The geometry of filtering

Frontiers in Mathematics

Advisory Editorial Board Leonid Bunimovich (Georgia Institute of Technology, Atlanta) Benoît Perthame (Université Pierre et Marie Curie, Paris) Laurent Saloff-Coste (Cornell University, Ithaca) Igor Shparlinski (Macquarie University, New South Wales) Wolfgang Sprössig (TU Bergakademie Freiberg) Cédric Villani (Ecole Normale Supérieure, Lyon)

K. David Elworthy Yves Le Jan Xue-Mei Li

The

Geometry of

Filtering

K. David Elworthy Institute of Mathematics University of Warwick Gibbet Hill Road CV4 7AL Coventry United Kingdom [email protected]

Yves Le Jan Laboratoire de Mathématiques Université Paris-Sud XI CNRS Orsay Cedex Bâtiment 425 France [email protected]

Xue-Mei Li Institute of Mathematics University of Warwick Gibbet Hill Road CV4 7AL Coventry United Kingdom

Mathematics Subject Classification: 58J65, 60H 07 ISBN 978-3-0346-0175-7 DOI 10.1007/978-3-0346-0176-4

e-ISBN 978-3-0346-0176-4

© Springer Basel AG 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. Cover design: deblik, Berlin Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media www.birkhauser-science.com

Contents Introduction 1

2

3

4

Diffusion Operators 1.1 Representations of Diffusion Operators . . 1.2 The Associated First-Order Operator . . . 1.3 Diffusion Operators Along a Distribution 1.4 Lifts of Diffusion Operators . . . . . . . . 1.5 Notes . . . . . . . . . . . . . . . . . . . .

vii

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1 1 4 5 7 10

Decomposition of Diffusion Operators 2.1 The Horizontal Lift Map . . . . . . . . . . . . . . . . . . . . . 2.2 Lifts of Cohesive Operators and The Decomposition Theorem 2.3 The Lift Map for SDEs and Decomposition of Noise . . . . . 2.3.1 Decomposition of Stratonovich SDE’s . . . . . . . . . 2.3.2 Decomposition of the noise and Itˆo SDE’s . . . . . . . 2.4 Diffusion Operators with Projectible Symbols . . . . . . . . . 2.5 Horizontal lifts of paths and completeness of semi-connections 2.6 Topological Implications . . . . . . . . . . . . . . . . . . . . . 2.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 11 17 23 24 25 26 28 30 31

Equivariant Diffusions on Principal Bundles 3.1 Invariant Semi-connections on Principal Bundles . . . . 3.2 Decompositions of Equivariant Operators . . . . . . . . 3.3 Derivative Flows and Adjoint Connections . . . . . . . . 3.4 Associated Vector Bundles and Generalised Weitzenb¨ock 3.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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33 34 36 41 46 58

Projectible Diffusion Processes and Markovian Filtering 4.1 Integration of predictable processes . . . . . . . . . 4.2 Horizontality and filtrations . . . . . . . . . . . . . 4.3 Intertwined diffusion processes . . . . . . . . . . . 4.4 A family of Markovian kernels . . . . . . . . . . . .

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61 62 66 66 70

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vi

Contents 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12

5

The filtering equation . . . . . . . . . . . . . . . . . . . . . . . . Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . Krylov-Veretennikov Expansion . . . . . . . . . . . . . . . . . . . Conditional Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . An SPDE example . . . . . . . . . . . . . . . . . . . . . . . . . . Equivariant case: skew-product decomposition . . . . . . . . . . . Conditional expectations of induced processes on vector bundles Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Filtering with non-Markovian Observations 5.1 Signals with Projectible Symbol . . . . 5.2 Innovations and innovations processes 5.3 Classical Filtering . . . . . . . . . . . 5.4 Example: Another SPDE . . . . . . . 5.5 Notes . . . . . . . . . . . . . . . . . .

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71 73 74 75 79 81 83 85

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87 88 91 94 95 99

6

The Commutation Property 101 6.1 Commutativity of Diffusion Semigroups . . . . . . . . . . . . . . . 103 6.2 Consequences for the Horizontal Flow . . . . . . . . . . . . . . . . 105

7

Example: Riemannian Submersions and Symmetric 7.1 Riemannian Submersions . . . . . . . . . . . 7.2 Riemannian Symmetric Spaces . . . . . . . . 7.3 Notes . . . . . . . . . . . . . . . . . . . . . .

8

Example: Stochastic Flows 121 8.1 Semi-connections on the Bundle of Diffeomorphisms . . . . . . . . 121 8.2 Semi-connections Induced by Stochastic Flows . . . . . . . . . . . 125 8.3 Semi-connections on Natural Bundles . . . . . . . . . . . . . . . . . 131

9

Appendices 9.1 Girsanov-Maruyama-Cameron-Martin Theorem . . . . . 9.2 Stochastic differential equations for degenerate diffusions 9.3 Semi-martingales and Γ-martingales along a Subbundle 9.4 Second fundamental forms and shape operators . . . . . 9.5 Intertwined stochastic flows . . . . . . . . . . . . . . . .

Spaces 115 . . . . . . . . . . . . 115 . . . . . . . . . . . . 116 . . . . . . . . . . . . 119

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135 135 139 145 147 148

Bibliography

159

Index

167

Introduction Filtering is the science of finding the law of a process given a partial observation of it. The main objects we study here are diffusion processes. These are naturally associated with second-order linear differential operators which are semi-elliptic and so introduce a possibly degenerate Riemannian structure on the state space. In fact, much of what we discuss is simply about two such operators intertwined by a smooth map, the “projection from the state space to the observations space”, and does not involve any stochastic analysis. From the point of view of stochastic processes, our purpose is to present and to study the underlying geometric structure which allows us to perform the filtering in a Markovian framework with the resulting conditional law being that of a Markov process which is time inhomogeneous in general. This geometry is determined by the symbol of the operator on the state space which projects to a symbol on the observation space. The projectible symbol induces a (possibly non-linear and partially defined) connection which lifts the observation process to the state space and gives a decomposition of the operator on the state space and of the noise. As is standard we can recover the classical filtering theory in which the observations are not usually Markovian by application of the GirsanovMaruyama-Cameron-Martin Theorem. This structure we have is examined in relation to a number of geometrical topics. In one direction this leads to a generalisation of Hermann’s theorem on the fibre bundle structure of certain Riemannian submersions. In another it gives a novel description of generalised Weitzenb¨ock curvature. It also applies to infinite dimensional state spaces such as arise naturally for stochastic flows of diffeomorphisms defined by stochastic differential equations, and for certain stochastic partial differential equations. A feature of our approach is that in general we use canonical processes as solutions of martingale problems to describe our processes, rather than stochastic differential equations and semi-martingale calculus, unless we are explicitly dealing with the latter. This leads to some new constructions, for example of integrals along the paths of our diffusions in Section 4.1, which are valid more generally than in the very regular cases we discuss here.

viii

Introduction

Those whose interest is mainly in filtering rather than in the geometry should look at Chapter 1, most of Chapter 2, especially Section 2.3, but omitting Section 2.6. Then move to Chapter 4 where Section 4.11 can be ignored. They could then finish with Chapter 5, though some of the Appendices may be of interest. A central role is played by certain generalised connections determined by the principal symbols of the operators involved. To describe this in more detail let M be a smooth manifold. Consider a smooth second-order semi-elliptic differential operator L such that L1 ≡ 0. In a local chart, such an operator takes the form L=

n X ∂ 1 X ij ∂ ∂ a + bi i 2 i,j=1 ∂xi ∂xj ∂x

(1)

where the aij ’s and bi ’s are smooth functions and the matrix (aij ) is positive semi-definite. Such differential operators are called diffusion operators. An elliptic diffusion operator induces a Riemannian metric on M . In the degenerate case we shall have to assume that the “symbol” of L (essentially the matrix [aij ] in the representation (1)) has constant rank and so determines a subbundle E of the tangent bundle T M together with a Riemannian metric on E. In Elworthy-LeJan-Li [35] and [36] it was shown that a diffusion operator in H¨ ormander form, satisfying this condition, induces a linear connection on E which is adapted to the Riemannian metric induced on E, but not necessarily torsion free. It was also shown that all metric connections on E can be constructed by some choice of H¨ormander form for a given L in this way. The use of such connections has turned out to be instrumental in the decomposition of noise and calculation of covariant derivatives of the derivative flows. A related construction of connections can arise with principal fibre bundles P . An equivariant differential operator on P induces naturally a diffusion operator on the base manifold. Conversely, given an equivariant or “principal” connection on P , one can lift horizontally a diffusion operator on the base manifold of the form of sum of squares of vector fields by simply lifting up the vector fields. It still needs to be shown that the lift is independent of choices of its H¨ormander form. Consider now a diffusion operator not given in H¨ormander form. Since it has no zero-order term we can associate with it an operator δ which sends differential one-forms to functions. In Proposition 1.2.1, members of a class of such operators are described, each of which determines a diffusion operator. Horizontal lifts of diffusion operators can then be defined in terms of the δ operator. This construction extends to situations where there is no equivariance and we have only partially defined and non-linear connections. We show that given a smooth p : N → M : a diffusion operator B on N which lies over a diffusion operator A on M satisfying a “cohesiveness” property gives rise to a semi-connection, a partially defined, non-linear, connection which can be characterised by the property that, with respect to it, B can be written as the direct sum of the horizontal lift of its

Introduction

ix

induced operator and a vertical diffusion operator. Of particular importance are examples where p : N → M is a principal bundle. In that case the vertical component of B induces differential operators on spaces of sections of associated vector bundles: we observe that these are zero-order operators, and can have geometric significance. This geometric significance, and the relationship between these partially defined connections and the metric connections determined by the H¨ormander form as in [35] and [36], is seen when taking B to be the generator of the diffusion given on the frame bundle GLM of M by the action of the derivative flow of a stochastic differential equation on M . The semi- connection determined by B is then equivariant and is the adjoint of the metric connection induced by the SDE in a sense extending that of Driver [25] and described in [36]. The zero-order operators induced by the vertical component of B acting on differential forms turn out to be generalised Weitzenb¨ ock curvature operators, in the sense of [36], reducing to the classical ones when M is Riemannian for particular choices of stochastic differential equations for Brownian motion on M . Our filtering then reproduces the conditioning results for derivatives of stochastic flows in [38] and [36]. Our approach is also applied to the case where M is compact and N is its diffeomorphism group, Diff(M ) , with P evaluation at a chosen point of M . The operator B is taken to be the generator of the diffusion process on Diff(M ) arising from a stochastic flow. However our constructions can be made in terms of the reproducing Hilbert space of vector fields on M defined by the flow. From this we see that stochastic flows are essentially determined by a class of semiconnections on the bundle p : Diff(M ) → M and smooth stochastic flows whose one-point motions have a cohesive generator determine semi- connections on all natural bundles over M . Apart from these geometrical aspects of stochastic flows we also obtain a skew-product decomposition which, for example, can be used to find conditional expectations of functionals of such flows given knowledge of the one-point motion from our chosen point in M . The plan of the book is as follows: In Chapter 1 we describe various representations of diffusion operators and when they are available. We also define the notion of such an operator being along a distribution. In Chapter 2 we introduce the notion of semi-connection which is fundamental for what follows, and we show how these are induced by certain intertwined pairs of diffusion operators and how they relate to a canonical decomposition of such operators. We also have a first look at the topological consequences on p : N → M of having B on N over some A on M which possesses hypo-ellipticity type properties. This is a minor extension of part of Hermann’s theorem, [51], for Riemannian submersions. In Chapter 3 we specialise to the case of principal bundles, introduce the example of derivative flow, and show how the generalised Wietzenbock curvatures arise. It is not really until Chapter 4 that stochastic analysis plays a major role. Here we describe methods of conditioning functionals of the B-process given information about its projection onto M . We also use our decomposition of B and resulting decomposition of the B-process to describe the conditional B-process.

x

Introduction

In the equivariant case of principal bundles the decomposition of the process can be considered as a skew-product decomposition. In Chapter 5 we show how our constructions can apply to classical filtering problems, where the projection of the B-process is non-Markovian by an appropriate change of probability measure. We can follow the classical approach, illustrated for example in the lecture notes of Pardoux [85], and obtain, in Theorem 5.9, a version of Kushner’s formula for non-linear filtering in somewhat greater generality than is standard. This requires some discussion of analogues of innovations processes in our setting. We return to more geometrical analysis in Chapter 6, giving further extensions of Hermann’s theorem and analysing the consequences of the horizontal lift of A commuting with B, thereby extending the discussion in [9]. In particular we see that such commutativity, plus hypo-ellipticity conditions on A, gives a bundle structure and a diffusion operator on the fibre which is preserved by the trivialisations of the bundle structure. This leads to an extension of the “skew-product” decomposition given in [33] for Brownian motions on the total space of Riemannian submersions with totally geodesic fibres. In fact the well-known theory for Riemann submersions, and the special case arising from Riemannian symmetric spaces is presented in Chapter 7. Chapter 8 is where we describe the theory for the diffeomorphism bundle p : Diff(M ) → M with a stochastic flow of diffeomorphism on M . Initially this is done independently of stochastic analysis and in terms of reproducing kernel Hilbert spaces of vector fields on M . The correspondence between such Hilbert spaces and stochastic flows is then used to get results for flows and in particular skew-product decompositions of them. In the Appendices we present the Girsanov Theorem in a way which does not rely on having to use conditions such as Novikov’s criteria for it to remain valid. This has been known for a long time, but does not appear to be as well known as it deserves. We also look at conditions for degenerate, but smooth, diffusion operators to have smooth H¨ ormander forms, and so to have stochastic differential equation representations for their associated processes. We also discuss semi-martingales and Γ-martingales along a subbundle of the tangent bundle with a connection. One section of the Appendix is a very brief exposition of the differential geometry of submanifolds, defining second fundamental forms and shape operators. This is used in the final section which analyses the situation of intertwined stochastic flows or essentially equivalently of diffusion operators which are not only intertwined but also have H¨ormander forms composed of intertwined vector fields. It is shown that the H¨ormander forms determine a decomposition of the operator B, which is not generally the same as the canonical decomposition described in Chapter 2. Here it is not necessary to make the constant rank condition on the symbol of A which plays an important role in Chapter 2. At the end of this section we show that having intertwined Brownian flows which both induce Levi-Civita connections can only occur given severe restrictions on the geometry of the submersion p : N → M . For Brownian motions on the total spaces of Riemannian submersions much of our basic discussion, as in the first two and a half chapters, of skew-product

Introduction

xi

decompositions is very close to that in [33] which was taken further by Liao in [69]. A major difference from Liao’s work is that for degenerate diffusions we use the semi-connection determined by our operators rather than an arbitrary one, so obtaining canonical decompositions. The same holds for the very recent work of Lazaro-Cami & Ortega, [62] where they are motivated by the reduction and reconstruction of Hamiltonian systems and consider similar decompositions for semi-martingales. An extension of [33] in a different direction, to shed light on the Fadeev-Popov procedure for gauge theories in theoretical physics, was given by Arnaudon &Paycha in [1]. Much of the equivariant theory presented here was announced with some sketched proofs in [34]. Acknowledgements The collaborative work for this book was carried out at a variety of institutions, especially Orsay and Warwick, with final touches at the Newton Institute. We gratefully acknowledge the importance of our contacts during its preparation with many people, notably including J.-M.Bony, D.O.Crisan, R.Dalang, M.Fuhrman, E.P. Hsu, and J.Teichmann. This research also benefited from an EPSRC grant (EP/E058124/1). K.D. Elworthy, Y. Le Jan and Xue-Mei Li

Chapter 1

Diffusion Operators Pn Pn i ∂ ∂ ∂ n Let i,j=1 aij ∂x i ∂xj + i=1 b ∂xi be a differential operator on R , with smooth coefficients. We can and will assume that (ai,j ) is symmetric. Its symbol is given by the matrix-valued function (ai,j ) considered as a bilinear form on Rn or equivalently as a map from (Rn )? to Rn . The operator is said to be semi-elliptic if the symbol is positive semi-definite, and elliptic if it is positive definite. More generally if L is a second-order differential operator on a manifold M , denote by σ L : T ∗ M → T M its symbol determined by
1 1 1 df σ L (dg) = L (f g) − (Lf )g − f (Lg), 2 2 2 for C 2 functions f, g. We will often write σ L (`1 , `2 ) for `1 σ L (`2 ) and consider σ L as a bilinear form on T ∗ M . Note that it is symmetric. The operator is said to be semi-elliptic if σ L (`1 , `2 ) > 0 for all `1 , `2 ∈ Tu M ∗ , all u ∈ M , and elliptic if the inequality holds strictly. Ellipticity is equivalent to σ L being onto. Definition 1.0.1. A semi-elliptic smooth second-order differential operator L is said to be a diffusion operator if L1 = 0. The standard example of a diffusion operator is the Laplace-Beltrami operator or Laplacian , 4, of a Riemannian manifold M. It is given as in Rn by 4 = div ∇ = −d∗ d where div is the negative of the adjoint of the gradient operator ∇. Its symbol σ 4 : T ? M → T M is just the isomorphism induced by the Riemannian metric. Conversely the symbol of any elliptic diffusion operator determines a Riemannian metric with respect to which the operator differs from the Laplacian by a firstorder term.

1.1

Representations of Diffusion Operators

Apart from local representations as given above there are several global ways to represent a diffusion operator L. One is to take a connection ∇ on T M . Recall that

K.D. Elworthy et al., The Geometry of Filtering, Frontiers in Mathematics, DOI 10.1007/978-3-0346-0176-4_1, © Springer Basel AG 2010

1

2

Chapter 1. Diffusion Operators

a connection on T M gives, or is given by, a covariant derivative operator ∇ acting on vector fields. For each C r vector field U on M it gives a C r−1 section ∇− U of L(T M ; T M ). In other words, for each x ∈ M we have a linear map v 7→ ∇v U of Tx M to itself. This covariant derivative of U in the direction v satisfies the usual rules. In particular it is a derivation with respect to multiplication by differentiable functions f : M → R, so that ∇v (f U ) = df (v)U (x) + f (x)∇v U . For example on Rn a connection is given by Christoffel symbols Γkij : Rn → R, i, j, k = 1, . . . , n. These define a covariant differentiation of vector fields by   n n k X X k  ∂U (x) + Γkij (x)U j (x) v i (1.1) (∇v U ) = ∂x i i=1 j=1 where, for example, U j denotes the j-th component of the vector field U and we are considering v as a tangent vector at x ∈ Rn . We can also consider the Christoffel symbols as the components of a map Γ : Rn → L (Rn ; L(Rn ; Rn )) and equation (1.1) becomes: (1.2) ∇v U = DU (x)(v) + Γ(x)(v)(U (x)). Given any smooth vector bundle τ : E → M over M a connection on E gives a similar covariant derivative acting on sections U of E. This time v 7→ ∇v U is in L(Tx M ; Ex ), where Ex is the fibre over x for x ∈ M . In our local representation the Christoffel symbol now has Γ(x) ∈ L (Rn ; L(Ex ; Ex )) where Ex := p−1 (x) is the fibre of E over x. A connection on T M determines one on the cotangent bundle T ∗ M , and on all tensor bundles over M . Such connections always exist; more details can be found in [19], [55]. In terms of a connection on T M we can write Lf (x) = traceTx M ∇− (σ L (df )) + df (V 0 (x))

(1.3)

for some smooth vector field V 0 on M . The trace is that of the mapping v 7→ ∇v (σ L (df )) from Tx M to itself. To see this it is only necessary to check that the right-hand side has the correct symbol since the symbol determines the diffusion operator up to a first-order term. For the Laplacian on Rn , when Rn is given its usual “trivial ” connection this reduces to the representation: 4f (x) = trace D2 f (x)

(1.4)

where D2 f (x) ∈ L(Rn , Rn ; R) is the second Frechet derivative of f at x. If a smooth ‘square root’ to 2σ L can be found we have a H¨ormander representation. The ‘square root’ is a smooth map X : M × Rm → T M with each X(x) ≡ X(x, −) : Rm → Tx M linear, such that 2σxL = X(x)X(x)∗ : Tx∗ M → Tx M

1.1. Representations of Diffusion Operators

3

where X(x)∗ : Tx∗ M → Rm is the adjoint map of X(x). Thus there is a smooth vector field A with m 1X L= L j L j + LA , (1.5) 2 j=1 X X where X j (x) = X(x)(ej ) for {ej } an orthonormal basis of Rm , and LV denotes Lie differentiation with respect to a vector field V , so LV f (x) = dfx (V (x)). On Rn , X ∂ LV f (x) = V i (x) f (x). ∂x i i For the Laplacian on Rn the simplest H¨ormander form representation is n

4=

1 X√ ∂ √ ∂ 2 j 2 j 2 j=1 ∂x ∂x

and the generator, 12 4, of Brownian motion has n

1 1X ∂ ∂ 4= . 2 2 j=1 ∂xj ∂xj If σ L has constant rank, such X may be found. A proof of this is given in the Appendix, Theorem 9.2.1. Otherwise it is only known that locally Lipschitz square roots exist (see the discussions in the Appendix, Section 9.2). In that case LX j LX j is only defined almost surely everywhere and the vector field A can only be assumed measurable and locally bounded. Nevertheless uniqueness of the martingale problem still holds (see below). Also there is still the hybrid representation, given a connection ∇ on T M , m

Lf (x) =

1X ∇ j (df )(X j (x)) + df (V 0 (x)) 2 j=1 X (x)

(1.6)

for V 0 locally Lipschitz. The choice of a H¨ ormander representation for a diffusion operator, if it exists, determines a locally defined stochastic flow of diffeomorphisms {ξt : 0 6 t < ζ} whose one-point motion solves the martingale problem for the diffusion operator. In particular on bounded measurable compactly supported f : M → R the associated (sub-) Markovian semigroup is given by Pt f = E(f ◦ ξt ). See also Appendix II. Despite the discussion above we can always write L in the form N 1 X ij a LX i LX j + LX 0 , L= 2 ij=1

(1.7)

where N is a finite number, aij and X k are respectively smooth functions and smooth vector fields with aij = aji .

4

1.2

Chapter 1. Diffusion Operators

The Associated First-Order Operator

Denote by C r Λp ≡ C r Λp T ∗ M , r > 0, the space of C r smooth differential p-forms on a manifold N . To each diffusion operator L we shall associate an operator δ L : C r+1 Λ1 → C r (M ; R), see Elworthy-LeJan-Li [35], [36] c.f. Eberle [27]. The horizontal lift of L will then be defined in terms of a lift of δ L . The existence of δ L comes from the lack of a 0-order term in L: Proposition 1.2.1. For each diffusion operator L there is a unique smooth linear differential operator δ L : C r+1 Λ1 → C r Λ0 such that (1) δ L (f φ) = df σ L (φ) + f · δ L (φ), (2) δ L (df ) = Lf. In particular in the Rn case with L = δL φ =

m X

aij

i,j=1

Pn

i,j=1

∂ ∂ aij ∂x i ∂xj +

P

∂ bi ∂x i it is given by

X ∂ φj (x) + bi φi (x) ∂xi

where φ has the representation φx =

X

φj (x) dxi .

(1.8)

Equivalently δ L is determined by either one of the following: δ L (f dg) = σ L (df, dg) + f Lg, 1 1 1 δ L (f dg) = L(f g) − gLf + f Lg. 2 2 2

(1.9) (1.10)

Proof. The statements are rather obvious in the Rn case. In general take a connection ∇ on T M , then, as in (1.3), L can be written as Lf = trace ∇σ L (df ) + LV 0 f for some smooth vector field V 0 . Set δ L φ = trace ∇(σ L φ) + φ(V 0 ). Then δ L (df ) = Lf and δ L (f φ) = trace ∇(f (σ L φ)) + f φ(V 0 ) = f δ L φ + df (σ L φ). Pk Note that a general C r 1-form φ can be written as φ = j=1 fi dgi for some C r function fi and smooth gi , for example, by taking (g 1 , . . . , g m ) : M → Rm to be an immersion. This shows that (1) and (2) determine δ L uniquely. Moreover since L is a smooth operator so is δ L .  Remark 1.2.2. If the diffusion operator L has a representation L=

m X j=1

aij LX i LX j + LX 0

1.3. Diffusion Operators Along a Distribution

5

for some smooth vector fields X i and smooth functions aij , i, j = 0, 1, . . . , m with aij = aji , then m X aij LX i ιX j + ιX 0 , δL = j=1

where ιA denotes the interior product of the vector field A with a differential form. [In particular if φ is a one-form, then ιA (φ) : M → R is given by ιA (φ)(x) = φx (A(x)).] One can check directly that δ L (df ) = Lf and that (1) holds.

1.3

Diffusion Operators Along a Distribution

Let N be a smooth manifold. By a distribution S in N we mean a family {Su : u ∈ N } where Su is a linear subspace of Tu N ; for example S could be a subbundle of T N . In Rn each Su can be viewed as a linear subspace of Rn . Given such a distribution S let S 0 = ∪u Su0 for Su0 the annihilator of Su in Tu∗ N . Definition 1.3.1. Let S be a distribution in T N . Denote by C r S 0 the set of C r 1-forms which vanish on S. A diffusion operator L on N is said to be along S if δ L φ = 0 for all φ ∈ C 1 S 0 . Example 1.3.2. For N = Rn − {0} let Su be the hyperplane orthogonal to u. The ∂2 spherical Laplacian, ( ∂θ 2 if n = 2), is along this distribution. Example 1.3.3. Let N = R3 with the C ∞ Heisenberg group structure induced by the central extension of the symplectic vector space R2 . This is defined by
1 (x, y, z) · (x0 , y 0 , z 0 ) = x + x0 , y + y 0 , z + z 0 + (xy 0 − yx0 ) . 2 This is isomorphic to the matrix group of 3 × 3 upper diagonal matrices:   1 x z (x, y, z) 7→ 0 1 y  . (1.11) 0 0 1 Let X, Y, Z be the left-invariant vector fields which give the standard basis for R3 at the origin. As operators: 1 ∂ ∂ − y , ∂x 2 ∂z ∂ Z(x, y, z) = . ∂z

X(x, y, z) =

Y (x, y, z) =

∂ 1 ∂ + x ∂y 2 ∂z

Let L be the operator LH given by 1 (LX LX + LY LY ) 2  2 1 ∂2 ∂2 ∂2 1 2 ∂2 2 ∂ = −y . + 2 + (x + y ) 2 + x 2 ∂x2 ∂y 4 ∂z ∂y∂z ∂x∂z

LH =

(1.12) (1.13)

6

Chapter 1. Diffusion Operators

This operator is clearly along the distribution S given by left translates of the (x, y)-plane. This distribution is not integrable: it is not tangent to any foliation / Su . of R3 . Indeed the Lie bracket of X and Y is Z so [X, Y ](u) ∈ In general suppose L is along S and take φ ∈ C r S 0 . By Proposition 1.2.1 and the symmetry of σ L , 0 = (df )(σ L (φ)) = φ(σ L (df ) giving φx ∈ Image[σxL ]0 . This proves Remark 1.3.4 (i): Remark 1.3.4. (i) if δ L φ = 0 for all φ ∈ C 1 S 0 , then σ L φ = 0 for all such φ and Image[σxL ] ⊂ ∩φ∈C 1 S 0 [ker φx ] for all x ∈ N . (ii) If S is a subbundle of T N , (essentially a smooth family of subspaces of constant dimension), and L is along S, then without ambiguity we can define δ L φ for φ a C 0 section of S ∗ by δ L φ := δ L φ˜ for any 1-form φ˜ extending φ. Recall that S ∗ is canonically isomorphic to the quotient T ∗ N/S 0 . An important case is where Su is the image of σuL assuming that the symbol σ L has constant rank. Definition 1.3.5. If Sx = ∩φ∈C 1 S 0 [ker φx ] for all x we say S is a regular distribution. Clearly subbundles are regular. As another example take the circle, M = S 1 , and suppose that Su = 0 except for finitely many points. Then C 1 S 0 consists of those C 1 forms which vanish at those points. This distribution is regular but is not a subbundle. However this would not hold in general if it were non-zero precisely on a countably infinite set of points. The notion is introduced in order to be able to consider the vertical distribution {V Tu N := ker Tu p : u ∈ N } of a smooth map p : N → M : Lemma 1.3.6. Let p : N → M be a smooth map, then {ker Tu p : u ∈ N } is a regular distribution. Proof. This is immediate since ker[T p] is annihilated by all differential 1-forms of the form θ ◦ T p for θ a C 1 1-form on M .  Proposition 1.3.7. (1) Let S be a regular distribution of N and L an operator written in H¨ ormander form: m

L=

1X L j L j + LY 0 2 j=1 Y Y

(1.14)

where the vector fields Y 0 and Y j , j = 1, . . . , m are C 0 and C 1 respectively. Then L is along S if and only if Y i are sections of S. (2) If B is along a smooth subbundle S of T N , then for any connection ∇S on S we can write B as
Bf = traceSx ∇S− σ B (df ) + LX 0 f.

1.4. Lifts of Diffusion Operators

7

Also we can find smooth sections X 0 , . . . , X m of S and smooth functions aij such that 1 X ij a LX i LX j + LX 0 . B= 2 i,j [Recall that ∇S is a covariant derivative operator defined only on those vector fields which take values in S, though one can differentiate in any direction.] Proof. For part (1), if Y i are sections of S, take φ ∈ C 1 S 0 , then m

δL φ =

1X L j φ(Y j ) + φ(Y 0 ) = 0 2 j=1 Y

and so L is along S. Conversely L is along S. Define a C 1 bundle map Y : Rm → T N by Pm suppose j m Y (x)(e) = j=1 Y (x)ej for {ej }m j=1 an orthonormal base of R . Then 2σxL = Y (x)Y (x)∗ and Image[Y (x)] = Image[σxL ] ⊂ S, by Remark 1.3.4. Now δL φ =

1X LY j (φ(Y j )) + φ(Y 0 ) = φ(Y 0 ), 2

which can only vanish for all φ ∈ C 1 S 0 if Y 0 is a section of S. Thus Y 1 , . . . , Y m , and Y 0 are all sections of S. For part (2), we use (1.3) and take ∇ there to be the direct sum of ∇S with an arbitrary connection on a complementary bundle, observing σ B has image in S by Remark 1.3.4(i). 

1.4

Lifts of Diffusion Operators

Let p : N → M be a smooth map and E a subbundle of T M . Recall that V Tu N := ker Tu p denotes the vertical distribution. If p(u) is a regular value of p, that is if Ty p : Ty N → Tp(u) M is surjective for all y ∈ p−1 (p(u)), then the fibre Ep (u) := p−1 (p(u)) is a submanifold with V Tu N as its tangent space at u. Let S be a subbundle of T N transversal to the fibre of p, i.e. V Tu N ∩ S = {0} all u ∈ N and such that Ty p maps Sy isomorphically onto Ep(y) , for each y. Example 1.4.1. Set N = R2 − {0}, M = R(> 0) with Ex = R for all x ∈ M . Take p(x) = |x|. Then for all u ∈ N the orthogonal complement to Ru is the vertical tangent spaced V Tu N and we may take Su = Ru. Here we are, as usual, identifying Tu {R2 − {0}} with R2 .

8

Chapter 1. Diffusion Operators

The example above was a special case of the situation when we have a Riemannian metric on N , and so an inner product, h−, −iu , on each tangent space Tu N and when E = T M . If, as in the example, p is a submersion , i.e. its derivative Tu p : Tu N → Tp(u) M is onto for each u, we can take each Su to be the orthogonal complement in Tu of V Tu N . If p were not a submersion, no transversal subbundle S would exist. Such submersions are described in detail in Chapter 7 with examples. Here is another one: Example 1.4.2. Let N = SO(3) − {Id}, be the special orthogonal group of R3 with identity element removed. Let M = RP2 , real 2-dimensional projective space, considered as the space of lines through the origin in R3 or equivalently as the 2sphere with antipodal points identified. Define p : N → RP2 by taking p(u) to be the axis of rotation of u, i.e. the line determined by the eigenvector with eigenvalue 1. The fibre through any u ∈ N is a copy of the rotation group SO(2), that is, of S 1 , with the identity removed. In fact it is part of a one-parameter subgroup {etα : t ∈ R}, say, of SO(3) through u. Here α is an element of the Lie algebra so(3). We can take Su to be the left translate of the orthogonal complement of α in so(3), identified with TId SO(3), to u. It is identified under T p with the plane in R3 orthogonal to p(u), which is naturally identified with Tp(u) RP2 . For another class of examples see Example 2.1.6 below. Lemma 1.4.3. Every smooth 1-form on N can be written as a linear combination of sections of the form ψ + λp∗ (φ) for λ : N → R smooth, φ a 1-form on M , and ψ annihilates S. In particular any 1-form annihilating V T N is of the form λp∗ (φ). If E = T M , then ψ is uniquely determined. Proof. Take Riemannian metrics on M and N such that the isomorphism between S and p∗ (E) given by T p is isometric. Fix y0 ∈ N . Take a neighbourhood V of p(y0 ) in M over which E is trivializable. Let v 1 , v 2 , . . . , v p be a trivialising family of sections over V . Set U = p−1 (V ). If φj = (v j )∗ , the dual 1-form to v j , j = 1 to p, over V , then {p∗ (φj )# , j = 1 to p} gives a trivialization of S over U . [Indeed p∗ (φj )y (−) = φjp(y) (Ty p−) = h(Ty p)∗ (v j ), −i.] Since any vector field over V can therefore be written as one orthogonal to S plus a linear combination of the p∗ (φj )# , by duality the result holds for forms with support in U . The global result follows using a partition of unity. For the uniqueness note that if E = T M , then T N = V T N + S.  By a lift of a diffusion operator A on M over p we mean a diffusion operator B on N such that B(f ◦ p) = (Af ) ◦ p (1.15) for all C 2 functions f on M . Proposition 1.4.4. Let A be T M . There is a unique lift transversal bundle S. Write

a diffusion operator on M along the subbundle E of of A to a smooth diffusion generator AS along the S ¯ δ = δ A . Then AS is determined by

1.4. Lifts of Diffusion Operators

9

¯ (i) δ(ψ) = 0 if ψ annihilates S. (ii) δ¯ (p∗ φ) = (δ A φ) ◦ p,

for φ ∈ Ω1 (M ).

Moreover (iii) for y ∈ N let hy : Ep(y) → Ty N be the right inverse of Ty p with image Sy . Then S

(a) σyA = hy σ A h∗y . (b) If A is given by A=

N 1 X ij a LX i LX j + LX 0 2 i,j=1

(1.16)

where X 1 , . . . , X N and X 0 are sections of E, then AS =

N 1 X ij (a ◦ p) LX¯ i LX¯ j + LX¯ 0 2 i,j=1

(1.17)

¯ j (y) = hy (X j (p(y)). for X Proof. Lemma 1.4.3 ensures that (i) and (ii) determine δ¯ uniquely as a smooth operator on smooth 1-forms if it exists. On the other hand we can represent A S as in (1.16) and define AS by (1.17). It is straightforward to check that then δ A satisfies (i) and (ii).  2

d Example 1.4.5. . In Example 1.4.1 above we can take A = dx 2 on M = R > 0, 2 then the lift to N = R − {0} along the given distribution in polar co-ordinates ∂2 2 we could use is ∂r 2 . On the other hand if we changed p to be given by p(u) = u 1 ∂2 the same distribution S but the lift would be changed to 2r ∂r2 .

Definition 1.4.6. When an operator B is along the vertical distribution ker[T p] we say B is vertical, and when there is a horizontal distribution such as {Hu : u ∈ N } as given by Proposition 2.1.2 below and B is along that horizontal distribution we say B is horizontal . Proposition 1.4.7. Let B be a smooth diffusion operator on N and p : N → M any smooth map, then the following conditions are equivalent: (1) The operator B is vertical. Pm ij (2) The operator B has an expression of the form of j=1 a LY i LY j + LY 0 ij j where a are smooth functions and Y are smooth sections of the vertical tangent bundle of T N . (3) B(f ◦ p) = 0 for all C 2 f : M → R.

10

Chapter 1. Diffusion Operators

Proof. (a). From (1) to (3) is trivial. From (3) to (1) note that every φ which vanishes on vertical vectors is a linear combination of elements of the form f p∗ (dg) for some smooth g : M → R by Lemma 1.4.3. To show that B is vertical we only need to show that δ B (f p∗ (dg)) = 0. But B(g ◦ p) = 0 implies δ B (p∗ (dg)) = 0 and also p∗ (dg)σ B (p∗ (dg)) = 12 B(g ◦ p)2 − (g ◦ p)B(g ◦ p) = 0. By semi-ellipticity of B, σ B (p∗ (dg)) = 0. Thus assertion (1) follows since δ B (f p∗ (dg)) = df σ B (p∗ (dg)) + f · δ B (p∗ (dg)) from Proposition 1.2.1(1), and so (1) and (3) are equivalent. Equivalence of (1) and (2) follows from Proposition 1.3.7.  Remark 1.4.8. (1) If B is vertical, then by Proposition 1.2.1, for all C 2 functions f1 on N and f2 on M , B (f1 (f2 ◦ p)) = (f2 ◦ p)Bf1 ; (2) If B and B 0 are both over a diffusion operator A of constant nonzero rank such that A is along the image of σ A , then B − B 0 is not in general vertical, although (B − B 0 )(f ◦ p) = 0 for all C 2 function f : M → R, since it may not be semi-elliptic. For example take p : R2 → R to be the projection ∂2 ∂2 ∂2 ∂2 ∂2 ∂2 0 p(x, y) = x with A = ∂x 2 , B = ∂x2 + ∂y 2 . Let B = ∂x2 + ∂y 2 + ∂x∂y . Then 2

∂ is not vertical. In particular this shows B 0 is also over A but B − B 0 = − ∂x∂y that Proposition 1.4.7 would not hold without the assumption that B is a diffusion operator.

1.5

Notes

On symbols What we have called the symbol of our diffusion operator is often called the principal symbol. In the semi-elliptic case it is related to the energy density used in Dirichlet form theory mainly for symmetric diffusions, and is also given in terms of the carr´e du champ or squared gradient operator Γ as used, for example, in Bakry-Emery theory, [3], by σ L (df, dg) = Γ(f, g).

Chapter 2

Decomposition of Diffusion Operators Consider again a smooth map p : N → M between smooth manifolds M and N and a lift B of a diffusion operator A on M . Definition 2.0.1. In this situation we say that B is over A, or that A and B are intertwined by p. In general a diffusion operator B on N is said to be projectible (over p), or p-projectible, if it is over some diffusion operator A. Recall that the pull-back p∗ φ of a 1-form φ is defined by p∗ (φ)u = φp(u) (T p(−)) = (T p)∗ φp(u) . In local co-ordinates p∗ (φ)i =

X ∂pk k

∂xi

φk ◦ p.

For our map p : N → M , a diffusion operator B is over A if and only if δ B (p∗ φ)) = (δ A φ)(p),

(2.1)

for all φ ∈ C 1 ∧1 T ∗ M .

2.1

The Horizontal Lift Map

Lemma 2.1.1. Suppose that B is over A. Let σ B and σ A be respectively the symbols for B and A. Then A (Tu p)σuB (Tu p)∗ = σp(u) ,

∀u ∈ N,

(2.2)

K.D. Elworthy et al., The Geometry of Filtering, Frontiers in Mathematics, DOI 10.1007/978-3-0346-0176-4_2, © Springer Basel AG 2010

11

i.e. the following diagram is commutative:

12

Chapter 2. Decomposition of Diffusion Operators σuB

Tu∗ N 6 ∗ (Tu p)

- Tu N Tu p

∗ Tp(u) M

? - Tp(u) M .

A σp(u)

Proof. Let f and g be two smooth functions on M . Then for u ∈ N , x = p(u), 1 1 1 A(f g)(x) − (f Ag)(x) − (gAf )(x) 2 2 2 1 1 1 = B ((f g) ◦ p) (u) − f ◦ pB(g ◦ p)(u) − g ◦ pB(f ◦ p)(u) 2 2 2 B = d (g ◦ p)u σu (d (f ◦ p)u )

(dfx ) σxA (dgx ) =

= (dg ◦ Tu p) σuB (df ◦ Tu p) , which gives the desired equality.



For x in M , set Ex := Image[σxA ] ⊂ Tx M . If σ A has constant rank, i.e. dim[Ex ] is independent of x, then E := ∪x Ex is a smooth subbundle of T M . Proposition 2.1.2. Assume σ A has constant rank and B is over A. Then there is a unique, smooth, horizontal lift map hu : Ep(u) → Tu N , u ∈ N , characterised by A hu ◦ σp(u) = σuB (Tu p)∗ .

(2.3)

hu (v) = σuB ((Tu p)∗ α)

(2.4)

In particular where α ∈

∗ Tp(u) M

Tu∗ N

satisfies

A σp(u) (α)

σ

= v.

B

Tu N hu

(Tu p)∗

∗ M Tp(u)

Tu p σA

Tp(u) M

Proof. Clearly (2.4) implies (2.3) by Lemma 2.1.1 and so it suffices to prove hu is well defined by (2.4). For this we only need to show σ B ((Tu p)∗ (α)) = 0 for every A α in ker[σp(u) ]. Now σ A α = 0 implies that (T p)∗ (α)σ B ((T p)∗ α) = 0,

2.1. The Horizontal Lift Map

13

by Lemma 2.1.1. Considering σ B as a semi-definite bilinear form this implies  σuB (Tu p)∗ α vanishes as required. Recall from Lemma 1.3.6 that the vertical distribution ker[T p] is regular. Let Hu = Image[hu ], the horizontal subspace at u, and H = tu Hu . Set Fu = (Tu p)−1 [Ep(u) ] so we have a splitting Fu = Hu + V Tu N

(2.5)

where V Tu N = ker[Tu P ] the ‘vertical’ tangent space at u to N .

