C°(0, T;H)) satisfying the energy equality
d\u(t)\2 = [2{f (t), u(t)> + Tr(g2(t)] dt + 2(g(t), u(i)) d«;(t)
(2.22)
see e.g. Gyongy and Krylov [15], Pardoux [19]. Proposition 2.3 (2-D existence). Let f, g and UQ be such that
fel^TjV),
g€LP(0,r;^(H)),
u0€H,
(2.23)
/or some p > 4. Then there is an adapted process u(t, x, uj] with the regularity u e ^(ft; C°(0, T; M)) n L2(fi; L2(0, T; V))
(2.24)
which solves the stochastic Navier-Stokes equation and the following a priori
bound holds E{ sup |u(t)| p + / 0
fT
|Vu(<)| 2 |u(i)| p ~
J0
<-" C1 Trf 111 /(IMP _L f / ^ Op.C/-S |U^Uj| T
I
(2 _ 25)
FlUV^MIP _L TW/»2/^\\p/2] j/\
I | | I \ " / H / ~r -l^Ag (."))
I'*"/)
v 7o /or some constant Cp = C(T,v,p) depending only on the numbers T > 0, i/ > 0 andp> 2. D
The proof of this result can be found in our previous work [17, 24] and also in Vishik and Fursikov [26] and Da Prato and Zabczyk [11]. Our proof is based on the L4-monotonicity of the nonlinear Navier-Stokes operator (and it generalizes to other cases, including multiplicative noise). Ewe denote by Br the (closed) L4-ball in V Br = {v e V ; ||v||L4(0iR2) < r},
(2.26)
then the nonlinear operator u i-4- Au + B(u] is monotone in the convex ball Br i.e.,
(Aw, w) + (B(u) - S(v), w) + Vu € V, v € Br and w = u - v.
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|w|2 > v- ||w||2,
(2.27)
Optimal Stopping Time for Stochastic Navier-Stokes Equations
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Proposition 2.4 (V—regularity). Let the above assumptions hold as in previous Proposition. If f e I/(0, T; H),
g € LP(Q, T;^(V)),
u0 6 V,
(2.28)
with p > 2, then the solution u(£) of the stochastic Navier-Stokes equation has the regularity u 6 C°(0, T; V) n L2(0, T; H2(C»; R 2 ))
(2.29)
with probability 1, and the following estimate holds rT
E{ sup [|Vu(£)|pe~r(*'u)] + / |Au( o
Olf(*)IIS 4 (2.30)
where r(t, u) = cv JQ ||u(s)||44 on p, T and v > 0.
ds for some constants Cp, cv depending only D
In general if u(i) belongs to H n H 2 (0,R 2 ) then Au(i) (respectively Vu(t))does not necessarily belong to HI (respectively V), however the norms | A • | (respectively | V • |) and | A • \ (respectively | .A1/2 • |) are equivalent due to the Cattabriga regularity theorem for the Stokes operator (Temam [25]). We are able to establish the above a priori estimate based on finite-dimensional approximations of the solution.
3
Markov-Feller Process
In what follows for the sake of simplicity we assume that the processes f(x,£,cj) and g(x,t,u} are independent of t, i.e.,
feV
and
g€^ 2 (EI)
(3.1)
and we denote by u(t; UQ) the semiflow, i.e., the solution of Navier-Stokes equation. Also usually we substitute UQ with v.
Proposition 3.1 (continuity). Under the previous conditions the stochastic semiflow u(t; v) is locally uniformly continuous in v, locally uniformly for t in [0, oo). Moreover, for any p > 0 and a > 0 there is a positive constant A sufficiently large such that the following estimate E{e-at(\ + \u(t- v)|2)"/2} < (A + |v|2f/2,
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(3.2)
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Vt > 0, v € H holds, also for any stopping time t = T. Furthermore, if f and g belong to H and 1% (V) respectively, then the semiflow is also locally uniformly continuous in t, locally uniformly for v in V.
D
The Navier-Stokes semigroup (®(t), t > 0) defined by $(t)/i(v) = E{h(u (t; v))}, is indeed a Markov-Feller semigroup on the space C7&(H) (of continuous and bounded real function on H endowed with the sup-norm). Since the base space H is not locally compact, the Navier-Stokes semigroup is not strongly continuous. In our approach, it is convenient to work with unbounded functions. Let CP(K) be the space of real uniformly continuous functions on any ball and with a growth bounded by the norm to the p > 0 power, in another words, the space of real functions h on H such that x i-4 /i(v)(l + |v|2)~p/2 is bounded and locally uniformly continuous, with the weighted sup-norm ,
(3.3)
where A is a positive constant sufficiently large to so that
a > oo(p),
p > 0.
(3.4)
It is clear that C6(H) = C0(H) and Cg(H) C CP(M.) for any 0 < q < p. Then for any a > 0, (linear) Navier-Stokes semigroup ($Q(*), * > 0) with an a-exponential factor is defined as follows
&a(t) : CP(M.) —+ CP(M),
$0(t)/i(v) = E{e~ath[u(t- v)]}.
(3.5)
Proposition 3.2 (semigroup). Under the above assumptions the NavierStokes semigroup ($a(i), t > 0) is a weakly continuous Markov-Feller semigroup in the space CP(M.). D Since the Navier-Stokes semigroup is not strongly continuous, we cannot consider the strong infinitesimal generator as acting on a dense domain in Cp(H). However, this Markov-Feller semigroup ($Q(i), t > 0) may be considered as acting on real Borel functions with p-polynomial growth, which is Banach space with the sup-weighted norm and denoted by BP(M). It is convenient to define the family of semi-norms on BP(W) (3.6) s>0
where A is sufficiently large. Now, if a sequence {hn} of equi-bounded functions in J5p(BI) satisfies po(hn — h,v) —>• 0 for any v in H, we say that hn -+ h boundedly pointwise convergence relative to the above family of semi-norms. It is clear that po(3>a(t)h — h, v) —> 0 as t —>• 0, for any function h in Cp(H) and any v in H.
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Definition 3.3. Let (7(H) be the subspace of functions h in B (W) such that the mapping given by t ^ h[u(t;v)] is almost surely continuous on [0, +00) for any v in H and satisfies P
p
]impQ(3?a(t)h - h, v) = 0, Vv € H
(3.7)
v—^U
where PQ(-, •) is given above.
D
This is the space of function (uniformly) continuous over the flow u(-, v), relative to the family of semi-norms and it is independent of a. Hence, we may consider the Navier-Stokes semigroup on the Banach space CP(BI), en-
dowed with the sup-weighted norm. The weak infinitesimal generator — Aa with domain T>p(Aa) (as a subspace of <7P(H)) is defined by boundedly pointwise limit [h — $a(t)h]/t —i Aah as t —>• 0, relative to the family of seminorms. Notice that po(3>a(t)h, v) < po(h, v) for any t > 0, h in CP(H) and vinH.
Proposition 3.4 (density). If the above assumptions hold, then C (SS) C CP(H), the Navier-Stokes semigroup leaves invariant the space CP(H) and for any function h in CP(M), there is a equi-bounded sequence {hn} of functions in the domain T>p(Aa) satisfying po(hn — h, v) -40 for any v in EL D P
^From above results it is clear that given a > 0, p > 0, A sufficiently large and a function h in CP(M.) there is another function u in T>p(Aa) such that AaU = h, where
u= / $a(t)hdt. Jo
(3.8)
The right-hand side is called _the weak resolvent operator and denoted by either 72.Q = A^1 or Tia = (Ao + a/)"1. Moreover, if ao = ao(A) then for any p > 0 we have ao(A) -4 0 as A —> oo, and for any stopping time T,
{
vv e fr —E{ I |Vu(t;2 Jo + E{e-aoT(\ + |u(r; v)|2f /2} < (A + |v
(3.9) Vv e H, and then for any a > ao we obtain ) < e - - p ^ h , v),
(3.10)
for any t > 0, and
ll^n < a—!— \\h\\, — ao
Po(nah,v)
for any v in H and where the norm || • || and the semi-norms po(-, v).
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(3.ii)
398
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Menaldi and Sritharan
Stopping Time Problem
Given two functions F and G in CP(M) and a > 0 we consider the cost functional
J(v, r) = E{ f F(u(t- v))e~ dt + l G(u(r;))e~ } Jo and the optimal cost ^(v)=infJ(v,r), T
at
ar
r<00
V
(4.1) (4.2)
where the infimum is taken over all stopping times r. Our purpose is to give a characterization of the optimal cost U and to exhibit an optimal stopping time f. A natural way of studying optimal stopping times is via a penalized problem. Given a,e > 0 and F and G in CP(M), we want to solve the nonlinear equation
U£ 6 Vp(Aa)
such that AaUs + -(U£ - G)+ = F,
(4.3)
where (•) denote the positive part and T)p(Aa) is the domain of the weak infinitesimal generator — Aa of the Navier-Stokes semigroup (<& Q (t), t > 0). The solution Ue of the penalized problem can be interpreted as an optimal cost (or valued function) of a stochastic optimal control problem. Proposition 4.1. Let the above conditions and
F, GeC p (H),
(4.4)
hold. Then, for any £ > 0, there is one and only one solution of the penalized problem. Moreover, if F and G belong to Cp(H) then Us also belongs to Cp(H). Furthermore, if G belongs to T>p(Aa) then we have the estimates
for any 0 < s < e' and v in EL
D
Let us consider the problem of finding U € <7P(H)
such that
U < G,
and AJU < G,
(4.6)
usually referred to as a sub-solution. Notice that since U does not necessary belongs T>p(Aa), the domain of the weak infinitesimal generator — Aa of the Navier-Stokes semigroup ($Q(t), t > 0), the last inequality AoU < G is understood in the semigroup sense, i.e.,
o Vt > 0, v e H. We have
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)ds,
(4.7)
Optimal Stopping Time for Stochastic Navier-Stokes Equations
399
Theorem 4.2 (VI). Under the above conditions, the optimal cost U is the maximum sub-solution of VI problem and it is given as the boundedly pointwise limit of the penalized solutions U£ as e goes to zero. Moreover the exit time of the continuation region f = f(v) defined by f (v) = inf { t > 0 U[u(t; v)] = G[u(t; v)] },
(4.8)
Vv € H, is optimal, i.e., U(v) = J(v,f). Furthermore, if G belongs to 'Dp(Aa) then the Lewy-Stampacchia inequality
F/\AaG
(4.9)
holds and Ue converges to U in the sup-norm ofCp(M), therefore U belongs to CP(M), whenever F and G are in CP(M). D
5
Impulse Control Problem
Let us now consider the problem of sequentially controlling the evolution of the stochastic process u(i;v) by changing the initial condition v. For this purpose we consider a controlled Markov chain q/t(i) in H with transition operator Q(k) and a control parameter k which belongs to a compact metric space K. For a sequence (£«, i = 1,2,...) of independent identically distributed H-valued random variables we have qk(i + 1) = q(q*(z), 0 I k),
V* = 1, 2, . . . ,
), Vv € M,
l
'
J
for any initial value q(l), any bounded and measurable real-valued function h on H and any k in K. For the sake of simplicity, this Markov chain (i.e., each random variable C,i) is assumed to be independent of the Wiener process w = (wi,W2, . ..) used to model the disturbances in dynamic equation. A sequence fa, k^ i = 1, 2, . . .} of stopping times n and decisions ki such that Ti approaches infinity is called an impulse control. At time t = TJ the system has an impulse described by the (controlled) Markov chain q/t(z) with k = ki. Between two consecutive times TJ < t < Ti+1, the evolution follows the Navier-Stokes equation: f u ( t ) = u(t,Ti;u(Ti)),
if
Ti
and U(TJ) = q(u(rj-), where u(t, s; v) is the solution of Navier-Stokes eqaution with initial value v at time s. Since TJ —>• oo, we can construct the process u(t) by iteration, for any impulse control {TJ, k^ i = 1,2,...} and initial condition v in H. It
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is clear that TJ is an stopping time with respect to the Wiener process enlarged by the cr-algebras generated by the random variables £1, £2, . . . , Ci-lAlso, the decision random variables ki are measurable with respect to the cr-algebra generated by TJ. To each impulse we associate a strictly positive cost known as cost-perimpulse and given by the functional L(v, k). The total cost for an impulse control {TJ, fc,; i = 1, 2, . . .} and initial condition v is given by
J(v, {n, ka}) = E{ r F(u(t))e~atdt + ^L(u(Ti-),ki)e-^}
Jo
(5.3)
i
and the optimal cost
tf(v)= inf J(v,{Ti,ki}),
(5.4)
{Ti,ki}
where the infimum is taken over all impulse controls, and u(t) is the evolution constructed as above, with initial condition v. Impulse control problems are very well know in finite dimensional setting, but only a few results are available for Markov processes on polish spaces which are not necessarily locally compact (Bensoussan [2, Chapter 8]). As mentioned before, these results do not apply directly to our problem. Here, we are following a hybrid control setting as in Bensoussan and Menaldi [6]. The dynamic programming principle yields to the following problem. Find U in Cp(H) such that
U<MU,
AaU
and AaU = F in [U < MU],
(5.5)
where A& is interpreted in the martingale (semigroup or weak) sense and M is the following nonlinear operator on C(H) given by
}, Vv e H,
(5.6)
J
where Q(k)h(v) = J5{/i(q(v, (j { k ) ) } is the transition operator. This prob-
lem is called a quasi-variational inequality (QVI) . To solve the QVI we define by induction the sequence of variational inequalities (VI) ( jjn+l
€
£,^M)
guch that
^n+1 < Mf jn ;
^J/n+1 < p>
and
(AaUn+1 = F in [Un+1<MUn], (5-7) where U° = U° solves the equation AaU° = F. This VI can be formulated as a maximum sub-solution problem
Un+1
€ CP(M)
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such that
Un+l < MUn,
AaUn+l < F,
(5.8)
Optimal Stopping Time for Stochastic Navier-Stokes Equations
401
for any n > 0. In view of the Theorem for the VI in the previous section, we need only to assume that M operates on the space CP(1B) to define the above sequence Un of functions. This means that first, we impose the condition
lim su.p{ ($ (t)L(; k) - L(-, k), v)} = 0, t-s-o k PO a Vv € M, and next
lim
p{p0(3>a(t)Q(k)h
t-»o SUk
- Q(k)h, v)} = 0,
-
Vk € K, v e H, V/i € Cp(H), for any m > 0, some positive constant Cm and where the norm || • || and the semi-norms po(-, v). One of the main differences between impulse and continuous type control is the positive cost-per-impulse, i.e., the requirement
L(v, k) > 4) > 0, Vv € H, k € K,
(5.11)
which forbids the accumulation of impulses. We also need
F e CP(H),
F(v) > 0, Vv € H,
(5.12)
to set up the above sequence. An important role is played by the function U° = U° which solves
AaU° = F, and by the function Uo = Uo, which is defined as the solution of the following variational inequality (Uo£ Cp(H) such that U0 < inf L(-, k), k < _ . . \AaU in [t/o< inf £(-,&)], Q = F V K
AaUo < F
and (5.13)
or as the maximum sub-solution of the problem
C/o e Cp(H) such that
U0
AaUo < F.
(5.14)
K
Consider the quasi-variational inequality (QVI)
( U e Cp(H) such that
\
_
(AaU = F
»
U < MU, »
AaU < F
and
in [U<MU],
(5.15)
or the maximum sub-solution of the problem U € Cp(H) such that
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U < MU,
AaU < F.
(5.16)
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Menaldi and Sritharan
Theorem 5.1 (QVT). Let the above assumptions hold. Then the VI defines a (pointwise) decreasing sequence of functions £7n(v) which converges to the optimal cost U(v), for any v in H. Moreover, if the condition there exists r € (0,1]
such that rU°(v) < UQ(v),
Vv € H
(5.17)
is satisfied then we have the estimate 0
(5.18)
the automaton impulse control {fj,fcj}, generated by the the continuation region [U < MU] and defined by TQ = 0, i = inf {t > fi_i U[u(t; ufa-i))] = MU[u(t; u(r<_i))] }, = argmin{L(u(Ti),fc) +g(fc)C7(u(r i ))}
(5>19)
rC t JrV
is optimal, i.e., U(v) = J(v, {fj,fej}), and i/ie optimal cost U is the unique solution of the QVI or the maximum sub-solution of the problem. D If we impose
L(v, k) > 4(1 + |v|2f/2 > 0,
Vv € H, jfe e -PT,
(5.20)
then assumption (5.17) holds for any 0 < r < 1 such that r ||F|| < lo(a—QO)-
References [1] A. Bensoussan, Stochastic Control by Functional Analysis Methods, North-Holland, Amsterdam, 1982. [2] A. Bensoussan, Stochastic Navier-Stokes Equations, Acta Appl. Mathematicae, 38, (1995), 267-304. [3] A. Bensoussan and J.L. Lions, Applications of variational inequalities in stochastic control, North-Holland, Amsterdam, 1982. [4] A. Bensoussan and R. Temam, Equations Stochastique du Type NavierStokes, J. Punct. Analysis, 13 (1973), 195-222. [5] A. Bensoussan and J.L. Lions, Impulse control and quasi-variational inequalities, Gauthier-Villars, Paris, 1984. [6] A. Bensoussan and J.L. Menaldi, Hybrid Control and Dynamic Programming, Dynamics of Continuous Discrete and Impulsive Systems, 3
(1997), 395-442.
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[7] M. Capinski and N. J. Cutland, Stochastic Navier-Stokes equations, Acta Appl. Mathematicae, 25 (1991), 59-95. [8] M. Capinski and N.J. Cutland, Nonstandard methods for stochastic fluid mechanics, Wolds Scientific, Singapur, 1995.
[9] P.L. Chow and R.Z. Khasminskii, Stationary Solutions of Nonlinear Stochastic Evolution Equations, Stochastic Anal. Appl., 15 (1997), 671699 [10] P.L. Chow and J.L. Menaldi, Variational Inequalities for the Control of Stochastic Partial Differential Equations, in Proceedings of the Stochastic Partial Differential Equations and Application II, Trento, Italy, 1988, Lecture Notes in Math., 1390 (1989), 42-51.
[11] G. Da Prato and J. Zabczyk, Ergodicity for infinite dimensional systems, Cambridge University Press, Cambridge, 1996. [12] V.P. Dymnikov and A.N. Filatov, Mathematics of Climate Modeling, Birkhauser, Boston, 1997. [13] F. Flandoli and D. Gatarek, Martingale and Stationary Solutions for the Stochastic Navier-Stokes Equation, Probab. Th. Rel. Fields, 102 (1995), 367-391
[14] F. Flandoli and B. Maslowski, Ergodicity of the 2-D Navier-Stokes Equation Under Random Perturbations, Comm. Math. Phys., 171 (1995), 119-141. [15] I. Gyongy and N. V. Krylov, On stochastic equations with respect to semimartingales Ito formula in Banach spaces, Stochastics, 6 (1982), 153-173. [16] J.L. Menaldi, Optimal Impulse Control Problems for Degenerate Diffusions with Jumps, Acta Appl. Math., 8 (1987), 165-198. [17] J.L. Menaldi and S.S. Sritharan, Stochastic 2-D Navier-Stokes Equation, Appl. Math. Optim., submitted.
[18] J.L. Menaldi and S.S. Sritharan, Impulse Control of Stochastic NavierStokes Equations, Nonlinear Analysis, Methods, Theory and Application, submitted. [19] E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes, Stochastics, 6 (1979), 127-167.
[20]
M. Robin, Controle Impulsionnel des Processus de Markov, Thesis INRIA, TE-035 (1977), Paris, France.
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[21] A.N. Shiryayev, Optimal Stopping Rules, Springer-Verlag, New York, 1978. [22] L. Stettner, On some stopping and impulsive control problems with a general discount rate criterion, Probab. Math. Statist., 10 (1989), 223245.
[23] S.S. Sritharan, (Editor) Optimal Control of Viscous Flow, SIAM, Philadelphia, 1998. [24] S.S. Sritharan, Deterministic and Stochastic Control of Navier-Stokes Equation with Linear, Monotone and Hyper Viscosities, Appl. Math.
Optim., 41 (2000), 255-308. [25] R. Temam, Navier-Stokes equations and nonlinear functional analysis (Second edition), CBMS-NSF 66, SIAM, Philadelphia, PA, 1995.
[26] M.J. Vishik and A.V. Fursikov, Mathematical Problems in Statistical Hydromechanics, Kluwer Acad. Publ., Dordrecht, The Netherlands, 1988.
[27] J. Zabczyk, Stopping Problems on Polish Spaces, Ann. Univ. Mariae Curie-Sklodowska, Sect. A, 51 (1997), 181-199. [28] J. Zabczyk, Bellmans's Inclusions and Excessive Measures, Preprint Scuola Normale Superiore, Pisa, Italy, 1999.
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On Martingale Problem Solutions for Stochastic Navier-Stokes Equation R. MIKULEVICIUS t and B. ROZOVSKII * Center of Applied Mathematical Sciences, USC, 90089-1113 Los Angeles, USA
1
Introduction
Equations of Fluid Mechanics driven by white noise type random force were studied by many authors (see, e.g. [1], [2], [3], [5], [7], [8], [12] etc.). In most of these papers, some form of stochastic Navier-Stokes (SNS) or Euler (EU) equation was postulated at the inception point. A somewhat different, approach was taken in the recent papers [9] and [10] where it was postulated that the dynamics of the fluid particle was given by the stochastic diffeomorphism
)=x
(1.1)
with undetermined local characteristics u (t, x) and a (t, x) . In this setting, a (t, x)oW models the turbulent part of the velocity field while u (t, x) models its regular component. Following the classical scheme of the Newtonian fluid mechanics (i.e. coupling (1.1) with Newton's second law), a very general SNS equation, was derived. It includes as special cases the classical deterministic Navier-Stokes and Euler equation as well as most of the variations of the SNS equation considered in the literature. In the present paper we will discus new results on the existence and uniqueness of martingale problem solutions to these and more general equations, including SNS equations in Rd with body forces and pressure being nonlinear functions of the velocity field. f
This work was partially supported by NSF Grant DMS-98-02423. *This work was partially supported by NSF Grant DMS-98-02423, ONR Grant N00014-97- 1-0229, and ARO Grant DAAG55-98-1-0418.
405
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Specifically, we will consider the following equation for the velocity u(t, x] =
(u1 (t, x)}1
' dtul (t, x) = (a1' v (t, x) 4. (t, x)} \
-uk (t, x) ulxk (t, x) / Xi
-Pxt (t, x) + bl'k(t, x)uXk(t, x) + pXk(t, x)hl>k(t, x) +fl (t, x, u (t, x)) + (/«(*, x, u(t, x)))Xj
+[(Tl'k(t, x)4k (t, x) + gl (t, x, u (t, x)) - pxi (t, x)} Wt, u (0, x) = uQ (x} € L2 (Rd) , / = 1, . . . d, div u (t, x) = 0 in Rd for all t (1.2) where W is a cylindrical Brownian motion in a separable Hilbert space Y. A less general Navier-Stokes equations in a bounded domain were considered in [12], [2] and [7] when /»'>* = 0, al'ij = <%,, /'•* = 0, /' = /'(*, x),gl = gl(t,x), and also by [3] when h1'* = 0,a'« = %,,/ M = O,^'* = 0, /' = fl(t,x). For a bounded domain, the existence of the solutions was proved in [8] under a certain condition allowing polynomial growth of coefficients (h1'1 = 0). We finish the section introducing several notation to be used throughout the paper. If u is a function on Rd, the following notational conventions will be used for its partial derivatives: d{U = du/dxi,d^ — d u/dxidxj, dtu = du/dt,Vu ~ du = (diu, . . . , ddu) , and d2u = (d^u) denotes the Hessian matrix of second derivatives. Let CQ° = Co°(Rd) be the set of all infinitely differentiable functions on d R with compact support. For p € [1, oo) and s e (—00, oo), we define the space H£ = Hp(Rd) as the space of generalized functions u with the finite norm \u\Sjp = \ (1 — A)s' u\p, where | • \p is the Lp norm. Obviously, H® = Lp. Let us fix a separable Hilbert space Y. The scalar product of x, y € Y
will be denoted by x-y. Up € [1, oo), and s € (-00, oo), Hp(Y] = H^(Rd, Y) denotes the space of Y— valued functions on Rd so that the norm \g s,p = ||(l-A) s / 2 5 |y \p < oo. We also write LP(Y) = Lp(Rd, Y) = H$(Y) =
Obviously, the spaces C^jHp (Rd} , and Hp(Rd,Y) can be extended to vector functions (denoted with bold-faced letters). For example, the space of all vector functions u = (u1,..., ud) such that (1 — A)s'2 u1 € Lp, I = l,...,d, with the finite norm |u|Sip = (^ \ul\ps,p)l/p we denote by H£ =
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Martingale Problem for Stochastic Navier-Stokes Equation
407
H£ (#<*). Similarly, we denote by H*(Y) = H*(#d, Y) the space of all vector functions g = (gl)i
Coo o •
When 5 = 0, M^(Y) == L p (y) = Lp(Rd, Y). Also, in this case, the norm
|g|o,p is denoted more briefly by |g|p.
2 2.1
Navier-Stokes equations in Hd Solenoidal and gradient projections of vector fields in R. d
To begin with we recall some facts regarding solenoidal decomposition of square integrable vector fields. Let <S(L2(y)) = {g G L2(y) :divg = 0}. Obviously, <S(L2(y)) is Hilbert subspaceof L2(F), andL 2 (y) = £(L 2 (y))® <S(L2(y)), where £/(L 2 (y)) is orthogonal complement of ^(L2(y)). A vector field from <S(L2(y)) is called solenoidal (divergence free). Lemma 2.1 LetQ be aprojection o/L2(y) onto ^(L2(y)). Thenforv €L 2 (y),
Q(v) = -V JTXi(x-y)vi(y}dy, 5(v)
(2.3)
= v-0(v), veL 2 (y).
Remark 2.2 Since the projection Q is defined by a singular integral (2.3), it can be continuously extended to W£(Y] for all p > l,n € (—00,00). For
each p and n there is a constant C such that
(2.4) 2.2
The Navier-Stokes equation
In this section we will consider equation (1.2) . Note that in contrast to the classical Navier-Stokes equation, the solenoidal projection of equation (1.2) does not eliminate the pressure components. However, since divu = 0, we have
Vp(t, x] = (Ll(u,t, Z))I
Lo(iM, x) = (L (u,t, x)) = G(-u (t) d u(t)), l
k
0
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1
k
Mikulevicius and Rozovskii
408
(i'(u,t,i))ls,
(2.6)
L*(u, t}hl*(t] + fl (t, u(t)) + ft ( So, equation (1.2) can be reduced to the following one:
' dtul (t, x) = ft (o''« (t, a;) 0,-«J (t, x)) - uk (t, x) dkul(t, x) -LlQ(u, t, x) - Ll(u, t, x} + bl*(t, x)diul(t, x) + L*(u, t, x^'^t, x)
+fl (t, x, u (t, x)) + 8^ (^ x, u (t, x)))
(2.7)
+[al'i(t, x)diul (t, x) + gl (t, x, u (t, x)) - Ll(u, t, x)] Wt, u (0, a;) = u0 (a;) .
We assume that: a1™ , bl>i are measurable functions on [0, T] x Hd, fl>i , fl are measurable functions on [0, T] x Rd x Rd; cr'1-7, h1^ are y-valued measurable functions on [0, T] x Rd, gl are F-valued functions on [0, T] x Rd x Rd, and matrices (a1™} are non-negative. In addition we assume the following: El) al'ij, bl'j, f l ' j , | A:|y are bounded, and there is 6 > 0 such that for all £ e Rd, [a'« (t, x) - (1/2)
fl(t,x,u) + fl'j(t,x,u) and
\flt,x,u)-fl(t,x,u)
+\fl>'(t,x,u))-fl>>(t,x,u)\
+ gl(t,x,u) - gl(t,x,u)
r
where
Jo 2.3
y
< C\u-u\,
t, x)2 dt dx < co.
Approximations of Navier-Stokes equation
Following the ideas of [4], we begin with a linearized equation, derive the existence and some estimates of the solution for it, and then, using a retarded molifier, construct an approximation by piecing together the "linearized solutions.