Tu p

hu

Fu ⊂ Tu N

Eu ⊂ Tp(u) M In the elliptic case p is a submersion, the vertical tangent spaces have constant rank, and F := tu Fu is a smooth subbundle of T N . In this case we have a splitting of T N , a connection in the terminology of Kolar-Michor-Slovak [57]. In general we will define a semi-connection on E to be a subbundle Hu of T N such that Tu p maps each fibre Hu isomorphically to Ep(u) . In the equivariant case considered in Chapter 3 such objects are called E-connections by Gromov. For the case when p : N → M is the tangent bundle projection, or the orthonormal frame bundle, note that the “partial connections” as defined by Ge in [45] are rather different from the semi-connections we would have: they give parallel translations along E-horizontal paths which send vectors in E to vectors in E, and preserve the Riemannian metric of E, whereas the parallel transports of our semi-connections do not in general preserve the fibres of E, nor any Riemannian metric, and they act on all tangent vectors. Lemma 2.1.3. Assume σ A has constant rank and B is over A. For all u ∈ N the image of σuB is in Fu . Proof. Suppose α ∈ Tu∗ N with σ
B (α) 6∈ Fu . Then there exists k in the annihilator of Ep(u) such that k Tu p σ B (α) 6= 0. However

A (k) k Tu p σ B (α) = α σ B ((Tu p)∗ (k)) = α hu σp(u) A ∗ by Proposition 2.1.2; while σp(u) (k) = 0 because for all β ∈ Tp(u) M, A A β σp(u) (k) = k σp(u) (β) = 0

giving a contradiction.



14

Chapter 2. Decomposition of Diffusion Operators

Proposition 2.1.4. Let A be a diffusion operator on M with σ A of constant rank. For i ∈ {1, 2}, let pi : N i → M be smooth maps and Bi be diffusion operators on N i over A. Let F : N 1 → N 2 be a smooth map with p2 ◦ F = p1 . Assume F intertwines B 1 and B 2 . Let h1 , h2 be the horizontal lift maps determined by A, B 1 and A, B 2 . Then u ∈ N 1; (2.6) h2F (u) = Tu F (h1u ), i.e. the diagram Tu F

Tu N 1 @ I @

- T 2 F (u) N 

h1u @ @

h2F (u) @ Ep1 (u)

commutes for all u ∈ N . Proof. Since F intertwines B 1 and B 2 , Lemma 2.1.1 gives 2

1

σFB (u) = Tu F ◦ σuB ◦ (Tu F )∗ . Now take α ∈ Tp∗1 (u) M with σpA1 (u) (α) = v, some given v ∈ Ep1 (u) . From (2.4) 2

h2F (u) (v) = σFB (u) ((T p2 )∗ α) 1

= Tu F ◦ σuB ◦ (Tu F )∗ (T p2 )∗ α 1

= Tu F ◦ σuB (Tu p1 )∗ α = Tu h1u (v) as required.



Definition 2.1.5. A diffusion operator B on N will be said to have projectible symbol for p : N → M if there exists a map η : T ∗ M → T M such that for all u ∈ N the diagram Tu∗ N 6 ∗ (Tu p) ∗ Tp(u) M

commutes, i.e. if

(Tu p)σuB (Tu p)∗

σuB

- Tu N Tu p

ηp(u)

? - Tp(u) M .

depends only on p(u).

2.1. The Horizontal Lift Map

15

In this case we also get a uniquely defined horizontal lift map as in Proposition 2.1.4 defined by equation (2.6) using η instead of the symbol of A. This situation arises naturally in the standard non-linear filtering literature as described later, see Chapter 5. Example 2.1.6.

1. Consider N = R2 , M = R, with p(x, y) = x. Take B = a(x)

∂2 ∂2 ∂2 + c(x, y) + 2b(x, y) ∂x2 ∂x∂y ∂y 2

2

∂ 2 so A = a(x) ∂x 2 . For semi-ellipticity of B we require a(x)c(x, y) > b (x, y) and 2 a(x) > 0 for all (x, y) ∈ R . For the constant rank condition on A and nontriviality we will assume a(x) > 0 for all real x. Define γ(x, y) = b(x,y) a(x) . Then the horizontal lift map, or more precisely its principal part, hu : R → R2 at u = (x, y), is given by

h(x,y) (r) = (r, γ(x, y)r).

(2.7)

To check this satisfies the defining criterion equation (2.3) observe that with our definition, for r ∈ R, A A A hu ◦ σp(u) r = (σp(u) r, γ(u)σp(u) r)

(2.8)

= (a(x)r, b(x, y)r);

(2.9)

while σuB (Tu p)∗ r =



a(x) b(x, y)

b(x, y) c(x, y)



= (a(x)r, b(x, y)r).

r 0

 (2.10) (2.11)

2. More generally take N = Rn × R, M = Rn , with p the projection. Now take n X

n

X ∂2 ∂2 ∂2 a (x) f+ bk (x, y) k f + c(x, y) 2 f (Bf )(x, y) = ∂xi ∂xj ∂x ∂y ∂y i,j=1 i,j

k=1

where a = (ai,j ) is a symmetric n×n-matrix-valued function, b = (b1 , . . . , bn ) takes values in Rn and c is real-valued. Then B lies over A for A=

n X i,j=1

ai,j (x)

∂2 . ∂xi ∂xj

For A to be semi-elliptic with symbol of constant rank we require a(x) to be positive semi-definite for all x ∈ M and to have constant rank. It is

16

Chapter 2. Decomposition of Diffusion Operators easy to see that, assuming this, B will be semi-elliptic if and only if for all ξ ∈ Rn we have hb(x, y)ξ, ξi2 6 c(x, y)ha(x)ξ, ξi (2.12) or equivalently, as matrices, b(x, y)t b(x, y) 6 c(x, y)a(x)

(2.13)

for all (x, y) ∈ N . For this to hold we see that b(x, y) must always lie in the image of a(x), otherwise there will be some ξ orthogonal to the image of a(x) for which hb(x, y)ξ, ξi2 > 0 contradicting condition (2.12). Moreover since a(x) is symmetric its kernel is orthogonal to its image, and so if v is in its image, ha(x)−1 b, vi is well defined. In fact it is simply the inner product hb, vix of v and b in the metric on E determined by A. In this notation the horizontal lift map h(x,y) : Ex → N × (Rn × R) is given by h(x,y) (v) = ((x, y), (v, hb, vix )).

(2.14)

3. More generally if A(x) is given by a positive definite matrix (m + p) × (m + p) matrix A of rank m, B(x, y) is an (m + p) × q matrix with B(x, y) in the image of A(x), and C(x, y) a q × q matrix. We have a horizontal lifting map u ∈ Image(A(x)) → (u, B T (x, y)A−1 (x)u). Example 2.1.7 (Coupling of diffusion operators). Consider diffusion operators A1 and A2 on manifolds M 1 and M 2 . Take N = M1 × M2 with p1 and p2 the corresponding projections. A diffusion operator B on N is a coupling of A1 with A2 if B and Aj are intertwined by pj for j = 1, 2. If so it is easy to see that there is a bilinear ΓB : T ∗ M 1 × T ∗ M 2 → R such that B(f ⊗ g)(x, y) = A1 (f )(x)g(y) + f (x)A2 (g)(y) + ΓB ((df )x , (dg)y )

(2.15)

where f ⊗ g : M 1 × M 2 → R denotes the map (x, y) 7→ f (x)g(y) and f, g are C 2 . Note that the symbol σ B : T ∗ M1 × T ∗ M2 → T M1 × T M2 is given by  1  2 1,2 2,1 B σ(x,y) (`1 , `2 ) = σxA (`1 ) + σ(x,y) (`2 ), σxA (`2 ) + σ(x,y) (`1 ) , `1 ∈ Tx∗ M1 , `2 ∈ Ty∗ M2

(2.16)

1,2 2,1 : Ty∗ M2 → Tx M1 and σ(x,y) : Tx∗ M1 → Ty M2 are defined by where σ(x,y) 1,2 (`2 ) = `1 σ(x,y)

1 B 2,1 Γ (`1 , `2 ) = `2 σ(x,y) (`1 ). 2 1

Now take A = A1 and p = p1 and assume σ A has constant rank.

2.2. Lifts of Cohesive Operators and The Decomposition Theorem

17

Lemma 2.1.8. For (x, y) ∈ M1 × M2 : 1,2 2,1 1. σ(x,y) = (σ(x,y) )∗ : Ty∗ M2 → Tx M1 , 1,2 2. σ(x,y) has image in Ex .

Proof. The first assertion is immediate from the definitions. For the second we use the semi-ellipticity of B with the argument in the proof of the existence of a horizontal lift, Proposition 2.1.2, to see that if ` ∈ ker σxA , then, for all y ∈ M2 2,1 we have σ B (T(x,y) p)∗ (`) = 0. This means that ` ∈ ker σ(x,y) . By the first part 1,2 ∗ ˜ = 0 for all `˜ ∈ T M2 . Thus ` ∈ ker σ A implies that ` this implies that `σ (`) (x,y)

y

x

1,2 annihilates the image of σ(x,y) . On the other hand since A is symmetric ` ∈ ker σxA if and only if ` annihilates Ex . 

By the lemma we can use the Riemannian metric on E induced by the symbol 1,2 # 1,2 ) : Ex → Ty M2 of σ(x,y) . We claim that the of A to define the adjoint (σx,y) horizontal lift of the semi-connection induced by our coupling is given by   1,2 # ) (v) ∈ Tx M1 × Ty M2 v ∈ Ex . (2.17) h(x,y) (v) = v, (σx,y) To check this, first note that from Proposition 2.1.2 we know   2,1 ((σxA )−1 (v)) . h(x,y) (v) = v, σ(x,y) Next take `1 ∈ Tx∗ M1 and `2 ∈ Ty∗ M2 . Write σx for σ A considered as a map from Tx∗ M1 → Ex . Then our claim follows from: 1,2 1,2 `2 [(σ(x,y) )# σx (`1 )] = `1 σx∗ (`2 ◦ (σ(x,y) )# ) 1,2 = h`1 |Ex , `2 ◦ (σ(x,y) )# iEx∗ 2,1 = `2 σ(x,y) (`1 ).

2.2

Lifts of Cohesive Operators and The Decomposition Theorem

A diffusion generator L on a manifold is said to be cohesive if (i) σxL , x ∈ X, has constant non-zero rank and (ii) L is along the image of σ L . Remark 2.2.1. From Theorem 2.1.1 in Elworthy-LeJan-Li [36] we see that if the rank of σxL is bigger than 1 for all x, then L is cohesive if and only if it has a representation m 1X L jL j L= 2 j=1 X X

18

Chapter 2. Decomposition of Diffusion Operators

where Ex = span{X 1 (x), . . . X m (x)} has constant rank. Let H be the horizontal distribution of the semi-connection determined by a cohesive diffusion generator A. We can now define the horizontal lift of A to be the diffusion generator AH on N given by Proposition 1.4.4. The equivalence of (i) and (ii) in the following proposition shows that AH can be characterised independently of any semi-connection. Proposition 2.2.2. Let B be a smooth diffusion operator on N over A with A cohesive. The following are equivalent: (i) B = AH . (ii) B is cohesive and Tu p is injective on the image of σuB for all u ∈ N . (iii) B can be written as m

B=

1X L ˜ j L ˜ j + LX˜ 0 2 j=1 X X

˜ 0, . . . , X ˜ m are smooth vector fields on N lying over smooth vector where X 0 ˜ j (u)) = X j (p(u)) for u ∈ N for all j. fields X , . . . , X m on M , i.e. Tu p(X Pm Proof. If (i) holds, take smooth X 1 , . . . , X m with A = 12 j=1 LX j LX j + LX 0 , ˜ j (u) = hu X j (p(u)) to see (iii) holds. Clearly (iii) by Proposition 1.3.7, and set X implies (ii) and (ii) implies (i), so the three statements are equivalent.  Definition 2.2.3. If condition (ii) of the proposition holds we say that B has no vertical part. Recall that if S is a distribution in T N , then S 0 denotes the set of annihilators of S in T ∗ N . Lemma 2.2.4. For ` ∈ Hu0 and k ∈ (Vu T N )0 , some u ∈ N we have: A. `σ B (k) = 0, H

B. σ B (k) = σ A (k), H

C. σ A (`) = 0. In particular Hu is the orthogonal complement of V Tu N ∩Image(σuB ) in Image(σuB ) with its inner product induced by σuB . Proof. Set x = p(u). For part A and part B it suffices to take k = φ ◦ Tu p some φ ∈ Tx∗ M . Then by (2.3), σuB (φ ◦ Tu p) = hu ◦ σxA (φ) giving part A, and also part B by Proposition 1.4.4 (iii)(a) since φ = h∗u (φ ◦ Tu p). Part C comes directly from Proposition 1.4.4 (iii)(a). 

2.2. Lifts of Cohesive Operators and The Decomposition Theorem

19

Theorem 2.2.5. For B over A with A cohesive there is a unique decomposition B = B1 + BV where B 1 and B V are smooth diffusion generators with BV vertical and B 1 over A having no vertical part. In this decomposition B 1 = AH , the horizontal lift of A to H. Proof. Set B V = B − AH . To see that BV is semi-elliptic take u ∈ N and observe that any element of Tu∗ N can be written as ` + k where ` ∈ Hu0 and k ∈ (V Tu N )0 by Lemma 2.2.4 and (` + k)σ B (` + k) = `σ B (`) > 0. Since B V (f ◦ p) = 0 any f ∈ C 2 (M ; R), Proposition 1.4.7 implies B V is vertical. Uniqueness holds since the semi-connections determined by B and B0 are the same by Remark 1.3.4(i) applied to B V and so by Proposition 2.2.2 we must have B 1 =  AH . Remark 2.2.6. For p a Riemannian submersion and B the Laplacian, BerardBergery and Bourguignon [9] define B V directly by B V f (u) = ∆Nx (f |Nx )(u) for x = p(u) and Nx = p−1 (x) with ∆Nx the Laplace-Beltrami operator of Nx . Definition 2.2.7. For a smooth p : N → M vector fields A˜ and A on N and M respectively are said to be p-related if   ˜ = A(p(u)) for all u ∈ N. Tu p A(u) Remark 2.2.8. When A and B are given in H¨ormander forms using p-related vector fields, another decomposition of B is described in the Appendix, Section 9.5 Remark 9.5. Example 2.2.9. Returning to Example 2.1.6 for p : R2 → R the projection and B = a(x)

∂2 ∂2 ∂2 + c(x, y) 2 + 2b(x, y) 2 ∂x ∂x∂y ∂y

the decomposition is given by  B = a(x)

∂ ∂ + h(x, y) ∂x ∂y

2 + d(x, y)

∂2 ∂y 2 2

b(x,y) where as in Example 2.1.6 h(x, y) = b(x,y) a(x) while d(x, y) = c(x, y) − a(x) . This follows from the fact  that from Example 2.1.6 we know that the horizontal subspace at (x, y) is just (r, h(x, y)r) : r ∈ R . This means that the first term in our decomposition is horizontal (while clearly a lift of A); the second term is clearly vertical.

20

Chapter 2. Decomposition of Diffusion Operators

Example 2.2.10. Take N = S 1 × S 1 and M = S 1 with p the projection on the first factor. Let 1 ∂2 ∂2 ∂2 B= . + 2 ) + tan α 2 2 ∂x ∂y ∂x∂y Here 0 < α < BV =

π 4

so that B is elliptic. Then A =

1 ∂2 (1 − (tan α)2 ) 2 , 2 ∂y

AH =

1 ∂2 2 ∂x2 ,

∂2 1 ∂2 ∂2 ( 2 + (tan α)2 2 ) + tan α . 2 ∂x ∂y ∂x∂y

This is easily checked since, with this definition AH has H¨ormander form AH =

∂ 1 ∂ ( + tan α )2 2 ∂x ∂y

and so is a diffusion operator which has no vertical part. Also B V is clearly vertical and elliptic. Note that this is another example of a Riemannian submersion: several more of a similar type can be found in [9]. In this case the horizontal distribution is integrable and if α is irrational the foliation it determines has dense leaves. Example 2.2.11. Take N = H, the first Heisenberg group with Heisenberg group action as in Example 1.3.3,
1 (x, y, z) · (x0 , y 0 , z 0 ) = x + x0 , y + y 0 , z + z 0 + (xy 0 − yx0 ) . 2 As before let X, Y, Z be the left-invariant vector fields which give the standard basis for R3 at the origin so that as operators: ∂ 1 ∂ ∂ 1 ∂ − y , Y (x, y, z) = + x , ∂x 2 ∂z ∂y 2 ∂z ∂ Z(x, y, z) = . ∂z Take B to be half the sum of the squares of X, Y , and Z. This is half the left invariant Laplacian:   1 ∂2 ∂2 ∂2 1 2 ∂2 ∂2 2 −y ) . + 2 + (1 + (x + y )) 2 + (x B= 2 ∂x2 ∂y 4 ∂z ∂y∂z ∂x∂z X(x, y, z) =

Take M = R2 and p : R3 → R2 to be the projection on the first 2 coordinates. Then the horizontal lift map, induced by (A, B), from R2 to H is 1 h(x,y,z) : (u, v) 7→ (u, v, (xv − yu)) 2 and 1 ∂2 ∂2 ( 2 + 2 ), 2 ∂x ∂y 1 1 ∂2 = Z2 = . 2 2 ∂z 2

A= BV

AH =

1 2 (X + Y 2 ), 2

2.2. Lifts of Cohesive Operators and The Decomposition Theorem

21

The decomposition of the operator is just the completion of squares. This leads back to the canonical left-invariant horizontal vector fields: ∂2 1 2 ∂2 ∂2 2 (x + + (1 + + y )) ∂x2 ∂y 2 4 ∂z 2 2 2 ∂ ∂ −y ) + (x ∂z∂y ∂z∂x ∂ y ∂ 2 x ∂ 2 ∂ ∂2 − ) +( + ) + 2. =( ∂x 2 ∂z ∂y 2 ∂z ∂z

2B =

This philosophy we maintain throughout the book. Note that the horizontal lift σ ˜ , of a smooth curve σ : [0, T ] → M with σ(0) = 0, is given by   Z
1 t 1 1 2 2 2 1 σ ˜ (t) = σ (t), σ (t), (2.18) σ (t)dσ (t) − σ (t)dσ (t) . 2 0 Thus the “vertical” component of the horizontal lift is the area integral of the curve. Equation (2.18) remains valid for the horizontal lift of Brownian motion on R2 , or more generally for any continuous semi-martingale, provided it is interpreted as a Stratonovich equation ( or equivalently an Itˆo equation in the Brownian motion case). This example is also that of a Riemannian submersion. In this case the horizontal distributions are not integrable. Indeed the Lie brackets satisfy [X, Y ] = Z and H¨ ormander’s condition for hypoellipticity: a diffusion operator L satisfies H¨ormander’s condition if for some (and hence all) H¨ormander form representation such as in equation (1.14) the vector fields Y 1 , . . . , Y m together with their iterated Lie brackets span the tangent space at each point of the manifold. For an enjoyable discussion of the Heisenberg group and the relevance of this example to “Dido’s problem” see [79]. See also [5],[11], and [49]. Example 2.2.12. For nontrivial connections on the Heisenberg group discussed ∂2 ∂2 above, consider A = 12 ( ∂x 2 + ∂y 2 ) as before, and for real-valued functions r1 , r2 , γ with γ > r12 + r22 , Br1 ,r2 =

∂2 ∂2 1 ∂2 1 ∂2 ∂2 ( 2 + 2 ) + r1 + r2 + γ 2 . 2 ∂x ∂y ∂x∂z ∂y∂z 2 ∂ z

The horizontal lift map is: h(x,y,z) (u, v) = (u, v, r1 u + r2 v), X1 := hu (

∂ ∂ ∂ )=( + r1 ), ∂x ∂x ∂z

and Br1 ,r2 =

X2 := hu (

∂ ∂ ∂ )=( + r2 ) ∂y ∂y ∂z

1 2 1 ∂2 (X1 + X22 ) + (γ − r12 − r22 ) 2 . 2 2 ∂ z

22

Chapter 2. Decomposition of Diffusion Operators

Example 2.2.13. Consider N = R3 × R3 with coordinates u = (v1 , v2 , v3 ; w1 , w2 , w3 )(u). Take M = R3 and p(u) = (v1 , v2 , w3 ). On N , for a fixed α ∈ R, consider the operator  3  X 1 ∂2 ∂ ∂ 1 2 2 ∂2 ∂ 2 B= + α (vi+1 + vi+2 ) 2 + α(vi − vi+1 ) , 2 ∂vi2 2 ∂wi ∂vi+1 ∂vi ∂wi+2 1 where the suffixes are to be taken modulo 3. It projects by p on the operator A=

1 ∂2 ∂ ∂ ∂2 1 ∂2 ∂ ( 2 + 2 ) + α2 (v12 + v22 ) 2 + α(v1 − v2 ) . 2 ∂v1 ∂v2 2 ∂w3 ∂v2 ∂v1 ∂w3

Note that B =

1 2

P3 1

Xi =

Xi2 and A = 12 (Y12 + Y22 ) with ∂ ∂ ∂ + α(vi+2 − vi+1 ) ∂vi ∂wi+1 ∂wi+2

and ∂ ∂ − αv2 , ∂v1 ∂w3 ∂ ∂ + αv1 . Y2 = ∂v2 ∂w3

Y1 =

(2.19) (2.20)

The horizontal lift is determined by the identities: AH = 12 (X12 + X22 ), B V = 12 X32 and YiH = Xi , i = 1, 2. Note that N is a group with multiplication given by u·u0 = u00 , with vi (u00 ) = vi (u) + vi (u0 ) and wi+2 (u00 ) = wi+2 (u) + α(vi (u)vi+1 (u0 ) − vi+1 (u)vi (u0 )), and p is a homomorphism of N onto the Heisenberg group. The diffusion generators B and A are invariant under left multiplication. Recall that F ≡ tu Fu = ∪u (Tu p)−1 [Ep(u) ], we can now strengthen Lemma 2.1.3 which states that Image[σuB ] ⊂ Fu . Corollary 2.2.14. If B is over A with A cohesive, then B is along F . Proof. Since Hu ∈ Fu and V Tu N ⊂ Fu both B 1 and B V are along F .



2.3. The Lift Map for SDEs and Decomposition of Noise

2.3

23

The Lift Map for SDEs and Decomposition of Noise

Let us consider the horizontal lift connection in more detail when B and A are given by stochastic differential equations. For this write A and B in H¨ormander ˜ X(x) ˜ ∗ for form corresponding to factorisations σxA = X(x)X(x)∗ and σxB = X(x) X(x) : Rm → Tx M,

x ∈ M,

˜ ˜ → Tu N, X(u) : Rm

u ∈ N.

Then X(x) maps onto Ex for each x ∈ M . Define Yx : Ex → Rm to be its right i−1 h . inverse: Yx = Y (x) = X(x) ker X(x)⊥ ˜ such that Lemma 2.3.1. For each u ∈ N there is a unique linear `u : Rm → Rm ker `u = ker X(p(u)) and the diagram

˜ ∗ X(u)

Tu∗ N

-

6 (Tu p)∗

Tx∗ M

˜ Rm 6

˜ X(u) -Tu N

Tu p

`u

-

? -Tx M

Rm

X(x)∗

X(x)

˜ ◦ `u . commutes, for x = p(u), i.e. σxA = Tu p ◦ σxB (Tu p)∗ and X(x) = Tu p ◦ X(u) ˜ In particular the horizontal lift map is given by hu = X(u)`u Y (p(u)). Proof. The larger square commutes by Lemma 2.1.1. For the rest we need to construct `u . It suffices to define `u on [ker X(x)]⊥ . Note that [ker X(x)]⊥ = Image X(x)∗ in Rm . We only have to show that α ∈ ker X(x)∗ implies ˜ ∗ (Tu p)∗ α = 0. X(u) In fact for such α the proof in Proposition 2.1.2 is valid and therefore (Tu p)∗ α ∈ ˜ ˜ ∗ we see ker σ B = is injective on the image of X(u) ker σuB . However since X(u) u . ˜ ker X(u) . Thus `u is defined with ker `u = ker X(x) and such that the left-hand square of the diagram commutes. Since the perimeter commutes it is easy to see from the construction of `u that the right-hand side also commutes. The uniqueness of `u with kernel equal to that of X(x) is therefore clear since, on [ker X(x)]⊥ , we ˜ ∗ (Tu p)∗ for any α ∈ T ∗ M with X(x)∗ α = e.  require `u (e) to be X(u) x From now on assume that X(x) has rank independent of x ∈ M . This ensures that `u is smooth in u ∈ N . Also assume that A(x) ∈ Ex for all x ∈ M , i.e. that A is cohesive. This is needed when we wish to consider the horizontal lift AH of A.

24

Chapter 2. Decomposition of Diffusion Operators

The horizontal lift of X(x), which can be used to construct a H¨ormander form representation of AH , as in Theorem 2.2.5 and Theorem 3.2.1 below is given by: X H (u) : Rm → Tu P, ˜ X H (u) = hu X(u) = X(u)` u since Yx X(x) is the projection onto ker X(x)⊥ . Now for x ∈ M let K(x) be the orthogonal projection of Rm onto the kernel of X(x)and K ⊥ (x) the projection onto [ker X(x)]⊥ , so K ⊥ (x) = Y (x)X(x).

(2.21)

˜ and X are p-related, i.e. Consider the special case that m ˜ = m and also that X ˜ Tu p(X(u)e) = X(p(u))e,

u ∈ N, e ∈ Rm .

Then `u = Y (p(u))X(p(u)) = K ⊥ (p(u)) giving ˜ hu = X(u)Y (p(u)) :

(Tu N )∗

Tu N

X˜ (u ∗ )

(Tu p)∗

˜ (u) X Rm

∗

X

(2.22)

)) (p(u

Tu p

X (p

(u)) Tp(u) M

(Tp(u) M )∗ `u = K ⊥ (p(u))

2.3.1

Decomposition of Stratonovich SDE’s

Suppose we have an SDE on N : ˜ t ) ◦ dBt + A(u ˜ t ) dt dut = X(u

(2.23)

˜ is as above and p-related to X on M while the vector field A˜ is p-related where X to a vector field A on M . Thus if xt = p(ut ) we have dxt = X(xt ) ◦ dBt + A(xt ) dt.

(2.24)

2.3. The Lift Map for SDEs and Decomposition of Noise

25

Then we can decompose our SDE for ut by: ˜ t )K ⊥ (p(ut )) ◦ dBt + A(u ˜ t ) dt + X(u ˜ t )K(p(ut )) ◦ dBt dut = X(u   ˜ t )K(p(ut )) ◦ dBt + A(u ˜ t ) − AH (ut ) dt = X H (ut ) ◦ dBt + AH (ut ) dt + X(u   ˜ t )K(p(ut )) ◦ dBt + A(u ˜ t ) − AH (ut ) dt. = hut ◦ dxt + X(u We shall come back to such decompositions in Sections 4.8, 5.4, and 8.2.

2.3.2

Decomposition of the noise and Itˆ o SDE’s

The decomposition of our SDE on N described above is closely related to the decomposition of the noise of an SDE into essential and redundant components. This was first described in [38] with a more general discussion in [36]. The latter allowed for infinite dimensional noise and incomplete SDE. Here we will review the situation for our SDE (2.24) assuming it is complete, and X has constant rank with A(x) ∈ Ex for all x ∈ M . The projections K and K ⊥ determine metric connections on the subbundles ker X and ker⊥ X of the trivial bundle Rm . Writing Rm − = ker X ⊕ ker⊥ X we have the direct sum connection on Rm . Let //˜t ∈ O(m) for t > 0, be the corresponding parallel translation along the sample paths of the solution to our SDE (2.24) starting at a given point x0 of M . Then //˜t [ker X(x0 )] = ker X(xt )] and //˜t [ker⊥ X(x0 )] = ker⊥ X(xt ). Define processes {Bte }t > 0 and {βt }t > 0 by Z t −1 //˜s K(xs ) dBs Bte = (2.25) 0 Z t −1 //˜s K ⊥ (xs ) dBs . βt = (2.26) 0

Then: 1. {Bte }t > 0 and {βt }t > 0 are independent Brownian motions, on ker⊥ X(x0 ) and ker X(x0 ) respectively; 2. The filtration of {Bte }t > 0 is the same as that of {xt }t > 0 ; 3. dBt = //˜t dBte + //˜t dβt . ˜. , the The process β. is the redundant noise and B e , sometimes denoted by B relevant or essential noise. Suppose now that M = Rn and N = Rk , and that we have an Itˆo SDE ˜ t ) dBt + A(u ˜ t ) dt dut = X(u on Rk whose solutions are such that if xt := p(ut ) for t > 0, then dxt = X(xt ) dBt + A(xt ) dt

(2.27)

26

Chapter 2. Decomposition of Diffusion Operators

˜ Then from above we have: where X is p-related to X. ˜ t )//˜t dB e + A(u ˜ t ) dt + X(u ˜ t )//˜t dβt dut = X(u t = X H (ut ) dBt + AH (ut ) dt   ˜ t )//˜t dβt . ˜ t ) − AH (ut ) dt + X(u + A(u However note that the solutions to dzt = X H (zt ) dBt + AH (zt ) dt will not in general be horizontal lifts of solutions to equation (2.27) and unless p is linear will not in general be lifts.

2.4

Diffusion Operators with Projectible Symbols

Given p : N → M as before, suppose now that we have a diffusion operator B on M with a projectible symbol, c.f. Definition 2.1.5. This means that σ B lies over some positive semi-definite linear map η : T ∗ M → T M . Assume that η has constant rank. We will show that in this case we also have a decomposition of B. To do this first choose some cohesive diffusion operator A on M with σ A = η. In general there is no canonical way to do this, though if η were non-degenerate we could choose A to be a multiple of the Laplace-Beltrami operator of the induced metric on M . From above we also have an induced semi-connection with horizontal subbundle H, say, of T N . Definition 2.4.1. We will say that B descends cohesively (over p) if it has a projectible symbol inducing a constant rank η : T ∗ M → T M , and there exists a horizontal vector field, bH , such that B − LbH is projectible over p. The following is a useful observation. Its proof is immediate. Proposition 2.4.2. If B descends cohesively, then for each choice of A satisfying A σp(u) = Tu pσuB (Tu p)∗ there is a horizontal vector field bH such that B − LbH lies over A. Lemma 2.4.3. Assume that η has constant rank. If f is a function on M let f˜ = f ◦ p. For any choice of A with symbol η the map ]) f 7→ B(f˜) − A(f is a derivation from C ∞ M to C ∞ N where any f ∈ C ∞ M acts on C ∞ N by multiplication by f˜.

2.4. Diffusion Operators with Projectible Symbols

27

Proof. The map is clearly linear and for smooth f, g : M → R we have ^ g) η(df, dg) = σ B (df˜, d˜ so by definition of symbols: ^ ])˜ ] f˜ B(f˜g˜) − A(f g) = B(f˜)˜ g + B(˜ g )f˜ − A(f g − A(g) as required.



Let D denote the space of derivations from C ∞ M to C ∞ N using the above action. Note that for p∗ T M → N the pull-back of T M over p, the space C ∞ Γp∗ T M of smooth sections of p∗ T M can be considered as the space of smooth functions V : N → T M with V (u) ∈ Tp(u) M for all u ∈ N . We can then define Θ : C ∞ Γp∗ T M → D by Θ(V )(f )(u) = dfp(u) (V (u)). Lemma 2.4.4. Assume that η has constant rank. The map Θ : C ∞ Γp∗ T M → D is a linear bijection. Proof. Let d ∈ D. Fix u ∈ N . The map from C ∞ M to R given by f 7→ df (u) is a derivation at p(u); here the action of any f ∈ C ∞ M on R is multiplication by f (p(u)), and so corresponds to a tangent vector, V (u) say, in Tp(u) M . Then df (u) = dfp(u) (V (u)). By assumption df (u) is smooth in u, and so by suitable choices of f we see that V is smooth. Thus Θ(V ) = d and Θ has an inverse.  From these lemmas we see there exists b ∈ C ∞ Γp∗ T M with the property that  
f (u) = dfp(u) b(u) (2.28) B f˜ − Af for all u ∈ N and f ∈ C ∞ M . Assume that b has its image in the subbundle E of T M determined by η. Using the horizontal lift map h determined by B, define a vector field bH on N :
bH (u) = hu b(u) . Proposition 2.4.5. Assume that η has constant rank and that b has its image in the subbundle E determined by η. The vector field bH is such that B − bH is over A, and so B descends cohesively. Proof. For f ∈ C ∞ M , f + df (b(−)) − df ◦ T p(bH (−)) = Af f (B − bH )(f˜) = Af
using the fact that T p bH (−) = b(−). We can now extend the decomposition theorem:



28

Chapter 2. Decomposition of Diffusion Operators

Theorem 2.4.6. Let B be a diffusion operator on N which descends cohesively over p : N → M . Then B has a unique decomposition: B = BH + BV into the sum of diffusion operators such that (i) BV is vertical, (ii) B H is cohesive and Tu p is injective on the image of σuB

H

for all u ∈ N .

With respect to the induced semi-connection B H is horizontal. Proof. Using the notation of the previous proposition we know that B − bH is over a cohesive diffusion operator A. By Theorem 2.2.5 we have a canonical decomposition B − bH = B1 + B V , leading to B = (bH + B 1 ) + B V . If we set B H = bH + B 1 we have a decomposition as required. On the other hand if we have two such decompositions of B we get two decompositions of B − bH . Both components of the latter must agree by the uniqueness in Theorem 2.2.5, and so we obtain uniqueness in our situation.  Extending Definition 2.2.3 we could say that a diffusion operator B H satisfying condition (ii) in the theorem has no vertical part. Note that if we drop the hypothesis that bH is horizontal, or equivalently that b in Proposition 2.4.5 has its image in E, we still get a decomposition by taking an arbitrary lift of b to be bH but we will no longer have uniqueness.

2.5

Horizontal lifts of paths and completeness of semi-connections

A semi-connection on p : N → M over a subbundle E of T M gives a procedure for horizontally lifting paths on M to paths on N as for ordinary connections but now we require the original path to have derivatives in E; such paths may be called E-horizontal. Definition 2.5.1. A Lipschitz path σ ˜ in N is said to be a horizontal lift of a path σ in M if • p◦σ ˜ = σ, • the derivative of σ ˜ almost surely takes values in the horizontal subbundle H of T N .

2.5. Horizontal lifts of paths and completeness of semi-connections

N

•

σ ˜ (t)

•

σ ˜ (t)

29

σ ˜ (t) •

σ(t)

M

Note that a Lipschitz path σ : [a, b] → M with σ(t) ˙ ∈ Eσ(t) for almost all a 6 t 6 b has at most one horizontal lift from any starting point ua in p−1 (σ(a)). To see this first note that any such lift must satisfy σ ˜˙ (t) = hσ˜ (t) σ(t). ˙

(2.29)

This equation can be extended to give an ordinary differential equation on all of N . For example take a smooth embedding j : M → Rm into some Euclidean space. Set β(t) = j(σ(t)). Let X(x) : Rm → Ex be the adjoint of the restriction of the derivative Tx j of j to Ex , using some Riemannian metric on E. Then σ satisfies the differential equation ˙ x(t) ˙ = X(x(t))(β(t))

(2.30)

and it is easy to see that the horizontal lifts of σ are precisely the solutions of ˙ u(t) ˙ = hu(t) X(p(u(t)))(β(t)) starting from points above σ(a) and lasting until time b. In the generality in which we are working there may not be any such solutions, for example because of “holes” in N . We define the semi-connection to be complete if every Lipschitz path σ with derivatives in E almost surely has a horizontal lift starting from any point above the starting point of σ. Note that completeness is assured if the fibres of N are compact, or if an X, with values in E, and β, can be found so that σ is a solution to equation (2.30) and there is a complete metric on N for which the horizontal lift of X is bounded on the inverse image of σ under p. In particular the latter will hold if p is a principal bundle and we have an equivariant semi-connection as in the next chapter. It will also hold if there is a complete metric on N for which the horizontal lift map hu ∈ L(Ep(u) ; Tu N ) is uniformly bounded for u in the image of σ.

30

2.6

Chapter 2. Decomposition of Diffusion Operators

Topological Implications

Although our set-up of intertwining diffusions with a cohesive A seems quite general it implies strong topological restrictions if the manifolds are compact and more general. Here we partially extend the approach Hermann used for Riemannian submersions in [51] with a more detailed discussion in Chapter 6 below. For this let D0 (x) be the set of points z ∈ M which can be reached by Lipschitz curves σ : [0, t] → M with σ(0) = x and σ(t) = z with derivative in E almost surely. Its closure D0 (x) relates to the propagation set for the maximum principle for A, and to the support of the A- diffusion as in Stroock-Varadhan [95], see Taira [99]. Theorem 2.6.1. For B and A as before with A cohesive, take x0 ∈ M and z ∈ D0 (x0 ). Assume the induced semi-connection is complete. Then if p−1 (x0 ) is a submanifold of N so is p−1 (z) and they are diffeomorphic. Also if z is a regular value of p so is x. Proof. Let σ : [0, T ] → M be a Lipschitz E-horizontal path from x to z. There is a smooth factorisation σxA = X(x)X(x)∗ for X(x) ∈ L(Rm ; Tx M ), x ∈ M . Take ˜ : Rm → T N of X. the horizontal lift X By the completeness hypothesis the time dependent ODE on N ,  −1 dys ˜ s )X σ(s)|[ker X(x )]⊥ = X(y (σ(s)) ˙ 0 ds will have solutions from each point above σ(0) defined up to time T and so a flow giving the required diffeomorphism of fibres. Moreover, by the usual lower semi-continuity property of the “explosion time”, this holonomy flow gives a diffeomorphism of a neighbourhood of p−1 (x) in N with a neighbourhood of the fibre above z. The diffeomorphism commutes with p. Thus if one of x and z is a regular value so is the other.  Corollary 2.6.2. Assume the conditions of the theorem and that E satisfies the standard H¨ ormander condition that the Lie algebra of vector fields generated by sections of E spans each tangent space Ty M after evaluation at y. Then p is a submersion all of whose fibres are diffeomorphic. Proof. The H¨ormander condition implies that D0 (x) = M for all x ∈ M by Chow’s theorem (e.g. see Sussmann [98] or [49]. In [49] Gromov shows that under this condition any two points of M can be joined by a smooth E-horizontal curve.  Corollary 2.6.3. Assume the conditions of the theorem and that D0 (x) is dense in M for all x ∈ M and p : N → M is proper. Then p is a locally trivial bundle over M. Proof. Take x ∈ M . The set Reg(p) of regular values of p is open by our properness assumption. It is also non-empty, even dense in M , by Sard’s theorem, and so since D0 (x) is dense, there exists a regular value z which is in D0 (x). It follows from the

2.7. Notes

31

theorem that x ∈ Reg(p), and so p is a submersion. However it is a well-known consequence of the inverse function theorem that a proper submersion is a locally trivial bundle.  Note that we only need Reg(p) to be open, rather than p proper, to ensure that p is a submersion. The density of D0 (x) can hold because of global behaviour, for example if M is a torus and E is tangent to the foliation given by an irrational flow.

2.7

Notes

Intertwined stochastic differential equations Consider p-related stochastic differential equations as in Section 2.3.1: ˜ t ) ◦ dBt + A(u ˜ t ) dt, dut = X(u dxt = X(xt ) ◦ dBt + A(xt ) dt. Then not only are their generators B and A interwined but also the operators B ∧ and A∧ , which they induce on differential forms, and their semigroups P˜.∧ and P.∧ : and P˜ ∧ ◦ p∗ = p∗ ◦ P ∧ . B ∧ ◦ p∗ = p∗ ◦ A∧ Recall that the pull-back f ∗ φ of a differential form φ by a differentiable map f : N → M is given by f ∗ φ(v 1 , . . . , v k ) = φf (u) (Tu f (v 1 ), . . . , Tu f (v k )) for v 1 , . . . , v k in Tu N when φ is a k-form on M . The operators are given by the same formulae as for functions: m

A∧ =

1X L j L j + LA , 2 j=1 X X

(2.31)

where now the Lie derivatives are acting on forms. Thus, for example, LA φ =


d (ηtA )∗ φ |t=0 dt

(2.32)

where η.A denotes the flow of the vector field A. The semi-groups are given by: Pt∧ φ = E(ξt )∗ φ

(2.33)

when the integrals exist, with modifications if the flow is only locally defined. There is a detailed discussion in [36]. That the operators are intertwined is clear from equation (2.32) and the fact that the flows of the vector fields involved are intertwined by p. Similarly the intertwining of the semi-groups comes from the fact that the flows of the SDE’s are intertwined.

32

Chapter 2. Decomposition of Diffusion Operators

In fact there is the stronger result, following in the same way, that the “codifferentials” δˆ and δˆ determined by the two SDE’s are intertwined. These take k-forms to (k − 1)-forms and are defined by: δˆ = −

m X

ιX j LX j

(2.34)

j=1

where ι denotes interior product. It is shown in [36], page 37, that A∧ = −

 1 ˆ δd + dδˆ + LA 2

where d is the usual exterior derivative. There is further discussion about this and its relationships with the more classical results in [71], [105], [106], and [48] in the Notes to Chapter 7, and the Appendix, Section 9.5.2.

Chapter 3

Equivariant Diffusions on Principal Bundles Let M be a smooth finite dimensional manifold and P (M, G) a principal fibre bundle over M with structure group G a Lie group. Denote by π : P → M the projection and Ra right translation by a. Consider on P a diffusion generator B, which is equivariant, i.e. for all f ∈ C 2 (P ; R), Bf ◦ Ra = B(f ◦ Ra ),

a ∈ G. a

Set f a (u) = f (ua). Then the above equality can be written as Bf a = (Bf ) . The operator B induces an operator A on the base manifold M . Set Af (x) = B (f ◦ π) (u),

u ∈ π −1 (x), f ∈ C 2 (M ),

(3.1)

which is well defined since a

B (f ◦ π) (u · a) = B ((f ◦ π) ) (u) = B ((f ◦ π)) (u). Example 3.0.1. One of the simplest examples is obtained from the map p : SO(n+ 1) → S n defined by choosing some point x0 on S n , considering the natural action of SO(n + 1) on the sphere by rotation, and setting p(u) = u.x0 for u ∈ SO(n + 1). This has the natural structure of a principal bundle with group SO(n) when we identify SO(n) with the subgroup of SO(n + 1) which fixes x0 . If we take the bi-invariant Riemannian metric on SO(n + 1) determined by the Hilbert-Schmidt inner product hA, BiH S := trace(B ∗ A) on the Lie algebra so(n + 1), identified with the space of skew-symmetric (n + 1) × (n + 1)-matrices, the projection√ p is a Riemannian submersion, see Chapter 7 below, but onto the sphere of radius 2. It therefore sends the Laplacian of SO(3), which is bi-invariant under the full group √ SO(3), onto the Laplacian of S n ( 2).