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Martingale Problem for Stochastic Navier-Stokes Equation
2.3.1
409
Linearization of the quadratic term
We consider the following equation for u = (u[}-,
' dtul (t, x) = di (al'ij (t, x) dju1 (t, x)) - ck (t, x) dkul(t, x) -Lo(u, t, x) - Ll(u, t, x) + &*>*(<, x)diul(t, x) + U(u, t, x}hl^(t, x)
+fl (t, x, u (t, x)) + di(f1^ (t, x, u (t, x}}}
(2.8)
+[<j^(t, x)diul (t, x) + gl (t, x, u (t, x)) - Ll(u, t, x)} Wt, u(0,a;) = UQ(X).
where c =(cl) is measurable bounded F-adapted random function such that dive = 0, and = Q(-ck (t) dku(t)). (2.9)
Proposition 2.3 Let Bl), B2) be satisfied. Assume |u |2 < oo P-a.s. and \c(s,x)\ is uniformly bounded. Then there is a unique continuous solution u of (2.8) such that P-a.s. 0
rT fi
I
Jo
|2 \dvi(s}\lds
Moreover, divu(t) = 0 for all t. Proof. Let V = H^, H = I®. We consider (2.8) as a stochastic evolution equation in Gelfand triple V C H C V':
du(t) = A(t, u) dt + B(t, u) dWt, u(0) = u0, where for each v € C§°, u € L2(0,T; V),
(A (t, u), v) = J{-al'^(t, x)diuldjvl - /'J (t, x, u) djV1 + [-ck(t, x)dkulvl+
b' (t, x)d u + f (t, x, u) + L (u) • h' (t, x) - L (u, t, x) - L(u, t, x)}v } dx, l
k
l
k
k
l
k
l
l
1
0
B (t, u) v = j (al"j (t, x) djUl + gl (t, x, u)) - Ll(u, t, x)} v1 dx. Our assumptions imply (see e.g. [11]) the existence and uniqueness of a continuous L2-valued u(t) such that J0 |u(s)|2
2.1 and definition of L(u,t, x), Z(u,i, x), it follows that divu(£) = 0 for all t (both drift and diffusion part of u(t) are divergence free).
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2.3.2
Construction of an approximating sequence
Following [4], we will use a retarded mollifier ^(u) as c. Let tp(t,x) e
C^(RxRd), ^ > 0 , f^dxdt = l,supp ^c{(t,x) : \x\2 Q and 0 otherwise. Let t, x) = «T4 I I/>(S/K, y/K)v(f -s, x-y) dyds. Obviously, the values of ^«(v) at time t depend only on the values of v at s € [t — 2K, t — K]. Now we define a sequence of approximations. For any large n, let K = T/n. We find u = (ul)i
' dtul (t, x) = dt (ol'« (t, x) djUl (t, x)) - VK(u)k(t, x)dkul(t, x) -Ll0(u, t, x) - Ll(u, t, x) + &*'*(*, x)diul(t, x) + U(u, t, x)hl>i(t, x) +fl (t, x, u (t, x)) + di(fl'{ (t, x, u (t, x))) +[(T^(t, x)diUl (t, x) + gl (t, x, u (t, x)) - Ll(u, t, x)} Wt, U (0, x) = UQ (x) .
(2.10) where Ll(u, t, x) is defined by (2.5), and LQ(U, t, x),Ll(u, t, x) are defined by (2.9), (2.6), only with # K (u)(t, x) instead of c(t, x). Proposition 2.4 Let Bl), B2) be satisfied, E|uo|2 < oo. Then for each n there exists a unique ~L^-valued continuous solution un(t) of (2.10) such that J0 \dun(s)\^ds < oo , divun(t) = 0 for all t P-a.s. Moreover, rT
sup E[ sup |un(t)|i + / n
t
JO
\dnn(t}\l dt] <00.
Proof. Since the values of \I/ (u)(i, x) at time t depend only on the values K
of u at s € (t — 2/t, t — K), step by step on the intervals of the length K = T/n, we have the unique continuous L2-valued solution u = un to (2.10) by Proposition 2.3. By Ito formula for |un(i)|2 (see [11]), we obtain easily the estimate. We can rewrite (2.10) in the following way:
dtu (t) = AK(t, un) + D(t, un) Wt, u (0, a:) = u0 (a:).
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where for each test function v € Co°(Rd) and L2-valued continuous u(i) such that J0 |u(i)|i)2 dt < oo,
(AK (t, u) , v) = /{-o'«(t, x)diuldjvl - f1* (t, x, u) djv1
+ [-V (n) (t, x)d uv + b' (t, x)d u + f (t, x, u) + L (u) • h*(t, x) k
l
K
l
l
k
l
k
k
l
k
— L0(u, t, x) — Ll(u, t, x)]v1} dx,
D(t, u) =
[[-diJ'tyJu1 (t) - a^(t}diVlul (t) + gl (t, un) v1 -di(ru(t)ul (t) Q(v}1 -
Let (Q(t, u)v, v) = \D(t, u)v|^. Fix U = H|, s > d/2 + 1. Obviously, we have the following estimates. Lemma 2.5 Assume Bl) and B2). Then for each 'Li-valued continuous vector field u(t) such that J |u(i)| ^ dt < oo, 0
truQ(t,u) 2.3.3
2
<
Weak compactness of approximations
Denote L2,/oc the space L2 with the topology of L2-convergence on compact subsets of Rd. It is denned by
|v| 2 -fl= /
\v\2dx, R>0.
J\x\
Denote by U[oc the space U' with a topology denned by seminorms \S\U',R = sup{|g(v)| : v € C§°, \v\u < 1, suppv C BR}, R>Q, where BR = {x : \x\ < R}.
Let C[0jT\ (Uioc) be the set of t//oc-valued trajectories with the topology 7i of the uniform convergence on [0,T]. Let Cr[oir](L2,u,) be the set of Upvalued weakly continuous functions with the topology To of the uniform weak convergence on [0, T]. Let L2,u> (0, T; H^) be the set of E^-valued square integrable functions fs on [0,T] with a topology ?2 of weak convergence on finite intervals, i.e. the topology defined by the maps fs -4 J0 (fs, gs) ds,where g /T!
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Mikulevicius and Rozovskii
is H^" - -valued such that J |
s
0
2
H2;T,.R — /
JO
/
oc
m^> x)\^dxdt, R > 0.
J\x\
Lemma 2.6 (cf. [12]) Let Z =q0,T] (U'loc) D C[QjT] (L2,«,) n L2)UJ (0, T; H*) n L2 (0, T; L,2,toc) ffln^ T be the supremum of corresponding topologies. Then K C Z is T-relatively compact if ds < CO,
,,.€K- supit-^i^^ |a;t - ^sly = 0s,t
Proof. It can be assumed that K is closed in T. The topologies 76, 7i , ?2 , 7a are metrisable on /f. Consider a sequence (xn) in IT. Since the imbedding L2 C U'ioc is compact (see [6]), there exist a subsequence (xn*=) , x € C lo,T}(V'loc}nL2tW (0,T;H£) such that xn*= -)• x in C[0,r|(l7/oc)nL2,w (0,T;1^) with respect to the supremum of 71 and 75- Since for each u € f7 (xw* , u)L2 = (x"fc, u) is uniformly convergent on finite intervals to (x., u) , we have sup sup (xs, u) < sup sup|x5|jy < oo,
and x is L,2-valued. Thus x"fe -> x in C[o,n (^2,w) as well- Obviously, for each p > 0, .R > 0 fT i \ s ^ I y/ R \ / 7o as A; -4- oo. Since the imbedding H^ —>• L2i;oc is compact, for each e > 0, -R there is a constant C = (7€i^ such that for all u G H^
Since supn/0T(|x£|i)2 + \y.B\\^}ds < oo, it follows by (2.11), (2.12) that
Let X(t) = X(t,x) = X(t,w) = w(t) = w(t,x), w € Z. Let T>t = 0(X(s], s < t), D = (X>t+)or- For each n = K.T, the solution u n (t) to (2.10) induces a measure Pn on (2, "D). By Proposition 2.4, Lemma 2.5 and Lemma 2.6, the following statement holds. Corollary 2.7 The set {P, n > 1} is relatively weakly compact. n
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Martingale Problem for Stochastic Navier-Stokes Equation
2.3.4
413
Pn as a solution of the martingale problem
For v e C§°(Rd), denote v(s,X) = ^(s,X) by taking *0(u) = u. Applying Ito formula, we derive easily the following
Lemma 2.8 For each n (K = T/ri), Pn is a measure on Z such that for each test function v e Co°(Rd) M
n,v (X ,v) _ / ^W,v> v^ j g M (P> P"). JO =
&i
t
> t e
v ?
Xj
dg
cloc
(We say P" is a solution of the martingale problem (uo,aK,Q)).
2.4
Existence of weak global solutions
Theorem 2.9 Assume B1)-B2) are satisfied. Then for each UQ £ L2 there is a measure P on Z solving the martingale problem (uo,a,Q) such that
• / \X(s)\l2ds] Jo
~t
< 00.
Proof. We can assume that a sequence of measures (Pn) converges weakly to a measure P on Z. Let v <E C§°(Rd) and Mtv = e^Xt' v>-/0f e^Xs'^• u in Z. The assumptions 51), 52) imply that
sup |Msn'v (yn) - MJ (w)| ->• 0. Thus for each compact set K C Z, |Msn'v (w) - Msv (w) | ->• 0,
sup
and therefore for each n > 0 n limP sup |Msn>v - MJ| > T? = 0. n
\s
(2.13)
J
Now we can simply repeat the lines of the proof in [8]
Remark 2.10 If P is a solution of the martingale problem (uo,a,Q), then, by Lemma 3.2 in [8], Xt satisfies (2.7) in a possibly extended probability space with some cylindrical Brownian motion Wt, i.e. Xt is a weak solution of (2.10) or (1.2).
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Mikulevicius and Rozovskii
Theorem 2.11 Assume B1)-B2) are satisfied and d = 2. Then there is a unique measure P on Z solving the martingale problem (UQ, a, Q) such that < oo. ~t
Moreover, P-a.s.
rT
I
\a(s,X)\2R-lds
and Xt is ^-valued continuous process Proof. Let P be a solution of (UQ, a, Q). Since for each v € Cg°
we have \X*dkXlt\^ < C\Xt\2\VXt\2. Then, obviously, ff \a(s,Xs)\^ ds < oo, P-a.s., and by [11] there is a L2-valued continuous modification of Xt (also, Ito formula holds for j-X't)^)- Uniqueness of measure P will follow from the pathwise uniqueness of solutions. Assume on some probability space (fi, F, P), with a right continuous filtration of cralgebras F = (Ft) and cylindrical Wiener process W in Y , we have two solutions C/i, C/2 to the Navier-Stokes equation. Let U = U\ — C/2- We apply Ito formula for noticing that
Since for each e there is a constant C£ such that
2 \U\2 dxY'\ j | W|2 dx)^( 1 1 V(7 2|
< e I \VU\2 dx + C£\U\2H I |VC/2|2 dx the pathwise uniqueness follows.
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Martingale Problem for Stochastic Navier-Stokes Equation
415
References [1] A. Bensoussan, R. Temam, Bquatios stochastique du type NavierStokes, J. Func. Anal. 13 (1973) 195-222. [2] Z. Brzezniak, M.Capinski, and F. Flandoli, Stochastic partial differential equations and turbulence, Mathematical Models and Methods in Applied Sciences 1(1) (1991), 41-59. [3] Z. Brzezniak, S. Peszat, Strong local and global solutions to stochastic Navier-Stokes equations, In: Infinite Dimensional Stochastic Analysis, Proceedings of the Colloquium of the Royal Academy of Arts and Sciences, Amsterdam, (1999), 85-98,
[4] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solution of the Navier-Stokes equations, Communications in Pure and Applied Math., 35 (1982), 771-831. [5] M. Capinski and N. J. Cutland, Stochastic Navier-Stokes equations, Ada Applicandae Mathematical 25 (1991), 59-85. [6] R. Temam, Navier-Stokes Equations. North-Holland, Amsterdam-New York, 1985. [7] F. Flandoli, D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Th. Related Fields, 105(1995) 367-391. [8] R. Mikulevicius and B. L. Rozovskii, Martingale problems for stochastic PDE's. In Stochastic Partial Differential Equations: Six Perspectives (^Editors: R. Carmona and B. L. Rozovskii), Mathematical Surveys and Monographs Series 64(1999), AMS, 243-325. [9] R. Mikulevicius, B. L. Rozovskii, On Equations of Stochastic Fluid Mechanics In Stochastics in Finite and Infinite Dimensions: In Honor of Gopinath Kallianpur" ('Editors: T. Hida, R. Karandikar, H. Kunita, B. Rajput, S. Watanabe and J. Xiong), Birkhauser (2000), (to appear). [10] R. Mikulevicius, B. L. Rozovskii, Stochastic Navier-Stokes Equations. Propagation of Chaos and Statistical Moments, In A. Bensoussan Festschrift, IOS Press, Amsterdam, ( 2000), (to appear). [11] B. L. Rozovskii, Stochastic Evolution Systems. Kluwer Academic Publ., Dordrecht, 1990. [12] M. Viot, Solutions faibles d'equation aux derives partielles stochastiquess non lineaires, These de doctorat, Paris VI, 1976.
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SPDEs Driven by a Homogeneous Wiener Process
SZYMON PESZAT Institute of Mathematics, Polish Academy of Sciences, Sw. Tomasza 30/7, 31-027 Krakow, Poland, e-mail: [email protected]
0. INTRODUCTION The aim of the paper is to present known results on the existence and regularity of solutions to the Cauchy problem for the stochastic parabolic equation
u(0)
= u0
(0.1)
and the stochastic hyperbolic equation
f)2v
w(t)
• = Au(t] + R(t, «(<)) + B(t, u(t))W(t),
u(0)
du = «Q, — (0) = v0. (0.2)
a jC>a with a
€
d
In (0.1) and (0.2), A = E|a|<2m « <* Cg>(R ) is a uniformly d eUiptic operator on R , F(t,u(t))(x) = f(t,x,u(t,x)), ( B ( t , u ( t ) ) f l ( x ) = b(t, x, u(t, z))f (ar), and R(t, u(t))(x) = f ( t , x, u(t, z))+EU ^ (/<(*, ', «(<, •))) (* are generalized Nemytskii operators corresponding to functions f , f i , b : [0, oo) x E.d x K -^ M and Cg* vector fields {%} on M d . In the formula for B, f belongs to the RKHS of W, see Section 1.1. For (0.2) we assume that A is a second order operator. Therefore m = I in (0.2). We deal with equations driven by a spatially homogeneous Wiener process W, which represents the influence of a random environment. W takes values in the space <S'(Rd) of tempered distributions. It is a generalization of a Gaussian random field W on [0, oo) x Rd such that for x € Rd, W(-, x) is a 1-dimensional Wiener process, and for all t,s and x,y, E*W(t, x)W(s, y) = min{t, s}r(o; — y). In general however, the space correlation F can be a 417
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418
Peszat
distribution. Typically T(x) = \x\~^ , or F = 60 in the case of W being a space-time white-noise. In this paper we are concerned with Markovian solution. Thus (0.1) or (0.2) will define a Markov family on a given function state space. We consider the scales of state spaces Ifp = Lp(E.d, e-f\x\ da;), p € R, p € [2, oo), and Cp, p e M; the space of all continuous i/j : E.d -» R such that and |^'(a;)|e~'>la;l ->• 0 as |a;| —>• oo.
0.1. Parabolic equation. The linear equations of the type (0.1),
(0.3)
were introduced by Dawson and Salehi [10] as a model of the growth of a population in a random environment. The same equation was considered by Nobel [20] as a continuous version of the lattice geophysical system studied by Carmona and Molchanov [5]. (0.3) was used by Kifer [16] and Handa [13] in the study of multidimensional Burgers equations via a Hopf-Cole type transformation, see also [26], [27]. [10] and [20] were a starting point for Peszat and Zabczyk paper [23] dealing with the general non-linear equation (0.1). They shown that the noise can be interpreted as a Wiener process on a Hilbert space continuously embedded into the space of tempered distributions on R d , see Theorem 1.1 of the present paper, and that (0.1) can be treated as an evolution equation. This approach enabled to obtain the existence of continuous in time, Markovian solutions in L2,, p € Md. The main assumption in [23] was formulated in terms of the spectral measure fj, = T of W,
') < oo, or n « dx and -^ e Lq(Rd, dx) with ( 1 - - } -£- < 1. da; \ qj 2m
(0.4) Next Brzezniak and Peszat [2] showed that (0.4) is in fact sufficient to obtain the existence of continuous in time solutions in If and Cp. To obtain the spatial and temporal continuity of solutions they used the theory of stochastic integration in Banach spaces. In [2] the case of the drift coefficient F of a polynomial type was also studied. More general case with a continuous onesided linearly bounded drift was treated by Manthey and Zausinger [17]. Tessitore and Zabczyk [26] investigated the long time behavior of solutions to (0.1) with A = A, F = 0, and time independent Lipschitz continuous B. They showed that there is an invariant measure v with support in any L? , p > 0 space provided that d > 3, (0.4) holds true, and
47^/2
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SPDEs Driven by a Homogeneous Wiener Process
419
where Q is the Euler gamma function, 7 = -^, and I is the Lipschitz constant of B. In particular, they proved that (0.4), d > 3 and (0.5) with I = a
guarantee the existence of a nontrivial invariant measures for (0.3). Then by the Cole-Hopf transformation they constructed nontrivial stationary solutions of a vector stochastic Burgers equation, see also [16], [13], [27]. The stochastic evolution equation with additive noise //"?/
^(t) = A«(t)
- m«(t)
•
+ W(t),
(0.6)
was studied by Da Prato and Zabczyk [9], under the assumption that the
spectral measure [i is absolutely continuous and 7 = -^ is integrable. They showed that (0.6)
defines a Markov family in any L?p, p > 0 space, and
that if m > 0, then there is an invariant measure. Here the assumption on the integrability of the spectral density 7 played an important role. Then Karczewska and Zabczyk [14],
[15] showed that < oo
(0.7)
is a sufficient and necessary condition for the existence of a function valued
solution to (0.6). Using ideas from [9], [14], [15] one can show that (0.7) is also a sufficient and necessary condition for the existence of an invariant measure
to (0.6) with m > 0 considered on any L^, p > 0 space. The same statement holds true for m = 0, provided that (0.7) is replaced by JRd Q^ < oo. Necessary and sufficient conditions for the existence of a function-valued solutions to a nonlinear stochastic heat equation du — (t, x) = A«(i, C/li
x) + f ( t , x, u(t, x)) + b(t, x, u(t, x))W(t)
(0.8)
were formulated by Peszat and Zabczyk [24]. They proved that if / and b are Lipschitz continuous with respect to the third variable, and that the space
correlation F of the noise satisfies 3C > 0: F + C da; is a non-negative measure, then (0.7)
(0.9)
is a sufficient condition for the existence of an I^-valued solutions
to (0.8). It is also a necessary condition if b is separated from 0. Karczewska
and Zabczyk [14] showed that if (0.9)
holds true then (0.7)
log(|y|- 1 )r(dy)
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ifd=2
is satisfied iff (0.10)
420
Peszat
and
oo
ifd>2.
(0.11)
If d = 1 then (0.7) follows from (0.9) and the fact that F is a tempered measure. The existence of martingale solutions to stochastic Navier-Stokes and Euler equations driven by a spatially homogeneous Wiener random fields was studied by Capiriski and Peszat [4] and Brzezniak and Peszat [3]. Karczewska and Zabczyk considered the linear equation (with A = A, / = 0, b = 1) on a torus, proving that (0.7) is a necessary and sufficient condition for the existence of a function-valued solution. Tindel and Viens [28] obtained necessary and sufficient conditions for the existence of space-time Holder continuous solutions to nonlinear heat equation on a compact Lie group. Finally the
heat and wave equations on a locally compact Lie group are treated in [22]. In fact our Theorems 3.1 and 3.2 (for m = I) follows ^from [22]. The case m 7^ 1 can be treated in Rd using methods from [22] and known estimates for the kernel corresponding to A. Theorem 3.1 says that if (0.10) is satisfied and the coefficients are Lipschitz continuous with respect to the third variable, then the following conditions are sufficient, and in a sense necessary for the existence of a function-valued solutions to (0.1), log(|y|- 1 )F(dy)
ifd = 2m
(0.12)
i\v\
L
(0.13)
0.2. Wave equation. The study of nonlinear wave equation driven by a spatially homogeneous Wiener process was initiated by Dalang and Frangos [7]. They proved the existence of a local in time solution to the nonlinear equation (0.2) with A = A, R(t, u(t))(x) = f ( t , x, u(t, x)), and d = 2. Dalang and Frangos assumed that the functions / and b are Lipschitz, and that the space correlation F is a non-negative function, continuous outside 0 and satisfies (0.10), which in fact was first formulated in [7]. Then Millet and Sanz-Sole [18] showed, still for dimension 2, that under (0.10) the solution is global and that (0.10) is also necessary for the existence. Peszat and Zabczyk [24] proved the existence of a solution in dimension d = 3 under the condition (0.11). Karczewska and Zabczyk [14], [15] showed that in the linear case (^4 = A, / = 0, b = 1) (0.7) is a necessary and sufficient condition for the existence of a function-valued solution for equations on Rd and on torus Td. The space-time Holder continuity of solutions to a linear wave
equation has been studied recently by Sanz-Sole and Sarra [25].
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It seems
SPDEs Driven by a Homogeneous Wiener Process
421
that similar results can be obtaining by considering Markovian solutions on the state space (H ,H ~) with a properly chosen r. The methods applied in [7], [18] rely on the integration theory with respect to martingale measures developed by Walsh. This theory was originally thought not to be applicable for the wave equation in dimension d = 3 and higher because the integrand is a distribution valued process. Dalang [6] l
p
p
has recently extended the theory of stochastic integration with respect to martingale measures to a class of distribution-valued processes, and applied
this to 3 dimensional wave equations. In the papers [6], [7], [18], [14], [15], [24] the solution to stochastic wave equation was defined as a process u satisfying the integral equation
d /"* u(t) = -^G(t) * H0 + G(t) *v0 + G(t-s)* R(s, u(s)) ds Jo
(0.14)
f G(t-s)*B(s,u(s)}dW(s), Jo where G is the fundamental solution to -j^ — A — 0. In this paper we present
a result on the existence of a Markovian solution, see Section 2, involving the evolution of the pair X = (u(t), ^u(t)) T . Clearly one obtains a solution
to (0.14) by taking the first coordinate of the Markovian solution. The problem of the existence of a function valued solution to (0.2) for an arbitrary
d, and without assuming (0.9) was solved by Peszat [21], see Theorem 3.3 from the present paper. He proved the existence of a Markovian solution
to (0.2) provided that the spectral measure n of W satisfies the following intergrability condition sup '
d (2/)
^
Similar result has been obtained for a nonlinear wave equation on a locally
compact Lie group, see [22]. Remark 0.1. It is easy to see that if (0.9) holds true, then (0.10) — (0.11), (0.7) and (0.15) are equivalent.
Remark 0.2. Consider fj, = ^^W (6k + 6_k), 0 < r? < 1. Then fj, is a positive symmetric tempered measure on R satisfying (0.7). Clearly, it does not satisfy (0.15). Hence, see Remark 0.1, its Fourier transform F does not satisfy (0.9).
1. DISTRIBUTION-VALUED WIENER PROCESSES 1.2. Spatially homogeneous Wiener process on R. In the paper we denote by {£, tp) the value of a distribution £ € <S'(Rd) on a test function V>
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Peszat
Definition 1.1. Let 21 = (fl, JF, P) be a complete probability space with a filtration (Ft)t>o- We say that an S^W )-valued process W denned on 21 is Wiener iff (i) for arbitrary finite sets {-01,... , V'n} C «S(Rd) and {ti,... ,i n } C [0,oo) the random vector ((W(ii), V"i}) • • • > {W(*n)j VVi}) ig Gaussian, (ii) for any test function ij) e <S(R d ), (W(t),i/>}, t > 0 is a real-valued )-adapted Wiener process. 1
Let W be an <S'(Rd) -valued Wiener process. Then one can find a unique real separable Hilbert space H C <S'(Rd) such for an arbitrary orthonormal basis {ffc} of H there is a sequence {Wk} of standard independent realvalued (jFt)- adapted Wiener processes for which W(t) = ^Wfc(i)ffc. The series converges in *S'(Rd) almost surely. We call H the Reproducing Kernel Hilbert Space of W, RKHS in short. Then W is a cylindrical Wiener process onH. Let [rx : x € Rd} be the group translations on <S'(Rd), that is for £ € V1 € S(Rd) and x,y € R d , (r^,^) = (£,r_xij;), where r_ Definition 1.2. An «S'(R ) -valued Wiener process W^ is called spatially homogeneous iff for any t > 0 the law of W(t) is invariant with respect to the group of translations {TX : x € !d}, that is P(W(£) € AT) = P(W(<) € for aU x e Rd and # d
Let A.W be the covariance form of an *S'(R) -valued Wiener process W, d
For • R we write (/?*(«) = y>(— #). If
Proof. Since W is Gaussian, it is spatially homogeneous iff AW is translation invariant, that is if for all 1/1, f € 5(Rd), and re € Rd, A(^>, y?) = A (r^^, r^^). This holds true, see [12, Th. 6, p. 169] iff there is a T € S'(Rd) such that for V>, f €
We will call /^ the spectral measure of a spatially homogeneous Wiener process W and F = fi its space correlation. Obviously, the law of W is determined by its spectral measure. Note that if the spectral measure of W is finite, then the space correlation F is a bounded continuous function, and W is a random field. In the case of the covariance form of W given by
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> f) — (^i f } one nas r = <5o, /u = (27r)~'5 'do;, and W = @jf is a white noise onL 2 ([0,oo) xRd). Let // be a symmetric non-negative tempered measure on R d . Write
Note that ^u, ip G L 2 (M d , fj,; C), are complex- valued tempered distributions. Thus in particular their Fourier transforms are well defined. Since fj, is symmetric V>jU, i/) = tj>* are real-valued distributions. The theorem below was stated and proved in [23, Prop. 1].
Theorem 1.2. Let W be a spatially homogeneous Wiener process with a spectral measure fj,. Then (H^., {•, -}H^) is RKHS of W.
1.2 Stochastic integration in weighted I/p-spaces. Let (O,B,6) be a measurable space, and let LP(O) = 17(0,3,9). Here 0 is a positive but not necessarily finite measure. In this section we briefly recall the contraction of the stochastic Ito integral in 1^(0), p > 2 spaces. It is worth noting that this is a part of a general theory of stochastic integration in Banach spaces. Let W be a Wiener process with RKHS H. Let us fix p € [2, oo) and an orthonormal basis {f^} of H, and a sequence {/3fc} of independent standard real-valued normal random variables defined on a probability base 21. We will integrate operator- valued processes. Recall that a bounded linear operator K : H —>• L (O) is called 'j-radonifying, iff the series JZfcLi A^ffc converges in L 2 (fi, JF,P; U>(O)). Let us denote by R(H,LP(O)) the class of all 7-radonifying operators from H into LP(O). Given a linear operator K from H into LPiO] we define its 7-radonifying norm P
fc=l
Then, see e.g. [N], the operator K is 7-radonifying iff
oo. Moreover, (R(H, LP(O)), \\ • \\R(H,LP(O))) '1S a Banach space. Note that R(H, L2(O)) is equal to the space of Hilbert-Schmidt operators L(ns)(H, L2(O)). Denote by P the space of all measurable (Jri)-adapted processes a with values in R(H, Lf(O}} such that the seminorms rri
aT= (E f \v(t)\qLP(0}dt}1 \ ^
JO
re(0,00),
'
are finite. The stochastic integral with respect to W can be defined in a natural way for simple processes a € P. Then it can be extended to the whole space P. In fact we have the following consequence of general theorems on stochastic integration in Banach spaces, see e.g. [1], [8], [19].