K.D. Elworthy et al., The Geometry of Filtering, Frontiers in Mathematics, DOI 10.1007/978-3-0346-0176-4_3, © Springer Basel AG 2010

33

34

Chapter 3. Equivariant Diffusions on Principal Bundles

n+1 be the elementary (n + 1) × (n + 1)- matrix with entries which Let E[p,q] are all zero except for that in the p-th row and q-th column which is 1. Set √1 (E n+1 − E n+1 ). Then {An+1 }1 6 p
√ If we want A to be the usual Laplacian for S n , rather than for S n ( 2), we can modify the metric of SO(3) to be 12 times that given by the Hilbert-Schmidt norm. Denoting the resulting Laplacian by B we then have X 1 2A∗[p,q] 2A∗[p,q] . B= 2 1 6 p
3.1

Invariant Semi-connections on Principal Bundles

Definition 3.1.1. Let E be a subbundle of T M and π : P → M a principal Gbundle. An invariant semi-connection over E, or principal semi-connection in the terminology of Michor, on π : P → M is a smooth subbundle H E T P of T P such that (i) Tu π maps the fibres H E Tu P bijectively onto Eπ(u) for all u ∈ P . (ii) H E T P is G-invariant. Notes. 1. Such a semi-connection determines and is determined by, a smooth horizontal lift: hu : Eπ(u) → Tu P such that (i) Tu π ◦ hu (v) = v, for all v ∈ Ex ⊂ Tx M ; (ii) hu·a = Tu Ra ◦ hu . 2. The action of G on P induces a homomorphism of the Lie algebra g of G with the algebra of left invariant vector fields on P : if A ∈ g, d ∗ u exp(tA), u ∈ P, A (u) = dt t=0 and A∗ is called the fundamental vector field corresponding to A. Note that for a ∈ G: (3.2) A∗ (ua) = ad(a)A∗ (u)

3.1. Invariant Semi-connections on Principal Bundles

35

for ad(a) : g → g the adjoint action of a. Using the splitting (2.5) of Fu our semi-connection determines, (and is determined by), a ‘semi-connection one-form’ $ ∈ L(H + V T P ; g) which vanishes on H has $(A∗ (u)) = A, and $ua T Ra (V ) = ad(a−1 )$u (V )

V ∈ Hu + V Tu P.

(3.3)

3. Let F be an associated vector bundle to P with fibre V . An E-semi-connection on P gives a covariant derivative ∇w Z ∈ Fx for w ∈ Ex , x ∈ M where Z is a section of F . This is defined, as usual for connections, by
˜ u (w)) , ∇w Z = u d(Z)(h u ∈ π −1 (x). Here Z˜ : P → V is ˜ Z(u) = u−1 Z (π(u)) considering u as an isomorphism u : V → Fπ(u) . This agrees with the ‘semiconnections on E’ defined in Elworthy-LeJan-Li, [36], when P is taken to be the linear frame bundle of T M and F = T M . Such semi-connections could be called ‘linear semi-connections’ since they arise from a principal semiconnection via a linear representation, but since we will only deal with such connections when the vector bundle structure is being used we shall usually take as read the qualification ‘linear’. Theorem 3.1.2. Assume σ A has constant rank. Then σ B gives rise to an invariant semi-connection on the principal bundle P whose horizontal map is given by (2.4). Proof. It has been shown that hu is well defined by (2.4). Next we show hu defines a semi-connection. As noted earlier, h defines a semi-connection if (i) Tu π◦hu (v) = v, v ∈ Ex ⊂ Tx M and (ii) hu·a = Tu Ra ◦ hu . The first is immediate by Lemma ∗ 2.1.1 and for the second observe π ◦ Ra = π. So T π ◦ T Ra = T π and (T π) = ∗ ∗ (T Ra ) · (T π) while the following diagram

Tu∗ P

σuB

- Tu P

6 (Tu Ra )∗

∗ Tu·a P

Tu Ra

B σu·a

? Tua P

36

Chapter 3. Equivariant Diffusions on Principal Bundles

commutes by equivariance of B. Therefore ∗

Tu Ra ◦ hu = Tu Ra · σuB (Tu π) ◦ σxA


−1

∗

∗

= Tu Ra · σuB ◦ (Tu Ra ) ◦ (Tu·a π) ◦ σxA
−1 ∗ B = σu·a ◦ (Tu·a π) ◦ σxA = hu·a .


−1 

Curvature forms and holonomy groups etc for semi-connections are defined analogously to those associated two connections, we note the following: Proposition 3.1.3. In the situation of Proposition 2.1.4, suppose A is elliptic, p1 , p2 are principal bundles with groups G1 and G2 respectively, and F is a homomorphism of principal bundles with corresponding homomorphism f : G1 → G2 . Let Γ1 and Γ2 be the semi-connections on N 1 , N 2 determined by B 1 and B 2 . Then: (i) Γ2 is the unique semi-connection on p2 : N 2 → M such that T F maps the horizontal subspaces of T N 1 into those of T N 2 . (ii) If ω j , Ωj are the semi-connection and curvature form of Γj , for j = 1, 2, then F ∗ (ω 2 ) = f∗ ◦ ω 1 and F ∗ (Ω2 ) = f∗ ◦ Ω1 for f∗ : g 1 → g 2 the homomorphism of Lie algebras induced by f . (iii) Moreover f : G1 → G2 maps the Γ1 holonomy group at u ∈ N 1 onto the Γ2 holonomy group at F (u) for each u ∈ N 1 and similarly for the restricted holonomy groups. Proof. Proposition 2.1.4 assures us that T F maps horizontal to horizontal. Uniqueness together with (ii), (iii) come as in Kobayashi-Nomizu [56] (Proposition 6.1 on p79). 

3.2

Decompositions of Equivariant Operators

Take a basis A1 , . . . , An of g with corresponding fields {A∗i }. P k fundamental vector k Write the semi-connection 1-form as $ = $ Ak so that $ are real-valued, partially defined, 1-forms on P . In our equivariant situation we can give a more detailed description of the decomposition in Theorem 2.2.5. Theorem 3.2.1. Let B be an equivariant operator on P and A be the induced operator on the base manifold. Assume that A is cohesive and let B = AH + B V be theX decomposition ofX Theorem 2.2.5. Then B V has a unique expression of the ij form α LA∗i LA∗j + β k LA∗k , where αij and β k are smooth functions on P ,

3.2. Decompositions of Equivariant Operators

37


given by αk` = $k σ B ($` ) , and β ` = δ B ($` ) for $ the semi-connection 1-form on P . Define α : P → g ⊗ g and β : P → g by X X α(u) = αij (u)Ai ⊗ Aj , β(u) = β k (u)Ak . (3.4) These are independent of the choices of basis of g and are equivariant: α(ug) = (ad(g) ⊗ ad(g)) α(u) and β(ug) = ad(g)β(u). Proof. Since every vertical vector field is a linear combination of the fundamental vertical vector fields, Proposition 1.4.7, shows that X X αi,j LA∗i LA∗j + β k LA∗k BV = H

for certain functions αij , β k . For f, g : P → R setting σ := σ B−A , 1 X i,j 1 X i,j α LA∗i LA∗j (f g) − gα LA∗i LA∗j (f ) 2 2 1 X i,j f α LA∗i LA∗j (g) − 2 X = αi,j LA∗i (f )LA∗j (g) X = αi,j df (Ai ∗ )dg(Aj ∗ ).

df (σ(dg)) =

Since $(A∗k ) = Ak , we see that $k (A∗` ) = δk` and $k (σ($` )) =

X

αi,j δik δj` = αk` .

H

Since AH is horizontal σ A has its image in the horizontal tangent bundle and so is annihilated by $k . Thus
(3.5) αk` = $k σ B ($` ) . Note that by the characterisation, Proposition 1.2.1, X X V δB = αi,j LAi ∗ ιAj ∗ + β k ιA k ∗ . V

H

Since $` (A∗ ` ) is identically 1, it follows that δ B ($) = β ` . Again δ A ($` ) = 0 and so (3.6) β ` = δ B ($` ) as required.

38

Chapter 3. Equivariant Diffusions on Principal Bundles

For the last part α and β can be considered as obtained from the extension of the symbol σ B and δ B to g-valued two- and one-forms respectively: α = $(−)σ B $(−) and β = δ B ($(−)). To make this precise consider σuB as a bilinear form and so as a linear map σuB : Tu∗ P ⊗ Tu∗ P → R. The extension is the trivial one given by σuB ⊗ 1 ⊗ 1 : Tu∗ P ⊗ Tu∗ P ⊗ g ⊗ g → R ⊗ g ⊗ g ' g ⊗ g using the identification of Tu∗ P ⊗ g with L(Tu P ; g). Similarly the extension of δ B is δuB ⊗ 1 : Tu∗ P ⊗ g → R ⊗ g ' g. Thus
α(u)(ω ⊗ ω) = σ B ⊗ 1 ⊗ 1 (P23 ω ⊗ ω) where P23 : T ∗ P ⊗ g ⊗ T ∗ P ⊗ g → T ∗ P ⊗ T ∗ P ⊗ g ⊗ g is the standard permutation and βu (ω) = (δuB ⊗ 1)(ω). The equivariance of $, (Rg )∗ $ = ad(g −1 )($),

g ∈ G,

is equivalent to the invariance of $ when considered as a section of T ∗ M ⊗g under T Rg ⊗ ad(g) : T ∗ M ⊗ g → T ∗ M ⊗ g,

g ∈ G.  P Remark 3.2.2. (a) For any equivariant operator of the form B = i,j αij LA∗i LA∗j P k + β LA∗k with (αij (u)) positive semi-definite for each u ∈ P we can define maps α and β by (3.4). Note that α(u) is essentially the symbol of B restricted to the fibre Pπ(u) through u: σuB |Pπ(u) : Tu∗ Pπ(u) → Tu Pπ(u) with $u identifying Tu Pπ(u) with g. Similarly β determines δ B on a basis of sections of (V T P )∗ . (b) Let {ut : 0 6 t D6 ζ} be a B-diffusion on PE. By (3.5), 2αkl (ut ) is the derivative R· k R· $us ◦ dus , 0 $ul s ◦ dus of the integrals of ω k and ω l along of the bracket 0 {ut : 0 6 t < ζ}. See Chapter 4 below for a detailed discussion. Thus α(ut ) is the derivative of the tensor quadratic variation: Z  1 d t $ut ◦ dut ⊗ $ut ◦ dut . α(ut ) = 2 dt 0 Rt Moreover by (3.6) and Lemma 4.1.2 below 0 β(us )ds is the bounded variaRt tion part of 0 $us ◦ dus .

3.2. Decompositions of Equivariant Operators

39

(c) If we fix u0 ∈ P and take an inner product on g we can diagonalise α(u0 ) to write X µn An ⊗ An α(u0 ) = n

where {An : n = 1, . . . dim(g)} is an orthonormal basis. The µn are the eigenvalues of α(u0 )# : g → g obtained using the isomorphism: g ⊗ g → L(g; g) a ⊗ b 7→ (a ⊗ b)# , where (a ⊗ b)# (v) = hb, via. P Note that for g ∈ G, α(u0 · g) = n µn ad(g)An ⊗ ad(g)An . When the dim(g) inner product is ad(G)-invariant, then {ad(g)An }n=1 is still orthonormal # and the {µn }n are the eigenvalues of α(u0 · g) . They are therefore independent of the choice of u0 in a given fibre, (but depend on the inner product chosen). Example 3.2.3. Using the notation of Example 2.2.11, let P be the special orthonormal frame bundle of the two-dimensional horizontal distribution of the Heisenberg group H, whose fibre at (x, y, z) ∈ H are orthogonal frames with values in E = span{X, Y }. Note that E, and so P , is trivialised by the left action of H and using this we shall consider P as the product H × SO(2) with projection (x, y, z, A) 7→ (x, y, z). The actual frames are compositions of the rotation A on (u, v) and the map (u, v) 7→ (u, v, 12 (xu − yv)). Identifying SO(2) with the circle S 1 the bundle P becomes a principal bundle with group S 1 acting on the right to be written as: π : H × S 1 → H. Then the Lie algebra g = s1 , which we shall identify with the imaginary axis iR. The tangent space to S 1 at a general point eiθ of S 1 will then be represented by {ieiθ a : a ∈ R} ⊂ C. An invariant semi-connection on P may be defined by its semi-connection 1form $, which here will be a section of the dual bundle to the bundle E × T S 1 → H × S 1 , but with values in iR. In our situation this corresponds to a horizontal vector field V0 = v1 X + v2 Y . (As a vector potential determines the connection in the gauge-theoretic interpretation of electromagnetism.) The vector field defines such an $ given on E × T S 1 by $(x,y,z,eiθ ) (w, a) = e−iθ a + ihw, V0 (x, y, z)iE . The horizontal subspace HT P for this connection is given at (x, y, z, eiθ ) by: H(x,y,z,eiθ ) T P = {(w, a) : e−iθ a + ihw, V0 (x, y, z)iE = 0}. Consider a path σ on H3 with σ˙ ∈ E. Denote its horizontal lift starting at (σ(0), 1) by σ ˜ (t) = (σ(t), eiθ(t) ), some θ(t). Then ˙ + ihV0 (σ(t)), σ(t)i. 0 = $(σ ˜˙ t ) = iθ(t) ˙

40

Chapter 3. Equivariant Diffusions on Principal Bundles

giving σ ˜ (t) = (σ(t), e−i

Rt 0

hV (σs ),σ(s)ids ˙

).

Note that an E-semi-connection on the principal bundle E allows one to covariantly diffferentiate sections of E in E-directions; see Note 3 at the beginning of Section 3.1. The covariant derivative of such a section U is given by ˆ w U = ∇L ∇ w U + ihV0 (x, y, z), wiE U (x, y, z),

w ∈ E(x,y,z)

(3.7)

where ∇L refers to the left invariant connection on our Lie group H and we are treating E as a complex line bundle via its trivialisation. In fact this shows that our semi-connection on E is the restriction of an Esemi-connection on the full tangent bundle T H of H since ∇L is defined on all C 1 vector fields on H and E ⊂ H so the right-hand side of the formula (3.7) is defined as an element of T H for all C 1 vector fields on H and w ∈ E. As for standard connections, semi-connections are determined by suitable covariant differentiation operators. Next consider A = 12 (X 2 + Y 2 ) and for real-valued functions r1 , r2 , γ on H with γ > r12 + r22 set B=

∂2 ∂2 ∂2 ∂r1 ∂r2 ∂ o ∂2 1n 2 (X +Y 2 )+r1 +r2 +γ 2 +(xr2 −r1 y) +( + ) . 2 ∂x∂θ ∂y∂θ ∂ θ ∂z∂θ ∂x ∂y ∂θ

∂ Now we are treating θ 7→ eiθ as giving charts for S 1 and ∂θ refers to the use of ∂ these local co-ordinates. Thus ∂θ corresponds to the tangent vector whose value at eiθ is ieiθ in our representation in C. We shall find the vector field V0 corresponding to the semi-connection induced by B. If we complete the squares for B, recalling ∂ ∂ ∂ ∂ − 12 y ∂z and ∂y + 12 x ∂z respectively, we obtain: that X and Y are given by ∂x   1  ˜ 2 ˜ 2 1 1 ∂r2 ∂r1 ∂ ∂2 B= X +Y + (γ − r12 − r22 ) 2 − (x −y ) (3.8) 2 2 ∂ θ 2 ∂z ∂z ∂θ

˜ = X + r1 ∂ and Y˜ = Y + r2 ∂ . As vector fields on H × S 1 this gives where X ∂θ ∂θ ˜ = (X, ir1 e−iθ ) and Y˜ = (Y, ir2 eiθ ). X ˜ 2 + Y˜ 2 ) is cohesive and Tu π is injective on the image of its symbol, Now 12 (X which is {(a, b, i(ar1 + br2 )eiθ ) : a, b, ∈ R} ⊂ E(x,y,z) × Teiθ S 1 at u = (x, y, z, eiθ ), ˜ 2 + Y˜ 2 ) is the lift of A. The second and so, by Proposition 2.2.2, we see 21 (X term in equation (3.8) is clearly vertical hence (3.8) is the decomposition for the semi-connection determined by (A, B), using Theorem 2.4.6. That is:   2 1 ∂r2 ∂r1 ∂ 1 ˜2 ˜2 1 H V 2 2 ∂ A = (X + Y ) and B = (γ − r1 − r2 ) 2 − (x −y ) . 2 2 ∂ θ 2 ∂z ∂z ∂θ ˜ := hu (X), Y˜ := (hu Y ) and the horizontal lift map is: We see that X 1 1 h(x,y,z,eiθ ) (a, b, (xb − ya)) = (a, b, (xb − ya), i(r1 a + r2 b)eiθ ). 2 2

3.3. Derivative Flows and Adjoint Connections

41

This horizontal lifting map is determined by the vector field V0 := −r1 X − r2 Y . In particular a couple of choices of r1 and r2 give some interesting semi-connections. Letting r1 = −y, r2 = x, we have ∂2 ∂2 1n 2 ∂2 ∂2 o (X + Y 2 ) − y +x + γ 2 + (x2 + y 2 ) B1 := 2 ∂x∂θ ∂y∂θ ∂ θ ∂z∂θ which determines the semi- connection 1 1 (a, b, (xb − ya)) 7→ (a, b, (xb − ya), i(−ya + xb)eiθ )). 2 2 Taking r1 = x, r2 = y, we have ∂2 ∂2 ∂ o 1n 2 ∂2 (X + Y 2 ) + x +y +γ 2 +2 B2 = 2 ∂x∂θ ∂y∂θ ∂ θ ∂θ with semi-connection (a, b, 12 (xb − ya)) 7→ (a, b, 12 (xb − ya), i(xa + yb)eiθ ). We return to this example in Example 3.4.3 below.

3.3

Derivative Flows and Adjoint Connections

Let A on M be given in H¨ ormander form m

A=

1X L j L j + LA 2 j=1 X X

(3.9)

for some smooth vector fields X 1 , . . . X m , A. As before let Ex be the linear span of {X 1 (x), . . . , X m (x)} and assume dim Ex is constant, denoted by p, giving a subbundle E ⊂ T M . The vector fields {X 1 (x), . . . , X m (x)} determine a vector bundle map X : Rm → T M with σ A = X(x)X(x)∗ . We can, and will, consider X as a map X : Rm → E. Let Yx be the right inverse [X(x)|ker X(x)⊥ ]−1 of X(x) and h, ix the inner product, induced on Ex by ˘ on E Yx . Then X projects the flat connection on Rm to a metric connection ∇ defined by ˘ v U = X(x)d[y 7→ Yy U (y)](v), ∇

U ∈ C 1 ΓE, v ∈ Ty M.

(3.10)

(In [36] we have studied the properties of this construction together with the SDE ˘ is referred as the LW connection for the SDE; see also induced by X, and there ∇ Section 9.5.2 in the Appendix.) Moreover any connection ∇ on a subbundle E of T M has an adjoint semi-connection ∇0 on T M over E defined by ∇0U V = ∇V U + [U, V ],

U ∈ ΓE, V ∈ ΓT M.

42

Chapter 3. Equivariant Diffusions on Principal Bundles

Remark 3.3.1. Note the converse of this is not true. In the discussion of Example 3.2.3 we noted shortly after equation (3.7) that our construction determined a semi-connection over E on the the whole tangent bundle. This could not be the adjoint of any connection on E because brackets of E-valued vector fields are not E-valued in general. In fact it is noted in [36] that parallel translation using the adjoint of a connection on a subbundle E of a tangent bundle can only preserve E if E is integrable. Let π : GLM → M be the frame bundle of M , so u ∈ π −1 (x) is a linear isomorphism u : Rn → Tx M . It is a principal bundle with group GL(n). If g ∈ GL(n) and π(u) = x, then u · g : Rn → Tx M is just the composition of u with g. Any smooth vector field A on M determines smooth vector fields AT M and GL on T M and GLM respectively as follows: Let ηt : t ∈ (−, ) be a (partial) A flow for A and T ηt its derivative. Then v 7→ T ηt (v) is a partial flow on T M and u 7→ T ηt ◦ u one on GLM , Let AT M and AGL be the vector fields generating these flows. In fact AT M is τ ◦ T A : T M → T T M where τ : T T M → T T M is the canonical twisting map: τ (x, v, w, v 0 ) = (x, v, v 0 , w) in local coordinates. Using this, the choice of our H¨ ormander form representation induces a diffusion operator B on GLM by setting B=

1X L(X j )GL L(X j )GL + LAGL . 2

This is invariant under the action of GL(n)and π intertwines B and A. For cohesive A, a principal semi-connection is therefore induced on GLM . In turn this induces a semi-connection on T M with covariant derivative ∇, say, as described in Note 3 of Section 3.1. We will also use ∇ to denote these semi-conections. For w ∈ Ex , set Z w (y) = X(y)Yx (w). Theorem 3.3.2. Assume the diffusion operator A given by (3.9) is cohesive and let B be the operator on GLM determined by A. Let E be the image of σ A , a vector bundle. ˘ given by (3.10). (a) The semi-connection ∇ induced by B is the adjoint of ∇ Consequently ∇w V = LZ w V for any vector field V and w ∈ Ex , (b) For u ∈ GLM , identifying gl(n) with L(Rn ; Rn ),    1 X  −1 ˘ u(−) X p ⊗ u−1 (−)∇ ˘ u(−) X p , u (−)∇ 2 1 −1 ˘ # 1 X −1 ˘ p ˘ u(−) A. u ∇∇ β(u) = − Ric u(−) + u−1 ∇ ˘ u(−) X p X − u 2 2

α(u) =

3.3. Derivative Flows and Adjoint Connections

43

˘ considered as an operator ˘ # : T M → E is the Ricci curvature of ∇ Here Ric from T M to E, defined by ˘ # (v) = Ric

m X


˘ v, X j (x) X j (x) R

j=1

˘ the curvature operator of ∇. ˘ for R Proof. The first part can be deduced from the stochastic flow results in Chapter 8 but we give a direct proof here. Let πte be the flow of X(·)(e). It induces a linear ˜ map X(u) : Rm → Tu GLM on the general linear bundle GLM : ˜ X(·)e = [X(·)(e)]GL , d ˜ X(u)(e) = (T Ste ◦ u)|t=0 , dt

u ∈ GLM.

˜ m = Rm and so `u = Y (p(u))X(p(u)). If We can apply Lemma 2.3.1 with R x = p(u) and e ⊥ ker[X(x)], then the horizontal lift map hu defined by Theorem 3.1.2 is d ˜ (T πte ◦ u) . (3.11) hu (X(x)(e)) = X(u) (`u (e)) = dt t=0 Note this will not hold in general if e ∈ ker[X(x)]. ˙ ∈ Eσ(t) each t. Then Let σ : [0, T ] → M be a C 1 curve with σ(t) ˙ Z σ(t) (x) := X(x)Yσ(t) σ(t). ˙ σ ˙ Let Ss,t be the flow, from time s to time t, of the time dependent vector field Z σ(t) . σ Now Ss,t (σ(s)) = σ(t) for 0 6 s 6 t 6 T . Also, for any torsion free connection and any v ∈ Tσ(s) M ,
D σ ˙ σ ˙ T Ss,t (v) = ∇Z σ(t) (v) |t=s = ∇v Z σ(s) . T Ss,t dt t=s

Thus

D σ(t) ˙ σ Z T S σ (v) = ∇T S0,t . dt 0,t If $ is the connection form of this torsion free connection, then  
−1 D D σ σ T S0,t T S σ (u0 (e))] $ ◦ u0 = [e 7→ T S0,t ◦ u0 dt dt 0,t
−1 σ σ(t) ˙ σ u (e) Z ◦ u0 ∇T S0,t ] = [e 7→ T S0,t 0   σ ◦u (σ(t)) ˙ = $ hT S0,t 0

44

Chapter 3. Equivariant Diffusions on Principal Bundles


d σ σ ◦u (σ(t)) ◦ u0 and hT S0,t T S0,t by (3.11), showing that the vertical parts of dt 0 ˙ equal. On the other hand, using this auxiliary connection, the horizontal parts of
d σ σ ◦u (σ(t)) ◦ u ˙ are both equal to the horizontal lift of σ(t). ˙ T S and hT S0,t 0 0,t 0 dt Thus
d σ σ ◦u (σ(t)) T S0,t ◦ u0 = hT S0,t ˙ 0 dt σ ◦ u0 : 0 6 t 6 T } is the horizontal lift of {σ(t) : 0 6 t 6 T } and so {T S0,t with respect to the semi-connection induced by B. However by Lemma 1.3.4 in σ σ (v) of S0,t is the parallel translation of v along σ by Elworthy-LeJan-Li [36], T S0,t ˆ the adjoint semi-connection ∇ of the LeJan-Watanabe connection on E associated σ to X and {T S0,t ◦ u0 : 0 6 t 6 T } is the horizontal lift of {σ(t) : 0 6 t 6 T } with ˆ This proves the first claim. And ∇w V = LZ w V by Lemma 1.3.4 of respect to ∇. Elworthy-LeJan-Li [36]. For the last part let $ : H ⊕ V T GLM → g = L(Rn ; Rn ) be the semiconnection 1-form. For u0 ∈ GLM , set ut = T ξt ◦ u0 where {ξt } is a local flow for the stochastic differential equation dxt = X(xt ) ◦ dBt + A(xt )dt

(3.12)

on M where {Bt } is a Brownian motion on Rm . (This defines the derivative flow on GLM .) As for ordinary connections $(◦dut ) = u−1 t

ˆ D (ut −) ∈ L(Rn ; Rn ). dt

Here, on the right-hand side ut is differentiated as a process of linear maps ut ∈ L(Rn ; Txt M ) over (xt ). [It suffices to check the equality for C 1 curves (ut ) with ˜t · gt for x ˜t a xt = π(ut ) having x˙ t ∈ Ext , t > 0. For this we can write ut = x ˆ −1 d D ˜t dt (˜ xt ut −).] horizontal lift of {xt } and gt ∈ G. Then observe that dt (ut −) = x However as in [36], u−1 t

ˆ D −1 ˘ ˘ (ut −) = u−1 t ∇ut − X ◦ dBt + ut ∇ut − Adt. dt

From this the formula for α(u) follows by Remark 3.2.2(b). For β(u) we need Rt to identify the bounded variation part of 0 $(◦dut ). For this write −1 −1 ˆ ˆ −1 ∇ ˘ ˘ T ξ ◦u X ◦ dBt u−1 t ∇ut − X ◦ dBt = u0 Tx0 ξt //t ◦ //t t 0

where //ˆt is the parallel translation along {ξs (x0 ) : 0 6 s 6 t} using our semi˘ by Theorem 3.3.2. As in [36] connection, which is the adjoint of ∇ −1 −1 −1 ˘ T ξ ◦u X ◦ dBt = //ˆt ∇ ˘ T ξ u XdBt − 1 //ˆt Ric# (T ξt ◦ u0 −)dt //ˆt ∇ t 0 t 0 2

3.3. Derivative Flows and Adjoint Connections

45

while −1 ˆ u−1 0 T ξt //t

=

u−1 0

Z − 0

t −1 ˘ u−1 ˆ −X 0 T ξs ∇// s

Z ◦ dBs − 0

t −1 ˘ u−1 ˆ − Ads 0 T ξs ∇//

giving the formula claimed for β.

s



Example: Gradient Brownian SDE An isometric immersion j : M → Rm of a Riemannian manifold M determines a stochastic differential equation on M : dxt = X(xt ) ◦ dBt where X(x) : Rm → Tx M is the orthogonal projection and B. is a Brownian motion on Rm . More precisely X(x)(e) = ∇[y 7→ hj(y), i](x). It is well known that the solutions of the SDE are Brownian motions on M , see [30], [92], [31], and the equation is often called a “gradient Brownian SDE” . Moreover the LW connection given by equation (3.10) is the Levi-Civita connection, (by the classical construction of the latter), see [36]. Since the adjoint of the LeviCivita connection is itself, Theorem 3.3.2, shows that our connection induced on GLM by the derivative flow of a gradient Brownian system is also the Levi-Civita connection. Almost by definition, h∇v X p , wiRm = ha(v, w), ep iRm

(3.13)

where a : T M × T M → Rm is the second fundamental form of the immersion with v ∈ Tx M, x ∈ M, e ∈ Rm (3.14) ∇v X(e) = A(v, nx e) for nx : Rm → Tx M ⊥ the projection and A : T M ⊕ T M ⊥ → T M the shape operator given by hA(v, e), wiRm = ha(v, w), ep iRm . Here T M ⊥ refers to the normal bundle of M and Tx M ⊥ to the normal space at x to M , though we are considering its elements as being in the ambient space Rm . Thus the vertical operator in the decomposition of the generator of the derivative flow on GLM for gradient flows is given by Theorem 3.3.2 with m−n 1 X −1 u A(u−, lj ) ⊗ u−1 A(u−, lj ), α(u) = 2 j=1

β(u) = −

m−n 1 X 1 A(A(u−, lj ), lj ) − u−1 Ric# (u−) 2 j=1 2

46

Chapter 3. Equivariant Diffusions on Principal Bundles

at a frame u over a point x. Here l1 , . . . , lm−n denotes an orthonormal base for Tx M ⊥ . For the standard embedding of S n in Rn+1 we have a(u, v) = hu, vix for u, v ∈ Tx S n . Also the Ricci curvature is given by Ric# (v) = (n − 1)v for all v ∈ T M . Thus for the standard gradient SDE on S n , at any frame u we have 1 Id ⊗ Id 2 1 β(u) = − n Id. 2

α(u) =

3.4

(3.15) (3.16)

Associated Vector Bundles and Generalised Weitzenb¨ ock Formulae

As before let π : P → M be a smooth principal G-bundle and ρ : G → L(V ; V ) a C ∞ representation of G on some separable Banach space V . There is then the (possibly weakly) associated vector bundle π ρ : F → M where F = P × V / ∼ for the equivalence relation given by (u, e) ∼ (ug, ρ(g −1 )e) for u ∈ P , e ∈ V , g ∈ G. If [(u, e)] ∈ F denotes the equivalence class of (u, e) we can identify any u ∈ P with a linear isomorphism ¯ : V → Fπ(u) u by ¯ (e) = [(u, e)]. u

(3.17)

Consider the set of smooth maps from P to V which are equivariant by ρ: Mρ (P ; V ) = {smooth Z : P → V, Z(ug) = ρ(g)−1 Z(u), u ∈ P, g ∈ G}. There is the standard bijective correspondence Fρ between Mρ (P, V ) and Γ(F ), the space of smooth sections of F defined by ¯ [Z(u)], Fρ (Z)(x) = u

u ∈ π −1 (x), Z ∈ Mρ (P ; V ).

Via this map, an equivariant diffusion generator B on P induces a differential operator B ρ ≡ Fρ (B) on Γ(F ), of order at most 2, by Fρ (B)(Fρ (Z)) = Fρ [B(Z)],

Z ∈ Mρ (P ; V ).

(3.18)

Here B has been extended trivially to act on V -valued functions. Note that the definition makes sense since,
B(Z)(ug) = B (Z ◦ Rg ) (u) = B ρ(g)−1 Z (u) = ρ(g)−1 B(Z)(u).

3.4. Associated Vector Bundles and Generalised Weitzenb¨ock Formulae

47

For such a representation ρ let ρ∗ : g → L(V ; V ) be the induced representation of the Lie algebra g (the derivative of ρ at the identity). Theorem 3.4.1. When B is a vertical equivariant diffusion generator the induced operator on sections of any associated vector bundle is a zero-order operator. With the notation of Theorem 3.2.1, the zero-order operator in Γ(F ) induced by B is represented by λρ : P → L(V ; V ) for
λρ (u) = ρ∗ (β(u)) + Comp (ρ∗ ⊗ ρ∗ )(α(u)) ,

u∈P

(3.19)

for Comp : L(V ; V ) ⊗ L(V ; V ) → L(V ; V ) the composition map A ⊗ B 7→ AB. Proof. The operator B ρ is a zero-order operator if F ρ (B)(S)(x0 ) = F ρ (B)(S 0 ) whenever two sections S and S 0 of F agree at x0 . This holds if B(f Z) = f B(Z) for any invariant function f : P → R and V -valued function Z on P . But this holds by Remark 1.4.8. For the representation (3.19), suppose Z : P → V is equivariant: Z(u ◦ g) = ρ(g)−1 Z(u),

g ∈ G.

Then d Z(u · eAj t ))|t=0 dt d ρ(e−Aj t )Z(u)|t=0 = dt = −ρ∗ (Aj )Z(u).

LA∗j (Z)(u) =

Iterating we have B(Z)(u) = proving (3.19).

X

αij (u)ρ∗ (Aj )ρ∗ (Ai )Z(u) +

X

βk ρ∗ (Ak )Z(u) 

From this theorem we easily have the following estimate, which combined with the discussion in Section 3.5 below, when applied to the associated bundle ∧F to the orthonormal bundle, shows that the Weitzenb¨ock curvature is positive if the curvature is. Corollary 3.4.2. If ρ is an orthogonal representation, i.e. (ρ∗ (α))∗ = −ρ∗ (α) for all α ∈ g, then λρ (v, v) 6 0 for all v ∈ V .

48

Chapter 3. Equivariant Diffusions on Principal Bundles

Proof. Write α = v ∈ F,

P

k

µk Ak ⊗ Ak where {Ak } is as in Remark 3.2.2(c). Then for

X hComp ◦(ρ∗ ⊗ ρ∗ )(α(u))(v), vi = h µk [ρ∗ Ak ]2 (v), vi X =− µk hρ∗ (Ak )(v), ρ∗ (Ak )(v)i 6 0, since µk > 0. The result follows from (3.19) since ρ∗ (β(u)) is skew symmetric.  The situation of Corollary 3.4.2 arises when considering the derivative flow for an SDE on a Riemannian manifold whose flow consists of isometries ; for example canonical SDE’s on symmetric spaces as in Section 7.2 below and [36]. Example 3.4.3. We use the notation in Example 3.2.3. Let P be the special orthonormal frame bundle for E over the Heisenberg group H, with group S 1 . We 1 use the left action of the Heisenberg group to trivialise it to H  × S . Denote by  cos t − sin t 1 2 2 1 it ρ : S → L(R ; R ) the representation of S given by ρ(e ) = . sin t cos t ∗ Let s = s1 X + s2 Y be a section of Γ(E), identified as s1 + is2 . Let ρ (s) be the induced equivariant map from P → R2 : ρ∗ (s)(p, eiθ ) = e−iθ s(p),

p ∈ H.

Define (B ρ s)(p) = eiθ B(ρ∗ (s))(p, eiθ ). As in Example 3.2.3 take B=

∂2 ∂2 ∂2 ∂r1 ∂r2 ∂ o ∂2 1n 2 (X +Y 2 )+r1 +r2 +γ 2 +(xr2 −r1 y) +( + ) , 2 ∂x∂θ ∂y∂θ ∂ θ ∂z∂θ ∂x ∂y ∂θ

and recall that then H

A

1 ˜2 ˜2 1 = (X + Y ) and B V = 2 2

  2 1 ∂r2 ∂r1 ∂ 2 2 ∂ (γ − r1 − r2 ) 2 − (x −y ) , ∂ θ 2 ∂z ∂z ∂θ

with semi-connection determined, as in Example 3.2.3, by the E-valued vector field V0 given by V0 (x, y, z) = −r1 X − r2 Y. From this B ρ = (B V )ρ + (AH )ρ with ∂r1 ∗ 1 1 ∂r2 −y )ρ (s)(p, eiθ ) B V (ρ∗ (s))(p, eiθ ) = − (γ − r12 − r22 )ρ∗ (s)(p, eiθ ) + i (x 2 4 ∂z ∂z and so ∂r1 1 1 ∂r2 −y )s(p). (B V )ρ (s)(p) = − (γ − r12 − r22 )s(p) + i (x 2 4 ∂z ∂z

(3.20)

3.4. Associated Vector Bundles and Generalised Weitzenb¨ock Formulae

49

We leave as an exercise the computation which verifies that 1 ˆ −∇ ˆ − (s) trace ∇ 2
1 L 2 4L s + i traceh∇L = − V0 , −iE + 2i∇V0 s − |V0 | s , 2

(AH )ρ (s) =

(3.21) (3.22)

recalling from Example 3.2.3 that the covariant derivative of our connection is given by ˆ w U = ∇L ∇ w U + ihV0 (x, y, z), wiE U (x, y, z),

w ∈ E(x,y,z)

L in terms of the left-invariant covariant derivative. Also 4L denotes trace ∇L − ∇− . Let us relate this to Theorem 3.4.1. Let   0 1 1 ρ∗ : t ∈ s → t −1 0

be the induced representation on the Lie algebra of S 1 . Consider B V , the vertical part of B. Now in the notation of Theorem 3.2.1 α1,1 (u) = 12 (γ − r12 − r22 ), and ∂r1 2 β(u) = 14 (x ∂r ∂z − y ∂z ) and hence     ∂r1 1 1 ∂r2 0 0 1 0 1 (γ − r12 − r22 ) −y ) + (x −1 −1 0 −1 0 2 4 ∂z ∂z ∂r1 1 ∂r2 1 −y ) = − (γ − r12 − r22 ) + i (x 2 4 ∂z ∂z

λρ (u) =

1 0



which is indeed the multiplication operator appearing in formula (3.20) for (BV )ρ .  We use the following conventions, as in [36]. Let V be an N -dimensional real inner product space. For 1 6 i 6 n, a1 ∧ · · · ∧ an =

ιv (u1 ∧ · · · ∧ uq ) =

1 X sgn (π)aπ(1) ⊗ · · · ⊗ aπ(n) , n! π

q X (−1)j+1 hv, uj iu1 ∧ · · · ∧ ubj ∧ · · · ∧ uq

(3.23)

j=1

h⊗ai , ⊗bi i = n!Πi hai , bi i, and h∧ai , ∧bi i = det(hai , bj i). Let ∧V stand for the exterior algebra of V and a∗j the “creation operator”on ∧V given by a∗j v = ej ∧v for (e1 , . . . , eN ) an orthonormal basis for ∧V . Let aj be its adjoint, the “annihilation operator” given by aj = ıej . Note the commutation law: ai a∗j + a∗j ai = δij .

(3.24)

50

Chapter 3. Equivariant Diffusions on Principal Bundles

If A : V → V is a linear map on V , there are the operators ∧A and (dΛ)(A) on ∧V , which restricted to ∧p V are: p

(dΛ )(A) (u1 ∧ · · · ∧ up ) =

p X

u1 ∧ · · · ∧ uj−1 ∧ Auj ∧ uj+1 ∧ · · · ∧ up ,

1

and also (∧p A)(u1 ∧ · · · ∧ up ) = Au1 ∧ · · · ∧ Aup . A useful formula for A ∈ L(V ; V ) is X dΛ(A) = Aij a∗i aj .

(3.25)

i,j

Note that since α(u) is symmetric, (ρ∗ ⊗ ρ∗ )α(u) : V ⊗ V → V ⊗ V has X αij (u)ρ∗ (Ai ) ⊗ ρ∗ (Aj )(v 1 ∧ v 2 ) (3.26) (ρ∗ ⊗ ρ∗ )α(u)(v 1 ∧ v 2 ) = i,j

=

X

αij (u) ρ∗ (Ai )v 1 ∧ ρ∗ (Aj )v 2 .

(3.27)

ij

and so (ρ∗ ⊗ ρ∗ )α(u) restricts to a map of ∧2 V to itself. Quantitative estimates can be obtained by some representation theory. For example suppose G = O(n) with ρ the standard representation on Rn . Consider the representation ∧k ρ on ∧k Rn . Corollary 3.4.4. Take the Hilbert-Schmidt inner product on so(n) and let 0 6 µ1 (x) 6 · · · 6 µ(x) 12 n(n−1) be the eigenvalues of α on the fibre p−1 (x), x ∈ M , as described in Remark 3.2.2(c). Then for all V ∈ ∧k Rn , D k E 1 1 − k(n − k)µ 12 n(n−1) (x) 6 λ∧ (u)V, V 6 − k(n − k)µ1 (x). 2 2 Proof. Following Humphreys [52], §6.2, consider the bilinear form β on so(n) given by   β(A, B) = trace (d∧k )(A)(d∧k )(B) =

(n − 2)! trace(AB) (k − 1)!(n − k − 1)!

by a short calculation using elementary matrices. By Remark 3.2.2(c) since our inner product on so(n) is ad(O(n))-invariant we can write 1 2 n(n−1)

α(u) =

X

µl (x)Al (u) ⊗ Al (u)

l=1

with x = p(u) and {Al (u)}l an orthonormal base for so(n) at each u ∈ P .

3.4. Associated Vector Bundles and Generalised Weitzenb¨ock Formulae

51

For each u ∈ P , set A0l (u) =

(k − 1)!(n − k − 1)! Al (u) (n − 2)!

to ensure β(A0l (u), Aj (u)) = δlj for each u. Then D E X D E k ∧k Comp ◦(ρ∧ = µl (x) (d∧k )Al (u) ◦ (d∧k )Al (u)V, V ∗ ⊗ ρ∗ )(α(u))V, V E X h (k − 1)!(n − k − 1)! i−1 D = (d∧k )Al (u) ◦ (d∧k )A0l (u)V, V (n − 2)! l

6−

 (n − 2)! c k V, V , (k − 1)!(n − k − 1)! ∧

where c∧k = (d∧k )Al (u) ◦ (d∧k )A0l (u), the Casimir element of our representation d∧k of so(n). Since the representation is irreducible, (for example see [12] Theorem 15.1 page 278), this element is a scalar, and we have, see Humphreys [52], c∧k =

n(n − 1) . . . (n − k + 1) dim so(n) 1 . = n(n − 1)/ k n dim ∧ R 2 k!

k

Thus λ∧ (u) 6 − 12 k(n − k)µ1 . The lower bound follows in the same way.



When B has an equivariant H¨ ormander form representation the zero-order operator F ρ (V ) can be given in a simple way by (3.28) below. This was noted for the classical Weitzenb¨ ock curvature terms using derivative flows in Elworthy [32]. Proposition 3.4.5. SupposePB lies over aPcohesive operator A and has a smooth H¨ ormander form: B = 21 βk LY 0 with the vector fields Y j , j = LY j LY j + j 1, . . . , m, being G-invariant. Let (ηt ) be the flow of Y j . For a representation ρ of G with associated vector bundle π ρ : F → M the zero-order operator F ρ (B V ) corresponding to the vertical component of B is given by F ρ (B V )(x0 ) =

m 1 X D2 j D 0 −1 η (u ) ◦ (¯ u ) + η (u ) ◦ (¯ u0 )−1 0 0 0 2 j=1 dt2 t dt t t=0 t=0

(3.28)

for any u0 ∈ π −1 (x0 ). Proof. Set ujt = ηtj (u0 ) ∈ P and σ(t) = π(ujt ) so u ¯jt ∈ L(V ; Fσ(t) ). From Remark 3.2.2(b), m 1X $(Y j (u0 )) ⊗ $(Y j (u0 )) α(u0 ) = 2 j=1

52

Chapter 3. Equivariant Diffusions on Principal Bundles

and so (ρ∗ ⊗ ρ∗ )α(u0 ) =

m D j D j 1X ¯t ¯ (¯ u0 )−1 u ⊗ (¯ u0 )−1 u 2 j=1 dt t=0 dt t t=0

as in the proof of Theorem 3.3.2. Also from equation (3.6) β(u0 ) =

m   1X 1 LY j $(Y j (−) (u0 ) + $(Y 0 (−) (u0 ). 2 j=1 2

Let (//t ) denote parallel translation in F along σ. Then    d ρ∗ LY j $(Y j (−) (u0 ) = ρ∗ $ Y j (ujt ) dt t=0 d  j −1 D j  (¯ ut ) u ¯ = dt dt t t=0  D  d (//t−1 ujt )−1 //t−1 ujt = dt dt t=0 D D D2 j j j = −¯ u−1 ut ◦u ¯−1 ut +u ¯−1 u 0 0 0 dt t=0 dt t=0 dt2 t t=0 leading to the required result via Theorem 3.4.1.



To examine particular examples we will need to have detailed information about the zero-order operators determined by a vertical diffusion generator. For this suppose B is vertical and given by X X B= αij LA∗i LA∗j + βk LA∗k for α : P → g ⊗ g and β : P → g as in Theorem 3.2.1 and (3.4). Motivated by the Weitzenb¨ ock formula for the Hodge-Kodaira Laplacian on differential forms, see Corollary 3.4.9 below, [93], [22], we shall examine in more detail the case of the exterior power ∧ρ : G → L(∧V ; ∧V ) of a fixed representation ρ showing that λ∧ρ has expressions in terms of annihilation and creation operators which are structurally the same as these of the Weitzenb¨ock curvature (which are shown to be a special case in Corollary 3.4.9). For notational convenience we give V an inner product in what follows. Lemma 3.4.6. If B is a vertical operator on P and (ei , i = 1, 2, . . . , N ) is an orthonormal basis of V , the zero-order operator on the associated bundle ∧F → M is represented by λ∧ρ : P → L(∧p V ; ∧p V ) with N X

λ∧ρ (u) =

h((ρ∗ ⊗ ρ∗ )α(u)) (ej ⊗ el ), ei ⊗ ek i a∗i aj a∗k al

i,j,k,l=1

+

N X

h(ρ∗ β(u))ej , ei ia∗i aj ,

i,j=1

u ∈ P.

3.4. Associated Vector Bundles and Generalised Weitzenb¨ock Formulae

53

Proof. Recall that if A ∈ L(V ; V ), then dΛ(A) =

N X

hAej , ei ia∗i aj ,

(3.29)

i,j=1

e.g. see Cycon-Froese-Kirsch-Simon [22]. Consequently dΛ(ρ∗ β(u)) =

N X

hρ∗ β(u)ej , ei ia∗i aj .

(3.30)

i,j=1

On the other hand by Theorem 3.2.1 and (3.4), we can represent α as: X an,m (u)An ⊗ Am α(u) = n,m

where {Ai }N i=1 is a basis of g. So Comp ◦(∧ρ∗ ⊗ ∧ρ∗ )(α(u)) X an,m (u) dΛ(ρ∗ Am ) ⊗ dΛ(ρ∗ An ) = Comp ◦ m,n

=

X

an,m (u)dΛ(ρ∗ Am ) ◦ dΛ(ρ∗ An )

m,n

=

X

an,m (u)

m,n

=

N X

hρ∗ Am ej , ei ihρ∗ An el , ek ia∗i aj a∗k al

i,j,k,l=1

N X 1X an,m (u) h(ρ∗ Am ⊗ ρ∗ An ) (ej ⊗ el ), ei ⊗ ek i a∗i aj a∗k al 2 m,n i,j,k,l=1

=

1 2

N X

h(ρ∗ ⊗ ρ∗ )α(u)(ej ⊗ el ), ei ⊗ ek i a∗i aj a∗k al ,

i,j,k,l=1

since our convention for the inner product on tensor products gives hu1 ⊗ v1 , u2 ⊗ v2 i = 2hu1 , u2 ihv1 , v2 i. The desired conclusion follows. 2

 2

Theorem 3.4.7. Let R(u) : ∧ V → ∧ V be the restriction of 2(ρ∗ ⊗ ρ∗ )α(u) : V ⊗ V → V ⊗ V , then X λ∧ρ (u) = − hR(u)(ej ∧ el ), ei ∧ ek i a∗i a∗k aj al i
+

N X

h(ρ∗ ⊗ ρ∗ )α(u)(ej ⊗ el ), ei ⊗ ej i a∗i al +

i,j,l=1

X i,j

hρ∗ β(u)ej , ei i (ai )∗ aj .

54

Chapter 3. Equivariant Diffusions on Principal Bundles

This can be rewritten as: X hR(u)(ej ∧ el ), ei ∧ ek i a∗i a∗k aj al + d ∧ (Z ρ (u)) + d ∧ (ρ∗ β(u)). λ∧ρ (u) = − i
(3.31) where Z ρ (u) ∈ L(V ; V ) is defined by N X

 hZ (v1 ), v2 i = (ρ∗ ⊗ ρ∗ )(α(u))(ej ⊗ v1 ), v2 ⊗ ej V ⊗V . ρ

j=1

Proof. This follows from Lemma 3.4.6 using the anti-commutation formula (3.24) since N X

h(ρ∗ ⊗ ρ∗ )α(u)(ej ⊗ el ), ei ⊗ ek i a∗i aj a∗k al

i,j,k,l=1

=−

N X

h(ρ∗ ⊗ ρ∗ )α(u)(ej ⊗ el ), ei ⊗ ek i a∗i a∗k aj al

i,j,k,l=1

+

N X

h(ρ∗ ⊗ ρ∗ )α(u)(ej ⊗ el ), ei ⊗ ej i a∗i al

i,j,l=1

=−

N X

hR(u)(ej ∧ el ), ei ∧ ek i a∗i a∗k aj al

j
+

N X

h(ρ∗ ⊗ ρ∗ )α(u)(ej ⊗ el ), ei ⊗ ej i a∗i al .



i,j,l=1

Remark 3.4.8. (a) Note that the second term in (3.31) in general depends on the symmetric part of (ρ∗ ⊗ ρ∗ )(α(u)) as well as on R. (b) If we write α(u) =

X

µk (u)Ak (u) ⊗ Ak (u)

as in Remark 3.2.2(c), Then Z ρ (u) in (3.31) has X Z ρ (u) = µk (u)ρ∗ (Ak (u))ρ∗ (Ak (u)). k

Corollary 3.4.9. For the derivative process in GLM of a cohesive generator A given in H¨ ormander form without a drift, the zero-order operator induced by the vertical diffusion on the exterior bundles ∧T M is minus one half times the generalised ˘ ∗ : ∧∗ T M → ∧∗ E, given by: Weitzenb¨ock curvature, R X ˘ q V = d ∧q (Ric# )(V ) − 2 Rikjl a∗l a∗j ak ai V (3.32) R 16i
3.4. Associated Vector Bundles and Generalised Weitzenb¨ock Formulae

55

for all VL∈ ∧q T M . Here Rikjl = hR(ei , ek )el , ej i , 1 6 i, k 6 n, 1 6 j, l 6 p for R : TM T M → L(E, E) the curvature transform of the associated connection on E, and if V ∈ ∧q Tx M the set {e1 , . . . , ep } is an orthonormal base for Ex which together with ep+1 , . . . , en forms an orthonormal base for some inner product on Tx M extending that of Ex . Proof. By Theorem 3.3.2, α(u) =



1 X −1 u ∇u(−) X p ⊗ u−1 ∇u(−) X p , 2

u ∈ GLM.

By Corollary C.5 in [36] the restriction 12 R(u) of α(u) to anti-symmetric tensors corresponds to one half of the curvature operator 12 R : ∧2 T M → ∧2 E composed with the inclusion of ∧2 E into ∧2 T M . By the relation between the curvature transform and the curvature operator: hR(v 1 ∧ v 2 ), w1 ∧ w2 i = hR(v 1 , v 2 )w2 , w1 i, the first term in λρ (u) of Theorem 3.4.7 corresponds to: X X hR(ei ∧ ek ), ej ∧ el i a∗j a∗l ai ak = − − 16i
Riklj a∗j a∗l ai ak

16i
=

X

Rikjl a∗l a∗j ak ai

16i
by the skew-symmetry of Riklj in i, k and in j.l and the anti-commutation of annihilation operators. By (ii) of Remark 3.4.8, the second term corresponds to   m X 1 d ∧q  ∇∇− X j X j  . 2 j=1 The required result follows since β(u) = −

m   1  1 X −1  u ∇∇u(−) X j X j − u−1 Ric# u(−) . 2 j=1 2



Corollary 3.4.9 reflects the results in [36], Theorem 2.4.2, concerning Weitzenb¨ock formulae for H¨ ormander form operators on differential forms. In particular it ˘ is the Levi-Civita gives another approach to the result that when E = T M and ∇ connection, as holds for gradient stochastic differential equations, the generator induced on differential forms by the derivative process is a constant times the HodgeKodaira Laplacian for the induced Riemannian structure on M up to a first-order ˘ q given by equation term, see [36]. This comes from identifying the adjoint of R ˘ (3.32) when ∇ is the Levi-Civita connection, with the standard Weitzenb¨ock term

56

Chapter 3. Equivariant Diffusions on Principal Bundles

ock formula for the Hodge-Kodaira LaplaRq , say, which appears in the Weitzenb¨ cian on q-forms: (3.33) 4q φ ≡ −(dd∗ + d∗ d)φ = ∇∗ ∇φ − Rq φ. and is the zero-order operator given by X Rq φ ≡ R• φ = Rijkl (ai )∗ aj (ak )∗ al φ see [22], or [93] where the opposite sign is used. Here the annihilation and creation operators acting on a form φ are denoted by ai and (ai )∗ so that (ai φ)(v) = φ(a∗i v) = φ(ei ∧ v) and ((ai )∗ φ)(v) = φ(ai v) = φ(ιei v) = (e∗i ∧ φ)(v). One of the standard ways of writing Rq is with the decomposition ˜ (4) φ Rq φ = φ ◦ d ∧q (Ric# ) + R for φ a q-form, where ˜ (4) = − R

X

Rijkl (ai )∗ (ak )∗ aj al .

This follows from the definition of Rq using the anti-commutation relation (3.24) and the formula (3.25). For example see [22] page 260. In [36] we showed that the second term in the right-hand side of equation ˜ (4) . There was an error in the sign of this term in the (3.32) corresponds to R statement of Theorem 2.4.2 of [36] and in the discussion of its relationship with ˜ (4) , and for completeness we repeat the argument. R Working with the Levi-Civita connection and using the anti-commutatitvity of the creation operators we have ˜ (4) ≡ − R =−

n X

Rijkl (ai )∗ (ak )∗ aj al

i,j,k,l=1 n n X X

[Rijkl − Rkjil ](ai )∗ (ak )∗ aj al

1 6 i
=−

XX [Rijkl − Rkjil − Rilkj + Rklij ](ai )∗ (ak )∗ aj al . i
However by Bianchi’s identity: Rijkl + Riljk + Riklj = 0 we have Rijkl − Rkjil = Rikjl , and interchanging j and l: −Rilkj + Rklij = −Riklj = Rikjl .

3.4. Associated Vector Bundles and Generalised Weitzenb¨ock Formulae

57

Thus ˜ (4) = −2 R

XX

Rikjl (ai )∗ (ak )∗ aj al .

i
˘ q V in equation Taking adjoints we obtain the second term in the expression for R (3.32), as claimed. Note that if B is the operator on GLM determined by the H¨ormander form (3.9) of A, then for a representation ρ : GL(M ) → L(V ; V ) with associated π ρ : GL(n) → L(V ; V ) the induced operator F ρ (B) on sections of π ρ is also given P 1 by the ‘H¨ormander form’ 2 j LX j LX j + LA , where for any C 1 vector field Y on M and any C 1 section U of π ρ the Lie derivative LY U ∈ ΓF is given by ¯ (LY U )(x) = u

−1   d Y Tηt ◦ u U ηtY (x) dt t=0

for x ∈ M , u a frame at x, and (ηtY ) the flow of Y , using the notation of (3.17). ¯ U (π(u)), so U = F ρ (Z), Indeed by (3.17), for Z(u) = u F ρ (B)(U ) = F ρ

while L(X j )GL (Z)(u) =

h 1 X 2

d Xj dt Z(T ηt

 i L(X j )GL L(X j )GL + LAGL (Z)

j

◦ u)

so that

t=0

i h j d ¯ Z(T ηtX ◦ u) F ρ L(X j )GL (Z) (x) = u = LX j (U )(x). dt t=0 This representation of F ρ (B) was noted in the case of the operator induced on differential forms by a stochastic flow in [36], and for the case of the Hodge-Kodaira Laplacian in Elworthy [32], see also Kusuoka [60]. Remark 3.4.10 (Elementary matrices, ∧2 Rn and so(n)). We always give so(n) its Hilbert-Schmidt inner product hA, Bi = trace B ∗ A. Using the standard basis n with E[p,q] (v) = vq ep . e1 , . . . , en of Rn let E[p,q] denote the elementary matrix E[p,q] 1 As in Example 3.0.1 set A[p,q] = √2 [E[p,q] − E[q,p] ] so that {A[p,q] : 1 6 p < q 6 n} forms an orthonormal basis for so(n). We have the isomorphism, but not isometry, of ∧2 Rn with so(n) given by mapping ep ∧ eq to 12 (E[p,q] − E[q,p] ) in accordance n n n with our √ usual interpretation of elements of R ⊗ R as linear operators on R . Then 2ep ∧ eq corresponds to A[p,q] . Example 3.4.11. Let P be the orthonormal frame bundle for a Riemannian metric on M . Let C : P → L(Rn ; Rn ) satisfy C(ug) = g −1 C(u)g for g ∈ O(n) with C(u) a symmetric map for all u ∈ P . Define α : P → so(n) ⊗ so(n) by X α(u) = tracehC(u)A[p,q] −, A[p0 ,q0 ] −iRn A[p,q] ⊗ A[p0 ,q0 ] , 1 6 p
58

Chapter 3. Equivariant Diffusions on Principal Bundles

where the {A[p,q] } are as in Remark 3.4.10 above. Then α corresponds to an equivariant operator B V , say, on P , namely, at u ∈ P , X tracehC(u)A[p,q] −, A[p0 ,q0 ] −iRn LA∗[p,q] LA∗[p0 ,q0 ] , BV = 1 6 p
and

1 1 Comp ◦α(u) = − (trace C(u))id + (2 − n)C(u). 4 4

Let Ric# : T M → T M be the Ricci curvature (for the Levi-Civita connection, −1 −) for u ∈ P . If the Ricci curvature is nonsay)and take C(u) = u Ric# π(u) (u V negative the operator B is a vertical diiffusion operator whose induced zero-order 1 term on vector fields is 14 (2 − n) Ric# π(u) − 4 k, where k is the scalar curvature. Proof. Since α(ug) = (ad(g) ⊗ ad(g)) α(u) for g ∈ O(n) the operator B V is equivariant. To compute Comp ◦α(u) first observe that tracehC(u)A[p,q] −, A[p0 ,q0 ] −iRn =

1 hC(u)A[p,q] + A[p,q] C(u), A[p0 ,q0 ] iso(n) 2

and so, because the {A[p,q] } form an orthonormal base, α=

1 X CA[p,q] ⊗ A[p,q] + A[p,q] C ⊗ A[p,q] . 2

Then we use the elementary fact about elementary matrices: E[p,q] CE[p0 ,q0 ] = Cqp0 E[p,q0 ]

3.5



Notes

G-invariant diffusion operators Suppose a diffusion operator on N is invariant under the action of a Lie group G. Even if the action is not principal in the sense that the quotient mapping N → N/G is not a principal bundle it may be possible to classify the orbits and give partial skew-product decompositions. This is discussed in L´azaro-Cami & Ortega’s article,[62], with special reference to stochastic Hamiltonian systems, and their analysis could easily be combined with the use of the connections described here. Canonical vertical diffusions on GLM We have seen in Corollary 3.4.9 that there is a zero-order operator on the associated bundle ∧F → M represented by the Weitzenb¨ock curvature of a given connection. On the other hand, given a curvature operator R of a metric connection, or more

3.5. Notes

59

generally an operator which has the same symmetry properties as a curvature tensor, is there a canonical vertical diffusion operator on GLM which induces zero-order operators on differential forms which have the form of the Weitzenb¨ock curvatures of R? A vertical operator with such a zero-order term always exists since we can take R in a diagonal form: R(u) =

N X

An (u) ∧ An (u),

(3.34)

n=1

for some An : GLM → gl(n) which are ad(G)-invariant, e.g. by taking an isometric embedding (e.g. see [36]). In this case let (ej ) be a basis of Eπ(u) and define N 1X An (u) ⊗ An (u), α(u) = 2 n=1

(3.35)

p N 1X 1X β(u) = − (An (u))2 − R(−, ej )ej , 2 n=1 2 j=1

see Remark 3.4.8(b). Then α is positive and we can define an operator with its coefficients these α and β. For a discussion of the representation of R in the form of (3.34) see Kobayashi-Nomizu [56] (Notes 17 and 18). In particular there is a discussion there of the number N required and of a rigidity theorem originating from Chern. See also Berger-Bryant-Griffiths [10]. When M is Riemannian with positive semi-definite curvature operator R : ∧2 T M → ∧2 T M there is a canonical construction. For this take the orthonormal frame bundle π : OM → M , with G = O(n). We will use the isomorphism of ∧2 Rn with so(n) described in Remark 3.4.10. Define α : OM → so(n) ⊗ so(n) by α(u) =

1 4

X

D E R(∧2 (u)(ep ∧eq )), ∧2 (u)(ep0 ∧eq0 ))

π(u)

16p6q6n,16p0 6q 0 6n

A[p,q] ⊗A[p0 ,q0 ] .

Our representation ρ is just the identity map and, by (3.27) and Bianchi’s identity, the restriction of α(u) : Rn ⊗ Rn → Rn ⊗ Rn to ∧2 Rn is just 12 R . In the notation of (3.31) we see 1 hZ ρ (v 1 ), v 2 i = − Ric(v 1 , v 2 ). 2 If we take β = 0, we obtain from (3.31) that X ρ λ∧ (u) = − Rjlik a∗i a∗k aj al − 2 (d∧)Ric# . i
To get the full Weitzenb¨ ock term, extend α over GLM by equivariance and define β(u), for u ∈ GLM , by β(u) = 32 u−1 Ric# (u−) as in (3.35).

Chapter 4

Projectible Diffusion Processes and Markovian Filtering S Let M + be the Alexandrov one-point compactification, M ∆, of a smooth manifold M . Consider the space CM + of processes (yt ) on M + with explosion time ζ, such that t → yt is continuous with yt = ∆ when t > ζ. We shall use Cy0 M + to denote those processes starting at a given point y0 of N . Let L be a diffusion operator on M and let {Py0 , y0 ∈ M + } be the family of L-diffusion measures in the sense of [53], i.e. the solution to the martingale problem on CM + so the canon+ ical process (yt , 0 6 t < ζ) with the system of diffusion measures {PL y0 , y0 ∈ N } + is a strong Markov process on M . Denote by E mathematical expectation with respect to the measure Py0 . We may add to these notations the relevant subscripts or superscripts indicating the diffusion operator or the Markov process concerned, L L,y0 or even Ey0 . e.g. {PL y0 }, ζ , E For y0 ∈ M and f ∈ Cc∞ M , the space of smooth functions on M with compact support, let Mtdf := Mtdf,L := f (yt∧ζ ) − f (y0 ) −

Z

t∧ζ

Lf (ys )ds.

(4.1)

0

Then (Mtdf : 0 6 t < ∞) is a martingale on the probability space (C(M ), PL y0 ) with respect to the {Fty0 }, where Fty0 = σ{ys ; 0 6 s 6 t}. Moreover it has bracket df

t∧ζ

Z

σ L ((df )ys , (df )ys )ds.

hM it = 2 0

This definition extends to the case of C 2 functions f but then Mtdf is only defined for 0 6 t < ζ L and is a local martingale.

K.D. Elworthy et al., The Geometry of Filtering, Frontiers in Mathematics, DOI 10.1007/978-3-0346-0176-4_4, © Springer Basel AG 2010

61

62

4.1

Chapter 4. Projectible Diffusion Processes and Markovian Filtering

Integration of predictable processes

Proposition 4.1.1. Let τ be a stopping time with τ < ζ and let {αt : 0 6 t < τ } be an F∗y0 predictable process in T ∗ M such that αt ∈ Ty∗t M for each t ∈ [0, τ ), and for each compact subset C of M we have Z τ χC (ys )αs (σ L αs ) ds < ∞ 0

almost surely. Then there is a unique local martingale {Mtα : 0 6 t < τ } such that for all f ∈ Cc∞ M , Z t 

α df σ L (αs , (df )ys ) ds, t < ζ. (4.2) M ,M t = 2 0

Proof. We can write αt =

m X

gtj · dfj (yt ),

(4.3)

j=1

where the functions g j are predictable real-valued processes, for example by taking m and gtj = αt (X j (yt )), for X(x) = (f P1m, . . . , ifm ) : M → R to be an embedding m to Tx M . Using a partition of unity, at the i=1 X (x)ei the projection from R cost of having an infinite, but locally finite sum, we can assume that the fj in the representation are all in Cc∞ M . Define XZ t gsj dMsdfj . (4.4) Mtα := 0

j

Clearly (4.2) holds. For uniqueness suppose K is a local martingale orthogonal to M df for all f ∈ Cc∞ M . Then K vanishes since the martingale problem for L is well posed by an argument attributed to Dellach´erie (see Rogers-Williams [92], the end of the proof of Theorem 2.5.1). In fact it it were not zero we could take a suitable stopping time τ to ensure (1 + Kτ ∧t )PL x0 solves the martingale problem up to time t since   Z s Kτ ∧t Msdf ≡ Kτ ∧t f (xs ) − f (x0 ) − Lf (xs )ds , 06s6t 0

is a uniformly integrable martingale.



We will often write Mtα =

Z

t

αs d{ys }

(4.5)

0

bringing out the fact that it is the martingale part of the Stratonovitch integral Rt α ◦ dy s s of (αt ) along the diffusion process (yt ) when that integral is defined, 0 e.g. when (αt ) is a continuous semi-martingale. Indeed

4.1. Integration of predictable processes

63

Lemma 4.1.2. Let α be a C 2 1-form, then Z t Z t
α αys ◦ dys − δ L α (ys )ds, Mt = 0

0 6 t < ζ.

(4.6)

0

Proof. This is clear for an exact 1-form. Suppose λ : M → R is C 2 and α is exact, then for t < ζ, Z ·  Z t Z t 1 λα α α α Mt = λ(ys )dMs = λ(ys ) ◦ dMs − dλ(ys )dys , M· 2 0 0 0 t Z t Z t
1 = λ(ys ) αys ◦ dys − λ(ys ) δ L α (ys )ds − hM·dλ , M·α it 2 0 0 Z t Z t = λ(ys ) αys ◦ dys − δ L (λα)(ys )ds 0

0

since M dλ is the martingale part of λ(ys ) and hM

dλ

α

Z

, M it = 2

t

σ L (dλs , αs )ds.

0

This proves the result for general α by taking a suitable representation.



Let Sx be the image of σxL in Tx M and let S := ∪x Sx . By a predictable S -valued process (αt ) over (yt : 0 6 t < ζ) we mean a process (αt : 0 6 t) such that ∗

(i) αt ∈ Sy∗t for all 0 6 t < ζ, (ii) (αt ◦ σyLt , 0 6 t < ζ) is a predictable process in T M , canonically identified with T ∗∗ M . Note that condition (ii) is equivalent to (ii)0 there exists a predictable (α ¯ t ) in T ∗ M over (yt ) such that α ¯ t |Syt = αt for all 0 6 t < ζ. That (ii0 ) implies (ii) is immediate. To see (ii) implies (ii0 ) first note that αt ◦ σyLt ∈ Syt for each t since αt ◦ σyLt = σyLt (α ˜ t ) for any extension α ˜ t of αt to Ty∗t M . We can then choose a measurable selection α ¯ t in Ty∗t M with σyLt (¯ αt ) = αt ◦ σyLt . This 0 process α ¯ t will satisfy the requirements of (ii ) since α ¯ t σyLt = σyLt α ¯ t = αt σyLt . Definition 4.1.3. If (αt ) satisfies (i) and (ii) we will say it is in L2L if Z 0

t

αs σyLs (αs )ds < ∞

(4.7)

64

Chapter 4. Projectible Diffusion Processes and Markovian Filtering

for all t > 0, and will say it is in L2L,loc if for any compact subset C of M t∧ζ

Z

χC (ys )αs (σ L αs ) ds < ∞

E 0

for all t > 0. Remark 4.1.4. Suppose the processes associated to diffusion operators L and L+Lb are both non-explosive, where b is a locally bounded measurable vector field on M . Assume that there exists a T ∗ M -valued measurable process b# . defined on the canonical probability space Cy0 M such that PL -almost surely: 1. 2σ L (b# s ) = b(ys ), Rt # L # 2. 0 bs σ (bs )ds < ∞. Then, by the GMCM-theorem, as in the Appendix, Section 9.1, we have on C([0, T ]; M ), PL+Lb = Zt PL Rt # L # where Zt = exp{Mtb − 0 b# s σ (bs )ds}. In an obvious notation, for suitable α, as canonical processes we have, almost surely, Z t Z t Z t L L+Lb αs d{ys } = αs d{ys } − αs (b(us ))ds. 0

0

0

Lemma 4.1.5. Suppose σ L has its image in a subset S of T M . Then (Mtα ) depends only on the restriction of αs in L(Tys M ; R) to Sys , 0 6 s < ζ. In particular (4.2) defines uniquely a local martingale for each predictable S ∗ -valued process (αt ) over (yt )for which the right-hand side of (4.2) is always finite almost surely. Proof. For T ∗ M -valued F∗y0 predictable processes (αt1 , 0 6 t < ζ) and (αt2 , 0 6 t < ζ) over (yt , 0 6 t < ζ) which agree on S we see hM

α1

−M

α2

Z

df

t∧ζ

σ(αs1 − αs2 , (df )ys ) ds = 0

, M it = 2 0 1

2

for all f ∈ Cc∞ M . Therefore M α = M α . On the other hand this also shows that if αs ∈ Sy∗s for all s, we can use condition (ii0 ) above to choose a predictable process {¯ αs : 0 6 s < ζ} with values in T ∗ M over (yt ) and set M·α = M·α¯ without ambiguity.  Example 4.1.6 (Canonical Brownian motion associated to a cohesive diffusion). For simplicity assume that our L-diffusion from a given point y0 is non-explosive. If L is cohesive with subbundle E of TM, take a metric connection Γ for E, using the metric determined by 2σ L . Let αs (σ) := (//sσ )−1 : Eσ(s) → Ey0

4.1. Integration of predictable processes

65

be the inverse of parallel translation, //sσ , along σ from Eσ(0) to Eσ(s) , for Py0 almost all paths σ in M . Each component of this with respect to an orthonormal basis for Ey0 clearly lies in L2L . With the obvious extension of our notation to the vector-space-valued case define an Ey0 -valued process Bt : t > 0 by Bt = Mtα =

t

Z

(//s )−1 d{ys }.

0

It is easy to check from its quadratic variation that it is a Brownian motion on the inner product space Ey0 . Moreover (as described in [36]) it has the same filtration as the canonical process on Cy0 M up to sets of measure zero. It is the martingale Rt part of the stochastic anti-development 0 (//s )−1 dys of our L-diffusion from y0 . The use of a different metric connection would change it bya random rotation, so this process is defined on the canonical probability space Cy0 M, F y0 , Py0 and up to such rotations depends only on it. We have, for α as usual: t

Z

Z αs d{ys } =

0

t

(αs ◦ //s ) dBs .

(4.8)

0

Using the definitions in Appendix 9.3 we see that if our diffusion process y. is a Γ-martingale, then Z t Z
t αs d{ys } = Γ αs dys . (4.9) 0

0

Note that there is always some metric connection Γ on E for which a cohesive diffusion process is a Γ-martingale, by section 2.1 of [36]. Example 4.1.7. In Example 2.2.13, the martingales Mtdvi associated with the process ut are independent Wiener processes which we will denote by Wti , and we have: vi (ut ) = vi (u0 ) + Wti Z wi (ut ) = wi (u0 ) + α

(4.10) t


vi+1 (us )dWsi+2 − vi+2 (u)dWsi+1 .

0

In particular, it follows that: Z

t


v2 (us )dWs3 − v3 (us )dWs2 , and by Itˆo’s formula, 0   Z t w1 (ut ) = w1 (u0 ) + α −v2 (ut )Wt3 − v3 (u0 )ν2 (ut ) + 2 v2 (us )dWs3 , 0   Z t 3 3 w2 (ut ) = w2 (u0 ) + α v1 (u0 )Wt + v3 (u0 )ν1 (ut ) − 2 v1 (us )dWs .

w3 (ut ) = w3 (u0 ) + α

0

66

Chapter 4. Projectible Diffusion Processes and Markovian Filtering

4.2

Horizontality and filtrations

We can characterise horizontality of a diffusion operator or process in terms of filtrations using the following lemma: Lemma 4.2.1. Suppose p : N → M is a smooth map, B a smooth diffusion operator over a smooth diffusion operator A, and also (i) σ A and σ B have constant rank and (ii) the filtration generated by u· and p(u· ) agree up to sets of PB u0 -measure zero for some u0 ∈ N . A for all u ∈ N . Then rank σuB = rank σp(u)

Proof. Set p = rank σxA and p˜ = rank σuB . By assumption p and p˜ do not depend on x ∈ M and u ∈ N . Take connections on Image σ B and Image σ A which are metric for the metrics induced by the symbols. Extend these connections to T N and T M . The martingale part of the stochastic anti-development of (u· ) will be a Brownian motion stopped at ζ B of dimension p˜ and that of (p(u· )) will be one of dimension p. By (ii) these have the same filtration up to sets of measure zero. But this implies p = p˜ by the martingale representation theorem, as required.  Proposition 4.2.2. The following are equivalent for B over A when A is cohesive: (a) B = AH , (b) B is cohesive and the filtration generated by its associated diffusion (u· ) agrees with that of p(u· ) up to sets of PB u0 -measure zero for given u0 in N . Proof. If (b) holds, Lemma 4.2.1 shows that Image[σuB ] = Hu for each u ∈ N , since by (2.3) we always have Hu ⊂ Image[σuB ]. Thus (b) implies criterion (ii) of Proposition 2.2.2. Also (b) follows from (iii) of Proposition 2.2.2 by considering ˜ m. ˜ 0, . . . , X  the stochastic differential equation driven by horizontal lifts X

4.3

Intertwined diffusion processes

Let p : N → M be a smooth surjective map. Suppose that B is over A. However A we do not assume σ A of constant rank. Let {PB u0 } and {Px0 } be, respectively, the solutions to the martingale problem for B and A on the canonical spaces CM + and CN + . Denote by (ut ) and (xt ) the corresponding canonical processes with explosion time ζ N and ζ M respectively. Note that ζ N 6 ζ M ◦ p almost surely with respect to PB u0 . We shall assume that the paths of the diffusion on N do not explode before their projections on M do, more precisely ζ M ◦ p = ζ N almost surely with respect to PB u0 for each u0 , equivalently, • Assumption S. Cup0 M + := {σ : [0, ∞) → M + : lim p(ut ) = ∆ when ζ N (u. ) < ∞} t↑ζ N

4.3. Intertwined diffusion processes

67

has full PB u0 measure for each u0 ∈ N . This assumption holds for all the examples considered earlier. There are two ways it may fail, given the standing assumption that p is surjective. These are exemplified by p : U → R, the projection p(x, y) = x of certain open sets U of R2 onto R. If B is the usual Laplacian on U , with N = U , then Assumption S fails if (i) U = {(x, y) ∈ R2 : x2 + y 2 > 1} and if (ii) U = {(x, y) ∈ R2 : y < 1}. Denote by the following the filtrations induced by the processes indicated: Ftu0 = σ(us , 0 6 s 6 t), Ftx0 = σ(xs , 0 6 s 6 t), p(u0 )

Ft

= σ(p(us ), 0 6 s 6 t),

F u0 = σ(ys , 0 6 s < ∞), F x0 = σ(xs , 0 6 s < ∞), F p(u0 ) = σ(p(us ), 0 6 s < ∞).

A B Proposition 4.3.1. Under Assumption S, p∗ (PB u0 ) = Pp(u0 ) and Pt (f ◦ p) = A ∞ Pt (f ◦ p) for all f ∈ Cc (M ).

Proof. If p(u0 ) = x0 , f ∈ Cc∞ (M ), we only need to show that Mtdf,A is a martingale with respect to p∗ (PB u0 ). Using Assumption S, Mtdf,A (p(u))

Z

t

= f (p(ut )) − f (p(u0 )) −

Af ◦ p(us ))ds 0

Z = f (p(ut )) − f (p(u0 )) −

t

(B(f ◦ p)) (us )ds 0

d(f ◦p),B

= Mt

x0 is a martingale with respect to (Ftu0 ) and PB u0 . Take s 6 t and let G be a Fs measurable function. Then o o n n B B Ep∗ (Pu0 ) Mtdf,A G = EPu0 Mtdf,A (p(u· ))G(p(u· )) o o n n B B d(f ◦p),B G ◦ p = EPu0 Msd(f ◦p),B G ◦ p) = EPu0 Mt  B = Ep∗ (Pu0 ) Msdf,A G

and the required result follows from the uniqueness of the martingale problem for A.  We will need the following elementary lemma: Lemma 4.3.2. Let (Ω, F, Ft , P} be a filtered probability space and G∗ a subfiltration of F∗ with the property that for all s > 0, E{A|Gs } = E{E{A|Fs }|G}, where G = ∨s Gs . Then

∀A ∈ F,

(4.11)

68

Chapter 4. Projectible Diffusion Processes and Markovian Filtering

(i) (E{Mt |G}, t > 0) is a G∗ -martingale whenever (Mt : t > 0) is an F∗ martingale; (ii) For all G-measurable and integrable H,  E H|Fs } = E{H|Gs };   (iii) E E{A|Fs }|G = E E{A|G}|Fs ,

∀A ∈ F.

Proof. For (i) set Nt = E{Mt |G}, 0 6 t < ∞. By (4.11), (Nt ) is Gt measurable. For s 6 t suppose that f is Gs -measurable and bounded. Then E(Nt f ) = E(Mt f ) = E(Ms f ) = E(Ns f ). For (ii), let H and F be bounded measurable functions with G-measurable and Fs -measurable representations. Then        E H|F = E H|E F |G = E H|E F |Gs = E H|E F |G  using (4.11). Thus  E H|Fs } = E{H|Gs } as required. Part (iii) follows from (ii) on taking H = E E{A|G} and using equation (4.3.2).  p(u0 )

Part (ii) of the following proposition says that the filtration F∗ mersed in the filtration F∗u0 in the terminology of Tsirelson [102]. Proposition 4.3.3. tion. Then

is im-

(i) For fixed t > 0 let f be a bounded Ftu0 -measurable funcn o n o p(u ) E f |F p(u0 ) = E f |Ft 0 .

p(u )

(ii) All F∗ 0 martingales are F∗u0 martingales. In fact if f = G ◦ p for G an integrable functional on C(M + ) with respect to PA , we have p(u0 )

Eu0 {f |Ft

} = Eu0 {f |Ftu0 }.

Proof. (i) Write f = F (us : 0 6 s 6 t) for F a bounded measurable function on CN + . Let G be bounded measurable functions of {p(us ) : 0 6 s 6 t} and g 1 , . . . , g k bounded Borel functions on M , with h1 , . . . , hk positive real numbers. By the Markov property of u· and of p(u· ),
E F (us : 0 6 s 6 t) G g 1 ◦ p(ut+h1 ) · · · · · g k ◦ p(ut+h1 +···+hk )

= E F (us : 0 6 s 6 t)G PhA1 g 1 PhA2 (g 2 . . . PhAk g k ) (p(ut )) . Therefore, n o n o p(u) E F (us : 0 6 s 6 t)|F p(u) = E F (us : 0 6 s 6 t)|Ft as required. Part (ii) is immediate from (i) by Lemma 4.3.2.



4.3. Intertwined diffusion processes

69

As in §2.1 set Ex = Image σxA with hu : Ep(u) → Tu N the horizontal lift defined by (2.3), although now we have no constant rank assumption and so no smoothness of h. Also let EuB = Image σuB . For an F∗x0 -predictable E ∗ -valued process φt := φt (σ· ), 0 6 t < ζ M along (σt : 0 6 t < ζ M ) let (p∗ (φt ) : 0 6 t < ζ N ) be the pull-back restricted to be an (E B )∗ -valued process along (ut : 0 6 t < ζ N ) defined by p∗ (φt )(u· ) = φt (p(u)) ◦ Tut p : E B ut → R. Since φt has a predictable extension φ¯t so does p∗ (φt ) and so the latter is predictable. Moreover p∗ (φt )σ B (p∗ φt ) = φt σ A (φt ) by Lemma 2.1.1 showing φ· is in L2A if and only if p∗ (φ· ) is in L2B . For such φ we have the following intertwining: Proposition 4.3.4. Let φ be a predictable L2A -valued process. B,p∗ (φ)

A,φ (1) For PB ◦ p = Mt u0 almost surely all sample paths, Mt

for t < ζ N .

(2) If α ∈ L2B with αt ◦hut = 0 almost surely for all t < ζ N , then hMtα , Mtdf ◦T p i = 0 and EB,u0 {Mtα |F p(u0 ) } = 0 for all C 1 functions f on M . Proof. For φ = df , (1) follows from p∗ (df )u = d(f ◦ p)u as in the proof of Proposition 4.3.1. For general φ, taking a predictable extension if necessary, write Pm i j j φt (x) = 1 gt (x· )(df )xt for smooth functions f : M → R and real-valued j M predictable {gt : 0 6 t < ζ }. Therefore m Z t X B,p∗ (df j ) B,p∗ (φ) gsj (p(u· )) dMt = Mt MtA,φ ◦ p = j=1

0

for all t < ζ N , giving (1). For (2) let F : N → R be a smooth measurable function with respect to F p(u0 ) . Then F = f (p(u· )) for some measurable function f : M → R. Z 1 B,u0 1 B,u0 t B df ◦T p B,u0 α α (Mt f (p(u· ))) = E hMt , Mt i= E σ (αs , df ◦ T p(us ))ds. E 2 2 0 If αt hut = 0 almost surely for all t, we apply (2.3) to see   A σ B (αs , df ◦ T p(us )) = αs σuBs T ∗ p(df ) = αs hus σp(u df = 0 s) and thus EB,u0 (Mtα f (p(u· ))) = 0 giving (2). L2B

B,u0



define βs ≡ E {αs ◦ hus |p(u· ) = x· }, 0 6 s < ζ to be the For α ∈ unique, up to equivalence, element of L2A such that     (4.12) EB,u0 αs ◦ hus σ A (φs (p(u· ))) = EA,p(u0 ) βs σ A (φs ) for any φ ∈ L2A . To see that such an element exists and is unique, recall that A αs ◦ hus σp(u = αs σuBs (tus p)∗ s)

70

Chapter 4. Projectible Diffusion Processes and Markovian Filtering

which is an F∗u0 -predictable process with values in Ep(us ) ⊂ Tp(us ) M at each time s, and by Proposition 4.3.3, (4.12) is equivalent to n o A B,u0 B ∗ p(u0 ) βs (p(u· ))σp(u = E σ (T p) |F α (4.13) s u u s ) s s in the sense of Elworthy-LeJan-Li [36]. The predictable projection theorem and the results of [36] show that there is a unique, up to indistinguishability, F p(u· ) predictable T M version {γt : 0 6 t < ζ} say, over {p(ut ) : 0 6 t < ζ}, of the righthand side of (4.13). By applying the uniqueness part of this projection theorem p(u ) to {φs (γs ) : 0 6 s < ζ} when φ· is F∗ · -predictable, T ∗ M -valued over p(u· ) and φt vanishes on Ep(ut ) for all 0 6 t < ρ with probability 1, we see γt ∈ Ep(uu ) for A ∗ ]−1 γs in Ep(u . all 0 6 t < ζ almost surely. Now set βs (p(u· )) = [σp(u s) s) Proposition 4.3.5. For any α· in L2B we have B,u0

E

{Mtα

Z | p(u· ) = x· } =

T

EB,u0 {αs ◦ hus | p(u· ) = x· } d{xs }.