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Peszat
Theorem 1.3. For any a € ~P the stochastic integral J0 &(s) dW(s], t > 0 is an Lp(0)-va]ued square integrable martingale with continuous modification and 0 mean. Moreover, for every q € [2, oo) there is a constant C such that for all T G (0, oo) and a € P, N9/2
E sup
) dW(s)
0
Li>U
'
2. DEFINITIONS OF SOLUTIONS TO STOCHASTIC PARABOLIC AND WAVE EQUATIONS Recall that the spaces V and Cp were defined in the introduction. Note that Cp with the norm l^lcp — suPrceMd I^K^)! e~p^ is a Banach space. It is know, see [11], [2], [23], that a uniformly elliptic operator A with Cg°coefficients generates semigroup S on <S'(Rd) and that its restriction to either Lpp, P € [2, oo), p € R or Cp, p € M is a Co-semigroup. This semigroup will be also denoted by S. By a solution to (0.1) we understand the so-called mild solution, that is, a solution to the following integral equation
u(t) = S(t)u(Q) + f S(t-s}F(s,u(s))ds+ Jo
I S(t - s)B(s,u(s}} dW(s), Jo (2-1)
where the stochastic integral is understand as the Ito integral in Lp with respect to the infinite dimensional Wiener process W, see Section 1.2. We set $p(x) = (i9(o;))'', where •$ e 5(Rd) is strictly positive and equals e~H for \x > 1. Define a Sobolev space Hr as the completion of <S(Rd) with
respect to the norm iV'lnr = / K d(l + x\2}r\'lf}^p/2(x)\2
A
dx. Let
y o
/o
=(A
Then, see [21], that A with the domain Dom*4 = Dp generates a Cosemigroup U = {U(t}} on Xp. We define an Xp-valued Markovian solution X to (0.2) as a process satisfying the following stochastic evolution equation
f U(t-s)V(s,X(s))ds+ Jo
f U(t-s)$(s,X(s})dW(s), Jo
where X(0) = (UQ, VO) T , and for X = (u, -u)T ,
, „)„
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3. EXISTENCE OP SOLUTION First we consider the case of a regular spectral measure, that is we assume
(0.9). Remark 3.1. Recall, see Theorem 1.1, that for any spatially homogeneous Wiener process its space correlation F is a tempered distribution. Thus (0.9) guarantees that F is a tempered measure, and hence /ria,|<1-i h(x)T( dx) < oo for any bounded measurable function h. In particular, (0.9) implies (0.13) in the case of d < 2m. Definition 3.1. Let p € [2, oo) and p e R. We say that a function h :
[0, oo) x Rd x R -4- R belongs to the class Lip (p, p) iff for any T < oo there are a constant L and a function IQ € lfp such that
\h(t,x,z)\
Theorem 3.1. Let p € [2, oo) and p € R. Assume (0.9) and that the
coefficients /, b are of the class Lip (p, p). (i) If (0.12) — (0.13) are fulfilled then for any UQ € Lp there is a unique mild solution u to the stochastic heat equation (0.1) such that for every T < oo,
sup E\u(t)\2Lp < oo. t€[0,T]
(3.1)
"
(ii) Assume that p > 0. If there are T > 0 and 60 > 0 such that \b(t, x, z}\ > b0 for all t <E [0,T], x € R d , and z € R, then (0.12) - (0.13) follow from the existence of a solution to (0.1) satisfying (3.1). Our next result deals with time and space-time continuous solutions to the stochastic heat equation, for its proof see [22, Th. 2.2]. i Theorem 3.2. Let p e R and p <E [2,oo). Assume (0.9) and that the coefficients f, b are of the class Lip (p, p). (i) If there is an a > 0 such that
fJ{\x\
(3.2)
then for any UQ € U> there is a unique mild solution u to the stochastic heat equation (0.1) having continuous trajectories in Lpp and satisfying E sup \u(t}\qJP < oo t€[0,T]
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"
for all T < oo, q € [2,oo).
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Peszat
(ii) If there is an a > 2d/p such that (3.2) holds, then for any UQ € 1 there is a unique mild solution u to the stochastic heat equation (0.1) having continuous trajectories in L? C\ Cp/p and satisfying
E sup (\u(t)\qc
te[o,T] ^
+ \u(t)\lp} < oo p
'
for all T < oo, q € [2, oo).
Next we consider the case of an irregular spectral measure. Recall, see [14] that if (0.9) holds, then (0.10) - (0.11) and (0.15) are equivalent. For a proof of Theorem 3.3 below see [21, Th. 1.1] and [22, Th. 2.3]. For a proof of Theorem 3.4 see [21, Remark 3.1]. Theorem 3.3. Let p € R. Assume that f , f i , b €. Lip(2,p) and that the
spectral measure /J,ofW satisfies (0.15). Then for any (uo,t>o) T *= "%-p ^ere is a a unique Markovian mild solution to (0.2) with continuous trajectories in Xp and such that E sup \X(t)\^ < oo te[o,T] "
for all T < oo, q £ [1, oo).
Theorem 3.4. Let p € R. Assume that f, b e Lip (2, p) and that there is an a > 0 such that the spectral measure /j,ofW satisfies
JR" Then for any UQ € L2 there is a unique mild solution u to the stochastic heat equation (0.1) having continuous trajectories in I? and satisfying
E sup u(t)\qL2 < oo t€[0,T]
for all T < oo, q € [2, oo).
"
REFERENCES [1]
Z. Brzezniak, On stochastic convolutions in Banach spaces and applications, Stochastics Stochastics Rep. 61 (1997), 245-295. [2] Z. Brzezniak and S. Peszat, Space-time continuous solutions to SPDEs driven by a homogeneous Wiener process, Studia Math. 137 (1999), 261-299. [3] Z. Brzezniak and S. Peszat, Stochastic two dimensional Euler equations, submitted. [4] M. Capiriski and S. Peszat, On the existence of a solution to stochastic Navier-Stokes equations, Nonlinear Anal., to appear. [5] R. Carmona and S. A. Molchanov, Parabolic Anderson problem and intermittency, Mem. Amer. Math. Soc. 108 (1994), 1-125. [6] R. C. Dalang, Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.'s, Electronic J. Probab. 4 (1999), 1-29.
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R. Dalang and N. Prangos, The stochastic wave equation in two spatial dimensions,
Ann. Probab. 26 (1998), 187-212. [8] [9] [10]
[11] [12]
[13]
[14]
[15]
E. Dettweiler, Stochastic integration relative to Brownian motion on a general Banach space, Doga Mat. 15 (1991), 6-44. G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge Univ. Press, Cambridge, 1996. D. A. Dawson and H. Salehi, Spatially homogeneous random evolutions, J. Multivariate Anal. 10 (1980), 141-180. T. Funaki, Regularity properties for stochastic partial differential equations of parabolic type, Osaka J. Math. 28 (1991), 495-516. I. M. Gel'fand and N. Ya. Vilenkin, Generalized Functions, vol. IV Applications of Harmonic Analysis, Academic Press, New York, 1964. K. Handa, On stochastic PDE related to Burgers' equation with noise, Collection: Nonlinear Stochastic PDES, Minneapolis 1994, IMA Vol. Math. Appl. Vol. 77, Springer, New York, 1996, pp. 147-156. A. Karczewska and J. Zabczyk, A note on stochastic wave equations, Evolution Equations and their Applications in Physical and Life Sciences (G. Lumre and L. Weis, eds.), Proceedings of the 6th International Conference, Bad Herrenhalb, 1998, Marcel Dekker, to appear. A. Karczewska and J. Zabczyk, Stochastic PDEs with function-valued solutions, Infinite Dimensional Stochastic Analysis (Ph. Clement, F. den Hollander, J. van Neerven
[16] [17]
[18]
[19] [20]
and B. de Pagter, eds.), Proceedings of the Colloquium (Amsterdam 1999). Verhandelingen, Afd. Natuurkunde, Vol. 52, Royal Netherlands Academy of Arts and Sciences, Amsterdam, 2000, pp. 197-216. Yu. Kifer, The Burgers equation with a random force and a general model for directed polymers in random environments, Probab. Theory Related Fields 108 (1997), 29-65. R. Manthey and T. Zausinger, Stochastic evolution equations in l?^, Stochastic Stochastic Rep 66 (1999), 37-85. A. Millet and M. Sanz-Sole, A stochastic wave equation in two space dimension: Smootheness of the law, Ann. Probab. 27 (1999), 803-844. A. L. Neidhardt, Stochastic integrals in 2-uniformly smooth Banach spaces, Ph.D. Thesis, University of Wisconsin, 1978. J. Nobel, Evolution equation with Gaussian potential, Nonlinear Anal. 28 (1997),
103-135. [21] S. Peszat, The Cauchy problem for a nonlinear stochastic wave equation in any dimension, submitted. [22] S. Peszat and S. Tindel, Stochastic heat and wave equations on a Lie group, in preparation. [23] S. Peszat and J. Zabczyk, Stochastic evolution equations with a spatially homogeneous Wiener process, Stochastic Processes Appl. 72 (1997), 187-204. [24] S. Peszat and J. Zabczyk, Nonlinear stochastic wave and heat equations, Probab. Theory Related Fields 116 (2000), 421-443. [25] M. Sanz-Sole and M. Sarra, Path properties of a class of martingale measures with applications to spde's, Canadian Mathematical Society Conference Proceedings. A volume in honor of Sergio Albeverio, to appear. [26] G. Tessitore and J. Zabczyk, Invariant measures for stochastic heat equations, Probab.
Math. Statist. 18 (1999), 271-287. [27] G. Tessitore and J. Zabczyk, Strict positivity for stochastic heat equations, Stochastic Processes Appl. 77 (1998), 83-98. [28] S. Tindel and F. Viens, On space-time regularity for the stochastic heat equations on a Lie groups, J. Funct. Anal. 169 (1999), 559-603.
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Applications of Malliavin Calculus to SPDE'S MARTA SANZ-SOLE Facultat de Matematiques, Universitat de Barcelona, Gran Via, 585, 08007 Barcelona, Spain, e-mail: [email protected]
1. INTRODUCTION The stochastic calculus of variations, also known as Malliavin Calculus, provides criteria for the law of Wiener functionals to possess a density. Moreover, it also furnishes appropiate tools to analyze properties of this density. The starting point of the theory, presented in [M], was motivated by the problem of giving a probabilistic proof of Hormander's theorem on hypoelliptic operators. As for its applications, the subsequent developments of the theory by Bismut, Kusuoka, Stroock, Watanabe and many others, focuss their attention mainly on stochastic differential equations. The purpose of this article is to review the above-mentioned applications of Malliavin Calculus to solutions of stochastic partial differential equations (spde's in the sequel). We will report on the following issues: existence of density, regularity properties and positivity. Some open questions will also be mentioned.
2. PRELIMINARIES ON MALLIAVIN CALCULUS We give in this section a very brief, self-contained introduction with the main ingredients on Malliavin Calculus which are needed. We have used the approach of [Nl] and [N2], where the proofs can be found, except otherwise specified. Let H be a real separable Hilbert space. We denote by (•, •) and \\-\\u the inner product and the norm in H, respectively. On a complete probability space (£2, A, P) we consider a centered, Gaussian process {W(h),h € H} with covariance defined by E(W(hi)W(h-2)) — (hi^h^n, hi,h-2 € H. Let S be the set of smooth random variables denned on (tl,A,P). That means, F € S if there exists / 6 C%°(R ), the set of bounded, infinitely differentiable n
429
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Sanz-Sole
functions with bounded derivatives of any order, and hi,...,hn € H, such that
F = f(W(hl),---,W(hn)).
(2.1)
For F as in (2.1) the Malliavin derivative DF is an H- valued random vector given by
DF = £ dif(W(h1), • • • , W(/in))/*i.
(2-2)
The identity (2.2) defines a closable operator from L p (fi) to Lp(f2;.H"), for any p > 1. The closure of 5 with respect to the norm
is the domain of D in L p (fi). It is denoted by D1>p. The k-th derivative of F € 5, Dfc.F, is defined by iteration of (2.2); it is an lZ"®fc-valued random vector. Then, by analogy with the definition of Dl'p , we set Dk'p the closure of S with respect to the norm
k€Z ,p€ [l,oo). We set L>°° = ngij. n £> ' . The derivative operator D is an unbounded operator from L 2 (fi) to L2(£l,H). The adjoint of D is an operator 5 whose domain is the set of If-valued square integrable random variables u such that fe
+
p
pe[1)0o)
\E((DF,U)H)\
E(F6(u)) = E((DF,u)H),
(2.3)
for F € -D1'2. The operator 8 is known as the divergence operator or the
Skorohod integral. Formula (2.3) is called the integration-by-parts formula. In the case of Brownian motion 6 is an extension of Ito's stochastic integral. Using the identity (2.3) one can prove the following Lemma. It will be needed in Section 3.2. Lemma 2.1 ([Mol]: Corollary 4.1) Let / € C£°(JR) and F be the anf
tiderivative of f. Let Z and £ be random variables such that Z £ D°°, . Then, for any r < r 0 ,
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where Cr is a constant depending on ||£||2(r+i),22('-+1) with some positive constant kr depending on r.
an
d E(\\D£\\H)
r
,
The domain of D in Lp(£l) can be localized, as follows. Let D^ be the set of random variables F such that there exists a sequence fin 6 A, n > 1, increasing to fi, and a sequence {Fn, n > 1} of elements in Dl>p such that on fi, F = Fn. Similarly, Dom<5 can also be localized. Let F : fi —> Rm be a random vector, F = (F 1 ,..., Fm). Assume Fi <E Dfo , i = 1,..., m. The Malliavin matrix of F is a symmetric nonnegative definite matrix defined by c
The study of the probability law P o F~l relies on nondegeneracy properties of JF and the duality relation (2.3). We first quote a result by Bouleau and Hirsch (see [Bo-H]) ensuring absolute continuity. Theorem 2.1 Assume that F € D^, i = l , . . . , m , for some p > 1. Suppose also that the Malliavin matrix 7^ is invertible a.s. Then, P o F"1 is absolutely continuous with respect to Lebesgue measure on Rm. 1
The smoothness of the density of P o F~l needs stronger hypotheses than those of the previous theorem, as follows (see for instance [M], [I-Wa], [St], among others).