0

¯t (p(u)) for {N ¯t } a Ftx0 Proof. Set Nt = E{Mtα | F p(u0 ) } and write Nt (u) = N p(u· ) measurable function. By Proposition 4.3.3, (Nt ) is an F∗ -martingale and we ¯t ) is an F x martingale. Take g ∈ C ∞ M , then by Proposition 4.3.4, see (N ∗ c n o ¯ , M A,dg it ◦ p(u) = EB,u0 hM α , M d(g◦p) it |Ftp(u0 ) (u) hN n o p(u ) = EB,u0 σuBt (αt , (Tut p)∗ (dg)) |Ft 0 n o p(u ) A (dg)|Ft 0 = EB,u0 αt ◦ hut σp(u t) by equation (2.3). By Proposition 4.1.1 and the definition  above of the conditional ¯t (p(u· )) = M A,β for β ◦ p(u· ) = EB,u0 αt ◦ hu |F p(u0 ) and so expectation, N t ¯t (x· ) = N

Z

t

E {αs ◦ hus |p(u· ) = x· } d{xs } 0

as required.

4.4



A family of Markovian kernels

For a probability measure µ0 on N + let the measures µt on N + be the distributions + of ut , t > 0, under PB µ0 and set νt = p∗ (µt ) on M . Let ηµ0 be the law of u· 7→ (p(u· ), u0 ) on CM + × N + under PB µ0 so Z y PA A ∈ B(M + ), Γ ∈ B(N + ) ηµ0 (A, Γ) = y (A) ρµ0 (Γ) ν0 (dy), y∈M +

4.5. The filtering equation

71

where ρyµ0 arises from a disintegration of µ0 , Z µ0 (Γ) = ρyµ0 (Γ) ν0 (dy),

Γ ∈ B(N + ).

y∈M +

For a measurable f : N + → R, integrable with respect to µt set πtµ0 ,σ f (v) = EB µ0 {f (ut )|p(u. ) = σ, u0 = v}.

(4.14)

It is defined for ηµ0 almost all (σ, v) in C(M + ) × N + and depends on the family of µ0 - measure zero sets rather than on µ0 itself. In particular for PA ν0 -almost all σ(0) + σ it is defined for ρµ0 -almost all v ∈ N . We could use the convention that πtµ0 ,σ f (v) = 0  if p(v) 6= σ(0). This enables us to choose a version of EB,v f (ut )|p(u. ) = σ which is jointly measurable in σ and v. With this convention, if we define θt σ(s) = σ(t+s) µt ,θt σ f (y) is defined for µt -almost we see that for PA v0 -almost all σ the map y 7→ πt + all y in N . Further for u0 ∈ N and f : N + → R bounded measurable define πt f (u0 ) : Cp(u0 ) M + → R, PA p(u0 ) -almost surely, by n o δ ,σ πt f (u0 )(σ) = E f (ut )|p(u· ) = σ = πt u0 f (u0 ).

(4.15)

This can be extended, as in [36], to the case of predictable processes in vector bundles over N . In particular if α. is a predictable σ B [T ∗ N ]∗ -valued process along u. , set h∗ (α. )t = αt ◦ hut . If σ A has constant rank this is a predictable process with values in the pull-back, p∗ (E), of E by p. In general it can be considered as an F∗N -predictable E ∗ -valued process because M. (αt ◦ hut ) ◦ σ A = αt ◦ σ B ◦ (Tut p)∗ ∈ Tx∗∗ t In any case we can define πt (h∗ (α. ). )(u0 ) : Cp(u0 ) M + → R as EB,u0 {h∗ (α. )t |p(u· ) = x· }.

4.5

The filtering equation

Theorem 4.5.1. (1) If f is Cc2 N , or more generally if f is C 2 with Bf and σ B (df, df ) bounded, then Z t Z t πt f (u0 ) = f (u0 ) + πs (Bf )(u0 )ds + πs (df ◦ hu· )(u0 )d{xs }. (4.16) 0

0 p(u0 )

In particular {πt f (u0 ) : t > 0} is a continuous F∗

semi-martingale.

72

Chapter 4. Projectible Diffusion Processes and Markovian Filtering

+ (2) For bounded measurable f : M + → R and PA v0 almost all σ in C(M ), for each s, t > 0, µ0 ,σ f (v) = πtµ0 ,σ πsθt σ, µt f (v) (4.17) πt+s σ(0)

for ρµ0

almost all v in N + .

(3) Moreover there exists a family of probability measures Qµν 0 ,σ on C(N + ) defining for ηu0 -almost surely all (σ, v) such that if F : CN + → R is of the form F (u· ) = f1 (ut1 ) . . . fn (utn ) some 0 6 t1 < t2 < · · · tn and bounded measurable fj : N + → R, j = 1, 2, . . . , n, then Z    µtn ,θtn−1 σ µ 1 ,θt1 σ F (u)Qµv 0 ,σ (du) = πtµ10 ,σ f1 πt2t−t . . . π f f (v) 2 n −t t 1 n n−1 u∈CN +

= EB µ0 {F (u· )|p(u· ) = σ, u0 = v}, ηµ0 -almost surely in (σ, v). Proof. (1) By definition of M df we have Z f (ut ) = f (u0 ) +

t

Bf (us )ds + Mtdf

0

so Z πt f (u0 ) = f (u0 ) +

t

n o πs Bf (u0 )ds + E Mtdf,B | p(u· ) = x

(4.18)

0

and part (1) follows from Proposition 4.3.5. (2) We observed above that the right-hand side of (4.17) is well defined for PA almost all σ. The equation then follows from the Markov property. µ0 (3) The existence of regular conditional probabilities in our situation implies the existence of the probabilities Qµv 0 ,σ as required, together with a standard use of the Markov property.  Remark 4.5.2. A description of the Qµv 0 ,σ is given in Section 4.8, in the case where A is cohesive. Recall that we have the decomposition Fu = Hu + V Tu N for each u ∈ N , and F = tFu . If ` ∈ Fu∗ there is a corresponding decomposition ` = `H + `V ∈ Fu∗ , where `H vanishes on V Tu N and `V on Hu . For ` ∈ Tu∗ N write `V = (`|Fu )V and `H = (`|Fu )H .

4.6. Approximations

73

Corollary 4.5.3. Suppose A is cohesive. If f is Cc3 N , then there is the Stratonovitch equation Z t Z t πs (B V f )(u0 )ds + πs (dfu0 ◦ huo ) ◦ dxs . (4.19) πt f (u0 )(x· ) = f (u0 ) + 0

0

Proof. We use (4.18). By Proposition 4.3.5, H

E{Mtdf | p(u· ) = x· } = E{Mtdf | p(u· ) = x· }. Note that Mtdf

H

Z =

t

(df H )us ◦ dus −

0

Z

t

δ B (df H )(us )ds

0

by Lemma 4.1.2. Furthermore V

H

H

H

δ B (df H ) = δ B (df H ) + δ A (df H ) = δ A (df H ) = δ A (df ) = AH (f ) since df H vanishes on vertical vectors and df = df H + df V while df V vanishes on H horizontal vectors, so δ A (df V ) = 0. This gives Z t  Z t πt f (u0 )(x· ) = f (u0 ) + πs (B V f )(u0 )(x· )ds + E (df )H ◦ du ) = x p(u s · · us 0

0

Finally (4.19) follows since dfuH = p∗ (df ◦ hu ) = df ◦ hu ◦ Tu p and Tu p ◦ dut =  ◦dxt .

4.6

Approximations

Assume now that the law of ut under PA u0 is given by Z PtA (u0 , A) = PtA (u0 , v)dv, A ∈ B(M ) A

for pA t (u0 , v) a smooth density with respect to some fixed, smooth, strictly positive measure on M to which ‘dv’ refers. This is the case if A is hypoelliptic. Consider the conditional probability qtu0 ,b (V ) = PB u0 {ut ∈ V |p(ut ) = b},

V ∈ B(N )

defined for pA t (u0 , −) almost surely all b in M . There is the disintegration of pB t (u0 , −), Z B pt (u0 , V ) = qtu0 ,b (V )pA t (p(u0 ), db), b∈M

and the formula −1 µµt 0 ,b (V ) = lim[pA t (p(u0 ), b)] ↓0

Z V

A pB t− (u0 , dv)p (p(v), b).

74

Chapter 4. Projectible Diffusion Processes and Markovian Filtering

Take a nested sequence {Π` }∞ l=1 of partitions of [0, t], Πl = {0 = tl0 < tl1 < · · · < tlkl = t}, say, with union dense in [0, t]. For any continuous bounded f : N + → R there is the following approximation scheme to complete πt f (u0 ): Proposition 4.6.1. Z

u ,σ(tl )

v ,σ(tl )

vk

−1 ,σ(t)

qtl0 1 (dv1 )qtl1−tl 2 (dv2 ) . . . qtl l−tl −1 (dvkl )f (vkl ) k k 1 2 1  B l l = lim Eu0 f (ut ) | p(utj ) = σ(tj ), 1 6 j 6 kl }.

πt f (u0 )(σ) = lim

l→∞ l→∞

Proof. The two versions of the right-hand sides are equal before taking limits. For l = 1, 2, . . . , set  S l f (σ) = Eu0 f (ut ) | p(utlj ) = σ(tlj ), 1 6 j 6 kl }. + l It is defined for PA x0 -almost all σ in C(M ), where x0 = p(u0 ). Let Q be the σ-algebra on C(M + ) generated by σ 7→ (σ(tl1 ), . . . , σ(tlj )). Directly from the definitions we see πtl f = E{πt (f )(u0 ) | Ql }, ∗ A and so {S l f }∞ l=1 is a Q -martingale. it is bounded and so converges Px0 -almost l  surely. Since ∨l Q is the Borel σ-algebra the limit is πt f (u0 ) as required.

4.7

Krylov-Veretennikov Expansion

Pm Suppose A = j=1 LX j LX j + LA for smooth vector fields {X j }m j=1 and A. We will now take {xt : 0 6 t < ζ} to be the solution to the stochastic differential equation (4.20) dxt = X(xt ) ◦ dBt + A(xt )dt, with x0 given, for a Brownian motion B· on Rm , rather than the canonical process. Here X(x) : Rm → Tx M is the map given by X(x)(a1 , . . . , am ) =

m X

aj X j (x),

x ∈ M.

j=1

Let {Pt : t > 0} be the sub-Markovian semi-group generated by B. Let f ∈ Cc∞ N . Assume Pt f ∈ C ∞ N . As in the proof of Theorem 4.5.1, from Z s d(Pt−r f )ur d{ur }, 0 6 s 6 t Pt−s f (us ) = Pt f (u0 ) + 0

4.8. Conditional Laws

75

we obtain Z

s

E {d(Pt−r f )ur ◦ hur | p(u· ) = x· } d{xr }

πs Pt−s (f )(u0 )(x· ) = Pt f (u0 ) + 0

Z

s

E {d(Pt−r f )ur ◦ hur | p(u· ) = x· } X(xr )dBr

= Pt f (u0 ) + 0

so that πs Pt−s f (u0 ), 0 6 s 6 t, is a continuous F∗x0 semi-martingale. Therefore Z t πt f (u0 ) − Pt f (u0 ) = ds (πs Pt−s f (us )) 0 Z t = E {d(Pt−r f ) ◦ hur | p(u· ) = x· } X(xr )dBr 0 Z t   = Sr d(Pt−r f ) ◦ h− ◦ X k (p(−)) (u0 )dBrk 0

giving a ‘Clark-Ocone’ formula for πt f (u0 ). Iterating this procedure formally, Z t   Sr d(Pt−r f ) ◦ h− ◦ X(p(−)) (u0 )dBr πt f (u0 ) = Pt f (u0 ) + 0 Z tZ r h i   + πs dPr−s d(Pt−r f ) ◦ h− ◦ X k (p(−)) h− ◦ X j (p(−) dBsj dBrk 0

0

= ..., we obtain the Wiener chaos expansion of πt f (u0 )(x· ).

4.8

Conditional Laws

It will be convenient to extend the notation of section 4.3. For 0 6 l < r < ∞ let C(l, r; N + ) and C(l, r; M + ) be respectively the space of continuous paths u : [l, r] → N + and x : [l, r] → M + which remain at ∆ from the time of explosion; and Cu0 (l, r; N + ) and Cx0 (l, r; M + ) the paths from u0 ∈ N + and x0 ∈ M + respectively, (l,r),B (l,r),A Let {Pu0 } and {Px0 } be the associated diffusion measures. The conditional law of {us : l 6 s 6 r} given {p(us ) : l 6 s 6 r} will (l,r);A almost surely from be given by probability kernels σ 7→ Ql,r σ,u0 defined P Cp(u0 ) (l, r; M + ) to Cup0 (l, r; N + ) for each u0 ∈ N , where Cup0 (l, r; N + ) is the subspace of Cu0 (l, r; N + ) whose paths satisfy Assumption S. The defining property is that, for integrable f : Cu0 (l, r; N + ) → R, Z f (y)dQl,r (4.21) E {f (u· ) | p(us ) = σs , l 6 s 6 r} = σ,u0 (y). y∈Cu0 (l,r;N + )

To obtain the conditional law take the decomposition B = AH + BV of Proposition 2.2.5. Represent the diffusion corresponding to A by a stochastic differential

76

Chapter 4. Projectible Diffusion Processes and Markovian Filtering

equation dx0t = X(x0t ) ◦ dBt + X 0 (x0t )dt.

(4.22)

Take a connection ∇V on V T N and let (∇V )

dzt = V (zt )dWt + V 0 (zt )dt

(4.23)

be an Itˆo equation whose solutions are B V -diffusions. Here (Wt ) is the canonical Brownian motion on Rm for some m, independent of (B· ), the map V : M × Rm → T M takes values in ker[T p], and V and V 0 are locally Lipschitz. For ˜ : such a representation of BV diffusions see the Appendix, Section 9.2.4. Let X m 0 0 ˜ N × R → H and X : N → H be the horizontal lifts of X and X respectively using Theorem 2.1.2. The solution to the equation of mixed type (∇V )

(l,r),B

has law Pu0

˜ t ) ◦ dBt + X ˜ 0 (yt )dt + V (yt )dWt + V 0 (yt )dt, dyt = X(y u0 ∈ N, l 6 t 6 r. yl = u0 , ˜ . Noting that X(u) = hu X(p(u)) for u ∈ M , (∇V )

dyt = hyt ◦ dx0t + V (yt )dWt + V 0 (yt )dt, yl = u0 , l 6 t 6 r,

(4.24)

where x0t = p(yt ) so that (x0t ) is a solution to (4.22) starting from p(u0 ) at time l. Without changing the law of y· we can replace x0 by the canonical process x· . Then Theorem 4.8.1. Consider the solution (yt ) as a process defined on the probability space Cp(u0 ) (l, r; M + ) × C0 Rm with product measure, y : [l, r] × Cp(u0 ) (l, r; M + ) × C0 Rm → N + , m → Cu0 (l, r; N + ). For bounded and define Ql,r σ,u0 to be the law of y(σ, −) : C0 R + measurable f : Cu0 (l, r; N ) → R,

Z E {f (u· ) | p(us ) = σs , l 6 s 6 r} = y∈Cu0 (l,r;N + )

f (y)dQl,r σ,u0 (y).

4.8. Conditional Laws

77

Proof. Take a measurable function α : Cp(u0 ) (l, r; M + ) → R. Then ! Z B

EPu0

α(p(u)) y∈Cu0 (l,r;N + )

PA p(u

=E

f (y) dQl,r p(u),u0 (y)

!

Z 0)

α(x) y∈Cu0 (l,r;N + )

f (y) dQl,r x,u0 (y)

  Z A f (y(x, ω)) dP(ω)) = EPp(u0 ) α(x) C0 Rm Z   = α(x)f (y(x, ω)) dPA dP(ω)) p(u0 ) Cp(u0 ) (l,r;M + )×C0 Rm

= Ef (u)α(p(u)), as required.



Note that Theorem 4.8.1 is equivalent to the statement that ω 7→ (l,r),B

ω ∈ Cu0 (l, r; N + ), is a regular conditional probability of Pu0

Ql,r p(ω),u0 ,

given p.

(ξtl (·, ·), l

Remark 4.8.2. Let 6 t < ∞) be a measurable flow for (4.22) and (ηtl (σ, ·, ), 0 6 t < ∞) one for (4.24) with x0 replaced by σ ∈ Cp(u0 ) (l, r; M + ). For ω ∈ Ω, the underlying probability space for the Brownian motion B, define + to itself, by Ql,r ω , from the space of bounded measurable functions on N
l l Ql,r ω (f )(u0 ) = Ef ηr (ξr (p(u0 ), ω), u0 ) . A direct calculation shows that r,s l,s Ql,r ω Qω = Qω

for 0 6 l 6 r 6 s < ∞. Thus their adjoints on a suitable dual space would form an evolution. More generally, letting Borel(X) stand for the Borel σ-algebra of a topological space X: Proposition 4.8.3. Let ϕ be a measurable map from Cx0 (l, r; M + ) to some measure space, and let : Cx0 (l, r; M + ) × Borel(Cx0 (l, r; M + )) → [0, 1] P(l,r),ϕ x0 (l,r)

be a regular conditional probability for Px0 given ϕ. For u0 with p(u0 ) = x0 set Z (l,r),ϕ (ω, A) = Ql,r (p(ω), dσ) Ql,r,ϕ◦p u0 σ,u0 (A)Px0 Cx0 (l,r;M + )

is a regular for ω ∈ Cu0 (l, r; N + ) and A ∈ Borel (Cu0 (l, r; N + )). Then Ql,r,ϕ◦p u0 (l,r),B given ϕ ◦ p. conditional probability of Pu0

78

Chapter 4. Projectible Diffusion Processes and Markovian Filtering

Proof. By definition n o l,r (l,r),A,x0 Q (ω, A) = E (A)|ϕ p(ω) Ql,r,ϕ◦p u0 p(−),u0 n o = E(l,r),A,x0 E(l,r),B,u0 {χA |p = −}|ϕ p(ω) = E(l,r),B,u0 {χA |ϕ ◦ p}(ω).



Corollary 4.8.4. For ϕ as in Theorem 4.8.3 suppose that the canonical process (0,T ),ϕ (σ, −) is a semi-martingale for almost all σ, in its own on M + with law Px0 (0,T ),A filtration Ftx0 , 0 6 t 6 T , for Px0 almost all σ. Then the solution y(σ, −) to the equation (∇V )

dyt = hyt ◦ dσt + V (yt )dWt + V 0 (yt )dt, yl = u0 , 0 6 t 6 T (0,T ),ϕ

where σt , 0 6 t 6 T is run with law Px0 from u0 conditioned by ϕ ◦ p.

(4.25)

(σ, −), is a version of the B-diffusion

Proof. That the law of the solution is as required follows from the discussion at the beginning of this section together with Proposition 4.8.3 and Fubini’s theorem.  Conditions under which conditioned processes are semi-martingales are discussed by Baudoin [4]. In particular bridge processes derived from elliptic diffusions are, so we obtain the following version of Carverhill’s result [16]: Corollary 4.8.5. Suppose A is elliptic and let bt : 0 6 t 6 T be a version of the A-bridge going from x0 to z in time T , some z ∈ M . Then the solutions to (∇V )

dyt = hyt ◦ dbt + V (yt )dWt + V 0 (yt )dt, y0 = u0 , 0 6 t 6 T

(4.26)

give a version of the B diffusion from u0 conditioned on p(uT ) = z. Example 4.8.6. In Example 2.2.13 and its continuation Example 4.1.7, for all t > 0 the conditional distribution of (us , s ≤ t) given u0 and the path (p(us ), s ≤ t) is determined by the conditional distribution of ((v3 , w1 , w2 )(us ), s ≤ t) given σ(Ws1 , Ws2 , s ≤ t) (Cf formula (4.10) for the definition of the processes W i ). It is the law of  Z t v2 (us )dWs , (v3 (u0 ) + Wt , w1 (u0 ) − v2 (u0 )Wt − v3 (u0 )ν2 (ut ) + 2 0

Z w2 (u0 ) + v1 (u0 )Wt + v3 (u0 )ν1 (ut ) − 2

t

 v1 (us )dWs ) ,

0

4.9. An SPDE example

79

where Wt is a Wiener process independent of W 1 and W 2 . In particular, the conditional distribution of ut given u0 and the path (p(us ), s ≤ t) is a Gaussian distribution with mean vector
v3 (u0 ), w1 (u0 ) − v3 (u0 )ν2 (ut ), w2 (u0 ) + v3 (u0 )ν1 (ut ) . Set g(s) = (1, 2v2 (us ) − v2 (u0 ), v1 (u0 ) − 2v1 (us )), then the covariance is given by Z

t

g(s)g T (s) ds .

0

4.9

An SPDE example

Consider now the SPDE dut (θ) = ∆ut (θ) + ut (θ)dWt + φ(θ)dBt

(4.27)

on the circle S 1 where W and B are independent one-dimensional Brownian motions. Suppose φ : S 1 → R is C ∞ . Let N be the Sobolev space H s (S 1 ; R), for sufficiently large s. Then for a smooth initial condition the solution to equation (4.27) will R t > 0. Take M = R and define p : N → M by R have ut ∈ N for p(u) = S 1 u(θ)dθ. Set m = S 1 φ(θ)dθ. The process xt := p(ut ) satisfies dxt = d{xt } = xt dWt + mdBt 2

∂ and is a diffusion process with generator A = 12 (x2 + m2 ) ∂x 2. xt t If m vanishes, Wt = log( x0 ) + 2 and conditionally to xs , 0 ≤ s ≤ t, ut is a Gaussian Markov process whose law is easily determined. 1 When m does not vanish, equation (4.16) applies. R As an example, for z ∈ S , to find the conditional expectation of ut (z) given { S 1 us (θ) dθ : 0 6 θ 6 t} take f : N → R to be the evaluation δz at z so f (u) = u(z) = δz (u). Then B(δz )(u) = ∆u(z) and therefore B(δz ) = δz ◦ ∆ : N → R. Also (dδz )u = δz for all u since δz is linear. Moreover, using Lemma 2.3.1, we see hu (1) = p(u)u+mφ p(u)2 +m2 ∈ N and so dδz ◦ hu :

R → R is multiplication by

p(u)u(z)+mφ(z) , p(u)2 +m2

(dδz ◦ hu )(λ) = λ

i.e.

pδz (u) + mφ(z) λ ∈ R. p2 + m2

Thus in equation (4.16) πs (dδz ◦ h)(λ) =

pπs (δz ) + mφ(z) (λ) p2 + m2

and

πs (Bδ z ) = πs (δz ◦ ∆)

80

Chapter 4. Projectible Diffusion Processes and Markovian Filtering

and equation (4.16) yields: Z t Z πs (δz ◦ ∆)(u0 )ds + πt (δz )(u0 ) = u0 (z) +

t

pπs (δz ) + mφ(z) (xs dWs + mdBs ). p2 + m2 0 0 (4.28) The SDE for the conditional process can be derived using the decomposition of the noise as in Section 2.3.2. For this write

dut = hut (1)dxt + dut − hut (1)dxt =

xt ut + mφ dxt + ∆ut dt + dMt x2t + m2

where x2t ut + mxt φ xt ut m + m2 φ dWt + φdBt − dBt 2 2 xt + m x2t + m2 m2 ut − mxt φ −xt ut m + x2t φ = dWt + dBt . 2 2 xt + m x2t + m2

dMt = ut dWt −

Note that as expected the bracket hM, xit vanishes. In fact Mt can be obtained by parallel translation of a Brownian motion βt , independent of xt , as explained in Section 2.3.2. The conditional expectation of ut given σ(xs , 0 ≤ s ≤ t) is easily obtained by taking its expectation against any stochastic integral with respect to x (which happens to be a Brownian martingale). We see that the two last terms  do not contribute so that, if zt denotes E ut |σ(xs , 0 ≤ s ≤ t) , we have: dzt =

xt zt + mφ dxt + ∆zt dt. x2t + m2

(4.29)

since πs (δz )(u0 ) = zs , in this case the use of the conditioned process is much simpler than the use of the filtering equation (4.16). The latter yields equation (4.28) which would traditionally be interpreted as an equation for a process with values in the space of measures on our Sobolev space H s . The former gives the rather straightforward linear SPDE (4.29). However note that the Fourier coefficients of ut verify linear SDE’s: db ut (k) =

b b bt (k) + mφ(k) bt (k) − mxt φ(k) xt u m2 u 2 dx − k u b (k)dt + dWt t t 2 2 xt + m2 xt + m2 b bt (k)m + x2t φ(k) −xt u + dBt . 2 2 xt + m

As before we see that the two last terms do not contribute under our conditioning, and for zt as above, db zt (k) =

b xt zbt (k) + mφ(k) dxt − k 2 zbt (k)dt. 2 2 xt + m

4.10. Equivariant case: skew-product decomposition

81

which is a linear equation from which zbt , and so zt , can be computed in the standard way. This formula can be recovered easily from equation (4.16) as, if we take f (u) = u b(k), we have Bf = −k 2 f and df ◦ hu =

b p(u)f (u)+mφ(k) , p(u)2 +m2

so that πs (Bf ) =

b pπs (f )+mφ(k) . p2 +m2

−k 2 πs (f ) and πs (df ◦ h) = More complicated equations can be given for the conditional expectations of functions of u bt (k). They involve the vertical part of the generator B.

4.10

Equivariant case: skew-product decomposition

In the equivariant case, when N is the total space P of a principal bundle π : P → M as in Chapter 3, a version of Theorem 4.8.1 is given in [34] which reflects the additional structure. In particular the following is proved there: Proposition 4.10.1. Let B be an equivariant diffusion operator on P which induces a cohesive diffusion operator A on M . Let {yt : 0 6 t < ζ} be a B-diffusion on P . Then ˜t · gtx˜· , yt = x where (i) {˜ xt : 0 6 t < ζ} is the horizontal lift of p(y· ), starting at y0 , using the semi-connection induced by B, (ii) {gtσ : 0 6 t < ζ(σ)} is a diffusion independent of {p(yt ) : 0 6 t < ζ} on G starting at the identity with time dependent generator Lσt given by Lσt f (g) =

X

αij (σ(t) · g)LA∗i LA∗j f (g) +

X

β k (σ(t)g)LA∗k f (g),

i,j

0 6 t < ζ(σ), for any σ ∈ CM + , where A∗1 , . . . , A∗k are the left invariant vector fields on G corresponding to a basis of g and the αij and β k are the coefficients for B V as in Theorem 3.2.1. Note that for each t the operator Lσt is conjugate to the restriction of B V to the fibre through σ(t) by the map G 7→ g

7→

p−1 (p(σ(t))), σ(t)g.

It is a right-invariant operator. Remark 4.10.2. Note that by the equivariance of Lσ· there will be no explosion of the process (gtσ ) before that of σ· . Consequently Assumption S of §4.3 holds automatically.

82

Chapter 4. Projectible Diffusion Processes and Markovian Filtering

Below we give the equivariant version of Proposition 4.8.1. We shall use the notation of §4.8. However we replace the one-point compactification P + of P by P¯ = P ∪ ∆ with the smallest topology agreeing with that of P and such that π : P¯ → M + is continuous. Also let G+ be the one-point compactification G ∪ ∆ of G with group multiplication and action of G extended so that u · ∆ = ∆, ∆ · g = g · ∆ = ∆,

¯ ∀u ∈ P¯ , g ∈ G.

For 0 6 l < r < ∞ if y ∈ C(l, r; P¯ ), we write ly = l and ry = r. Let C(∗, ∗; P¯ ) be the union of such spaces C(l, r; P¯ ). It has the standard additive structure under concatenation: if y and y 0 are two paths with ry = ly0 and y(ry ) = y 0 (ly0 ) let y + y 0 be the corresponding element in C(ly , ry0 ; P¯ ). The basic σ-algebra of C(∗, ∗, P¯ ) is defined to be the pull-back by π of the usual Borel σ-algebra on C(∗, ∗; M + ). : Given an equivariant diffusion operator B on P consider the laws {P(l,r),B a a ∈ P } as a kernel from P to C(l, r; P¯ ). The right action Rg by g in G+ extends to give a right action, also written Rg , of G+ on C(∗, ∗, P¯ ). Equivariance of B is equivalent to = (Rg )∗ P(l,r),B P(l,r),B ag a (l,r),B

for all 0 6 l 6 r and a ∈ P . Therefore π∗ (Pa ) depends only on π(a), l, r and gives the law of the induced diffusion A on M . We say that such a diffusion B is basic if for all a ∈ P and 0 6 l < r < ∞ the basic σ-algebra on C(l, r; P¯ ) contains (l,r),B all Borel sets up to Pa negligible sets, i.e. for all a ∈ P and Borel subsets B of
(l,r),B π −1 (A)∆B = 0. C(l, r; P¯ ) there exists a Borel subset A of C(l, r, M + ) s.t. Pa For paths in G it is more convenient to consider the space C˜id (l, r; G+ ) of cadlag paths σ : [l, r] → G+ with σ(l) = id such that σ is continuous until it leaves G and stays at ∆ from then on. It has a multiplication C˜id (s, t; G+ ) × C˜id (t, u; G+ ) −→ C˜id (s, u; G+ ) (g, g 0 ) 7→ g × g 0 where (g × g 0 )(r) = g(r) for r ∈ [s, t] and (g × g 0 )(r) = g(t)g 0 (r) for r ∈ [t, u]. Given probability measures Q, Q0 on C˜id (s, t; G+ ) and C˜id (t, u; G+ ) respectively this determines a convolution Q ∗ Q0 of Q with Q0 which is a probability measure on C˜id (s, u; G+ ). (l,r),B

Theorem 4.10.3. Given the laws {Pa : a ∈ P, 0 6 l < r < ∞} of an equivariant : a ∈ P } from diffusion B over a cohesive A there exist probability kernels {PH,l,r a l,r a.s. from C(l, r; P¯ ) to P to C(l, r; P¯ ), 0 6 l < r < ∞ and y 7→ Ql,r y , defined P C˜id (l, r; G+ ) such that : a ∈ P } is equivariant, basic and determining a cohesive generator. (i) {PH,l,r a (ii) y 7→ Ql,r y satisfies ly ,r

0

l

0 ,ry 0

Qy+yy0 = Qlyy ,ry ∗ Qyy0

for Ply ,ry ⊗ Ply0 ,ry0 almost all y, y 0 with ry = ly0 .

4.11. Conditional expectations of induced processes on vector bundles (iii) For U a Borel subset of C(l, r; P¯ ), Z Z Pl,r (U ) = a C(l,r;P¯ )

+) ˜ C(l,r;G

83

H,l,r χU (y· · g· )Ql,r (dy). y (dg)Pa

H,l,r ¯ The kernels PH,l,r are uniquely determined as are the {Ql,r a y : y ∈ C(l, r; P )}, Pa a.s. in y for all a in P . Furthermore Ql,r y depends on y only through its projection π(y) and its initial point yl .

The proof of this theorem is as that of Theorem 2.5 in [34] (although there the processes are assumed to have no explosion). Stochastic differential equations can be given for (˜ xt ) and (gtσ ) as in §4.8, from which the decomposition can be proved via Itˆo’s formula; see Theorem 8.2.5 below for details of a special case. Proposition 4.10.1 extends results for Riemannian submersions by ElworthyKendall [33] and related results by Liao[69]. A rich supply of examples of skewproduct decomposition of Brownian motions, with a general discussion, is given in Pauwels-Rogers[87]. For a special class of derivative flows, considered as a GLM -valued process as in §3.3, there is a different decomposition by Liao [70], see also Ruffino [94].

4.11

Conditional expectations of induced processes on vector bundles

In the notation of §3.4 let ρ : G → L(V, V ) be a C ∞ representation with Πρ : F → M the associated bundle. A B-diffusion {yt : 0 6 t < ζ} on P determines a family of {Ψt : 0 6 t < ζ} of random linear maps Wt from Fx0 → Fxt , where xt = π(yt ). By definition, Ψt [(y0 , e)] = [(yt , e)]. Assuming A is cohesive we have the parallel translation //t : Fx0 → Fxt along {xt : 0 6 t < ζ} determined by our semi-connection. This is given by //t [(y0 , e)] = [(˜ xt , e)] where x ˜· is the horizontal lift of x, starting at y0 . When taken together with Corollary 3.4.9 the following extends results for derivative flows in Elworthy-Yor[38], Li[68], Elworthy-Rosenberg [37], and Elworthy-LeJan-Li[36]. See the Notes below, Section 4.12. Theorem 4.11.1. Let ρ : G → L(V ; V ) be a representation of G on a Banach space V and Πρ : F → M the associated vector bundle. Let {yt : 0 6 t < ζ} be a B-diffusion for an equivariant diffusion operator B over a cohesive diffusion operator A. Set xt = p(yt ) and let Ψt : Fx0 → Fxt , 0 6 t < ζ be the induced

84

Chapter 4. Projectible Diffusion Processes and Markovian Filtering

transformations on F . Then the local conditional expectation {Ψt : 0 6 t < ζ}, for Ψt = E{Ψt |σ{xs : 0 6 s < ζ} exists and is the solution of the covariant equation along {xt : 0 6 t < ζ}: D Ψt = Λρ ◦ Ψt ∂t with Ψ0 the identity map, Λρ : F → F given by λρ in Theorem 3.4.1 and where refers to the semi-connection determined by B.

D ∂t

Proof. From above and Proposition 4.10.1 we have xt ◦ gtx˜ , e)] = [(˜ xt , ρ(gtx˜ )−1 e] Ψt [(y0 , e)] = [(˜ and so //t−1 Ψt [(y0 , e)] = [(y0 , ρ(gtx˜ )−1 e)]. Now from the right invariance of Gtσ , for fixed path σ and time t, we can apply Baxendale’s integrability theorem, [7], for the right action G × L(V ; V ) → L(V ; V ) (g, T ) → 7 ρ(gtσ )−1 ◦ T to see E|ρ(gtσ )−1 |L(V ;V ) < ∞ for each σ, t and we have E(σ)t ∈ L(V ; V ) given by E(σ)t e = Eρ(gtσ )−1 e. By considering //t−1 Ψt we see that the local conditional expectation Ψt exists in L(Fx0 ; Fxt ) and Ψt [(y0 , e)] = [(˜ xt , E(x· )t e)]. The computation in Theorem 3.4.1 shows that d −1 d // Ψt [(y0 , e)] = [(y0 , E(x· )t e)] = [(y0 , λρ (˜ xt )E(x· )t e)] dt t dt giving D Ψt [(y0 , e)] = [(˜ xt , λρ (˜ xt )E(x· )t e)] = Λρ (x· )Ψt [(y0 , e)] dt as required.



Remark 4.11.2. Theorem 4.11.1 could also be used to identify the generator of the operator induced on sections of F ∗ , re-proving Theorem 3.4.1, since if φ ∈ γF ∗ , then Eφ ◦ Ψt χt<ζ = Eφ ◦ Ψt χt<ζ if the expectations exist, by Corollary 3.3.5 of [36]. The extra information in Theorem 4.11.1 is the existence of the conditional expectation. Baxendales’ integrability theorem used for this applies in sufficient generality to give corresponding results for infinite dimensional G, for example in the situation arising in Chapter 8 below.

4.12. Notes

4.12

85

Notes

Noise-free observations From the filtering viewpoint we could be considered to be dealing with noise-free observations as discussed by Joannides & LeGland in [54], though our assumption in this Chapter that the observations form a diffusion process make the situation much simpler. Krylov-Veretennikov formula The Krylov-Veretennikov formula, see Section 4.7, giving a Wiener chaos expansion for solutions of stochastic differential equations appeared in the 1976 article [103]. A version was described later by Elliott &Kohlmann in [29] in a form which is easily used for stochastic differential equations on manifolds, as described briefly in [32]. For an application to stochastic pde see [72]. It is the basis of the generalised stochastic flow theory of LeJan & Raimond, [63], [64]. In filtering theory it appeared in [77],[81], and other articles such as: [58], [74], and [73]. Skew-product decompositions, regular conditional probabilities and SDE A basic measurability result concerning skew product decompositions obtained via by SDE’s is given by John Taylor in [101]. Operators on differential forms Suppose we have the stochastic differential equation dxt = X(xt ) ◦ dBt + A(xt ) dt. on M with a solution flow ξ. . The results of Theorem 4.11.1 combine with Corollary 3.4.9 to show that the operator on differential forms, m

A∧ :=

1X L j L j + LA , 2 j=1 X X

and its semi-group P.∧ , described in the Notes on Chapter 2, Section 2.7, can be written as A∧ φ

=

1 ˘ .∇ ˘ k P ∧ φ(V ) = Eφ(W k (V )) ˘ . (φ) − 1 φ ◦ R trace∇ t t 2 2

(4.30)

for φ a k-form and V an element of ∧k Tx0 M some x0 ∈ M . We are assuming the ˘ k : ∧k T M → ∧k T M is the generalised Weitzenb¨ock curexpectation exists. Here R vature defined in Corollary 3.4.9 and Wtk : ∧k Tx0 M → ∧k Tξt M is the “damped” parallel translation given by
D k ˘ k W k (V ) W (V ) = −R t dt t

(4.31)

86

Chapter 4. Projectible Diffusion Processes and Markovian Filtering

˘ in the sense of using the covariant derivative of the connection adjoint to ∇ Section 3.3. This follows from the argument in Remark 4.11.2 since Wtk V =  E ∧k Tx0 (V )|ξs (x0 ), 0 6 s 6 t by Theorem 4.11.1 and Corollary 3.4.9. For a detailed discussion of this argument see [36], especially page 43, but beware of some incorrect signs there.

Chapter 5

Filtering with non-Markovian Observations So far we have considered smooth maps p : N → M with a diffusion process u. on N mapping to a diffusion process x. = p(u. ) on M . From the point of view of filtering we have considered u. as the signal and x. as the observation process. However the standard set-up for filtering does not assume Markovianity of the observation process. Classically we have a signal z. , a diffusion process on Rd or a more general space, and an observation process x. on some Rn given by an SDE of the form dxt = a(t, xt , zt )dt + b(t, xt , zt )dBt

(5.1)

where B. is a Brownian motion independent of the signal. To fit this into our discussion we will need to assume that the noise coefficient of the observation SDE does not depend on the signal other than through the observations, as well as the usual cohesiveness assumptions. We can take N = Rd × Rn and M = Rn with p the projection and ut = (zt , xt ). To reduce to our Markovian case we can use the standard technique of applying the Girsanov-Maruyama theorem. Here we first carry this out in the general context of diffusions with basic symbols, as discussed in Section 2.4 and then show how it fits in with the classical situation. For simplicity we shall assume that the signal is a time-homogeneous diffusion, and that the coefficients in the observation SDE are also independent of time. The state spaces are taken to be smooth manifolds and the standard non-degeneracy assumptions on the observation process somewhat relaxed. For other discussions about filtering with processes which have values in a manifold see Duncan,[26], Pontier &Szpirglas, [88], Davis &Spathopoulos, [24], Estrade,Pontier,& Florchinger, [41], and Gy¨ongy, [50].

K.D. Elworthy et al., The Geometry of Filtering, Frontiers in Mathematics, DOI 10.1007/978-3-0346-0176-4_5, © Springer Basel AG 2010

87

88

5.1

Chapter 5. Filtering with non-Markovian Observations

Signals with Projectible Symbol

Using the notation and terminology of Section 2.4, suppose that our diffusion operator B on N is conservative and descends cohesively over p : N → M so that for a horizontal vector field bH on N the diffusion operator B˜ := B − bH lies over ˜ and so A, is also conservative: we some cohesive A. Choose such an A so that B, assume that this is possible. Also choose a locally bounded one-form b# on N with 2σ B (b# ) = bH . This is possible since bH is horizontal, and we can, and will, choose b# to vanish on vertical tangent vectors and satisfy H B # # H H b# y (b (y)) = 2σy (by , by ) = hb (y), b (y)iy

y∈N

(5.2)

where h−, −iy refers to the Riemannian metric on the horizontal tangent space H induced by 2σ A . This can be achieved by first choosing some smooth ˜b : N → A T ∗ M such that, in the notation of equation (2.28), σp(y) (˜b(y)) = b(y) for y ∈ N ; # and then taking b to be the pull-back of ˜b by p: ˜ b# y (v) = b(y)(Ty p(v))

y ∈ N.

Now set

1 hM α it } 2 for αt (u. ) = b# ut where u ∈ C([0, T ]; N ), our canonical probability space furnished ˜ := PB˜ and corresponding expectation operators E with measures P := PB and P ˜ and E. α Here and below we are using the notation of Proposition 4.1.1 with R t M etc α ˜ referring to taking martingale parts with respect to P while M and 0 αs d{ys } ˜ From the Girsanov-Maruyana-Cameron-Martin theorem are with respect to P. (see the Appendix, Section 9.1), we know that Z. is a martingale under P and the two measures are equivalent with Zt = exp{−Mtα −

˜

dPB y0 = ZT . dPB y0 Suppose f : N → R is bounded and measurable. We wish to find πt (f ) : N → R, 0 6 t 6 T where  πt (f )(y0 ) = Ey0 f (ut )|p(us ), 0 6 s 6 t . Following the approach due to Zakai, consider the unnormalised filtering process π ˆt (f ) : N → R given by  ˜ u f (ut )Z −1 | p(us ), 0 6 s 6 t . π ˆt (f )(u0 ) = E t 0 For completeness we state and prove the Kallianpur-Striebel formula, a version of Bayes’ formula:

5.1. Signals with Projectible Symbol Lemma 5.1.1. πt (f )(u0 ) =

89

π ˆt (f )(u0 ) π ˆt (1)(u0 )

Pu0 − as.

Proof. Set x0 = p(u0 ). Let g : Cu0 ([0, T ]; N ) → R be Ftx0 -measurable. Then ˜ 1 f (ut )g(u. )} Eu0 {f (ut )g(u. )} = E{ Zt ˜ E{ ˜ 1 f (ut )|F u0 }g(u. )} = E{ t Zt ˜ 1 f (ut )|F u0 }g(u. )}. = E{Zt E{ t Zt

(5.3)

Thus πt (f )(u0 ). πt (f )(u0 ) = E{Zt |Ftu0 }ˆ Taking f constant shows that E{Zt |Ftuo }ˆ πt (1)(u0 ) = 1 and the result follows.  We can now go on to obtain the analogue of the Duncan-Mortensen-Zakai (DMZ) equation for the unnormalized filtering process, using the results of Section 4.8 on conditional laws: ˜ Theorem 5.1.2. For any C 2 function f : N → R, under P, Z t Z t

π ˆt f (u0 ) = f (u0 ) + π ˆs Bf (u0 ) ds + π ˆs f b# (−)h− (u0 )d{xs } 0 0 Z t
+ π ˆs df− h− (u0 )d{xs }; (5.4) 0 Z t Z t
π ˆt f (u0 ) = f (u0 ) + π ˆs Bf (u0 ) ds + hˆ πs (f b)(u0 ), d{xs }ixs 0 0 Z t
π ˆs df− h− (u0 )d{xs } + (5.5) 0

where xs = p(us ), 0 6 s 6 ∞ is the projection to M of the canonical process from u0 on N , and h the horizontal lift map for the induced semi-connection. Using an alternative notation:

Z t π f b# ◦hu. ,A

π ˆt f = π ˆ0 f + Mt

π ˆ . df ◦hu. ,A

+ Mt

+

π ˆs (Bf )ds.

(5.6)

0

˜ we will write M b# for M b# ,B˜, etc. Also Z −1 Proof. Since we are working with P t satisfies: dZt−1 = Zt−1 dMtb

#

while ˜ )(ut )dt df (ut ) = dMtdf + B(f

90

Chapter 5. Filtering with non-Markovian Observations

giving
˜ )(ut )dt d Zt−1 f (ut ) = Zt−1 dMtdf + Zt−1 B(f #

+ f (ut )Zt−1 dMtb + Zt−1 + dfut (bH (ut ))dt
# ˜ since dMtdf dMtb = σ B dfut , b# dt = dfu. (bH (ut ))dt. Thus
# d Zt−1 f (ut ) = Zt−1 dMtdf + Zt−1 B(f )(ut )dt + f (ut )Zt−1 dMtb + Zt−1 . We can now take conditional expectations using Proposition 4.3.5 since B − LbH is over the cohesive operator A to complete the proof.  Lemma 5.1.3. There are the following formulae for angle brackets: πt (b), π ˆt (b)iE dhˆ π (1)it = hˆ xt dt,

(5.7)

dhˆ π (1), π ˆ (f )it = hˆ πt (f b), π ˆt (b)iE ˆt (df ◦ hu. ) ◦ π ˆt (b(u. ))dt. xt dt + π

(5.8)

Proof. From the previous theorem π ˆ (f b# ◦h),A

hˆ π (1), π ˆ (f )idt = dMt

π ˆ (df ◦hu. ),A


+ dMt .

π(b# ◦h),A

dMt



= 2σ A π ˆt (b# ◦ h) dt ˆt (f b# ◦ h), πˆt (b# ◦ h) dt + 2σ A π ˆt (df ◦ hu. ), π ˆt (b)ixt dt + π ˆt (df ◦ hu. ) ◦ π ˆt (b(u. ))dt, = hˆ πt (f b), π since for any one-form φ on M we have:
∗
ˆt hφ|E , b# ◦ hiE ˆt (b# ◦ h) = π σ A φ, π .
1 ˆt φ(b) = π 2 1 πt (b)). = φ(ˆ 2 This gives the second formula, from which comes the first.



We can now give a version of Kushner’s formula in our context: ˜ Theorem 5.1.4. In terms of the probability measure P Z πt f = π0 f +

t

t
πs B(f )ds + πs df ◦ hu. [d{xs } − πs (b)ds] 0 0 Z t + hπs (f b) − πs (f )πs (b), d{xs } − πs (b)ixs ds.

Z

0

(5.9)

5.2. Innovations and innovations processes

91

Proof. From the definition and then Itˆ o’s formula:   π ˆt (f ) dπt (f ) = d π ˆt (1) dˆ πt (f ) π πt (1) dˆ πt (1) ˆt (f )dˆ πt (f )dˆ = − − π ˆt (1) (ˆ πt (1))2 (ˆ πt (1))2 πt (1)dˆ πt (1) π ˆt (f )dˆ + . (ˆ πt (1))3 Now substitute in the second formula of Theorem 5.1.2 and use the previous lemma.  ˜ depend on the choice of A. We would like to Note that π ˆt (f ), b, and P, have a version of formula (5.9) which is independent of such choices. First note H that if B − bH 1 is over A1 , and B − b2 is over A2 , then the difference of the two vector fields on N descends to a vector field on M : if g : M → R is smooth and g˜ = g ◦ p : N → R, then H H ˜ g = (B − bH g = (A1 − A2 )g. (bH 2 − b1 )˜ 1 )f − (B − b1 )˜ H Therefore if we set b0 (z) = Ty p(bH 2 (y) − b1 (y)) for p(y) = z, z ∈ M , then A1 = A2 + Lb0 , and by Remark 4.1.4,

d{xs }A2 = d{xs }A1 + b0 ds,

(5.10)

From this we see immediately that the symbols d{xs } − πs (b)ds, and πs (f b) − πs (f )πs (b) in formula (5.9) are in fact independent of the choice we made of A. To relate to now classical concepts we next discuss the first of these in more detail.

5.2

Innovations and innovations processes

Keeping the notation above, for α ∈ L2A , so αt ∈ Tx∗t M for 0 6 t < ∞, define a real-valued process Itα : 0 6 t < ∞, the α-innovations process by Itα =

Z

t


αs d{xs }A − πs b(u. )ds

(5.11)

0

A generalisation of a standard result about innovations processes is: Proposition 5.2.1. The process I.α is independent of the choice of A. Under PB,u0 it is an F∗x0 martingale. Proof. The observations just made show that it is independent of the choice of A. It is clearly also adapted to F∗x0 . To prove the martingale property note first that

92

Chapter 5. Filtering with non-Markovian Observations

by Proposition 4.3.4 and formula (5.10) Z t Z t αs d{xs }A = p∗ (αs )d{us }B−LbH 0 0 Z t Z t = p∗ (αs )d{us }B − p∗ (αs )bH (us )ds 0 0 Z t Z t = p∗ (αs )d{us }B − αs (b(us ))ds. 0

0

From this we see that if 0 < r < t and Z ∈ σ{xs : 0 6 s 6 r}, then (Z ) t
B A αs d{xs } − πs b(u. ) ds E χZ r B

(Z

)

t

αs (b(us ) − πs b(u. )) ds

= E χZ

=0

r

giving the required result.



If we fix a metric connection, Γ, on E, as described in Example 4.1.6 we can take the canonical Brownian motion, B Γ,A say, on Ex0 determined by A and 0 Γ. Then, by equation (4.8), we can write d{xs }A − πs (b(u. ))ds = //s dB Γ A − πs (b(u. ))ds. In terms of the the P Brownian motion, B Γ , on Ex0 , which is the martingale part under P of the Γ- stochastic anti-development of x. we can define an Ex0 -valued process, ztΓ : 0 6 t < ∞, by Z t ztΓ = BtΓ + (//s )−1 (b(us ) − πs (b(u. ))ds. (5.12) 0

A candidate for the innovations process of our signal-observation system is the stochastic development , ν.Γ say, of z.Γ under Γ. This can be defined by using the canonical SDE on the orthonormal frame bundle of E, namely νt )(ν˜0 )−1 ◦ dzt d˜ νt = X(˜ for a fixed frame ν0 for Ex0 . Here X(µ)(e) = hΓµ (µ(e)) for µ : Rp → Em a frame in at some point m ∈ M , and e ∈ Rp , for p the fibre dimension of E. The process ν.Γ is then the projection of ν˜. on M . For example see [31]. It will satisfy the Stratonovich equation dνtΓ = //t ◦ dzt

(5.13)

where the parallel translation is now along the paths of ν.Γ . Let Θ : C0 (M ) → C0 (M ) be the map given by Θ(σ)t = ν Γ (σ)t , treating z.Γ as defined on C0 (M ).

5.2. Innovations and innovations processes

93

Let D = DΓ : C0 (Tx0 M → Cx0 M be the stochastic development using Γ with inverse D−1 . We will continue to assume that there is no explosion so that these maps are well defined. For example, z(x. ) = D−1 Θ(x. ). We define a semi-martingale, on M to be a Γ-martingale if it is the stochastic development using Γ of a local martingale, see the Appendix, Section 9.3. Theorem 5.2.2. For each metric connection Γ on E the innovations process ν Γ is a Γ-martingale. If Γ is chosen so that the A-diffusion process is a Γ-martingale under PA , then for α : [0, τ ) × Cx0 M → T ∗ M which is predictable and lives over x. , provided the integrals exist, Z Z .
. I α ◦ Θ(x. ) = Γ α(ν Γ (x. ). )sdν Γ (x. )s − α(x. )s b(x. )s ds (5.14) 0

0

where b(−)s : Cx0 → T M is the conditional expectation, bs = E{b(us )|p(u. ) = x. }, and has b(x. )s ∈ Txs M almost surely for all s. Proof. The fact that ν Γ is a Γ-martingale is immediate from the definition and Proposition 5.2.1. To prove the claimed identity note that our extra assumption on Γ implies that //s−1 d{xs }A = d(D−1 (x. ))s . Therefore Z . Z . α −1 αs (x. )//s dD (x. )s − αs (x. )¯b(x. )s ds (5.15) I (x. ) = 0

0

while by definition Z Z .
.
ν Γ (x ) Γ Γ αs dνs (x. ) = αs (νsΓ (x. ))//s . . d D−1 (ν Γ (x. )) s 0

(5.16)

0

where the superscript on the parallel translation symbol indicates that it is along the paths ν.Γ (x. ). Our identity follows.  Remark 5.2.3. (1) For Γ such that the A-process is a Γ martingale we can easily see that Θ has an adapted inverse. Indeed its inverse is defined almost surely by A Θ−1 = D ◦ MartP ◦ D−1 A

where MartP denotes the operation of taking the martingale part under the probability measure PA . (2) If we are given a connection Γ on E we could make our choice of A so that its diffusion process gives a Γ martingale. This specifies A uniquely and might be more natural sometimes, for example in the classical case with M = Rn .

94

Chapter 5. Filtering with non-Markovian Observations

(3) The results and earlier discussion still hold if Γ is not a metric connection. However then B.Γ,A cannot be expected to be a Brownian motion. The connection could even be on T M rather than on E in which case B.Γ,A will be a local martingale in Tx0 M . This will be a natural procedure when N = Rn , using the standard flat connection.

5.3

Classical Filtering

For an example of the situation treated above consider a signal process (zt , 0 6 t 6 T ) on Rd satisfying an SDE dzt = V (zt , xt )dWt + β(zt , xt )dt

(5.17)

with (xt , 0 6 t 6 T ), the observation process, taking values in Rn and satisfying: dxt = X (1) (xt )dBt + X (2) (xt )dWt + b(zt , xt )dt.

(5.18)

Here B. and W. are independent Brownian motions of dimension q and p respectively. We then take N = Rd × Rn and M = Rn , with p : N → M the projection. We set ut = (zt , xt ) so that Bf (z, x) =

1 2 D f (V i (z, x), V i (z, x)) + D1 f (β(z, x)) 2 1,1 1 2 1 2 f (X (1),i (x), X (1),i (x)) + D2,2 f (X (2),j (x), X (2),j (x)) + D2,2 2 2 2 f (z, x)(V i (z, x), X (1),i (z, x)) (5.19) + D2 f (z, x)(b(z, x)) + D1,2

using the repeated summation convention where i goes from 1 to p and j from 1 to q, with the V j referring to the components of V and similarly for X (1),i and 2 refers to the second partial Frechet derivative, mixed if l 6= m, X (2),j . Also Dl,m etc. The filtering problem would be to find E{g(zt ) | xs : 0 6 s 6 t} for suitable g : Rd → R. This would fit in with the discussion above by defining f : Rd ×Rn → R by f (z, x) = g(z). Note that we have allowed feedback from the signal to the observation; usually only the special case where V and β are independent of x is considered. Also we have allowed the noise driving the signal to also affect the observations (“correlated noise”). This can give a non-trivial connection, in which case the terms involving horizontal derivatives of f will not vanish even for f independent of x. This vanishing would occur otherwise (i.e. for uncorrelated noise) so that in that case the formula in Theorem 5.1.2 reduces to the usual DMZ equation, for example as in [84] or [85]. For an up-to-date account of filtering theory see [2]. For an approach to dealing with the situation when the noise in the observation process has coefficients depending on the signal see [21].

5.4. Example: Another SPDE

95

Our basic assumptions are smoothness of the coefficients, non-explosion (for simplicity of exposition), and the cohesiveness of our observation process. By the latter we mean that for all x ∈ Rn and z ∈ Rd the image of the map (e1 , e2 ) 7→ X 1 (x)(e1 ) + X 2 (X)(e2 ) from Rq × Rp to Rn contains b(z, x) and has dimension independent of x. Some bounds are needed on b to ensure the existence of its conditional expectations. To carry out the procedure for the signal and observation given above we must first identify the horizontal lift operator determined by B. For this for each x ∈ M let Yx : Rn → Rp+q be the inverse of the restriction of the map (e1 , e2 ) 7→ X 1 (x)(e1 )+X 2 (X)(e2 ), from Rq ×Rp to Rn , to the orthogonal complement of its kernel. Then from Lemma 2.3.1 we see that the horizontal lift hu : Rn → Rd × Rn is given by hu (v) = (V (z, x) ◦ Yx (v), v)

u = (z, x) ∈ Rd × Rn .

(5.20)

A natural choice of A is  1   1 2  (1),i 2 f X (x), X (1),i (x) + D2,2 f X (2),j (x), X (2),j (x) . A(f )(x) = D2,2 2 2 Having done that, the ‘b’ of our general discussion is just the drift b : Rd × Rn → Rn of our observation’s stochastic differential equation. Moreover for suitable T ∗ M -valued processes α. we have the α-innovations process Z t  Z t 
α (1) (2) α(xs ) X (xs )dBs + X (xs )dWs + α(xs ) b(zs , xs ) − ¯b(x)s ds, It = 0

0

where ¯b(σ) = EB {b(zs , xs ) | xr = σr 0 6 r 6 s}. From Theorem 5.1.4, Kushner’s formula, given smooth g : Rd → R, one has Z t
1 2 g(−)(V i (−), V i (−)) + πs D1 g(−)(β(−)) ds πs D1,1 πt g = g(z0 ) + 2 0
πs dg(−)V (−)◦Y

+ It

¯ −¯ ¯ hgb s gs bs ,−ixs

+ It

.

This can be compared, for example, with the formula given in the remark on page 85 of [85], following the proof of Proposition 2.2.5 there. Alternatively see [84]. Using the standard flat connection of Rn we get the innovations process given by Z t  Z t
(1) (2) b(zs , xs ) − ¯b(x)s ds. X (xs )dBs + X (xs )dWs + νt = x0 + 0

5.4

0

Example: Another SPDE

Consider the stochastic partial differential equation on L2 ([0, 1]; Rp ): dut (x) = ∆ut (x) +

m X i=1

Φi (x, ut (x))dBti

96

Chapter 5. Filtering with non-Markovian Observations

where (Bti ) are independent Brownian motions. For p > 1 it can be considered as a system of equations. One natural question is to find the law of ut given that of us (x0 ), 0 6 s 6 t for some given point x0 , or to find the conditional law of ut given us (x0 ), 0 6 s 6 t. Here we indicate briefly how the approach we have been following may sometimes be applied to this or similar problems. A discussion of a similar but more straightforward problem was given in Section 4.9. For simplicity we take p = 1, so our “observations” process is one-dimensional: M = R. As in Section 4.9 the state space N is taken to be a Sobolev space of sufficiently high regularity, N = H s ([0.1]; R) for large s. The projection p is just evaluation at x0 . Let yt = ut (x0 ). It satisfies: dyt = (∆ut )(x0 )dt +

m X

Φi (x0 , yt )dBti .

i=1

Because of the drift term we cannot expect this to be Markovian so we will have to remove the term (∆ut )(x0 )dt by a Girsanov transformation. Let (ei ) be the standard orthonormal base of Rm . Define Φ : L2 ([0, 1]; R) × Rm → L2 ([0, 1]; R) and ˜ : R → L(Rm ; R) Φ by Φ(u)(e)(x) =

m X

Φi (x, u(x))he, ei i

i=1

and ˜ Φ(z)(e) :=

m X

Φi (x0 , z)he, ei i,

i=1

respectively. Consider Tz R, identified with R and furnished with the metric in˜ duced by Φ(z): v1 v2 . hv1 , v2 iz = Pm 2 ˜ i=1 (Φi (z)) To have cohesivity and to be able to apply the Girsanov-Maruyama-CameronMartin theorem this must be well defined, i.e. the denominator must never vanish, and it must determine a non-explosive Brownian motion. If these conditions hold, we still have to be sure that the Girsanov transformed SPDE has solutions existing for all time and that we can apply the martingale method approach used in the proof of Theorem 9.1.4. Alternatively we can try to apply one of the standard tests to show that the local martingale which arises is a true martingale. First we apply Lemma 2.3.1 to obtain the horizontal lift map. For this we need the dual ˜ ∗ (z) : R → Rm is given by: map Φ m X 1 Φj (x0 , z)ej . 2 i=1 (Φi (x0 , z)) j=1

˜ ∗ (z)(1) = Pm Φ

5.4. Example: Another SPDE

97

Then from equation (2.22) the horizontal lift hu : Tu(x0 ) R → L2 ([0, 1]; R) at a function u is given by ˜ ∗ (u(x0 )). hu (1)(x) = Φ(x, u(x)) ◦ Φ In particular a natural choice of drift bH to be removed by the Girsanov-Maruyama-Cameron-Martin theorem, namely bH (u) = hu (4u(x0 )), is given by Pm j=1 Φj (x0 , u(x0 ))Φj (x, u(x)) H Pn 4u(x0 ). (5.21) b (u)(x) = 2 k=1 (Φk (x0 , u(x0 ))) ˜ we see that our SPDE becomes Making the change of probability to P Pm j=1 Φj (x0 , ut (x0 ))Φj (x, ut (x)) Pn dut (x) = ∆ut (x) − 4ut (x0 ) 2 k=1 (Φk (x0 , ut (x0 ))) m X ˜i + Φi (x, ut (x))dB t i=1

˜ 1, . . . , B ˜ m and has the decomposition for new, independent Brownian motions B Pm m X j=1 Φj (x0 , ut (x0 ))Φj (x, ut (x)) ˜ i] P dut (x) = [ Φi (x0 , ut (x0 )) dB n t 2 k=1 (Φk (x0 , ut (x0 ))) i=1 ! Pm j=1 Φj (x0 , ut (x0 ))Φj (x, ut (x)) Pn 4ut (x0 ) dt + [ 4ut (x) − 2 k=1 (Φk (x0 , ut (x0 ))) ! Pm m X j=1 Φj (x0 , ut (x0 ))Φj (x, ut (x)) ˜ i ]. Pn Φi (x0 , ut (x0 )) dB Φi (x, ut (x)) − + t 2 (Φ (x , u (x ))) k 0 t 0 k=1 i=1 In this decomposition the term in the first square brackets relates to the horizontal lift of the A-process , while that in the second is the vertical component. They are ˜ given u at x0 . independent (under P), We could continue by applying the Kallianpur-Striebel formula, Lemma 5.1.1 or go directly to our version of Kushner’s formula, Theorem 5.1.4. In that formula the operator B will be the infinite dimensional diffusion operator on L2 ([0, 1]; R) which is the generator of the solution of our SPDE, so there are extra analytical problems. There are cases where this machinery is not needed and the situation is fairly straightforward. For example: (1) Φi (z, u) = φi (z), where the vector {φ1 (z), . . . , φm (z)} never vanishes for any z. In this case yt is basically Gaussian. (2) The noise is one dimensional. In this case we see that the vertical noise term in our decomposition vanishes, so ut (z) is determined by {xs : 0 6 s 6 t}. In particular if Φ(z, u) = u with one-dimensional noise Wt , the SPDE has

98

Chapter 5. Filtering with non-Markovian Observations t2

solution ut (z) = eWt − 2 Pt u0 (z) where Pt refers to the Dirichlet heat kernel on [0, 1]. For our cohesiveness assumption to hold in this case we would need to take N to consist of functions in H s which are strictly positive. For u0 in 0) this space we have Wt = 12 t + log( Put ut (x ) which can be substituted back 0 (x0 ) into the formula for ut (z). However, the example dut = ∆ut + ut dWt + φ(x)dBt is considerably more complicated even though it has the explicit solution: Z t t−s Wt − 2t Pt u0 + eWt −Ws − 2 Pt−s φdBs , ut = e 0

Pt denoting the Dirichlet heat kernel as before. For this example let us see, as an exercise, what terms will arise in Kushner’s formula, Theorem 5.1.4, Z t Z t
πs B(f )ds + πs df ◦ hu. [d{xs } − πs (b(u. ))s ds] πt f = π0 f + 0 0 Z t + hπs (f b) − πs (f )πs (b), d{xs } − πs (b)ixs , 0

for the conditional expectation of f : N → R when f (u) = u(z), for z our chosen point in [0, 1]. Note first that Bf (u) = ∆u(z). As in Section 4.9 denote u(z) by δz (u), (so that now f = δz ). Recall from Section 4.9 that Bδz (u) = ∆u(z).and Bδz = δz ◦ ∆. Similarly, since b(u) = ∆u(x0 ), we have b = δx0 ◦ ∆. Moreover, hu (1) =

φ(x0 )φ + u(x0 )u φ(x0 )2 + u(x0 )2

and

bH (u) = ∆u(x0 )hu (1).

Recalling that the generator on the base is given by Ag(y) = (φ(x0 )2 + y 2 )

∂2g (y) ∂y 2

we have: d{ys }A = φ(x0 )dBs + ys dWs + ∆u(x0 )ds. Also h1, 1iy = φ(x0 )12 +y2 and πs (b) = πs (δx0 ◦ ∆). The innovation increment d{ys }A − πs (b)ds is therefore given by d{ys }A − πs (b)(u0 )ds = φ(x0 )dBs + ys dWs + ∆us (x0 )ds − ∆us (x0 )ds where ut = E(ut |σ(ys, 0 ≤ s ≤ t)), since then πs (δz )(u0 ) = us (z) and πs (∆)(u0 ) = ∆us (x0 ).

5.5. Notes

99

We have πs (Bδz ) = πs (δz ◦ ∆). Also πs (δz b) − πs (δz )πs (b), which is πs ((δz )(δx0 ◦ ∆)) − πs (δz )πs (δx0 ◦ ∆), when applied to u0 can be written (πs (δz b) − πs (δz )πs (b))(u0 ) = u(z)∆us (x0 ) − us (z)∆us (x0 ). Finally, πs (df ◦ h− )(u0 ) = hus (1)(z) =

5.5

φ(x0 )φ(z)+us (x0 )us (z) . φ(x0 )2 +us (x0 )2

Notes

Noise-free observations In this chapter we continued to deal with “noise-free observations”, [54], but with the simplifying assumption that the quadratic variation of the observations process depends only on the signal through the observations. The more general case is discussed in [54] and [21]. We also reduced the standard filtering set up to our “noise-free” situation.

Chapter 6

The Commutation Property In certain cases the filtering is in a sense trivial: the process decomposes into the observable and an independent process. From the geometric point of view this means the commutation of the vertical operator B V and the horizontal operator AH . See Theorem 6.2.9 below. For p a Riemannian submersion (defined in Chapter 7 below) with totally geodesic fibres and B the Laplacian, Berard-Bergery & Bourguignon [9] show that AH and B V commute. Their proof is based on the result of R.Hermann [51] Theorem 6.0.1 (R. Hermann). A Riemannian submersion p : N → M has totally geodesic fibres iff the Laplace-Beltrami operator of N commutes with all Lie derivations by horizontal lifts of vector fields on M . From this, and the H¨ ormander form representation of AH , it follows immediH V ately that A with B will commute in their situation. In this section we consider some extensions of this and their consequences. First, for p : N → M with a diffusion operator B over a cohesive A, as usual, we will say that a vector field on N is basic if it is the horizontal lift of a section of E. From our H¨ ormander form representation of AH we get the following extension of Berard-Bergery & Bourguignon’s result: Theorem 6.0.2. For a diffusion operator B over a cohesive diffusion operator A the following are equivalent: (i) B V commutes with all Lie derivations by smooth basic vector fields of N ; (ii) the operators B, B V , and AH commute (on C 4 functions); (iii) the operator B V commutes with the horizontal lifts of the vector fields which appear in one H¨ ormander form representation of A. Proof. It is clear that (i) implies (iii), and (iii) implies To
show (ii) implies (i) Pm (ii). j λ df since every one-form observe that every sectionPof E has the form σ A j 1 m on M can be written as 1 λj dfj for λj : M → R and fj : M → R and some

K.D. Elworthy et al., The Geometry of Filtering, Frontiers in Mathematics, DOI 10.1007/978-3-0346-0176-4_6, © Springer Basel AG 2010

101

102

Chapter 6. The Commutation Property

integer m. By definition of the connection this shows that every basic vector field Pm H on N has the form 1 λj ◦ p σ A (p∗ dfj ). It will therefore suffice to show that if H (ii) holds, then B V commutes with Lie differentiation by λ ◦ p σ A (p∗ df ) for all smooth λ, f : M → R. For this assume (ii) holds and take a smooth g : N → R. By definition of the symbol and Remark 1.4.8:   H 2BV dg λ ◦ p σ A (p∗ df )  H  = 2λ ◦ p B V dg σ A (p∗ df )
= λ ◦ p B V AH (f ◦ p g) − f ◦ p AH (g) − gAH (f ◦ p )
= λ ◦ p AH (f ◦ p )B V g − f ◦ p AH B V g − (Af ) ◦ p B V g H

= 2λ ◦ p d(B V g)σ A (p∗ df ) H

= 2d(B V g)σ A (λ ◦ p p∗ df ) as required.



For the special case of an equivariant diffusion on a principal bundle as considered in Chapter 3 we can obtain a working criterion for commutativity: see also Example 6.2.13. Corollary 6.0.3. In the notation of Theorem 3.2.1, commutativity of BV and AH holds if and only if both α and β are constant along all horizontal curves. This holds if and only if AH (αi,j ) = 0 and AH (β k ) = 0 for all i, j, k. Proof. First note that each vector field A∗k commutes with all basic vector fields. Indeed if V is basic it is equivariant and so (Rexp tAk )? (V ) = V

t > 0.

Differentiating in t at t = 0 gives the required commutativity. Thus the operators LA∗k are invariant under flows of basic vector fields and so for B V to commute with basic vector fields the coefficients α and β must be constant along their flows. By the theorem this gives the first result since any horizontal curve can be considered as an integral curve of a (possible time dependent) basic vector field. Clearly, from the H¨ ormander form of AH , if this holds both α and β are H A -harmonic. The converse holds since, from above, AH commutes with all of  the vertical vector fields L∗Ak . The Corollary is applied to derivative flows in Example 6.2.13 of Section 6.2 below. Hermann proved that a Riemannian submersion with totally geodesic fibres , see Section 9.4, has the natural structure of a fibre bundle with group the isometry group of a typical fibre:

6.1. Commutativity of Diffusion Semigroups

103

Theorem 6.0.4 (R. Hermann). If N is a complete Riemannian manifold and φ : N → M is a C ∞ Riemannian submersion, then φ is a locally trivial fibre space. If in addition the fibres of φ are totally geodesic submanifolds of N , φ is a fibre bundle with structure group the Lie group of isometries of the fibre. An analogous result given the hypothesis of Theorem 6.0.2 together with some completeness and hypoellipticity conditions is proved in Theorem 6.2.9 below. Before that we consider when the associated semi-groups commute.

6.1

Commutativity of Diffusion Semigroups

It is well known that in general the commutativity of two diffusion generators (on C 4 functions) does not imply that of their associated semi-groups. One reference is [90] page 273 where an example they ascribe to Nelson is given. Here is a minor modification of that construction: Cut R2 along the positive x-axis and remove the origin. Take a copy C, say, of (0, ∞) × [0, ∞) and glue it along the cut to the lower part of the cut plane, identifying (0, ∞) × {0} in C with the positive x-axis. This gives a version of the plane but with two copies of the positive quadrant, the original one now incomplete along the positive x-axis and the second one, C, incomplete along the positive yaxis, and with the origin missing. On this we have naturally defined vector fields ∂ ∂ and X 2 given by ∂y . These certainly commute. However their X 1 given by ∂x associated semi-groups do not, as can be seen by starting at the point (−1, −1) moving along the X1 -trajectory for time 2 and then along the X 2 trajectory for the same amount of time. We end up at the point (1, 1) of copy C. However if we had changed the order of the vector fields we would be at (1, 1) of the original positive quadrant. A more geometrically satisfying construction would be, as Nelson, to use the double covering of the punctured plane as state space with similarly behaved vector fields. Though it will not be used in the sequel here is an easy positive result: Proposition 6.1.1. Let A1 and A2 be diffusion operators with associated semigroups {Pt1 }t>0 and {Pt2 }t>0 acting as strongly continuous semi-groups on a Banach space E of functions which contains the C 2 functions with compact support. Let G1 and G2 be the corresponding generators, (closed extensions of the restrictions of A1 and A2 to the space of C 2 functions with compact support). Assume there is a core C2 for G2 consisting of bounded C ∞ functions such that for f ∈ C2 : (i) For all t > 0 the function Pt1 f is C 4 . (ii) A2 1t (Pt1 f −f ) is uniformly bounded in t ∈ (0, 1) and in space, and it converges pointwise to A2 A1 Pt1 f as t → 0+. (iii) A2 Pt1 f is uniformly bounded in t ∈ (0, 1) and in space.

104

Chapter 6. The Commutation Property

Then commutativity of Pt1 with Ps2 , 0 6 s, 0 6 t follows from commutativity of A1 with A2 on C 4 functions. Moreover if this holds, the semi-group {PtA1 +A2 }t>0 associated to A1 + A2 satisfies PtA1 +A2 = Pt1 Pt2 . Proof. Let f : M → R be in C2 . We show first that A2 Pt1 f = Pt1 A2 f.

(6.1)

For this set Vt = A2 Pt1 f . Then, by hypothesis (ii), ∂ Vt = A2 A1 Pt1 f ∂t = A1 Vt

(6.2)

by commutativity. By assumption (ii) we know Vs is bounded uniformly in s ∈ [0, t] for any t > 0. However there is a unique C 2 and uniformly bounded solution, P 1 V0 , to any diffusion equations such as (6.2) with given smooth bounded initial condition V0 (as is easily seen by the standard use of Itˆo’s formula applied to Vt−s acting on a diffusion process with generator A1 ). This gives A2 Pt1 f = Pt1 V0 = Pt1 A2 f as required. Now suppose f ∈ Dom(G2 ). By assumption there is a sequence {fn }n of functions in C2 converging in G2 -graph norm to f . Then Pt1 A2 fn → Pt1 G2 f and Pt1 fn → Pt1 f . Equation (6.1) therefore shows that Pt1 f ∈ Dom(G2 ) and we have G2 Pt1 ⊃ Pt1 G2 .

(6.3)

Next, for f ∈ Dom(G2 ), and our fixed t > 0 set Ws = Pt1 Ps2 f . Since the conver2 f − Ps2 f } to G2 Ps2 f is in E we see, using equation (6.3), gence of 1 {Ps+ ∂ Ws = Pt1 G2 Ps2 f = G2 Pt1 Ps2 f = G2 Ws ∂s since Ps2 f ∈ Dom(G2 ). In particular Ws ∈ Dom(G2 ). Although now it is not clear that W is C 2 we see from this that 0 for 0 < u < s, giving

∂ 2 ∂u Pu Ws−u

=

Pt1 Ps2 f = P02 Ws = Ps2 W0 = Ps2 Pt1 f for 0 6 s 6 t. For s > t it is now only necessary to use the semigroup property of P 2 , to commute with Pt1 portion by portion. Finally since Pt2 f ∈ Dom(G2 ) the above gives ∂ 1 2 P P f = A1 Pt1 Pt2 f + Pt1 A2 Pt2 f ∂t t t = (A1 + A2 )Pt1 Pt2 f and we can repeat the second arguement showing uniqueness of solutions of the diffusion equation to obtain PtA1 +A2 f = Pt1 Pt2 f . 

6.2. Consequences for the Horizontal Flow

105

Remark 6.1.2. Condition (i) does not always hold. A simple example is when the ∂2 state space is R2 − {(0, 0)} and the operator is ∂x 2 . The standard positive result for degenerate operators on Rn is due to Oleˇinik, [82].

6.2

Consequences for the Horizontal Flow

For our standard set-up of p : N → M with diffusion operator B over a cohesive A, let P V and P H denote the semi-groups , generated by the vertical and horizontal components of B, and let pVt (u, −), t 6 0, u ∈ N , be the transition probabilities of P V . If we set Nx = p−1 (x) for x ∈ M , then pVt (u, −) will be a probability measure + , the union of Np(u) with ∆. on Np(u) + for each x0 ∈ M there are measurable maps, For PA x0 -almost all σ ∈ Cx0 M the stochastic holonomy or stochastic parallel translation //tσ : Nx+0 → Nσ+t such that for each u ∈ Nx0 the process (t, σ) 7→ //tσ (u) is an AH -diffusion and is over σ. These can be obtained, for example, by taking a stochastic differential equation, such as equation (4.22), dxt = X(xt ) ◦ dBt + A(xt )dt for our A-diffusion. Let Yx : Ex → Rm be the adjoint (and right inverse) of X(x), each x ∈ M . Then consider the SDE on N , ˜ t )Y (σt ) ◦ dσt dyt = X(y and let (t, σ) 7→ //σt be the restriction of its flow to Nx0 , augmented by mapping the coffin state, ∆, to itself. This SDE is canonical since by equation (2.22) it can be rewritten as dyt = hyt ◦ dσt for h the horizontal lift map of Proposition 2.1.2. We will often need to assume that the lifetime of this diffusion is the same as that of its projection on M : Definition 6.2.1. The semi-connection induced by B is said to be stochastically complete if Cup0 M + := {σ : [0, ∞) → M + : lim p(ut ) = ∆ when ζ(u) < ∞} t→ζ

H

has full PA u0 measure for each u0 ∈ N or equivalently if the lifetimes satisfy ζ M (σ) = ζ M (//.σ u0 ) for PA u0 -almost all paths σ, and all u0 ∈ Nσ(0) .

106

Chapter 6. The Commutation Property

The semi-connection is said to be strongly stochastically complete if also we can choose a version of //σt : Nσ(0) → Nσ(t) which is a smooth diffeomorphism whenever σ(0) is a regular value of p and t < ζ(σ). Note that strong stochastic completeness of the connection will hold whenever the fibres of p are compact by the basic properties of the domains of local flows of SDE, [59], [30]. It also holds if the stochastic horizontal differential equation is strongly p-complete in the sense of Li [68] for p = dim(N ) − dim(M ). Proposition 6.2.2. Suppose the semi-groups P V and P H commute and stochastic completeness of the connection holds. Then the horizontal flow preserves the vertical transition probabilities in the sense that for all positive s and 0 < t < ζ(σ), (//σt )∗ pVs (u0 , −) = pVs (//σt u0 , −)

(6.4)

for all u0 ∈ Nσ for PA -almost all σ. Equivalently for any bounded measurable h : N → R we have PA -almost surely; PsV (h ◦ //σt ) (u0 ) = PsV h (//σt (u0 )) .

(6.5)

Proof. It suffices to show that given any finite sequence 0 6 t1 6 t2 6 · · · 6 tk < t, bounded measurable fj : M → R, j = 1, . . . , k and bounded measurable h : N → R, if u0 ∈ Nx0 , then  Ex0 f1 (σt1 ) . . . fk (σtk )χt<ζ(σ) · PsV (h ◦ //σt ) (u0 )  (6.6) = Ex0 f1 (σt1 ) . . . fk (σtk )χt<ζ(σ) · PsV (h) (//σt (u0 )) where χZ denotes the indicator function of a set Z. To see this set f˜j = fj ◦ p : N → R. Then the left-hand side of (6.6) is n o Ex0 f˜1 (//σt1 (u0 )) . . . f˜k (//tk (u0 )) · χt<ζ(σ) · PsV (h ◦ //σt )(u0 ) n   o = Ex0 PsV f˜1 (//σt1 (u0 )) . . . f˜k (//σtk (u0 )) · χt<ζ(σ) · h ◦ //σt (u0 )     H h (u0 ) = PsV PtH1 f˜1 . . . PtHk −tk−1 f˜k Pt−t k   H V P h (u0 ) = PtH1 f˜1 . . . PtHk −tk−1 f˜k Pt−t s k which reduces to the right-hand side of (6.6).



Remark 6.2.3. Assuming strong stochastic completeness of our semi-connection let {zt : 0 6 t < ζ(p(u. )} be a semi-martingale in N with p(zt ) = xt := p(ut ) : 0 6 t < ζ(p(u. )). If x0 is a regular value of p we have the Stratonovich equation: d//t−1 zt = T //t−1 ◦ dzt − T //t−1 (hzt ◦ dxt )

(6.7)

6.2. Consequences for the Horizontal Flow

107

where //t refers to //tx. . To see this, for example set bt = //t−1 zt and observe that dzt = d(//t bt ) = T //t ◦ dbt + h//t bt ◦ dxt . Now assume that our induced semi-connection is strongly stochastically complete. For a regular value x0 of p and u0 ∈ Nx0 define a process αu0 : [0, ∞) × Cu0 N + → Nx+0 by p(u) (6.8) αtu0 (u) = αt (u) = (//t )−1 ut if u ∈ Cuo N with t < ζ(u) and define αtu0 (u) = 4 if t > ζ(u). Note that αt may not go out to infinity in Nx0 as t increases to its extinction time. Also define //s∗ (B V )(f ) = B V (f ◦ //s ) ◦ //t−1 to obtain a random time-dependent diffusion operator //s∗ (B V ) on each fibre over a regular value of p. Lemma 6.2.4. In the notation of equation (4.24) we have the Itˆ o equation for αt := αtu0 : dαt = T //t−1 V (//t αt )dWt + T //t−1 V 0 (//t αt )dt

∇V which we can write as: ∇V

dαt = //t∗ V (αt )dWt + //t∗ V 0 (αt )dt.

In particular for f : N → R in C 2 Z t df,α Mt := f (αt ) − //s∗ (B V )(f )(αs )ds

(6.9)

(6.10)

0

is a local martingale. Proof. Formula (6.9) is immediate from equations (4.24) and (6.7). That M.df,α is a local martingale follows immediately using the properties of pull-backs under diffeomorphisms of Lie derivatives when V is C 1 , and by going to local co-ordinates otherwise.  Lemma 6.2.5. Suppose A has H¨ ormander form m

A=

1X L j L j + L0X . 2 j=1 X X

fj for j = 0, . . . , m ˜ j of the vector field X Let //.j be the flow of the horizontal lift X and represent //. by the flow of the SDE: dzs =

m X j=1

fj (zs ) ◦ dB j + X f0 (zs ). X

108

Chapter 6. The Commutation Property

d (//sj )∗ B V is the commutator [LX j , (//sj )∗ B V ]. Consequently for any C 3 Then ds function f : N → R and u0 ∈ N we have the Stratonovich formula: m
X d (//s )∗ (B V )(f )(u0 ) = [LX j , (//sj )∗ B V ](f )(u0 ) ◦ dB j j=1

+ [LX 0 , (//s0 )∗ B V ](f )(u0 ) dt.

(6.11)

Proof. From a standard result for any smooth vector field V on N we have d i ∗ j V ds (//s ) LV = [X , V ]. We can take a representation for B , as in equation (1.7) of Section 1.1, N 1 X ij a (·)LV i LV j + LV 0 BV = 2 ij=1 with smooth aij and vector fields V j . If the aij all vanish, so that we have a first-order operator, we can apply the standard result. Otherwise we can suppose V 0 = 0 and just consider B V = a(·)LV 1 LY 2 . Then
d j ∗ V d (//s ) B = a(//sj (·))(//sj )∗ (LV 1 )(//sj )∗ (LV 2 ) ds ds = LX j (a)(//sj (·))(//sj )∗ (LV 1 )(//sj )∗ (LV 2 )
+ a(//sj (·)) (//sj )∗ ([LX j , LV 1 ])(//sj )∗ LV 2 + LV 1 (//sj )∗ [LX j , LV 2 ] . Now (//sj )∗ [LX j , LV j ] = [LX j , (//sj )∗ LV j ] since Lie brackets commute with diffeomorphisms and any vector field is invariant under its flow. Expanding the brackets yields d j ∗ V (// ) B = LX j (a)(//sj (·))(//sj )∗ (LV 1 )(//sj )∗ (LV 2 ) ds s + a(//sj (·))[LX j , (//sj )∗ (LV 1 LV 2 )] = [LX j , (//sj )∗ B V ] as required.

 (//sj )∗ B V

Remark 6.2.6. Since is determined by the operators A and B, at least for fibres over regular values of p, it is independent of the choice of H¨ormander
form for A. That means that the stochastic differential d (//s )∗ (B V )(f )(u0 ) should also be expressible in terms of the operators using the approach of Section 4.1. Definition 6.2.7. For a regular value x0 of p. We say B V is stochastically holonomy invariant at x0 if on Nx0 we have //t∗ (B V ) = B V for all 0 6 t < ζ M with probability 1. If this holds for all regular values x0 , then we say B V is stochastically holonomy invariant. Similarly we say B V is holonomy invariant at x0 if the corresponding result holds for parallel translation along any piecewise C 1 curve starting at x0 in M , and is holonomy invariant if this holds for all regular values x0 .