Theorem 2.2 Suppose that (i) FieD°°,foreveryi = l,...,m, (ii) (det^p)-1 € np6[i,co)£p(fi). Then, Po F~l is absolutely continuous with respect to Lebesgue measure on Rm and its density belongs to C00(Rm). Let N > I be an integer, p € [2, oo). The following Lemma gives a useful tool in checking that a random variable G belongs to DN'P. The proof can be read in [R-SS1]. By convention D°X = X. Lemma 2.2 Let {X ,n > 1} be a sequence of random variables in D ' , N > 1, p & [2,oo). Assume that there exists X € DN~:'P such that {D ~X ,n > 1} converges to D ~ X in LP(tt,H®( -V) as n -» oo and moreover, the sequence {DNXn,n > 1} is bounded in Lp(£l,H®N). Then X 6 DN>*. n
N
i
N
N
1
P
N
n
3. LAWS OF SOLUTIONS OF SPDE'S We are interested in stochastic partial differential equations of the following type,
£u(t,x) = a(u(t,x))F(t,x) + b(u(t,x)),
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(3.1)
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t > 0, x € D, where D is a domain of Rd. £ is a second order differential operator defined on (0, oo) x D. The process F = {F(4>},(f) 6 T>(Rd+1)} is an L2(fi)-valued, Gaussian process with zero mean and covariance functional given by
}= I ds f JR+ JR where if>(s, x) := V>( s > —x) and F is a nonnegative and nonnegative definite tempered measure. F gives the correlation of the noise in the x-variable. If
F is the Dirac <5{0} distribution, F is called the space-time white noise. Associated with equation (3.1) we can consider, as in the deterministic case, two kind of problems:
(i) An initial value problem with a given functional value at t = 0. (ii) A boundary value problem with a given initial value and additional conditions to be satisfied at the boundary of D. We give a rigourous meaning to these problems through their evolution form. More precisely, the solution is defined to be a stochastic process
satisfying some measurability requirements and the equation
u(t,x) = uo(t,x) + I I G(t,s,x,y)cr(u(s,y))F(ds,dy) Jo JD + f
f G(t,s,x,y)b(u(s,y)}dsdy, Jo JD
(3.2)
where G(t, s, x, y), 0 < s < t, x, y € D, is the fundamental solution of Lu = 0 or the Green function for the problem, respectively. The term Uo(t,x) is the (deterministic) contribution of the initial data. In (3.2) D is either Rd or a bounded domain in Rd. This formulation leads to stochastic Volterra equations with kernels which are, in most of the cases, singular. For the meaning of the stochastic integral in (3.2) we refer the reader to
[W], [D-F], [D], [P-Z1], [P-Z2]. As has been made precise in these references, there are two sources of difficulties: the singularity of G(t, •, x, *) and the non smoothness of the noise. This forces to restrict C to particular examples. However, interesting cases can be covered. The case d = 1 is the only one where space-time white noise is allowed. For d > 1 the noise must be more regular; this is the reason why correlation in space is introduced. The existence of the stochastic integral is assured
through integrability conditions involving the measure F and the kernel G. Fix t > 0, x i , . . . , xm distinct points in D and assume that there exists a solution of (3.2) (see for instance Theorem 13 in [D] for an appropiate statement). Our first questions read as follows:
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(Ql) Does the law of u(t, x) := (u(t, x i ) , . . . , n(t, x m )) possess a density with respect to the Lebesgue measure on Rm? If the answer is yes, (Q2) Is the density pt,x(-} a C°° function on Rm? Fix y £ Rm and let x = (x\,..., xm). (Q3) What kind of regularity does the function (t, x) —> pt,x(y) have? In this article we will focuss our attention on operators £ either of parabolic or hyperbolic type. Moreover, we will assume that G(t, -, x, *) is a function. This excludes, for example, the wave equation in dimension d > 3. The elliptic case has been so far less explored (see [T]). 3.1 Existence and smoothness of density We will give here answers to (Ql) and (Q2), simultaneously, for several examples of spde's. The essential tool to be used is Theorem 2.2 in the following context. Let £ denote the inner product space of measurable functions (:Rd^R such that
endowed with the inner product
R*
Let H be the completion of £. Set H = L 2 ([0,r];H). The Gaussian process F can be identified with a Gaussian process (W(h), h € H) as follows. Let (BJ,J > 0) C £ be a complete orthonormal system of H. Then (Wj(t) = /o fRdej(x}F(ds,dx),j > 0) is a sequence of standard Brownian motions such that 3=0
e £([0,T] x Rd). For hE H set
3=0
Notice that for space-time white noise, H = L2([0,T] x Rd), since F = <5{o}We assume two kind of assumptions on the coefficients of the spde: (hi)Regularity: a, b are C°° functions with bounded derivatives of any order.
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(h2) Non degeneracy on a. This last condition will be formulated more explicitely on the examples. The differentiability of the random vector u(t, x) is usually checked by means of the Picard iterative scheme associated with (3.2). More precisely, for any t > 0, x € D, set u(0\t,x) = u0(t,x)
uW(t,x) = uW(t,x) + f f G(t,s,x,y}a(U^-1\s,y})F(ds,dy) Jo JD + [ [ G(t,s,x,y)b(u(n-l\s,y))dsdy,
n > 1.
Jo JD
For any integer N > I and T > 0, set (HN) For any t € [0, T], x 6 D, p € [2, oo), (i) {uW(t,x},n>Q}cDN'f, (ii) supxeD sup00 supx&D sup0 1. Then Lemma 2.2 applied to Xn := u^n\t,x) yields the required differentiability. The second property to be proved is a nondegeneracy condition on the Malliavin matrix. For the sake of simplicity we will assume m = 1. The Malliavin derivative Drt € H is given by
Dri
I
Jr JD
(3.3)
if r < t. Itr>t, Drtu(t, x} = 0. We want to prove that
E(\\Du(t,x)\\HP) 2, where Cp is a finite positive constant. This is equivalent to the following property: For any p > 2, there exists CQ > 0 such that
pe := P{ f \\Dr,.u(ttx)\^dr < e} < Jo for any e < €Q.
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Let e, 6 > 0 such that es < t. Using the triangle inequality we obtain Pe
P,6+Pe,S
\\Dr,u(t,x)-G(t,r,x,-)a(u(r,-))\\2ndr>e},
6
\\G(t,r,x,-)a(u(r,-))\\2Hdr
(3.5),
(3.6)
t-e
The term p\ s can be worked out using first (3.3), Chebychev's inequality and then Lp estimates on the solution of (3.3). The core of the problem is
p2s.
Here is where the assumption (h2) enters in the scene, as well as the
properties of G in each particular example.
3.1.1 Examples in the one-dimensional case
In this section {F(t, x),t > 0, x € R} is a space-time white noise. We give several examples where the answer to (Ql), (Q2) is affirmative. Stochastic heat equation ([B-P]) Let £ = ^ - |^, t € [0,T], x 6 [0,1]. Consider equation (3.1) with either Neumann or Dirichlet boundary conditions. Assume (hi) and
(h2) a(z)\ > C > 0, for any z e R. In this case, the key properties used to study (3.5), (3.6) are the following: (i) G(t,s,x,y) < Kiet,s(x,y) < K2G(t,s,x,y),
(lii] /_£ J,, G2<>(t, s, x, y}dyds < where K, K\,K 0, q < | and
Stochastic wave equation ([C-N]). Let L = -jj^ — -^2, t > 0, x € R. Consider equation (3.1) with initial condition u(0, x) = f ( x ) and ^r(0, x) an absolute continuous measure with density g(x). Assume (hi), that / and g are a-Holder continuous for some a > 0 and (h2) One of the following conditions hold: (a) 1 and either
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(bl) cr(/(£o)) 7^ 0 for some £0 € (x — t, x + f) and a — I or (b2) cr(/(£)) = 0 for all £ € [x - t, z + t] and either g e C2, #'(x - t) ^ 0 or 3 5'(x + t) ^ 0, or g e C , p"(x - t) + 6(y 0 ) or 5"(x + t) + 6(2/0) + 0. Besides the technicalities, let us notice that (h2) in the first example and (h2) (a) above are assumptions of elliptic type, while (h2) (b) is of hypoelliptic type.
3.1.2 Examples in the multi-dimensional case The case d > 2 is a less explored territory and at the present time, it is restricted to particular examples where the fundamental solution of £u = 0 is a function. In the recent paper [Ma-Me-Sa] a quite general approach is presented which applies to the stochastic heat equation on Rd and recovers partially the results of [Mi-SS3] on the stochastic wave equation on R2. We give a brief sketch here. Assume that the fundamental solution of Lu = 0 is a stationary function in time and space, that is, G(t,s,x,y) = S(t — s,x — y). Let n denote the spectral measure of F and T the Fourier transform operator. We introduce the following set of hypotheses: (hi), and (h2) \a(z)\ > C > 0 for any z € -R, (h3) JRd S(t, y}dy < CT < oo, 0 < t < T,
(h4) £dsfRd n(dt)\rS(s, -)(6|2 < oo. Let {u(t, x), (t, x) € [0, oo) x Rd} be the real- valued solution to the initial problem denned by (3.1) with zero initial condition.
Theorem 3.1 ([Ma-Me-Sa]: Theorem 2.1 ) Fix t > 0, x e Rd. Assume (hl)-(h4) and that there exist 9t > 0, i = 1, 2, 3, &2<&i,0i< (202) A(<9 2 +6> 3 ), positive constants d, i = 1, 2, 3, and t € [0, T] such that, for all p € [0, t ], 0
CV1 < f^s f JO
f ds f JO
|^S(S,-)(
JRd
S(s,y)dy
JRd
Then the law ofu(t, x) has a C°° density on R. The approach given in the previous theorem may likely be extended to random vectors u(t,x), to non null initial conditions and possibily, to boundary value problems. Notice that ^8(5, •) may be a function in cases where 5(s, •) is distribution-valued. This happens, for instance, to the fundamental solution of the wave equation in any dimension d. 3.1.3 Remarks
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As has been pointed at the begining of Section 3. The stochastic processes we are considering satisfy stochastic Volterra equations. This framework allows to cover new examples. Consider the second order differential operator defined on R^_ by r\2
f\
r\
Consider the spde (3.1) corresponding to this £, with given initial condition XQ (either deterministic or random) on the axes {(s,t) € R2 : st = 0}. The evolution form is
Xa,t = XQ+ t G(a, t, u, v} (a(Xu,v)F(du, dv) + b(XUtV}dudv] , JR*,t
(3.7)
where G(s, t, u, v), 0 < u < s, 0 < v < t, is the associated Green function. In [R-SS1], [Ma-SSl], we have proved the existence and smoothness of density for the law of X t, st ^ 0, under hypoelliptic type assumptions, exploiting the smoothness of the Green function. An abstract approach for Volterra equations has been initiated in [R-SS2]. Equation (3.7) is motivated by the construction of path- valued processes in a Riemannian manifold (see [No]). In addition, in a recent work by Dalang and Walsh [D-W] it is proved that time-reversed Brownian sheet in one and two coordinates satisfy equations like (3.7). a<
3.2 Further properties of the density Let {X(t,x),(t,x) € [0, T] x D} be an .Rm-valued stochastic process. Assume that for any (t, x) £ (0,T) x Int-D, the random vector F := X(t,x) satisfies the assumptions of Theorem 2.2. Denote by Pt, (y] the density of the law P o (X(t, x)) at y € Rm. We are interested here in regularity properties of (t,x) —>• Pt, (y]i for instance Holder continuity. We quote the approach of [Mol] and [Mi-Mo] for the stochastic heat equation in dimension one and the stochastic wave equation in dimension two, respectively. For the sake of simplicity we assume m = 1. Suppose we can establish the following: x
x
+ \h2D,
(3.8)
where hi > 0, h^ E Rd, hi, \h?\ small enough; Ff denotes the antiderivative of / and ai,a:2 € (0,1). Then the funcion (t, x) —> pt, (y) is ai-H61der continuous in t, a2-H61der continuous in x, for any fixed y. Indeed, let { f n , y , n > 1} be a sequence of smooth functions converging to the Dirac x
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delta distribution 6{yy and such that supn ||-F/n,J|oo < °°- F°r example, a sequence of Gaussian kernels centered at y with variance decreasing to zero as n —» oo. Then, it suffices to apply (3.8) to / := fn
E [ f ( X ( t + h1,x))-f(X(t,x))]=Tl+T2, with
T2 = E[ [ (l Jo where Y t t X ( h i , X ) = XX(t + hi,x) + (I - \}X(t,x). Lemma 2.1 applied to £ = Yt<x(hi,X), Z = (X(t + hi,x) - X(t,x)f yields
T2 < CHF/HoopX* + h!,x) - X(*,z)||I.2. < Chf, 21
for some 71 € (0, 1). The same method yields T\ < Ch^ . However this estimate can be improved taking into account (3.2) and using, in the corresponding stochastic integral term, the duality relation (2.3). We finish this section by mentioning two additional problems on densities of solutions of spde's that can be analized using Malliavin Calculus. The first one is the positivity of the density. An abstract result on the characterization of points of positive density has been proved in [A-K-St]. Applications to hyperbolic equations are given in [Mi-SSl], to the heat equation in [B-P] and recently, to a two-dimensional wave equation in [Ch-SS]. The second problem concerns the effect on the densities produced by small perturbations of the noise in (3.1): logarithmic estimates and Taylor expansions. A brief description goes far beyond the goal of this paper. We only give some references on it: [Mi-SS2], [Ma-SSl], [Ma-SS2], [K-Ma-SS]. 3.3 Nonlinear operators
Consider the spde
Ciu(t, x) = £ (g(u(t, a;))) + a(u(t, x))F(t, x) + b(u(t, z)) , 2
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(3.9)
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t > 0, x € D, where L\ is a second order differential operator defined on (0, oo) x D and £2 is a differential operator defined on D. The noise F and the coefficients a and b are as have been described at the begining of Section 3. We mention two examples of equations like (3.9) : (a) The stochastic Burgers equation, with L\ = -^ — A and £2 = ;gj(b) The Cahn-Hilliard stochastic equation, with L\ = -j^ — A and £2 = A2. The solution to these kind of equations are given by means of a truncation argument (see [G], [G-N] for case (a) in dimension d = 1 and [CW] for case (b) in dimension d = 1,2,3). In a rather natural way the construction of the solution provides the following property: u(t,x] belongs to D^ , at any fixed (t,x). Then, with some extra work, one can prove existence of density using Theorem 2.1 (see [La-N], [CW]). The smoothness of the density for the Burgers equation has been proved under very restrictive assumptions in [Le-N-Pe] c
The applications of the Malliavin Calculus to nonlinear spde's like those considered in this section is a field with promising new developments and interactions with other areas in Mathematics. On the other hand, for spde's like those considered in Sections 3.1, 3.2 the applications of Malliavin Calculus have been restricted to examples where the fundamental solution G is a function. The results of [D], [Pe] and [Ma-Me-Sa] open the door to further extensions.
Acknowledgements. The work has been partially supported by the grant PB 960088 from the Subdireccion General de Formation y Promotion del Conocimiento and the grant ERBF MRX CT960075A of the EU. References [A-K-St] S. Aida, S. Kusuoka and D. Stroock: On the support of Wiener Functionals, Asymptotic problems in probability theory. In: Wiener functionals and asymptotics, Pitman Research L.N. Series 94, pp. 3-34. Longman Sue Tech. New York, 1993 [B-P] V. Bally and E. Pardoux: Malliavin calculus for white noise driven
parabolic spde's. Potential Anal. 9, 27-64 (1998). [Bo-H] N. Bouleau and F. Hirsch: Dirichlet Forms and Analysis on Wiener Space, de Gruyter Studies in Math. 14, Walter de Gruyter, 1991. [C-N] R. Carmona and D. Nualart: Random nonlinear wave equations: smoothness of the solutions. Probab. Theory Related Fields 79, 469-508 (1988).
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[CW]
C. Cardon-Weber: Cahn-Hilliard stochastic equation: Existence of the solution and of its density. Prepublication du Laboratoire de Probabilites et Modeles Aleatoires 543, 1999.
[Ch-SS] M. Chaleyat-Maurel and M. Sanz-Sole: Positivity of the density for the stochastic wave equation in two spatial dimensions. Math. Preprint Series 287, Universitat de Barcelona, 2000. [D] R. Dalang: Extending the martingale measure stochastic integrals to spatially homogeneous spde's. Electronic J. of Probability 4, (1999) [D-F] [D-W]
R. Dalang and N. Frangos: The stochastic wave equation in two spatial dimensions. Ann. Probab. 26, 187-212 (1998). R. Dalang and J.B. Walsh: Time-reversal in hyperbolic s.p.d.e's. Preprint, 2000.
[G] I. Gyongy: On the stochastic Burgers equation. Stochastic Process. Appl. 73, 271-299 (1998).
[G-N] I. Gyongy and D. Nualart: On the stochastic Burgers equation in the real line. Ann. Probab. 27, 782-802 (1999). [I-Wa] N. Ikeda and S. Watanabe: Stochastic Differential Equations and Diffusion Processes, 2nd edition, North-Holand, 1989. [K-Ma-SS] A. Kohatsu-Higa, D. Marquez-Carreras and M. Sanz-Sole: Asymptotic behaviour of the density in a parabolic spde. J. Theoret. Probab., to appear. [La-N] N. Lanjri and D. Nualart: Burgers equation driven by space-time white noise: Absolute continuity of the solution. Stochastics and
Stochastics Reports 66, 273-292 (1999). [Le-N-Pe] J. Leon, D. Nualart and R. Petterson: The stochastic Burgers equation: Finite moments and smoothness of the density. Math. Preprint Series 274, Universitat de Barcelona, 2000, [M] P. Malliavin: Stochastic calculus of variations and hypoelliptic operators. In: Proc. Inter. Symp. on Stock. Diff. Equations, Kyoto 1976, Wiley 1978, pp. 195-263. [Ma-Me-Sa] D. Marquez-Carreras, M. Mellouk and M. Sarra: On stochastic partial differential equations with spatially correlated noise: smoothness of the law. Math. Preprint Series 279, Universitat de Barcelona, 2000. [Ma-SSl] D. Marquez-Carreras and M. Sanz-Sole: Small perturbations in a
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hyperbolic stochastic partial differential equation. Stochastic Process. Appl. 68, 133-154 (1997). [Ma-SS2] D. Marquez-Carreras and M. Sanz-Sole: Taylor expansion of the density in a stochastic heat equation. Collect. Math. 49, 399-415 (1998). [Mi-SSl] A. Millet and M. Sanz-Sole: Points of positive density for the solution to a hyperbolic Spde. Potential Analysis 7, 623-659 (1997).
[Mi-SS2] A. Millet and M. Sanz-Sole: Varadhan estimates for the density of the solution to a parabolic stochastic differential equation. In:
(Eds.), A. Truman, I.M. Davies, K.D. Elworthy Stochastic Analysis and Applications, pp. 330-342, World Scientific, Singapore, 1996. [M1-SS3] A. Millet and M. Sanz-Sole: A stochastic wave equation in two space dimension: smoothness of the law. Ann. Probab. 27, 803844 (1999). [Mi-Mo] A. Millet and P.-L. Morien: On a stochastic wave equation in two space dimensions: regularity of the solution and its density.
Stochastic Process. Appl. 86, 141-162 (2000). [Mol]
P.-L. Morien: The Holder and Besov regularity of the density for the solution of a parabolic stochastic partial differential equation. Bernoulli 5, 275-298 (1999).
[Mo2]
P.-L. Morien: On the density for the solution of a Burgers-type
SPDE. Ann. Inst. Henri Poincare 35, 459-482 (1999).
[Nl]
D. Nualart: Malliavin Calculus and Related Topics. Springer, New York, 1995.
[N2]
D. Nualart: Analysis on the Wiener space and anticipating calculus. In: Ecole d'ete de Probabilites de Saint Flour XXV. Lecture Notes in Math. 1690, pp. 863-901, Springer, New York, 1998.
[No]
J. Norris: Twisted sheets. J. Fund. Anal. 132, 273-334 (1995).
[P- T] E. Pardoux and Z. Tusheng: Absolute continuity of the law of the
solution of a parabolic spde. J. Fund. Anal. 112, 447-458 (1993). [Pe]
S. Peszat: The Cauchy problem for a nonlinear stochastic wave equation in any dimension. Preprint, 2000.
[Pe-Zl] S. Peszat and J. Zabczyk: Stochastic evolution equations with a spatially homogeneous Wiener process, Stochastic Process. Appl. 72, 187-204 (1997).
[Pe-Z2] S. Peszat and J. Zabczyk: Nonlinear stochastic wave and heat equa-
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tions. Probab. Theory Related Fields 116, 421-443 (2000). [R-SS1] C. Rovira and M. Sanz-Sole: The law of a solution to a nonlinear hyperbolic spde. J. Theoret. Probab. 9, 863-901 (1996). [R-SS2] C. Rovira and M. Sanz-Sole: Stochastic Volterra equations in the plane: Smoothness of the law. Stoch. Analysis and Appl., to appear [St]
D. Stroock: Some applications of stochastic calculus to partial differential equations. In: Ecole d'ete de Probabilites de Saint Flour. Lecture Notes in Math. 976, pp. 267-382, Springer, New York, 1983.
[T] S. Tindel: Quasilinear stochastic elliptic equations with reflection:
the existence of density. Bernoulli 7, 445-459 (1998). [W] J.B. Walsh: An introduction to stochastic partial differential equa-
tions. In: Ecole d'ete de Probabilites de Saint Flour XIV. Notes in Math. 1180, pp. 266-437, Springer, Berlin, 1986. [Wa]
Lecture
S. Watanabe: Lectures on stochastic differential equation and
Malliavin Calculus. Tata Inst. Fund. Res. Springer, Berlin, 1984.
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Stochastic Curvature Driven Flows
NUNG KWAN YIP Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA, [email protected]
This article surveys some results concerning motion by mean curvature (MMC) for curves and surfaces. Emphasis will be put on methods of incorporating stochastic noise. In addition, we will demonstrate the idea of using noise to provide a selection principle for non-unique solutions. We pursue the study of MMC for various reasons. The equations of MMC and related evolutions involve interesting and difficult nonlinear partial differential equations. Classical solutions to these equations do not exist globally in time due to the formation of singularities and topological changes of the surfaces. Various new mathematical devices are introduced to tackle these difficulties. Furthermore, MMC (and its anisotropic version) has wide range of applications in describing interfacial motions in materials science. Many solidification processes such as dendritic crystal growth are modeled by coupling the interfacial curvature flows with the diffusion of heat and other chemicals. These lead to various types of free boundary value problems.
Due to the physical origins of these evolutions, it appears that the consideration of noise is an important area of investigation. The noise can come from thermal fluctuations, impurities or the intrinsic instabilities of the deterministic evolutions. The mathematical formulations then involve stochastic partial differential equations. This subject is relatively new compared with its deterministic counterpart. One of the main difficulties in the study of these stochastic equations is how to combine the nonlinear effects of the geometric motions together with the stochastic perturbations. In the following we will first introduce the definition of MMC and its methods of solutions. Then several results for the stochastic motions are described. Next we explain the selection principle brought out by noise when the deterministic equations have non-unique solutions. This gives an example that stochasticity can indeed change the behavior of the evolution. 443
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Some open problems will be discussed at the end.
1
Motion by Mean Curvature
In this section, we briefly describe MMC and its methods of solution. This motion has already been considered in some early works of materials science [49, 1]. We refer to [55, 56] for more detailed discussions.
A hyper-surface M(t] of dimension n — 1 in Rn is said to evolve by its mean curvature for t > 0 if the inward normal velocity Vn at every point of the surface equals the mean curvature K: Vn = «.
(1)
In the classical setting, the mean curvature is defined as the sum of the principle curvatures. It can also be viewed as the first variation of the area functional (see (4)). The sign convention for K, is that a sphere will shrink. If the surface is parametrized by ~F(p, t) e R™, the motion of M(t) can be described as: where n is the unit inward normal. In this article, we will only consider the case when M(t) is the boundary of some n-dimensional domain D(t), i.e. M(t) = dD(t}. Then M(t) is a curve or surface if n = 2 or 3 respectively. Heuristically if M(t) is a curve, it will be smoothed by the evolution and become rounder and smaller. However, for general surfaces, singularities can develop. For a circle or sphere, the motion can be completely described by the evolution of the radius R(t) which satisfies
dR(t) dt
1 R(t)
The solution is given by R(t) = i/.R(0)2 — It. This shows that the circle(sphere) will disappear in finite time. More generally, the motion of a graph M(t) = {(x, f ( x , t } ) : x € JRn-1} is described by the following quasilinear parabolic equation:
The definition of MMC can be extended to some anisotropic version. This generalization has wide range of applications in modeling materials
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interfacial motions. For this purpose, we define the ^-surface energy £$ of a surface S to be:
e*(S)= I 3>(v}da (3) Js where $ : {y 6 .Rn : \v\ — 1} — > R+ is called the surface energy integrand] v = — n is the unit outward normal; a is the surface measure of S. We
further define the $-mean curvature K$ to be the variational derivative of £$. It has the property that for all smooth vector field g : Rn — > Rn, = s=0
K {g, v) dcr
(4)
JS
where for small s, Ss is the image of S under the diffeomorphism Gs(x) = x + sg(x). In this sense, K$ can be interpreted as the rate of change of £$ per volume swept out by the deformation Gs. We now define the motion by <&- (weighted) mean curvature as
Vn = ft*
(5)
Note that the original definition of MMC (1) is the special case of (5) when $ is isotropic, i.e. $(z/) = 1 for all v. In order for equation (5) to be wellv posed, it is usually required that the function 3?(v) = |v|
convex on R*1. However, this condition can be greatly relaxed (see [56]). One special feature of MMC is that the surface energy is monotonically decreasing in time:
= - I 4dtr. JMt
In fact, the motion can be viewed as the negative gradient flow for the surface energy with respect to some suitably chosen inner product on the space of velocity vector fields defined on M(t) (see (8)).
1.1
Methods of Solution
One of the main difficulties in the investigation of MMC is that singularities and topological changes can happen. Existence of local in time classical solutions are proved in [25]. See also [6, 20]. Global in time solutions are proved to exist for embedded curves in the plane [25, 28]; convex surfaces [29] and surfaces given by a graph [16]. Long time behaviors of these solutions are also investigated in these papers. In [30], the self-similarity structures of the singularities are studied. We refer to [2] for a survey on this type of questions. On the other hand, various mathematical techniques have been introduced
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to construct solutions which can handle the singularity formations and topological changes. Here we will mention some of these approaches which are relevant to this article. The readers are referred to [55] for other methods of solution. The papers [50, 51] also contain comprehensive references. The first global in time solutions were constructed by Brakke [10]. He considered the space of surfaces in the setting of varifolds. His construction also works in the case of higher co-dimensional flows such as those of curves in space. Unfortunately, the technique so far is only applicable in the isotropic version of the mean curvature flows (1). In addition, there is a high degree of non-uniqueness in the solutions constructed by his method. Nevertheless, his notion of solutions is very often regarded as the "definition" of MMC. Another approach is using the level set formulation. This method constructs a time dependent function u(x, t) such that its zero-level set (or any c-level set) M(t) defined as
M(t) = {x € .ft" : u(x, t) = 0} or M(t) = {x e Rn : u(x, t) = c} evolves by MMC. This function would then be a solution to the following equation: du
The above is a degenerate parabolic equation. The solution is interpreted in the viscosity sense. In this setting, solution for exists globally in time and is unique [19, 12]. However, the level sets of u can fatten, i.e. M(t) = {x E Rn : u(x, t) = 0 or c} can have positive Lebesgue n-dimensional measure. When this happens, the location of the actual sharp interface cannot be determined easily. (This issue will be the main topic of Section 3.) Reaction diffusion equations and singular perturbation techniques are also used to study MMC. A function called the phase- ordering parameter is introduced. It evolves according to the following Allen-Cahn equation [1]:
|e = A¥,-^>)
(7)
The function F is the double-well potential: F(
domains {x e Rn : (x, t) w —1}. As e -4 0, it is proved in [13, 15] that the zero-level set of if> evolves according to MMC when the curvature flow has smooth solution. This result has been extended to the setting of viscosity and varifold solutions in [18, 50, 51] and [31]. The paper [8] describe a general technique to tackle the problem of front propagations using viscosity and singular perturbation methods.
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In [6, 48], variational minimization and time discretization procedures are used to construct solutions for MMC. This method can also handle a model for dendritic crystal growth [5, 47]. The main idea is to write the evolu-
tion {X(t),t > 0} as a negative gradient flow with respect to some energy functional £:
±X(t) = -VS(X(t))t
t>0
(8)
where X(t) is a generic symbol for an element of some state space X. The solution for (8) is constructed by time stepping: given X(iAt), X((i +1) At) is chosen to be a minimizer of the following functional:
?(X)=e(X)
+ ±\\X-X(i*t)\\a,
X<=X
(9)
where ||-|| is the norm for X which is compatible with the gradient V in (8). Energy estimates and compactness for the function space are then proved in order to extract a convergent subsequence as Ai —> 0. The mathematical framework to handle the above minimization problem is geometric measure theory [23]. In Sections 2.1 and 2.2, this approach will be combined with stochastic perturbations to construct random evolutions of subsets of Rn.