6.2. Consequences for the Horizontal Flow

109

Remark 6.2.8. 1. If the A-diffusion on M is represented by a stochastic differential equation we can lift that equation to N and obtain a local flow ηtH : 0 6 t < ζ H (−) where ζ H (y) : y ∈ N gives its explosion times; so that with probability 1 ηtH is defined and smooth on the open set {y ∈ N : t 6 ζ H (y), see [59] or [30]. We can say that B V is invariant under the horizontal flow if for all C 2 functions f : N → R we have B V (f ) ◦ ηt = BV (f ◦ ηt ) on {y ∈ N : t 6 ζ H (y), almost surely, for all t > 0. This does not require strong stochastic completeness of the semi-connection, nor do we have to restrict attention to fibres over regular values. On the other hand if it holds, and given such strong stochastic completeness, if x0 is a regular value it follows that Nx0 lies in {y ∈ N : t 6 ζ H (y) for all t < ζ M (x0 ) and that we have stochastic holonomy invariance at x0 . 2. Assume completeness of the semi-connection. If A satisfies the standard H¨ormander condition, or more generally if the space D0 (x0 ), as in Section 2.6, is all of M , then holonomy invariance at x0 implies holonomy invariance. This follows since concatenation of paths gives composition of the corresponding parallel translations and the conditions imply that any two points can be joined by a smooth path with derivatives in E. Moreover by Theorem 2.6.1 every point is a regular value and so given also strong stochastic completeness of the connection, from the theorem below we see that holonomy invariance of B V at one point implies it is invariant under the horizontal flow induced by any SDE on M which gives one-point motions with generator A. The same holds for stochastic holonomy invariance: see Theorem 6.2.9 below. Theorem 6.2.9. Suppose the induced semi-connection is complete and strongly stochastically complete, and x0 is a regular value of p. Then the following are equivalent: (ix0 ) For all u0 ∈ Nx0 and for any F α -stopping time τ with τ (α(u)) < ζ(p(u)), the process {αt : 0 6 t < τ } is independent of F x0 ; (iix0 ) B V is stochastically holonomy invariant at x0 ; (iiix0 ) B V is holonomy invariant at x0 ; (ivx0 ) B V and AH commute at all points of D0 (x0 ); (vx0 ) P.V and P.H commute at all points of D0 (x0 ). If the above hold at some regular value x0 they hold for all elements in D0 (x0 ). Moreover α.u0 is a Markov process on Nx0 with generator BV . Proof. We will show that (ix0 ) is equivalent to (iix0 ) which implies (ivx0 ). Then (ivx0 ) implies (iiiy) for all y ∈ D0 (x0 ) which implies (vx0 ). Finally we show (vx0 ) implies (iiy) for all y ∈ D0 (x0 ).

110

Chapter 6. The Commutation Property

Assume (ix0 ) holds. Let f : Nx0 → R be smooth with compact support. Then the local martingale M df,α given by formula (6.10) is a martingale and from equation (6.9) we see that E{M df,α |F x0 } = f (u0 ). Therefore, for Px0 -almost all σ in Cx0 M , Z E{f (αt )} = E{f (αt )|p(u. ) = σ} = f (u0 ) +

t

E{(//sσ )∗ (B V )(f )(αs )}ds. (6.12)

0

Also, in the notation of equation (6.9), with the obvious notation for the filtrations generated by our processes, we have Ftα. ⊂ FtW. ∧ Ftx0 and FtW. ⊂ Ftα. ∧ Ftx0 so our assumption implies that FtW. = Ftα. , for all positive t, after stopping W. at the explosion time of α. . From this, and equation (6.9), we see that if we set ¯ tdf,α = E{Mtdf,α |F αt } we obtain a martingale with respect to F α and M ∗ Z t ¯ tdf,α + //s∗ B V (f )(αs )ds (6.13) f (αt ) = M 0

where //s∗ B V = E{//s∗ B V }. Thus by the usual martingale characterisation of Markov processes we see that α. is Markov with (possibly time dependent) generator //s∗ B V at time s. However equation (6.12) then implies, for example by [91] Proposition(2.2), Chapter VII, that the generator is given by (//sσ )∗ (B V ) for arbitrary σ in a set of full measure in Cx0 M . Thus (ix0 ) implies the stochastic holonomy invariance (iix0 ). Conversely if (iix0 ) holds, equation (6.10) gives Z t df,α + B V (f )(αs )ds. f (αt ) = Mt 0

Then M.df,α is an F∗α. -martingale and again we see that α. is Markov, with generator BV . It is therefore independent of x. giving (ix0 ). Moreover, in an obvious notation, if 0 6 s 6 t, by the flow property of parallel translations, on Nx0 , B V = //t∗ (B V ) = //s∗ (//ts )∗ (B V ), and so, almost surely, at all points of Nxs we have (//ts )∗ (B V ) = (//s∗ )−1 B V = BV . ∗ Since (//ts )∗ (B V ) has the same law as //t−s (B V ) and is independent of Fsx0 this A shows that (iiy) holds for ps (x0 , −)-almost all y ∈ M for all s > 0. On the other hand (iiy) implies that B V and AH commute on Ny by Lemma 6.2.5 and Theorem 6.0.2. Thus by continuity of [B V , AH ], and the support theorem we see that (iix0 ) implies (ivx0 ).

6.2. Consequences for the Horizontal Flow

111

Furthermore as in Theorem 6.0.2 we see that (ivx0 ) implies that B V commutes with basic vector fields at all points over D0 (x0 ). From this the holonomy invariance (iiiy) holds for all y ∈ D0 (x0 ). Now assume (iiix0 ) and so by Remark 6.2.8(2.) we have (iiiy) for all y ∈ D0 (x0 ). Since //tσ (u0 ) stays above D0 (x0 ) for any suitable piecewise smooth σ we find the solution to the martingale problem of BV for any point u0 of Nx0 is holonomy invariant at u0 , i.e. along piecewise smooth curves σ in M starting at x0 , PtV (f ◦ //sσ −)(u0 ) = PtV (f )(//sσ u0 ). By Wong-Zakai approximations we see that stochastic holonomy invariance V of P B holds over x0 and hence on taking expectations we get (vx0 ). As observed we also get (vy) for all y ∈ D0 (x0 ) and hence by continuity for all y ∈ D0 (x0 ). Thus (iiix0 ) implies (vx0 ). Finally assuming (vx0 ) we can apply Proposition 6.2.2, observing that the proof still holds since it only involves points in D0 (x0 ). Differentiating equation (6.5) in s at s = 0 gives the stochastic holonomy invariance (iiy) for all y ∈  D0 (x0 ) Remark 6.2.10. From the proof and Theorem 6.0.2 we see that the stochastic completeness of the connection is not needed to ensure that (ivx0 ) and (iiix0 ) are equivalent. We can now go further than our Theorem 2.6.1 in extending Hermann’s result, Theorem 6.0.1. For this we will need some extra hypoellipticity conditions to deal with the case of non-compact fibres. Take a H¨ormander form A corresponding to a smooth factorisation σxA = X(x)X(x)∗ with X(x) ∈ L(Rm : Tx M ) for x ∈ M as usual. Let H denote the usual Cameronm Martin space of finite energy paths H = L2,1 0 ([0, 1]; R ). For h ∈ H and x ∈ M h let φt (x), 0 6 t 6 1 be the solution at time t ∈ [0, 1] to the ordinary differential equation ˙ z(t) ˙ = X(z(t))(h) (6.14) with φh0 (x) = x. In particular we assume such a solution exists up to time t = 1. h For each x ∈ M this gives a smooth mapping φ− 1 (x) : H → M , namely h 7→ φ1 (x). h,x : Ex → Ex be the deterministic Malliavin covariance operator, see [11], Let C given by − ∗ C h,x = Th φ− 1 (x)(Th φ1 (x)) . h,x Then φ− is non1 (x) is a submersion in a neighbourhood of h if and only if C degenerate. It is shown in [11] that this condition is independent of the choice of H¨ormander form for A, and follows from the standard H¨ormander condition that X 1 , . . . , X m and their iterated Lie brackets span Tx M when evaluated at the point x. A more intrinsic formulation of it can be made in terms of the manifold of E-horizontal paths of finite energy, as described in [79].

112

Chapter 6. The Commutation Property

Theorem 6.2.11. Consider a smooth map p : N → M with diffusion operator B on N over a cohesive diffusion operator A. Suppose that the connection induced by B is complete. Also assume that D0 (x) is dense in M for all x ∈ M and that either the fibres of p are compact or that the solutions to equation (6.14) exist up to time 1 and there exists h0 ∈ H and x0 ∈ M such that C h0 ,x0 is non-degenerate. Then p : N → M is a locally trivial bundle. If also B and AH commute we can take Nx0 , the fibre over x0 , to be the model fibre and choose the local trivialisations τ : U × Nx0 → p−1 (U ) to satisfy τ (x, −)∗ (B V |Nx ) = B V |Nx0 . Proof. The local triviality given compactness of the fibres is a special case of Corollary 2.6.3 so we will only consider the other case. For this set y = φh1 0 (x0 ). Our assumption on the covariance operator together with the smoothness of h 7→ φh1 (x0 ) implies by the inverse function theorem that there is a neighbourhood Uy of y in M and a smooth immersion s : Uy → H with s(x) s(y) = h0 and φ1 (x0 ) = x for x ∈ Uy . We know from Theorem 2.6.1 that p is a submersion so all its fibres are submanifolds of N . Define τUy : Uy × Nx0 → p−1 (Uy ) by using the parallel translation s(x) along the curves φt : 0 6 t ≤ 1 that is: φs(x)

τUy (x, v) = //1 .

(v),

(x, v) ∈ (Uy × Nx0 ).

(6.15)

For a general point x of M we can find an x0 ∈ Uy ∩ D0 (x) and argue as in the proof of Theorem 2.6.1 to obtain open neighbourhoods Ux of x in M and Ux0 0 of x0 in Ux0 and a fibrewise diffeomorphism of p−1 (Ux0 0 ) with p−1 (Ux ) obtained from parallel translations. This can be composed with a restriction of τUx0 to give a trivialisation near x. This proves local triviality. The rest follows directly from Remark 6.2.10 since our trivialisations came from parallel translations.  Remark 6.2.12. Set G(BxV0 ) = {α ∈ Diff(Nx0 ) : α∗ (B V |Nx0 ) = B V |Nx0 }.

(6.16)

Then assuming the commutativity in the theorem we can consider G(BxV0 ) as a structure group for our bundle, though unless the fibres of p are compact it is not clear if we have a smooth fibre bundle with this as group in the usual sense, since this requires smoothness into G(BxV0 ) of the transition maps between overlapping trivialisations. See the next section and Michor [76] section 13. Note that elements of G(BxV0 ) preserve the symbol of B V and so if that symbol V has constant rank they preserve the inner product induced on the image of σ B . In particular if B V is elliptic they are isometries of the Riemannian structure induced

6.2. Consequences for the Horizontal Flow

113

on the fibre Nx0 . This is the situation arising from Riemannian submersions as in Hermann’s Theorem 6.0.4 and described in detail in Chapter 7 below. The space of isometries of a Riemannian manifold with compact- open topology is well known to form a Lie group, for example see [55]. However there appears to be no detailed proof that the same holds in degenerate cases even when the H¨ormander condition holds at each point. When H¨ ormander’s condition holds, the Caratheodory metric on the manifold determines the standard manifold topology, e.g. see [79] Theorem 2.3, which is locally compact, and the group of isometries of a connected locally compact metric space is locally compact in the compact-open topology, see [55], Chapter 1, Theorem 4.7. Thus in this case G(BxV0 ) will be locally compact. In general, preserving the possibly degenerate Riemannian structure determined by its symbol will not be enough to characterise G(BxV0 ). Even in the elliptic case there may be a “drift vector” which needs to be preserved as well and this may ∂ lead to G(BxV0 ) being very small. For example if Nx0 is R2 and B V = 12 4 − |x|2 ∂x 1 the group is trivial. Example 6.2.13. 1. As an example consider the situation described in Section 3.3 of the derivative flow of a stochastic differential equation (3.12) on M acting on the frame bundle GLM to produce a diffusion operator B on GLM . Assume that M is Riemannian and complete, and that the one-point motions are Brownian motions, so that A = 12 4 . Assume also that the connection induced is the Levi-Civita connection. Then if B and AH commute, by Corollary 6.0.3, we see that the co-efficients α and β of B V described in Theorem 3.3.2 must be constant along horizontal curves. However as pointed out in the proof of Corollary, 3.4.9 the restriction of α(u) for u ∈ GLM to anti-symmetric tensors is essentially (one half of) the curvature operator. It follows that the curvature is parallel, ∇R = 0. In turn this implies, [55] page 303, that M is a local symmetric space and so, if simply connected, a symmetric space. In Section 7.2 we show how such stochastic differential equations arise on any symmetric space. Also from the example in Section 3.3 we see that the standard gradient SDE for Brownian motion on spheres also give derivative flows with this property. 2. For the apparently weaker property of commutativity for the derivative flow T ξt of our SDE (3.12) acting directly on the tangent bundle T M , recall first that if the generator A is cohesive (and even if it just happens that the symbol of A has constant rank, see [36]), then for vt = T ξt (v0 ) and some v0 ∈ Tx0 M we have the covariant SDE ˘ # (vt )dt + ∇ ˆ t=∇ ˘ v XdBt − 1 Ric ˘ v Adt. Dv t t 2

(6.17)

˘ v Xetc. refer to the LW connection of the SDE (3.12) and D ˆ refers Here ∇ t to the adjoint connection, both described in Section 3.3, see also Appendix 9.5.2.

114

Chapter 6. The Commutation Property From this we see that if A is cohesive the process α. defined by αt = −1 //ˆt T ξt (v0 ) satisfies the SDE   −1 ˘ # (//ˆt αt )dt + ∇ ˘ ˆ Adt . ˘ ˆ XdBt − 1 Ric ∇ dαt = //ˆt //t αt //t αt 2 Suppose also that A = 0. We see that α. is independent of ξ. (x0 ) if and ˘ − X and Ric ˘ # are holonomy invariant. If M is Riemannian and only if both ∇ the solutions of the SDE are Brownian motions and the induced connection is the Levi-Civita connection we can deduce, as above, using Theorem 6.2.9, that commutativity of the the vertical and horizontal diffusion operators on T M holds only if M is locally symmetric .

Chapter 7

Example: Riemannian Submersions and Symmetric Spaces 7.1

Riemannian Submersions

Recall that when N and M are Riemannian manifolds, a smooth surjection p : N → M is a Riemannian submersion if for each u in N the map Tu p is an orthogonal projection onto Tp(u) M , i.e. restricted to the orthogonal complement of its kernel, it is an isometry. Note that if p : N → M is a submersion and M is Riemannian we can choose a Riemannian structure for N which makes p a Riemannian submersion. If a diffusion operator B on N which has projectible symbol for p : N → M is also elliptic, its symbol induces Riemannian metrics on N and M for which p becomes a Riemannian submersion. A well-studied situation is when p is a Riemannian submersion and B is the Laplacian, or 12 4N , on N . The basic geometry of Riemannian submersions was set out by O’Neill in [83]; he ascribes the term ‘submersion’ to Alfred Gray. In this section we shall mainly be relating the work of B´erard-Bergery & Bourguignon [9], Hermann, [51], Elworthy & Kendall, [33], and Liao, [69], to the discussion above. The book [42] shows the breadth of geometric structures which can be considered in association with Riemannian submersions. A simple example of a Riemannian submersion is the map p : Rn − {0} → R given by p(x) = |x|. Then, for n > 1, Brownian motion on Rn − {0} is mapped n−1 d d2 to the Bessel process on (0, ∞) with generator A = 12 dx 2 + 2x dx . Thus in this 1 1 case 2 4N is projectible but its projection is not 2 4M . The well-known criterion for the latter to hold is that p has minimal fibres as we show below. See also [30], and [69].

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Chapter 7. Example: Riemannian Submersions and Symmetric Spaces

To examine this in more detail we follow Liao [69]. Suppose that p is a Riemannian submersion. The horizontal subbundle on N is just the orthogonal complement of the vertical bundle. Working locally, take an orthonormal family of vector fields X 1 , . . . , X n in a neighbourhood of a given point x0 of M . Let ˜ n be their horizontal lifts to a neighbourhood of some u0 above x0 , and ˜ 1, . . . , X X let V 1 , . . . , V p be a locally defined orthonormal family of vertical vector fields around u0 . Then near u0 , using the summation convention over j = 1, . . . , n, α = 1, . . . , p, we have α ˜ jX ˜ j + V α V α − ∇N˜ j X ˜ j − ∇N 4N = X V αV X

(7.1)

j 4M = X j X j − ∇M Xj X .

(7.2)

while Here ∇M , ∇N refer to the Levi-Civita connections on M and N , and we are identifying the vector fields with the Lie differentiation in their directions. ˜ jX ˜ j lies over X j X j while V α V α is vertical. Also the horizontal comNow X ponent of the sum ∇V α V α at a point u ∈ N is the trace of the second fundamental form, see the Appendix, Section 9.4, of the fibre Np(u) of p through u, denoted by j ˜j TV α V α in O’Neill’s notation, while ∇N ˜ j X lies over ∇X j X by Lemma 1 of [83]. X 1 Thus we see that 2 4N is projectible if and only if the trace of the second fundamental form, trace T , of each fibre p−1 (x) is constant along the fibre in the sense of being the horizontal lift of a fixed tangent vector, 2A(x) ∈ Tx M . If so, 1 1 2 4N lies over 2 4M − A. In particular A = 0, or equivalently p maps Brownian motion to Brownian motion, if and only if p has minimal fibres. In general to relate to the discussion in Section 2.4, we can set bH (u) = 1 − 2 trace T (u), with b(u) = Tu pbH (u) in Tp(U ) M . Let 4V be the vertical operator on N which restricts to the Laplacian on each fibre, and let 4H be the horizontal lift of 12 4M . Our decomposition in Theorem 2.4.6 becomes   1 H 1 1 1 4N = 4 − trace T + 4V (7.3) 2 2 2 2 since the vertical part of ∇V α V α is just ∇VV α V α where ∇V refers to the connection on the vertical bundle which restricts to the Levi-Civita of the fibres, and also the ˜ jX ˜ j vanishes because by Lemma 2 of [83] the vertical part of vertical part of X j k ˜ j, X ˜ k ]. ˜ X ˜ is the vertical part of 1 [X X 2

7.2

Riemannian Symmetric Spaces

Let K be a Lie group with bi-invariant metric and let M be a Riemannian manifold with a symmetric space structure given by a triple (K, G, σ). This means that there is a smooth left action K × M → M, (k, x) 7→ Lk (x) of K on M by isometries such that, if we fix a point x0 of M and define p : K → M by p(k) = Lk (x0 ), then p is

7.2. Riemannian Symmetric Spaces

117

a Riemannian submersion and a principal bundle with group the subgroup Kx0 of K which fixes x0 . Write G for Kx0 . Thus M is diffeomorphic to K/G. Moreover if g denotes the Lie algebra of G, and k that of K, (identified with the tangent spaces at the identity to G and K respectively), there is an orthogonal and adG invariant decomposition k=g+m where m is a linear subspace of Tid K. Further σ is an involution on K and g and m are, respectively, the +1 and the −1 eigenspaces of the involution on Tid K induced by σ. See Note 7, page 301, of Kobayashi & Nomizu Volume I, [55], for definitions and basic properties, and Volume II, [56], for a detailed treatment. We shall also let σ denote the involutions induced by σ on k and on M , and by differentiation on T M and OM . On M it is an isometry, so it does act on OM . Note that on Tx0 M it acts as v 7→ −v. Since G fixes x0 the derivative of the left action Lk at x0 gives a representation of G by isometries of Tx0 M . The linear isotropy representation. We shall assume it to be faithful, i.e. injective. As a consequence the action of K on M is effective, so that K can be considered as a subgroup of the diffeomorphism group of M , and also the action of K on the frame bundle of M is free, i.e the only element of K which fixes a frame is the identity element. See page 187 and the remark on page 198 of [56] for a discussion of this, and how the condition can be avoided. Taking a fixed orthonormal frame u0 : Rn → Tx0 M , say, at x0 , we can consider G as acting by isometries on Rn by g · e = u−1 0 T Lg u0 (e).

(7.4)

Let ρ : G → O(n) denote this representation. We then have the well-known identification of K as a subbundle of the orthonormal frame bundle of M : Proposition 7.2.1. Let Φ : K → OM be defined by Φ(k)(e) = T Lk (u0 e) for e ∈ Rn . Then Φ is an injective homomorphism of principle bundles. Moreover Φ is equivariant for the actions of σ on K and OM . Proof. To see that Φ is a bundle homomorphism it is only necessary to check that Φ commutes with the actions of G. For this take e ∈ Rn and g ∈ G. Then, for k ∈ K, Φ(k · g)(e) = T Lk T Lg u0 (e) = Φ(k)T Lg u0 (e) = Φ(k)u0 (g · e) as required. For the equivariance with respect to σ, observe that by definition, σ(Lk x0 ) = Lσ(k) x0 so that acting on the frame Φ(k) we have σ(Φ(k)) = σ(T Lk ◦ u0 ) = T Lσ(k) u0 = Φ(σ(k)). 

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Chapter 7. Example: Riemannian Submersions and Symmetric Spaces

It is easy to see that p : K → M has totally geodesic fibres. We can therefore take B = 12 4K to have B lying over 12 4M . Moreover in the decomposition of B the vertical component 12 4V restricts to one half the Laplacian of G on the fibre p−1 (x0 ). The induced connection has horizontal subspace m at the identity element of K. It is clearly left K-invariant and so Hk = T Lk [m] for general k ∈ K. From the equi-variance under the right action of G it is a principle connection: T Rg [Hk ] = Hkg . Since Hkg = T Lk T Lg [m] = T Rk T Lk adg [m] this holds because of the adG -invariance of m. This is the canonical connection. The connection on K extends to one on OM as described in Proposition 3.1.3. This is known as the canonical linear connection. Since the connection on K is invariant under σ, by the equi-variance of Φ so is the canonical linear connection. As in [56] we have: Proposition 7.2.2. The canonical linear connection is the Levi-Civita connection. Proof. It is only necessary to check that its torsion T vanishes. By left invariance it is enough to do that at the point x0 . Let u, v ∈ Tx0 M . However by invariance under σ we see T (u, v) = σT (σ(u), σ(v)) = −T (−u, −v) = −T (u, v), as required.



Let kt , t > 0 be the canonical Brownian motion on K starting at the identity, id, and let Bt be the Brownian motion on the Euclidean space k given by the right flat anti-development: Z t

Bt = 0

T Rk−1 d{ks }. s

Define ξt : M → M by ξt (x) = Lkt x, for t > 0, x ∈ M . Proposition 7.2.3. The diffeomorphism group-valued process ξt , t > 0 is the flow of the SDE dxt = X(xt ) ◦ dBt where

d Lexp tα x|t=0 . dt Proof. Observe that k. satisfies the right invariant SDE, X(x)α =

dkt = T Rkt ◦ dBt , which is p-related to the given SDE on M .



Remark 7.2.4. The last two propositions relate to the discussion of connections determined by stochastic flows in the next Chapter, and to the discussion about canonical SDE on symmetric spaces in [36]. In [36] it was shown that the connection determined by our SDE is the Levi-Civita connection. In Theorem 8.1.3 below,

7.3. Notes

119

and in Theorem 3.1 of [34], it is shown that the connection determined by a flow (in this case the canonical linear connection) is the adjoint of that induced by its SDE. This is confirmed in our special case since the adjoint of a Levi-Civita connection is itself. We can also apply our analysis of the vertical operators and Weitzenb¨ock formulae to our situation, For this it is simplest to assume the symmetric space is irreducible. This means that the restricted linear holonomy group of the canonical connection on p : K → M is irreducible, i.e. for every g ∈ G there is a nullhomotopic loop based at x0 whose horizontal lift starting at id ∈ K ends at the point g. The definition in [56] is that [m, m] acts irreducibly on m via the adjoint action, and it is shown there, page 252, that this implies that g = [m, m]. As a consequence the linear isotropy representation of G on Tx0 M is irreducible, and equivalently so is our representation ρ. The vertical operators determined by B V on the bundles associated to p via our representation ρ and its exterior powers ∧k ρ are given in Theorem 3.4.1 by the k function λ∧ ρ : K → L(∧k Rn ; ∧k Rn ). By Corollary 3.4.9 and the discussion above they correspond to the Weitzenb¨ ock curvatures of the Levi-Civita connection, and so in particular are symmetric. To calculate them using Theorem X 3.4.1, first use the fact that B V restricts to 21 4G on p−1 (x0 ) to represent it as 12 LA∗j LA∗j for A∗j as in Section 3.2. The computation in the proof of Corollary 3.4.4 shows that k

λ∧ ρ (u) = −

(n − 2)! c k (u), (k − 1)!(n − k − 1)! ∧

(7.5)

for c∧k (u) = (d∧k )Al (u) ◦ (d∧k )A0l (u) the Casimir element of our representation ∧k ρ of G. If ∧k ρ is irreducible, then c∧k (u) is constant scalar. As remarked in Corollary 3.4.4 this happens when G = SO(n), given our irreducibility hypothesis on the √ . Thus for the sphere S n ( 2) of ρ and then it is just 12 n(n − 1)/ n(n−1)...(n−k+1) k! √ radius 2, considered as SO(n + 1)/SO(n) we have k 1 λ∧ ρ (u) = − k(n − k). 4

7.3

(7.6)

Notes

Intertwining of Laplacians etc. on functions and forms The result given in Section 7.1, that a Riemannian submersion commutes with the Laplacian if and only if the submersion has minimal fibres, is due to B. Watson, [105]. He also showed that if a map p commutes with the Hodge-Kodaira Laplacian on k-forms, i.e. ∆p∗ φ = p∗ ∆φ

for all k-forms φ

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Chapter 7. Example: Riemannian Submersions and Symmetric Spaces

for some fixed k with 0 6 k 6 dim M , then p must be a Riemannian submersion. Goldberg & Ishihara, [48] proved that if this holds for some k > 1 it holds for all k and p is a Riemannian submersion with minimal fibres and integrable horizontal distribution. Earlier, Watson [106] had proved that a C 2 map p : N → M between compact oriented Riemannian manifolds commutes with the usual co-differential d∗ on k-forms for some k > 2 if and only if p is a totally geodesic Riemannian submersion. Recall that a map is totally geodesic if it maps geodesics to geodesics. This corrected earlier work by Lichnerowicz, [71]. If the co-differential is defined to be the trace of the relevant covariant derivative operator, this is a local result so compactness and orientability are not needed. It was shown by Vilms [104] that a Riemannian submersion is totally geodesic if and only if it has totally geodesic fibres and integrable horizontal distribution. Extending Hermann’s result, [51] , Vilms showed that this implies that then, if N is complete so are M and the fibres, and the submersion is a fibre bundle with structure group consisting of isometries of the fibre and an equivariant flat connection. In particular if also M is simply connected the N is just the Riemannian product of M with another Riemannian manifold and p is the projection. It is worth noting the result from the book of Falcitelli, Ianus, and Pastore, [42], that if p is a Riemannian submersion with totally geodesic fibres and N has non-positive sectional curvatures, then so has M and the horizontal distribution is integrable. For results concerning intertwining of eigenforms of Hodge-Kodaira Laplacians see the book by Gilkey, Leahy, &Park, [46]. For intertwining of certain elliptic pseudo-differential operators, such as powers of the Laplacian, see Furutani [44]. Note that a Riemannian submersion with minimal fibres maps Brownian motion α to Brownian motion and so by subordination it will intertwine the powers (−∆) 2 for 0 < α 6 2, [78]. See also Appendix, Section 9.5.2.

Chapter 8

Example: Stochastic Flows Before analysing stochastic flows by the methods of the previous paragraphs we describe some purely geometric constructions which will enable us to identify the semi-connections which arise in that analysis.

8.1 Semi-connections on the Bundle of Diffeomorphisms Assume that M is compact. For r ∈ {1, 2, . . . } and s > r + dim M/2, let Ds = Ds (M ) be the space of diffeomorphisms of M of Sobolev class H s . See, for example, Ebin-Marsden [28] and Elworthy [30] for the detailed structure of this space. Elements of Ds are then C r diffeomorphisms. The space is a topological group under composition, and has a natural Hilbert manifold structure for which the tangent space Tθ Ds at θ ∈ Ds can be identified with the space of H s maps v : M → T M with v(x) ∈ Tθ(x) M , all x ∈ M . In particular Tid Ds can be identified with the space H s Γ(T M ) of H s vector fields on M . For each h ∈ Ds the right translation Rh : D s → Rh (f ) =

Ds f ◦h

is C ∞ . However the joint map Ds+r × Ds → Ds

(8.1)

is C r rather than C ∞ for each r in {0, 1, 2, . . . }. For x0 ∈ M fixed, define π : Ds → M by π(θ) = θ(x0 ).

(8.2)

The fibre π −1 (y) at y ∈ M is given by: {θ ∈ Ds : θ(x0 ) = y}. Set Dxs 0 := π −1 (x0 ). Then the elements of Dxs 0 act on the right as C ∞ diffeomorphisms of Ds . We can

K.D. Elworthy et al., The Geometry of Filtering, Frontiers in Mathematics, DOI 10.1007/978-3-0346-0176-4_8, © Springer Basel AG 2010

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Chapter 8. Example: Stochastic Flows

consider this as giving a principal bundle structure to π : Ds → M with group Dxs 0 , although there is the lack of regularity noted in equation (8.1). A smooth semi-connection on π : Ds → M over a subbundle E of T M consists of a family of linear horizontal lift maps hθ : Eπ(θ) → Tθ Ds , θ ∈ Ds , which is smooth in the sense that it determines a C ∞ section of L(π ∗ E; T Ds ) → Ds . In particular we have hθ (u) : M → T M with hθ (u)(y) ∈ Tθ(y) M, u ∈ Eθ(x0 ) , θ ∈ Ds , y ∈ M . It is a principal semi-connection if it also has the equivariance property: hθ◦k (u)(y) = hθ (u)(k(y))

for k ∈ Dxs 0 .

In this chapter we will only consider principal semi-connections and semi-connections induced by them on associated bundles. We shall relate principal semi-connections on Ds → M to certain reproducing kernel Hilbert spaces. For this let E be a smooth subbundle of T M and H a Hilbert space which consists of smooth sections of E such that the inclusion H → C 0 ΓE is continuous (from which comes the continuity into Hs ΓE for all s > 0). Such a Hilbert space determines and is determined by its reproducing kernel k, a C ∞ section of the bundle L(E ∗ ; E) → M × M with fibre L(Ex∗ ; Ey ) at (x, y), see [6]. By definition, k(x, −) = ρ∗x : Ex∗ → H where ρx : H → Ex is the evaluation map at x, and so k(x, y) = ρy ρ∗x : Ex∗ → Ey . Assume H spans E in the sense that for each x in M , ρx : H → Ex is surjective. It then induces an inner product h, iH x on Ex for each x via the isomorphism ρx ρ∗x : Ex∗ → Ex . Using the metric on E the reproducing kernel k induces linear maps k # (x, y) : Ex → Ey ,

x, y ∈ M,

with k # (x, x) = id. Proposition 8.1.1. A Hilbert space H of smooth sections of a subbundle E of T M which spans E determines a smooth principal semi-connection hH on π : Ds → M over E by   # θ(x0 ), θ(y) (u), θ ∈ Ds , u ∈ Eθ(x0 ) , y ∈ M, (8.3) hH θ (u)(y) = k

8.1. Semi-connections on the Bundle of Diffeomorphisms

123

for k # derived from the reproducing kernel of H as above. In particular the horizontal lift α ˜ starting from α ˜ (0) = id, of a curve α : [0, T ] → M , α(0) = x0 with α(t) ˙ ∈ Eα(t) for all t, is the flow of the non-autonomous ODE on M ,   z˙t = k # α(t), zt α(t). ˙

(8.4)

The mapping H 7→ (hH , h, iH ) from such Hilbert spaces to principal semi-connections over E and Riemannian metrics on E is injective. Proof. From the definition of k # we see hH θ (u)(y), as given by (8.3), takes values in Tθ(y) M , is linear in u ∈ Eθ(x0 ) into Tθ Ds , and is Dxs 0 -invariant. Moreover,   H # Tθ π ◦ h H θ(x0 ), θ(x0 ) (u) = u θ (u) = hθ (u)(x0 ) = k for u ∈ Eθ(x0 ) and so hH θ is a ‘lift’. That h is C ∞ as a section of L(π ∗ E; T Ds ) → Ds essentially comes from the smoothness of the map x → ρx . More precisely note that for each r ∈ {0, 1, 2, . . . } the composition map Tid Dr+s × Ds (V, θ)

→ T Ds 7 → T Rθ (V )

is a C r−1 vector bundle map over Ds , being a partial derivative of the composition Dr+s × Ds → Ds . Therefore it induces a C r−1 vector bundle map Z 7→ T Rθ ◦ Z, for Z : Eθ(x0 ) → H and for H the trivial H-bundle over Ds , by composition L(π ∗ E; H) Q Q Q

- L(π ∗ E; T Ds )    QQ s +  Ds

On the other hand y 7→ k(y, −) can be considered as a C ∞ section of L(E; H) → M and so θ 7→ k(θ(x0 ), −) as a C ∞ section of L(π ∗ E; H). This proves the regularity of h. That the horizontal lift α ˜ is the flow of (8.4) is immediate. To see that # the claimed injectivity holds, given hH θ observe that (8.3) determines k : this is ∞ because given any x in M there exists a C diffeomorphism θ such that θ(x0 ) = x and for such θ, −1 k # (x, z)(u) = hH z). (8.5) θ (u)(θ 

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Chapter 8. Example: Stochastic Flows

Remark 8.1.2. We cannot expect surjectivity of the map H → hH into the space of principal semi-connections on π : Ds → M . Indeed for k # given by (8.5) to correspond to the reproducing kernel for some Hilbert space of sections of E we need some specific conditions. 1) hH θ (u)(y) ∈ Eθ(y) for u ∈ Eθ(x0 ) , y ∈ M , and a metric h, i on E with respect to which the following holds: 2) for x, y ∈ M ,  ∗ k # (x, y) = k # (y, x) , 3) for any finite set S of points of M and {ξa } ∈ Ea , a ∈ S, E XD k # (a, b)ξa , ξb > 0. For each frame u0 : Rn → Tx0 M there is a homomorphism of principal bundles Ψu0 : Ds → GLM (8.6) θ 7→ Tx0 θ ◦ u0 . As with connections such a homeomorphism maps a principal semi-connection on Ds over E to one on GLM . The horizontal lift maps are related by Tθ Ψu0

Tθ Ds kQ Q Q hθ Q Q

- TΨu0 (θ) GLM 3     hΨu0 (θ) 

Q  Eθ(x0 ) and if α ˜ : [0, T ] → Ds is a horizontal lift of α : [0, T ] → M , then Ψu0 (˜ α(t)) = Tx0 α ˜ (t) ◦ u0 ,

06t6T

is a horizontal lift of α to GLM . Theorem 8.1.3. Let hH be the semi-connection on π : Ds → M over E determined by some H as in Proposition 8.1.1. Then the semi-connection induced on ˆ of the metric GLM , and so on T M , by the homeomorphism Ψu0 is the adjoint ∇ H connection which is projected on (E, h, i ) by the evaluation map (x, e) 7→ ρx (e) from M ×H → E, c.f. (1.1.10) in [36]. In particular every semi-connection on T M with metric adjoint connection arises this way from some, even finite dimensional, choice of H.

8.2. Semi-connections Induced by Stochastic Flows

125

˙ ∈ Eα(t) for each t. By PropoProof. Let α : [0, T ] → M be a C 1 curve with α(t) sition 8.1.1 its horizontal lift α ˜ to Ds starting from θ ∈ π −1 (α(0)) is the solution to   d˜ α = k # α(t)(x ˜ ), α ˜ (t) − α(t), ˙ (8.7) 0 dt α ˜ (0) = θ. (8.8) ˜ (t) ◦ u0 and to T M through v0 ∈ Tθ(x0 ) M , The horizontal lift to GLM is t 7→ Tx0 α i.e. the parallel translation {//t (v0 ) : 0 6 t 6 T } of v0 along α, is given by   ˜ ◦ θ−1 (v0 ). //t (v0 ) = Tx0 α ˜ (t) ◦ (Tx0 θ)−1 (v0 ) = Tα(0) α(t) However this is Tα(0) πt (v0 ) for {πt : 0 6 t 6 T } the solution flow of   dzt = k # α(t), z(t) α(t) ˙ dt which by Lemma 1.3.4 of [36] is the parallel translation of the adjoint of the associated connection (in [36] k # is denoted by k). The fact that all such semi-connections on T M arise from some finite dimensional H comes from Narasimhan-Ramanan [80] as described in [36], or more directly from Quillen [89] 

8.2

Semi-connections Induced by Stochastic Flows

From Baxendale [7] we know that a C ∞ stochastic flow {ξt : t > 0} on M , i.e. a Wiener process on D∞ := ∩s Ds , can be considered as the solution flow of a stochastic differential equation on M driven by a possibly infinite dimensional noise. Its one-point motions form a diffusion process on M with generator A, say. The noise comes from the Brownian motion {Wt : t > 0} on Hs Γ(T M ) determined by a Gaussian measure γ on Hs Γ(T M ). (In our C ∞ case they lie on H∞ (T M ) := ∩s Hs Γ(T M ).) We will take γ to be mean zero and so we may have a drift A in H∞ (T M ). The stochastic flow {ξt : t > 0} can then be taken to be the solution of the right invariant stochastic differential equation on Ds , dθt = T Rθt ◦ dWt + T Rθt (A)dt

(8.9)

with ξ0 the identity map id. In particular it determines a right invariant generator B on Ds . For fixed x0 in M the one-point motion xt := ξt (x0 ) solves dxt = ◦dWt (xt ) + A(xt )dt.

(8.10)

dxt = ρxt ◦ dWt + A(xt )dt.

(8.11)

We can write (8.10) as

126

Chapter 8. Example: Stochastic Flows

Thus π(ξt ) = ξt (x0 ) = xt . For a map θ in Ds , the solution ξt ◦ θ to (8.9) starting at θ has π(ξt ◦ θ) = ξt (π(θ)), the solution to (8.11) starting from π(θ), and we see that the diffusions are π-related (c.f. [30]), and A and B are intertwined by π. The measure γ corresponds to a reproducing kernel Hilbert space, Hγ say, or equivalently to an abstract Wiener space structure i : Hγ → Hs Γ(T M ) with i the inclusion (although i may not have a dense image). Then σθB : (Tθ Ds )∗ → Tθ Ds is right invariant and determined at θ = id by the canonical isomorphism Hγ∗ ' Hγ through the usual map j = i∗ , j

i

(Hs Γ(T M ))∗ ,→ Hγ∗ ' Hγ ,→ Hs Γ(T M ), i.e. B σid = i ◦ j. B This shows that Hγ is the image of σid with induced metric. In this situation our cohesiveness condition on A becomes the assumption that there is a C ∞ subbundle E of T M such that Hγ consists of sections of E and spans E, and A is a section of E. Let h, iy be the inner product on Ey induced by Hγ . The reproducing kernel k of Hγ is the covariance of γ and: Z D E v ∈ Ex ; x, y ∈ M. U (x), v U (y) dγ(U ), k # (x, y)v = x

U ∈Hs Γ(E)

Analogously to Lemma 2.3.1 we have the commutative diagram (Tθ Ds )∗

j ◦ (T Rθ )∗ - Hγ

6 (Tθ π)∗

6 `θ k(θ(x0 ), −)

∗ ∗ Tθ(x M → Eθ(x s) 0)

T Rθ ◦ i

ρθ(x0 ) -

Hγ

-Tθ Ds Tθ π = ρx0 ? Eθ(x0 ) ,→ Tx0 M

with `θ uniquely determined under the extra condition ker `θ = kerρθ(x0 ) . Writing K : M → L(Hγ ; Hγ ) for the map giving the projection K(x) of Hγ onto ker ρx for each x in M , and letting K ⊥ (x) be the projection onto [ker ρx ]⊥ , we have
`θ = K ⊥ θ(x0 ) ,

8.2. Semi-connections Induced by Stochastic Flows

127

(agreeing with the note following Lemma 2.3.1), and so   U ∈ Hγ . `θ (U ) = k # θ(x0 ), − U (θ(x0 )), Note that the formula K ⊥ (y)(U ) = k # (y, −)U (y) for U in Hγ determines an extension K ⊥ (y) : ΓE → Hγ . We then define K(y)U = U − K ⊥ (y)U . Note that ρy (K(y)U ) = 0 for all U in ΓE. The horizontal lift map determined by B as in Proposition 2.1.2 is therefore given by hθ : Eθ(x0 ) → T Rθ (Hγ ) ⊂ Tθ D s h i (8.12) hθ (u) = T Rθ `θ k # (θ(x0 ), −)u , for θ ∈ Ds . Consequently   hθ (u)(y) = k # θ(x0 ), θ(y) (u).

(8.13)

Comparing this with formula (8.3) we have Proposition 8.2.1. The semi-connection h determined on π : Ds → M by the equivariant diffusion operator B is just that given by the reproducing kernel Hilbert space Hγ of the stochastic flow which determines B, i.e. h = hHγ . The horizontal lift {˜ xt : t > 0} of the one-point motion {xt : t > 0} with x ˜0 = id is the solution to   ˜t (x0 ), x ˜t − ◦ dxt ; (8.14) d˜ xt = k # x which in a more revealing notation is:     d˜ xt = T Rx˜t K ⊥ (˜ xt (x0 )) ◦ dWt + T Rx˜t K ⊥ (˜ xt (x0 ))A dt.

(8.15)

Equivalently {˜ xt : t > 0} can be considered as the solution flow of the nonautonomous stochastic differential equation on M ,
dyt = k # xt , yt ◦ dxt , i.e.

  dyt = K ⊥ (xt ) ◦ dWt (yt ) + K ⊥ (xt )(A)(yt ) dt.

(8.16)

The standard fact that the solution to such an equation as (8.16) starting at x0 is just {xt : t > 0}, i.e. that x ˜t (x0 ) = xt reflects the fact that x ˜· is a lift of x· . xt ◦ φ : t > 0}. The lift through φ ∈ Dxs 0 is just {˜

128

Chapter 8. Example: Stochastic Flows

Remark 8.2.2. If our solution flow is that of an SDE, dxt = X(xt ) ◦ dBt + A(xt )dt for X(x) : Rm → T M arising, for example, from a H¨ormander form representation of A as in §4.7 above, the relationships with the notation in this section is as follows: Hγ = {X(·)e : e ∈ Rm } with inner product induced by the surjection Rm → Hγ . If Yx = [X(x)|ker X(x)⊥ ]−1 , then k # (y, −) : Ey → Hγ is k # (y, −)u = X(−)Yy (u),

u ∈ Ey .