2
Stochastic Curvature Driven Flows
In this section, we mention some mathematical results which can incorporate stochastic noise into curvature driven flows. This type of flows includes the previous MMC ((1) or (5)) and also some mathematical models for dendritic crystal growth. As mentioned earlier, due to the underlying physical motivations of our equations, it is natural to consider noise which can come from the background fluctuations or the intrinsic instabilities of the deterministic evolutions. One excellent example is the dendritic pattern formation in crystal growth. There exists many models to explain the side-branching events, the shape of the tips and their speed of propagation. Noise is found to be an important factor in the process. However, its consequences have not been completely understood yet [34, 44]. Noise can also come from some microscopic models for these growth processes. It would be interesting to investigate in what form the noise survives (if at all) by going to a larger spatial scale descriptions.
A natural mathematical framework of studying stochastic evolution is by means of stochastic partial differential equations (SPDEs). There are several existence results for linear and semi-linear equations [58, 24, 14, 41]. The regularity of the solutions depends on the competition between the smoothing and roughening effects of the deterministic equations and the
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stochastic driving forces. However, for highly nonlinear equations such as those used in MMC and other geometric motions, not much is known.
A simple stochastic version of MMC (1) or (5) is given by
Vn = K + W(x,t).
(10)
Here W(x, t) is a white noise which is required to be sufficiently correlated in the spatial variables in order to give solutions in reasonable function spaces. A simpler version of (10) is the following evolution equation for a graph:
where o refers to the Stratonovich interpretation of stochastic calculus. In [37, 38], equations similar to the above are derived from a stochastic version of the Allen-Cahn equation (7). Equation (11) can be solved using the techniques developed by Lions and Souganidis [45, 46] (see Section 2.3). However, even when W(x, t) is spatially constant, it seems not clear that the solution will remain smooth for all time.
There are many other phenomenological equations used to describe similar growth processes. These include the KPZ equation [33] and some higher order equations to model surface diffusion:
where the surface is described by a height function /; A\ and A% are positive numbers; TH and 772are some stochastic driving forces. The work [57] formally derives these equations from microscopic models. Questions of particular interests include the dynamical scaling behaviors of the interfacial structures. We refer to [7, 40] for more detail discussions and references. Even though stochastic effects are known to exist for a long time, there are still many outstanding open problems concerning the existence and statistical properties of the solutions. In the following, we will describe some results that can give global in time solutions for some stochastic versions of MMC.
2.1
Stochastic Motion by Mean Curvature
In [59], the author constructed a random evolution of a subset K(t) of Rn such that its boundary dK(t) evolves continuously in time formally according
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to the following stochastic version of (5):
Vn(p) = «*(p) + /W(p, t), n\ , p e <9# (t)
(12)
where {W(-, t) : Rn —>• .Rn, t > 0} is a white noise vector field (to be defined later).
We consider the space of subsets of FS1 in the setting of sets of finite perimeter. S=\KCRH: I div(G)dnx < C f sup||G(aj)||l I JK x )
(13)
where G : Rn —>• Rn is any (71 vector field. The space S is endowed with the ^-metric: d(K^K^) = Ln(K}_&K 0. At the beginning of each interval, we choose K(iAt+) to be a minimizer of the following functional:
dist(z,dtf(*Ar))d£ n ,
KzS (14)
where £$ is the ^-surface energy (3) and dist(x, A) = inf{||a; — p|| :p € ^4} is the distance function from x to A C fC1. This change from K(i/\t~) to -K"(z'At+) can be shown to be an approximation to the deterministic motion law Vn = K$. During (z'At, (i + l)At), we use a white noise vector field W(a?,t) to deform the set K(i&t+). This vector field is smooth in space but white in time. It can be constructed by using the following representation:
(15) where the Fj(#)'s and Wi(t)'s are smooth vector fields and independent Wiener processes. The summation can be over a finite or countably infinite collections. From this W(x, t), we can define a two-parameters family of diffeomorphisms of Rn: pStt(x) which are the solutions to the following stochastic differential equation:
/** • W(<pa>r(x), r) dr,
Js
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x € Rn, t > s > 0.
(16)
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The machinery for this construction can be found in [42]. Using this <^ s ,t(')> we define:
K(t) = viAt,t(K(i/\t+)),
t € (zAt, (t + 1) At).
(17)
The two procedures (14) and (17) are then alternated to produce a
stochastic evolution of a subset .PfAt(t) of FC1 for t > 0. It is shown in [59] that the probability measures PA* induced by {KAt(t) € S : t > 0} on some appropriate function space has a weak limit (upon taking a subsequence) as At -4- 0. Furthermore, any limiting measure concentrates on the space C(R+,S) of K(t) which evolves continuously in time (t > 0).
2.2
Stochastic Dendritic Crystal Growth
The previous approach of combining variational minimization and stochastic perturbation has also been used in a similar fashion to prove the existence of a solution to a model of stochastic dendritic crystal growth. The evolution describes the motion of a crystal interface dK(t) coupled with an underlying stochastic heat equation. This is an example of a free boundary value problem as the location of the interface is one of the unknown variables. For many solidification processes, both the curvature of the crystal interface and the diffusion of heat or chemicals are important factors in determin-
ing the overall patterns of the crystal shape. We refer to [43] for a description of various physical systems and the associated mathematical models. In [60] , the author constructed stochastic processes {K(t) e S,T(t) € L?(D) : t > 0} which describe the evolution of a crystal shape and temperature field in a domain T>. These processes satisfy the
following statements: • Stochastic Heat Equation:
ck(c )T(x, t)) = div(E VT(x, t)) dt + c f(T(x, t))dW(x, t) K(t
K(t)
K(t)
t
(18) where c-K(f) d ^K(t) the specific heat capacity and conductivity matrix respectively. Both of them can take on different values in K(t) (the solid phase) and T>\K(t) (the liquid phase). W(x, t) is a spatially correlated Wiener process in L2(D). / is some function which controls the noise in regions of extreme temperature values. an
are
Gibbs-Thomson Relation:
= H(T(p,t)),
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PedK(t)
(19)
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which says that the curvature at any point of the crystal interface balances with a function H of the temperature value T. For practical purposes, H(T) is approximately equal to — T.
Furthermore, the heat distribution denned as Q = c#T evolves continuously in time in the Monge-Kantorovich norm which is a natural notion of distance for many heat transfer problems.
The Gibbs- Thomson relation (19) describes the equilibrium situation for the following more general kinetic undercooling condition:
Vn(p) = v(K*(p) - H(T(p,t))),
P€OK(t)
(20)
when [i (which is called the mobility function) tends to infinity. Not many mathematical results are known for the above motion law. In the classical setting, [11] proves the convergence of phase-field models (which are similar to the Allen-Cahn equation (7)) to various sharp interfacial limits related to (20) . The work [52] has constructed a weak solution.
2.3
Stochastic Viscosity Solution
Recently, Lions and Souganidis in [45, 46] have extended the viscosity method to solve the following fully nonlinear SPDE:
= F(D2u, Du, x, t)dt + T Hi(Du, x, t) o dWi(x, t)
(21)
where F is degenerate elliptic; H^s are functions satisfying some smoothness assumption; Wi(x,t)'s are independent Wiener processes. The above equation can be used to formulate a wide range of front propagation problems in random environments. In particular, it can handle the following stochastic version of the level set equation (6):
dtu = | V«|div
i—r
\\vu\J
+ a(a:)| V«| o dtW(t)
(22)
where a(x) is some space dependent function.
One of the main results in [45, 46] states that under any reasonable smoothing of the white noise term in (21), the corresponding approximated solutions will converge to the same limit. A notion of weak solution for (21) and some uniqueness results are also given. The methods outlined in the previous sections give the first type of results which can incorporate stochastic noise into these geometric motions. However, many questions concerning the properties of the stochastic evolutions remain open. In the next section, we will tackle the issue of non-uniqueness of the deterministic solutions.
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3
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Noise and Uniqueness of Motion by Mean Curvature
In this section, we demonstrate the possibility of using stochastic noise to provide a selection principle when the deterministic equations have nonunique solutions. We illustrate this idea in the case of MMC. We first give some examples of non-uniqueness and then motivate the need to have a selection method.
Two Touching Circles. Given the pah- of circles as in Figure 1, they can evolve according to different solutions such as (A) or (B) if the intersection point I is broken vertically or horizontally. If I is kept fixed, the solution (C) can be produced. All three evolutions satisfy the definition of MMC (1) in the varifold and viscosity formulations.
Figure 1. We remark that both (.A) and (B) are stable solutions while (C) is unstable.
Two Separating Circles [9]. Consider a pair of circles which are initially apart from each other (Figure 2). They evolve according to the following MMC with a driving force:
Vn = K- g(t}.
(23)
The g(t) is a time dependent function which decreases from positive to negative values. Its exact form is chosen in such a way that the two circles will enlarge at the beginning and then touch at some time t*. It is the time when g(t) becomes zero and then negative. At this moment, the two circles will try to separate again. Multiple solutions such as (D) or (E) can now develop. Presumably it is also possible to have an unstable solution similar to (C) in Figure 1.
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The two circles touch at some time.
... Two initial circles.. Figure 2. The works [3, 39, 4] also contain some non-uniqueness examples in higher dimensions.
The above non-uniqueness phenomenon corresponds to the fact that the solutions of MMC do not depend continuously on the initial data (in the L1 topology). If we approximate the initial data from the inside and outside using smooth curves, the approximating solutions will not stay close to each other. In the language of maximum principle, it means that there is a gap between the sub- and super-solutions. In the level set formulation, if the curves are the zero-level set of an initial data for u(x, t) which solves (6), then M(t) = {x € R2 : u(x,t) = 0} will instantaneously have positive Lebesgue two-dimensional measure for t > 0. It can be shown that any solution to MMC (for example, the varifold solutions in [10]) is trapped in this fattened region. However, its exact location cannot be easily determined. This non-uniqueness issue hinders many of the mathematical statements concerning MMC. Brakke in [10] proved that his varifold solution almost everywhere is a smooth manifold. But he needs to assume that his solutions satisfy a unit density hypothesis which is not known to be true by his construction. This appears to be related to the non-uniqueness phenomenon. The works [21, 22, 32] explain this in more detail. In addition, even though the convergence of the Allen-Cahn equation (7) to MMC has been established in the viscosity and varifold sense ([18, 50, 51] and [31]), again it is not clear which sharp interfacial limit it converges to when fattening occurs. From the gradient flow point of view of MMC (8), it seems that non-uniqueness happens exactly at the critical points of the energy functional (3). It would be interesting to investigate the behavior of the solutions near these critical points, especially in such a geometric setting. Furthermore, since we have in mind the application of MMC to some physical processes such as solidifications, non-uniqueness could mean that the model is incomplete. Hence the formulation of a selection principle would have both mathematical and physical interests. We now give two examples illustrating that the addition of stochastic noise can provide a selection principle. The results are joint work with
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Souganidis [53].
Consider the following stochastic version of MMC (1) and (23):
Vn = K + eW(t) and Vn = K-g(t) + eW(t]
(24)
where W(t) is a Wiener process and e is a small parameter. We apply these motion laws to the curves in Figures 1 and 2. They lead to some random evolutions of curves {Ce(t) : t > 0} for each e > 0. When we let e —>• 0, the following selection principles are obtained. Almost Sure Uniqueness. Starting from the initial curve in Figure 1, almost surely (with respect to the Wiener measure), the curve C€(t) converges to the solution (A) of the deterministic MMC (1).
Uniqueness in Probability Law. Starting from the initial curve in Figure 2, the curve Ce(t) converges with probability 0.5 to solution (D) and probability 0.5 to solution (E). The first result seems to be the strongest possible in terms of the convergence of stochastic processes. The second statement in effect has constructed a unique probability measure on the space of solutions. We feel that this latter formulation has wider applications.
4
Open Problems
In this section, we will discuss some open problems.
The results in the previous section on the selection principle highly rely on the specific geometries involved, especially the tangency condition of the two circles when they meet or come together. How can this be generalized to other types of fattening? In addition, as already seen in Section 1.1, there are several methods of constructing solutions for MMC. These solutions satisfy the following hierarchy:
Smooth \ / Limits of \ / Brakke \ / Levet-set flows ) ~\ Allen-Cahn (7) ) ~ \ flows [10] ) ~\ flows (6) The flows constructed by the variational method [6, 48] seem to lie in between the smooth flows and the Allen-Cahn limits as they deal with sets of finite perimeter. It would be interesting to understand this hierarchy more quantitatively. For example, can any Brakke flow be a limit of the Allen-Cahn equation? Can the space of all Brakke flows fill the whole fattening region in the level set formulation? The answers to these questions can further
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our understanding of the differences between the methods of constructing solutions for MMC.
Another question is to understand the curvature flows from a more microscopic point of view. This falls into the regime of bridging the descriptions from different length scales. Such question appears frequently in physics and in particular materials science. MMC is proved to be the hydrodynamic limit of interactive particle systems under Glauber-Kawasaki dynamics [35] and Glauber dynamics with Kac potential [36]. Usually these approaches produce deterministic evolution equations. Stochastic equations come from by considering the fluctuations. However, there is still not yet a derivation of a fully nonlinear stochastic MMC. We refer to [54, 26] for more discussions and references. In addition, what role do these microscopic descriptions play in the selection of non-unique solutions? Furthermore, the asymptotic limits of the following stochastic Allen-Cahn equation is not completely understood yet:
^ = A^ - 1 (*» + eiW(x, 0)
(25)
where 7 is some postive number. It seems that this can be handled by the method developed in [45, 46]. However, the regularity and statistical properties of the surfaces are not known. So far, all the theories require some spatial correlation for W(x, t). What would happen if it is replaced by spacetime white noise (or some regularized version W^fa, t) where the parameter r\ can depend on e)? Such noise is considered in many works (see for example [37, 38]). The main difficulty in tackling this problem is that the deterministic planar standing or traveling front used in approximating curve fronts can be destroyed by the roughness of the noise. How to define the structures and locations of the stochastic fronts can be challenging. The answers to these questions can help us relate the randomness in the interfacial motions to the noise in the background environment.
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