Also K ⊥ (y) : ΓE → Hγ is K ⊥ (y)U = X(−)Yy (U (y)). Remark 8.2.3. The reproducing kernel Hilbert space Hγ determines the stochastic flow and so by the injectivity part of Proposition 8.1.1 the semi-connection together with the generator A of the one-point motion determines the flow, or equivalently the operator B. This is because the symbol of A again gives the metric on E which together with the semi-connection determines Hγ by Proposition 8.1.1. The generator A then determines the drift A. A consequence is that the horizontal lift AH of A to Ds determines the flow (and hence B, so B V really is redundant). To see this directly note that given any cohesive A on M and Dxs 0 -equivariant A on Ds over A, with no vertical part, there is at most one vertical B V such that AH + B V is right invariant. This results from the following lemma. H

Lemma 8.2.4. Suppose B 1 is a diffusion operator on Ds which is vertical and right invariant, then B 1 = 0. 1

Proof. By Remark 1.3.4 (i) the image Eθ , say, of σθB lies in V Tθ Ds for θ ∈ Ds and so if V ∈ Eθ . On the other hand, by right invariance Eθ = T Rθ (Eid ). Therefore if V ∈ Eid , then V (θ(x0 )) = T Rθ (V )(x0 ) = 0 all θ ∈ Ds and so V ≡ 0. Thus Eid = {0} and by right invariance, B1 must be first order and so given by some vector field Z on Ds . But Z must be vertical and right invariant, so again we see Z ≡ 0.  Proposition 3.1.3 applies to the homomorphism Ψu0 : Ds → GL(M ) of (8.6). From this and Theorem 8.1.3 we see that the semi-connection ∇ on GLM deterˆ of the conmined by the generator of the derivative flow in §3.3 is the adjoint ∇ ˘ so giving an alternative proof of the first part of Theorem 3.3.2 above. nection ∇, Proposition 3.1.3 also gives a relationship between the curvature and holonomy x0 ˆ and those of the connection induced by the flow on Ds ρ→ M. group of ∇ We can summarize our decomposition results as applied to these stochastic flows in the following theorem. The skew-product decomposition was already described in [34] for the case of solution flows of SDE of the form (4.20), and in particular with finite dimensional noise: however the difference is essentially that of notation, see Remark 8.2.2 above.

8.2. Semi-connections Induced by Stochastic Flows

129

Theorem 8.2.5. Let {ξt : t > 0} be a C ∞ stochastic flow on a compact manifold M . Let A be the generator of the one-point motion on M and B the generator of the right invariant diffusion on Ds determined by {ξt : t > 0}. Assume A is strongly cohesive. Then there is a unique decomposition B = AH + B V for AH a diffusion operator which has no vertical part in the sense of Definition 2.2.3 and B V a diffusion operator which is along the fibres of ρx0 , both invariant under the right action of Dxs 0 . The diffusion process {θt : t > 0} and {φt : t > 0} corresponding to AH and B V respectively can be represented as solutions to     dθt = T Rθt K ⊥ (θt (x0 )) ◦ dWt + T Rθt K ⊥ (θt (x0 ))A dt (8.17) and

    dφt = T Rφt K(z0 ) ◦ dWt + T Rφt K(z0 )A dt

(8.18)

for z0 = φ0 (x0 ) = φt (x0 ). There is the corresponding skew-product decomposition of the given stochastic flow ˜t gtx· , 0 6 t < ∞ ξt = x where {˜ xt : t > 0} is the horizontal lift of the one-point motion {ξt (x0 ) : t > 0} σ s with x ˜0 = idM and for PA x0 -almost all σ : [0, ∞) → M , {gt : t > 0} is a Dx0 -valued process independent of {˜ xt : t > 0} and satisfying     ˜t−1 ρ(˜ ˜t−1 ρ(˜ σt gtσ −) K(σt ) ◦ dWt + T σ σt gtσ −) K(σt )A dt, dgtσ = T σ g˜0σ = idM where σ ˜ is the horizontal lift of σ to Ds with σ ˜0 = idM . Remark 8.2.6. We could rewrite the terms such as K(σt ) ◦ dWt and K ⊥ (σt ) ◦ dWt above in tems of Itˆ o differentials. As in [36], see Section 2.3.2, these can be written as K(σt )dWt = /˜/t (σ· ) dβt ˜t K ⊥ (σt )dWt = /˜/ (σ· ) dB t

where //˜t (σ· ) : Hγ → Hγ , 0 6 t < ∞, is a family of orthogonal transformations mapping ker ρx0 → ker ρσt defined for PA x0 -almost all σ : [0, ∞) → M and {βt : ˜t : t > 0} are independent Brownian motions, (βt could be cylindrical), t > 0}, {B on ker ρx0 and [ker ρx0 ]⊥ respectively. Proof. Our general result gives the decomposition B = AH + B V into horizontal and vertical parts. We have just proved the representation (8.17) for AH . To show that B − AH corresponds to (8.18) take an orthonormal base {X j } for Hγ . Then, on a suitable domain, 1X B= L j L j + LA , (8.19) 2 j X X

130

Chapter 8. Example: Stochastic Flows

for Xj (θ) = T Rθ (X j ) and A = T Rθ (A), while, by (8.17), AH =

1X L j L j + LB 2 j Y Y

(8.20)


for Yj (θ) = T Rθ K ⊥ (θ(x0 ))X j , B = T Rθ (K ⊥ (θ(x0 ))A). Define vector fields Zj , C on Ds by
and Zj (φ) = T Rφ K(φ(x0 ))X j , for φ ∈ Ds . C(φ) = T Rφ (K(φ(x0 ))A) , Then A = B + C and Xj = Yj + Zj each j. Moreover X X LYj LZj + LZj LYj = 0 j

j

by Lemma 8.2.7 below. This shows that BV =

1X L j L j + LC . 2 j Z Z

(8.21)

Thus the diffusion process from φ0 corresponding to BV can be represented by the solution to dφt = T Rφt (K(φt (x0 ) ◦ dWt )) + T Rφt (K(φt (x0 )A)) dt.

(8.22)

If we set zt = ρx0 (φt ) = φt (x0 ), we obtain, via Itˆo’s formula, dzt = ρzt (K(zt ) ◦ dWt ) + ρzt (K(zt )A) dt, i.e. dzt = 0. Thus φt (x0 ) = z0 and (8.18) holds. The skew-product formula is seen to hold by calculating the stochastic differential of x ˜t gtx˜ using (8.15) to see that it satisfies the SDE (8.9) for {ξt : t > 0}.  Lemma 8.2.7. X

LYj LZj + LZj LYj = 0.

j

Proof. Since, for fixed θ, we can choose our basis {X j }, such that either Yj (θ) = 0 or Xj (θ) = 0, and since for f : Ds → R we can write      df Zj (θ) = df ◦ T Rθ K(θ(x0 ))X j and

    df Yj (θ) = (df ◦ T Rθ ) K ⊥ (θ(x0 ))X j ,

θ ∈ Ds ,

8.3. Semi-connections on Natural Bundles

131

it suffices to show that   Xn
o (dK ⊥ )θ(x0 ) Zj (θ)(x0 ) X j + (dK)θ(x0 ) Yj (θ)(x0 ) X j = 0,

(8.23)

j

for all θ ∈ Ds . Now K ⊥ (y)K(y) = 0 for all y ∈ M . Therefore (dK ⊥ )y (v)K(y) + K ⊥ (y)(dK)y (v) = 0,

∀v ∈ Tx M, x ∈ M.

Writing

X j = K θ(x0 ) X j + K ⊥ θ(x0 ) X j this reduces the right-hand side of (8.23) to    X
dK ⊥ θ(x0 ) Zj (θ)(x0 ) K ⊥ (θ(x0 ))X j j

   + (dK)θ(x0 ) Yj (θ)(x0 ) K(θ(x0 ))X j = 0; with our choice of basis this clearly vanishes, as required.

8.3



Semi-connections on Natural Bundles

Our bundle π : Diff M → M can be considered as a universal natural bundle over M , and a connection on it induces a connection on each natural bundle over M . Natural bundles are discussed in Kolar-Michor-Slovak [57]), they include bundles such as jet bundles as well as the standard tensor bundles. For example let Grn be the Lie group of r-jets of diffeomorphisms θ : Rn → Rn with θ(0) = 0 for a positive integer r. The diffeomorphisms need only be locally defined so θ can be taken to be a diffeomorphism of an open set Uθ of Rn onto an open subset of Rn , mapping 0 to 0. Its r-jet at 0, denoted by j0r (θ), is the equivalence class of θ where C ∞ maps θi : Uθi → Rn for i = 1, 2 are equivalent if θ1 (0) = θ2 (0) and their first r derivatives at 0 are the same. Using local co-ordinates there is a corresponding definition for r-jets of smooth maps φ : Uφ → M where Uφ is an open neighbourhood of some given point x0 of a manifold N . The equivalence class is then denoted by jxr0 (θ). An r-th order frame u at a point x of M is the r-jet at 0 of some Ψ : U → M which maps an open set U of Rn diffeomorphically onto an open subset of M with 0 ∈ U and Ψ(0) = x. Clearly Grn acts on the right of such jets, by composition. From this we can define the r-th order frame bundle Grn M of M with group Grn as the collection of all r-frames at all points of M . If we fix an r-th order frame u0 at x0 we obtain a homomorphism of principal bundles Ψu0

:

Ds → Grn M θ 7→ jxr0 (θ) ◦ u0 ,

132

Chapter 8. Example: Stochastic Flows

as for GLM (which is the case r = 1) with associated group homomorphism r Dxs 0 → Gn given by θ → u−1 0 ◦ jx0 (θ) ◦ u0 . As for the case r = 1 there is a diffusion operator induced by the flow on Grn M and we are in the situation of Theorem 8.1.3. The behaviour of the flow induced on G2n M is essentially that of jx20 (ξt ) and so relevant to the effect on the curvature of submanifolds of M as they are moved by the flow, e.g. see Cranston-Le Jan [20], Lemaire [67]. Alternatively rather having to choose some u0 we see that Grn M is (weakly) associated to π : Ds → M by taking the action of Dxs 0 on (Grx M )x0 by (θ, α) 7→ jxr0 (θ) ◦ α. As a geometrical conclusion we can observe: Theorem 8.3.1. Any classifying bundle homomorphism OM

M

Φ

Φ0

V (n, m − n)

G(n, m − n)

for the tangent bundle to a compact Riemannian manifold M , (where G(n, m − n) is the Grassmannian of n-planes in Rm and V (n, m − n) the corresponding Stiefel manifold) induces not only a metric connection on T M as the pull-back of Narasimhan and Ramanan’s universal connection $U , but also a connection on Π : Ds → M . The latter induces a connection on each natural bundle over M to form a consistent family; that induced on the tangent bundle is the adjoint of Φ∗ ($U ). The above also holds with smooth stochastic flows replacing classifying bundle homomorphisms, and the resulting map from stochastic flows to connections on π : Ds → M is injective. Proof. It is only necessary to observe that Φ determines and is determined by a surjective vector bundle map X : M × Rm → T M (e.g. see [36], Appendix 1). This in turn determines a Hilbert space H of sections of T M as in Remark 8.2.2, so we can apply Proposition 8.1.1 and Theorem 8.1.3.  Some of the conclusions of Theorem 8.3.1 are explored further in [39]. Remark 8.3.2. This injectivity result in Theorem 8.3.1 implies that all properties of the flow can, at least theoretically, be obtainable from the induced connection on Ds .

8.3. Semi-connections on Natural Bundles

133

Flows on Non-compact Manifolds In general if M is not compact we will not be able to use the Hilbert manifolds Ds , or other Banach manifolds without growth conditions on the coefficients of our flow. One possibility could be to use the space Diff M of all smooth diffeomorphisms using the Fr¨ olicher-Kriegl differential calculus as in Michor [76]. In order to do any stochastic calculus we would have to localize and use Hilbert manifolds (or possibly rough path theory). The geometric structures would nevertheless be on Diff M . This was essentially what was happening in the compact case. However it is useful to include partial flows of stochastic differential equations which are not strongly complete, see Kunita [59] or Elworthy [30]. For the partial solution flow {ξt : t < τ } of an SDE as in Remark 8.2.2 we obtain the decomposition in Theorem 8.2.5 but now only for ξt (x) defined for t < τ (x, −). This can be proved from the compact versions by localization as in Carverhill-Elworthy [17] or Elworthy [30].

Chapter 9

Appendices 9.1

Girsanov-Maruyama-Cameron-Martin Theorem

To apply the Girsanov-Maruyama theorem it is often thought necessary to verify some condition such as Novikov’s condition to ensure that the exponential (local) martingale arising as Radon-Nikodym derivative is a true martingale. In fact for conservative diffusions this is automatic, and we give a proof of this fact here since it is not widely appreciated. The proof is along the lines of that given for elliptic diffusions in [30] but with the uniqueness of the martingale problem replacing the uniqueness of minimal semi-groups used in [30]. See also [65]. On the way we relate the expectation of the exponential local martingale to the probability of explosion of the trajectories of the associated diffusion process: a special case of this appeared in [75]. Let B be a conservative diffusion operator on a smooth manifold N . For fixed T > 0 and y0 ∈ N let PB y0 denote the solution to the martingale problem + for B on Cy0 ([0, T ]; N ) and let {PtB }t denote the corresponding Markov semigroup acting on bounded measurable functions. Choose an increasing sequence {Dn }n of connected open domains in N with smooth boundary which cover N . Let τn denote the first exit time from Dn . For n = 1, . . .and y0 ∈ Dn let PB,n y0 be the probability measure on Cy0 ([0, T ]; Dn+ ) giving the solution to the martingale problem for the restriction of B to Dn , and let {PtB,n }t be its Markov semi-group. It corresponds to Dirichlet boundary conditions on Dn . B Remark 9.1.1. The measure PB,n y0 is the push-forward of Py0 by the mapping

Cn : Cy0 ([0, T ]; N + ) → Cy0 ([0, T ]; Dn+ ) given by Cn (y. )t = c(yt∧τn ) where c : D¯n → Dn+ is the continuous map of the closure D¯n to Dn+ which sends the boundary of Dn to the point at infinity, leaving the other points unchanged. Moreover if f : Dn → R is bounded and measurable with compact support in Dn , then   PtB,n f (y0 ) = EB (9.1) for all y0 ∈ Dn . y0 f (yt )χ{t<τn }

K.D. Elworthy et al., The Geometry of Filtering, Frontiers in Mathematics, DOI 10.1007/978-3-0346-0176-4_9, © Springer Basel AG 2010

135

136

Chapter 9. Appendices

Using the notation of Chapter 4, let b be a vector field on N for which there is a T ∗ N -valued process α in L2B,loc such that 2σ B (αt ) = b(yt )

06t6T

for Py0 almost all y. ∈ Cy0 ([0, T ]; N + ). Set Zt = exp{Mtα −

1 hM α it } 2

0 6 t 6 T.

This exists by the non-explosion of the diffusion process generated by B, and is a local martingale with EZt 6 1. For bounded measurable f : N → R define Qt f (y0 ) = EB y0 [Zt f (yt )] for y0 ∈ N . Since the pair (y. , Z. ) is Markovian this determines a sub-Markovian semigroup on the space of bounded measurable functions. Also for bounded measurable f : Dn → R define Qnt f (y0 ) = EB y0 [Zt∧τn f (yt )χ{t<τn } ]. Lemma 9.1.2. For any T > 0 the measures {Qny0 }y0 ∈N given by Qny0 = ZT ∧τn PB,n y0 solve the martingale problem up to time T for B +b restricted to Dn . In particular, if the martingale problem for B +b on Dn has a unique solution, then for f : Dn → R with compact support, PtB+b,n f (y0 ) = Qnt f (y0 ).

(9.2)

Proof. First note that Z. satisfies the usual stochastic equation which in our notation becomes: 0 6 t 6 T. Zt = 1 + MtZα , It becomes a martingale when stopped at any τn . Let f : N → R be C 2 with compact support. Use Itˆo’s formula and the definition of M α to see that

f (yt )Zt = f (y0 ) +

Z(df )y. Mt

+

Mtf Zα

Z + 0

t

 Bf (ys )Zs ds + M df , M Zα t .

(9.3)

Now Z t

df
 Zα M ,M =2 df σyBs (Zs αs ) ds t 0 Z t
= df Zs b(ys ) ds. 0

(9.4) (9.5)

9.1. Girsanov-Maruyama-Cameron-Martin Theorem

137

Thus Z

t

f (yt )Zt − f (y0 ) −

t

Z


df Zs b(ys ) ds,

Bf (ys )Zs ds − 0

0 ≤ t 6 T,

0

is a local martingale under PB y0 . It is localised by the stopping times {τn }n . Now suppose the support of f is in Dn . Let φ : Cy0 ([0, T ]; Dn+ ) → R be Fry0 -measurable and bounded, so φ ◦ Cn will also be Fry0 -measurable. Then, using Remark 9.1.1 above, and the martingale properties just mentioned together with Fubini’s theorem, if 0 6 r 6 t 6 T ,   Z t Z r Z  f (zt ) − (B+b)(f )(zs )ds − f (zr ) − (B+b)(f )(zs )ds φ ZT ∧τn dPB,n y0 (z. ) 0 0    Z t∧τn B = Ey0 f (yt∧τn ) − f (yr∧τn ) − (B + b)(f )(ys )ds ZT ∧τn φ ◦ Cn r∧τn t

  = (B + b)(f )(ys ∧ τn )ds ZT ∧τn φ ◦ Cn f (yt∧τn ) − f (yr∧τn ) − r    Z t∧τn = EB (f (y ) − f (y )) Z − (B + b)(f )(y )Z ds φ ◦ C t∧τn r∧τn t∧τn s s n y0 EB y0



Z

r∧τn

=0 where the first integral is over the path space Cy0 ([0, T ]; Dn+ ). To obtain the second and third equalities we have used the fact that (B + b)f has compact support in Dn in order to remove and re-insert the stopping time τn : for example first remove and modify the upper limit of the range of integration by χ{sr∧τn } = χ{sr} . Thus {Qny0 }y0 ∈Dn is a solution to the martingale problem. It follows that for such f , if we have uniqueness of the martingale problem, then f (y0 ) = EB ZT ∧τn f (yt )χ{t<τn } pB+b t = EB Zt∧τn f (yt )χ{t<τn } as required.



In what follows in this section the assumption that b is locally Lipschitz is only used to ensure that uniqueness of the martingale problem holds for B + b together with its restrictions to each Dn , e.g. see [53], and could be replaced by assuming such uniqueness.

138

Chapter 9. Appendices

Theorem 9.1.3. Suppose b is locally Lipschitz, Then: (i) For all bounded measurable f : N → R, PtB+b f (y0 ) = EB y0 Zt f (yt )χ{t<ζ} .

(9.6)

(ii) The probability that the diffusion process from y0 generated by B + b has not exploded by time t is EB y0 Zt . (iii) If B + b is conservative, then {Zt }t is a martingale under PB y0 for all y0 ∈ N . Proof. First suppose f has compact support. Then f has support in Dn for sufficiently large n. Trivially we have Zt∧τn χ{t<τn } = Zt χ{t<τn } . From these facts and the dominated convergence theorem: PtB+b f (y0 ) = EB+b y0 f (yt )χ{t<ζ} = lim EB+b y0 f (yt )χ{t<τn } n

= lim PtB+b,n f (y0 ) n

= lim Qnt f (y0 ) n

= lim EB y0 [Zt∧τn f (yt )χ{t<τn } ] n

= lim EB y0 [Zt f (yt )χ{t<τn } ] n

= EB y0 [Zt f (yt )χ{t<ζ} ]. This can be extended to a general bounded measurable function f on N by writing f = limn χDn f to prove (i). In particular applying this when f (x) = 1 for all x ∈ N we obtain (ii). But then (iii) is immediate.  From this we immediately obtain our version of the GMCM Theorem: Theorem 9.1.4. Suppose the diffusion operator B and its perturbation B + b by a locally Lipschitz vector field b on N are both conservative. Assume that B + b is cohesive or more generally that there is a locally bounded, measurable one-form b# on N such that y ∈ N. 2σyB (b# y ) = b(y), Then #

exp Mtb −

1 b# 
M , t 2

06t6T

B+b is a martingale under PB and for each y0 ∈ N the measures PB on y0 and Py0 Cy0 ([0, T ]; N ) are equivalent with

dPB+b # 1 #
y0 = exp MTb − M b T . B dPy0 2

9.2. Stochastic differential equations for degenerate diffusions

9.2

139

Stochastic differential equations for degenerate diffusions

Let B be a (smooth) diffusion operator on N . If its symbol σ B : T ∗ N → T N does not have constant rank there may be no smooth, or even C 2 , factorisation X∗

X

T ∗ N → Rm → T N of 2σxB into X(x)X ∗ (x) for X : N ×Rm → T N , as usual, for any finite dimensional m or even with Rm replaced by a separable Hilbert space. For completeness we first give the details of the constant rank case. Following that we describe a counterexample for the general situation. It is taken from Choi & Lam, [18], which also includes a brief history of the algebraic aspects of the problem. Then we look at variations of the problem and some positive, and other negative, results. However the discussion here is far from complete. For more concerning the expression of non-negative real-valued functions as sums of squares see Bony, Broglia, Colombini, & Pernazza, [13].

9.2.1

Square roots of symbols of constant rank

We have made much use of the following standard result: Theorem 9.2.1. Let π : V → M be a C r vector bundle modelled on a Hilbert space G, for some 0 6 r 6 ∞, with a C r vector bundle map Λ : V ∗ → V . Suppose Λ satisfies the symmetry and positive semi-definiteness conditions: (i) Λ∗ = Λ; (ii) if ` ∈ V ∗ , then `(Λ(`)) > 0. If the image of Λ is a C r subbundle of V , then there exists a Hilbert space H and a C r vector bundle map X : H → V of the trivial H-bundle to V such that Λ = X ∗ X. If the fibre space G is finite dimensional and Λ has constant rank, then the image of Λ is automatically such a subbundle and if also M is finite dimensional (and paracompact) we can take H to be finite dimensional. Proof. Denote the image of Λ by W with inclusion i : W → V . The basic idea of the proof is to show that Λ induces a Riemannian metric on W . Then embed W in a trivial bundle H 1 , by some c : W → H 1 , extend the Riemannian metric over H 1 and then take a metric preserving isomorphism Ψ : H 1 → H where H denotes the trivial H-bundle over M with trivial Riemannian structure. The map X is then the adjoint Ψ ◦ c composed with the inclusion i. The details follow: Let Z = ker Λ. Since Λ maps onto W it follows that Z is a C r subbundle ∗ of V . Note that Z is the annihilator W ⊥ of W because of the symmetry of Λ. Moreover i∗ : V ∗ → W ∗ is surjective with kernel W ⊥ so gives a vector bundle ¯ : W∗ → W isomorphism α : V ∗ /Z → W ∗ . This way we obtain an isomorphism Λ

140

Chapter 9. Appendices

as the composition of α−1 with the map Λ0 : V ∗ /Z → W obtained from Λ by quotienting out its kernel. ¯ is symmetric and positive semi-definite. To see this take a We claim that Λ section s : V ∗ /Z → V ∗ of the projection (for example by giving V ∗ a Riemannian ¯ = Λ0 ◦ α−1 with α = i∗ ◦ s and i ◦ Λ0 = Λ ◦ s. Also if η ∈ W ∗ structure). Then Λ set η 0 = s(α−1 (η)) ∈ V ∗ , so that η = i∗ (η 0 ) = η 0 ◦ i. Thus if also ξ ∈ W ∗ we have ¯ = η 0 ◦ i(Λ0 ◦ α−1 (ξ)) η(Λξ) = η 0 (Λ(ξ 0 )), which is symmetric in ξ and η by the symmetry of Λ. This also shows the required positivity by taking ξ = η. ¯ is an isomorphism and so is strictly positive. It therefore induces a In fact Λ Riemannian structure on W for which it is the canonical isomorphism, given by the Riesz transform. For suitable H 1 we can then find a C r embedding c : W → H onto a subbundle, see for example [61]. Now extend the Riemannian structure of c[W ] to one on the whole of H 1 . This determines a C r map A : M → P os(H 1 ), where P os(H 1 ) is the space of positive definite symmetric linear automorphisms of H 1 , such that the metric over x ∈ M is given in terms of the inner product h−, −i of H 1 by hu, vix = hA(x)u, vi. √ see [61]. The map S 7→ S from P os(H 1 ) → P os(H 1 ) is C ∞ , for example p Let H be a copy of H 1 and define Ψ : H 1 → H by Ψ(x, h) = (x, A(x)h). Then hΨ(x, h), Ψ(x, k)i = hA(x)h, kix for h, k ∈ H 1 and x ∈ M . Thus Ψ ◦ c : W → H is ¯ = (Ψ ◦ c)∗ : H → W , an isometric into the product Riemannian structure. Let X ¯ : H → V . A similar ¯ =X ¯X ¯ ∗ . Now set X = i ◦ X orthogonal projection. Then Λ calculation to that done to check the symmetry above shows that XX ∗ = Λ. The claims about the case of finite dimensional G follow from results in [61] or other standard texts which cover vector bundle theory. 

9.2.2 A smooth diffusion operator with no smooth H¨ ormander form Let Q be the quartic polynomial in four variables given by Q(x, y, z, w) = (w4 + z 2 x2 + x2 y 2 + y 2 z 2 ) − 4xyzw. By the inequality of the geometric and arithmetic mean we see that Q takes only non-negative values so that ∂2f (9.7) ∂x2 defines a diffusion operator on R4 of non-constant rank. We will show this has no H¨ormander form consisting of a sum of squares of twice differentiable vector fields possibly with a drift . A(f )(x, y, z, w) := Q(x, y, z, w)

9.2. Stochastic differential equations for degenerate diffusions

141

Lemma 9.2.2. Suppose P is a polynomial in x, y, z, w satisfying P (x, y, z, w)2 6 Q(x, y, z, w) for all real x, y, z, w. Then P (0, 0, 0, w) = 0 for all w ∈ R. Proof [18]. Clearly P must be quadratic, with no linear or constant terms, and cannot contain x2 , y 2 or z 2 . Following from this we see it cannot contain any of xw, yw, zw. This leaves only zx, xy, yz and w2 . Consideration of the order of the terms for small x, y, z finishes the proof.  Pm Now suppose that Q(x, y, z, w) = j=1 (f j (x, y, z, w))2 where the functions f j are twice differentiable at the origin. We easily see that each f j vanishes together with its first derivative at the origin. It follows that each (f j )2 is 4times differentiable at the origin with D4 (f j )2 (0)(h, h, h, h) = (D2 f j (0)(h, h))2 for h ∈ R4 . Observe that D4 Q(0)(h, h, h, h) = 4!Q(h) and consider the Hessian of f j , a quadratic form, evaluated at h as a polynomial in the components of h. Hence (D2 f j (0)(h, h))2 = D4 (f j )2 (0)(h, h, h, h) 6 D4 Q(0)(h, h, h, h) = 4!Q(h). The Lemma then implies that each D2 f j (0)(h, h) vanishes when h is on the w-axis contradicting the fact that this is not true for Q(h). This proves our non-existence assertion. Note thatP the same proof shows that we cannot write Q as an infinite sum ∞ Q(x, y, z, w) = j=1 (f j (x, y, z, w))2 of squares of functions which converges pointwise on R4 and which are twice differentiable at the origin with second derivative at the origin also converging. Moreover: Proposition 9.2.3. The symbol of A given by formula (9.7) has no factorisation into X(x)X ∗ (x) for X : R4 × H → R4 which is twice differentiable at (0, 0), for any separable Hilbert space H. Proof. If such a factorisation existed it would lead to X(x, y, z, w)X ∗ (x, y, z, w)((1, 0, 0, 0)) = (Q(x, y, z, w), 0, 0, 0) for

(x, y, z, w) ∈ R4 .

Take an orthonormal base {ej }j for H. Set X j (x, y, z, w) = X(x, y, z, w)(ej ) ∈ R4 . Let fj be the first component of X j . Then X (fj (x, y, z, w))2 = Q(x, y, z, w)

for (x, y, z, w) ∈ R4 .

j

Also since p 7→ X(p) : R4 → H ∗ is twice differentiable at the origin we see that the second derivatives of the partial sums at the origin, which is Q(h, h), also converge. However we have just observed that that cannot happen.  The proof of the proposition also follows from the result in the next section.

142

9.2.3

Chapter 9. Appendices

Non-existence of C 2 flow-like couplings

Of particular interest is the question of the existence of a, possibly locally defined, continuous or differentiable, stochastic flow whose one-point motions have as generator a given diffusion operator A. We do not know of any definitative treatment of this question. The existence of such a local flow would follow from the existence of a sufficiently smooth H¨ ormander form for A. For example a C 2 Stratonovich stochastic differential equation has a C 1 local flow. On the other hand the generator A(2) of the two point motion of such a flow ξ gives a coupling of A with itself in the sense of Example 2.1.7. It is given by A(2) (f ⊗ g)(x, y) = A(f )(x) + A(g)(y) + Γξ ((df )x , (dg)y ) where Γξ : T ∗ M × T ∗ M → R has 1 (f (ξt (x) − f (x)) (g(ξt (y)) − g(y)) t→0 t

Γξ ((df )x , (dg)y ) = lim

and, [6], [7], [66], is essentially the reproducing kernel k which appears in Section 8.2. Indeed Γξ ((df )x , (dg)y ) = (dg)y (k(x, y)(df )x )) . For the coupling coming from such a flow the generator agrees with A when restricted to the diagonal, A(2) (f ⊗ g)(x, x) = A(f g)(x), so that on the diagonal 1 ξ 2 Γ agrees with the symbol of A: Γξ ((df )x , (dg)x ) = 2(df )x σ A ((dg)x )

for x ∈ M.

The existence of a smooth flow like coupling A(2) therefore corresponds to the question of smooth extendability of symbols: given a smooth positive semi-definite bilinear σ : T M ⊕ T ∗ M → R is there a C r bilinear Γ : T ∗ M × T ∗ M → R which agrees with 2σ on the diagonal, is symmetric, and satisfies Γ(u, v)2 6 4σ(u, u)σ(v, v),

u, v ∈ T ∗ M.

(9.8)

Note that this is weaker than the full positivity needed to have a reproducing kernel in order to obtain a flow. Choi & Lam’s example described above show that this is not in general possible for extensions for which the mixed second derivatives D22 D12 Γ exist on the diagonal. To see this consider A given by equation (9.7). To show its symbol is not smoothly extendable, suppose γ : R4 × R4 → R satisfies γ(a, a) = Q(a) and γ(a, b)2 6 Q(a)Q(b) for all a, b in R4 . Set b = (0, 0, 0, w) for some w 6= 0. Then  2 γ(a, b) √ 6 Q(a) Q(b) and we see that γ(0, b) and the partial derivative D1 γ(0, b) both vanish and moreover the second partial derivative must satisfy (D12 γ(0, b))2 6 Q(b)Q(−). Lemma

9.2. Stochastic differential equations for degenerate diffusions

143

9.2.2 then implies that D12 γ(0, b)((0, 0, 0, w0 ), (0, 0, 0, w0 )) = 0 for all w0 ∈ R. Differentiation with respect to w then shows that the quartic determined by D22 D12 (0, 0) vanishes on the w-axis. Thus taking a = (0, 0, 0, 1) we have Q(a) = 1 and so Q(a) =

1 d4 1 d4 1 Q(ta)| = γ(ta, ta)|t=0 = D12 D22 γ(0, 0)((a, a), (a, a)) t=0 4 4 4! dt 4! dt 4!

which vanishes by the argument above, contradicting the fact that Q(a) = 1.

9.2.4

Locally Lipschitz square roots and Itˆ o equations

The following is well known: Theorem 9.2.4. Let σ : Rd → L+ (Rm ; Rm ) √be a C 2 map into the symmetric positive semi-definite (m × m)-matrices, then σ : Rd → L+ (Rm ; Rm ) is locally Lipschitz . For a proof see Freidlin [43], page 97 in [96] or Ikeda-Watanabe [53]. Corollary 9.2.5. For a C 2 diffusion operator B on N there is a locally Lipschitz X : Rm → T N with σ B = XX ∗ for some m. i

Proof. For some m, take a smooth inclusion T N → Rm as a subbundle
(e.g. by embedding N in Rm ) and extend σ B trivially to σxB : N → L (Rm )∗ ; Rm by i∗

x Tx∗ N (Rm )∗ →

B σx

→ Tx N

i

x → Rm

identifying (Rm )∗ with Rm and take the square root.

 m

Let ∇ be a connection on a subbundle G of T N and let X : R → G be a locally Lipschitz bundle map. Let A be a locally Lipschitz vector field on N . As in Elworthy [30] (p184) we can form the Itˆ o stochastic differential equation on N , (∇)

dxt = X(xt )dBt + A(xt )dt

where (Bt ) is a Brownian motion on Rm . For given x0 ∈ N there will be a unique maximal solution {xt : 0 6 t < ζ x0 } as usual, where by a solution we mean a sample continuous adapted process such that, for all C 2 functions f : N → R, Z t Z t f (xt ) = f (x0 ) + (df )xs X(xs )dBs + (df )xs A(xs )ds 0

=

Z tX m

0

∇X j (xs ) (df |G )X j (xs )ds.

0 j=1

Indeed in a local coordinate (U, φ) system the equation is represented by m

dxφt = Xφ (xφt )dBt −

   1X Γφ (xφt ) Xφj (xφt ) Xφj (xφt ) dt + Aφ (xφt )dt, 2 j=1

144

Chapter 9. Appendices

where Xφ , Xφi , and Aφ are the local representations of X, X i and A, and Γφ is the Christoffel symbol. Note that the generator of the solution process has symbol σx = X(x)X(x)∗ , x ∈ N , and so a Lipschitz factorisation of σ B together with a suitable choice of A will give a diffusion process with generator B. If in addition we have another generator G on N given in H¨ormander form G=

p X

LY k LY k + LY 0

k=1

for Y 0 , Y 1 , . . . , Y k vector fields of class C 2 we can consider an SDE of mixed type (∇)

dxt =

p X

˜ k + X(xt )dBt + (Y 0 (xt ) + A(xt ))dt Y k (xt ) ◦ dB t

k=1

˜ k independent Brownian motions on R independent of (Bt ). For a ˜ 1, . . . , B for B 2 C map f : N → R, a solution {xt : 0 6 t < ζ x0 } will satisfy Z t Z tX n ˜k (df )xs X(xs )dBs + (df )xs (X k (xs ))dB f (xt ) = f (x0 ) + s 0

Z =

0 k=1

t

(B + G)f (xs )ds,

t < ζ x0 ,

0

giving the unique solution to the martingale problem for B+G. These SDE’s fit into the general frame work of the ‘Itˆ o bundle’ approach of Belopolskaya-Dalecky [8], see [47] and the Appendix of Brzezniak-Elworthy [15]; also see Emery [40](section 6.33, page 85) for a more semi-martingale oriented approach.

9.2.5

Miscellaneous results

A. A factorisation with X : N × H → T N , for H a separable Hilbert space, can be found following Stroock and Varadhan, Appendix in [97], with the property that X is continuous and each vector field X j is C ∞ , where X j (x) = X(x)(ej ) for an orthonormal basis (ej )∞ j=1 of H. However it seems unclear if such an X can be found with each x 7→ X(x)e, e ∈ H, smooth. B. A non-negative C 3,1 -function can be written as a finite sum of squares of C -functions by results of Fefferman & Phong and Guan, see [100]. Thus our operator A given by equation (9.7) is the generator of the solution of an Pm j j j 1 SDE on R4 of the form dxt = j=1 λ (xt )e1 dBt where the λ are C with j locally Lipschitz derivatives, the B are independent one-dimensional Brownian motions, Pm j and e1 = j(1, 0, 0, 0). Equivalently we have a Stratonovich SDE dxt = j=1 λ (xt )e1 ◦ dBt + A(xt ) dt where A is locally Lipschitz. See also Bony et al., [13]. It seems that this does not extend to more general A, see [86], [14]. 1,1

9.3. Semi-martingales and Γ-martingales along a Subbundle

9.3

145

Semi-martingales and Γ-martingales along a Subbundle

Several of the concepts we have defined for diffusions also have versions for semimartingales, and these are relevant to the discussion of non-Markovian observations in Chapter 5. Only continuous semi-martingales will be considered. Let S denote a subbundle of the tangent bundle T M to a smooth manifold M . Definition 9.3.1. A semi-martingale ys , 0 6 s < τ is said to be along S if whenever φ is a C 2 one-form on M which annihilates S we have vanishing of the Stratonovich integral of φ along y. : Z t

φys ◦ dys = 0

0 < t < τ.

0

For simplicity take y0 to be a point of M . Proposition 9.3.2. The following are equivalent: 1. the semi-martingale y. is along S; 2. if αs : 0 6 s < τ is a semi-martingale with values in the annihilator of S in T ∗ M , lying over y. , then Z t αys ◦ dys = 0 0 < t < τ ; 0

3. for some, and hence any, connection Γ on S the process y. is the stochastic development of a semi-martingale ysΓ , 0 6 s < τ on the fibre Sy0 of S above y0 . If L is a diffusion operator, then the associated diffusion processes are all along S if and only if L is along S in the sense of Section 1.3. Proof. Let //. denote the parallel translation along the paths of y. using Γ. If (3) holds, then dy. = //. ◦ dy.Γ and it is immediate that (2) is true. Also (2) trivially implies (1). Now suppose that (1) holds. Let Γ be a connection on E and Γ0 some extension of it to a connection on T M , so that the corresponding parallel translation 0 //0 will preserve S and some complementary subbundle of T M . Let y Γ be the stochastic anti-development of y. using this connection. To show that (3) holds it 0 suffices to show that y Γ takes values in Sy0 . For this choose a smooth vector bundle map Φ : T M → M × Rm whose kernel is precisely S and let φ : T M → Rm denote its principal part and φj , j = 1, . . . , m the components of φ. These are one-forms, which annihilates S. Then, for each j, Z t Z t 0 0= φs ◦ dys = φs //0s ◦ dysΓ 0 < t < τ. 0

0

146

Chapter 9. Appendices 0

By the lemma below we see that ysΓ ∈ Sy0 for each s, almost surely, and the result follows . Finally suppose that y. is a diffusion process with generator L. By Lemma 4.1.2 we have Z t Z t
Mtα = αys ◦ dys − 0 6 t < ζ. (9.9) δ L α (ys )ds, 0

0

S. Then if y. is along S
both the for any C 2 one-form α. Suppose α annihilates R. martingale and finite variation parts of 0 αys ◦ dys vanish and so δ L α (ys ) = 0 almost surely for almost all 0 6 s < τ . If this is true for all starting points we see L is along S. On the other hand if L is along S and α annihilates S we see that L M α vanishes by its characterisation in Proposition 4.1.1, since R . σ takes values in S. Thus both the martingale and finite variation parts of 0 αys ◦ dys vanish, and so the integral itself vanishes and the diffusion processes are along S.  Lemma 9.3.3. Suppose z. and Λ. are semi-martingales with values in a finite dimensional vector space V and the space of linear maps L(V ; W ) of V into a finite dimensional vector space W , respectively. Let V0 denote the kernel of Λs which is assumed non-random and independent of s > 0 . Assume Z . Λs ◦ dzs = 0. 0

Then z. lies in V0 almost surely. Proof. We can quotient out by V0 to assume that V0 = 0, so we need to show that z. vanishes. Giving W an inner product, let Ps : W → Λs [V ] be the orthogonal projection. Compose this with the inverse of Λs considered as taking values in ˜ . formed by left inverses of Λs [V ], to obtain an L(W ; V )-valued semi-martingale Λ Λ. . By the composition law for Stratonovich integrals Z zt =

Z dzs =

0

as required.

t

0

t

˜ s Λs ◦ dzs = Λ

Z 0

t

˜s ◦ d Λ

s

Z


Λr ◦ dzr = 0

(9.10)

0



Let Γ be a connection on S. Note that by the previous proposition any semimartingale y. which is along S has a well-defined anti-development y Γ , say , which is a semi-martingale in Sy0 . Definition 9.3.4. An M -valued semi-martingale is said to be a Γ-martingale if its anti-development using Γ is a local martingale. Also we can make the following definition of an Itˆo integral of a differential form, using the analogue of a characterisation by Darling, [23], for the case S = TM;

9.4. Second fundamental forms and shape operators

147

Definition 9.3.5. If α. is a predictable process
Rwith values in T ∗ M , lying over our t o integral, Γ 0 αs dys along the paths of y. with semi-martingale y. , define its Itˆ respect to Γ by Z t Z
t αs dys = αs //s dy Γ (9.11) Γ 0

0

whenever the (standard) Itˆ o integral on the right-hand side exists. As usual this Itˆ o integral is a local martingale for all suitable integrands α. if and only if the process y. is a Γ-martingale.

9.4

Second fundamental forms and shape operators

For a detailed treatment of the differential geometry of submanifolds of Riemannian manifolds see Chapter VII of Kobayashi &Nomizu Volume II, [56]. Here we just recall some basic formulae. Let p be a point of a submanifold P of a Riemannian manifold Q. Let ∇Q and ∇P denote the Levi-Civita connections of Q and P respectively. We consider T P as a subbundle of T Q|P and let T P ⊥ denote the normal bundle. Suppose U and V are vector fields on P and Z is a smooth section of the normal bundle. These can all be extended smoothly to vector fields on Q and we let ∇Q V (p) U and ∇Q V (p) Z denote the covariant derivatives of these extensions in the direction V (p). It is easy to see that they do not depend on the extensions. Define αp (V (p), U (p)) to be the normal component of ∇Q V (p) U . It depends only on the values of U and V at p, and gives a symmetric bilinear map α : T P ⊕ T P → T P ⊥ , This is the second fundamental form of P . Define AZ(p) (V (p)) to be the negative of the tangential component of ∇Q V (p) Z. It depends only on the values of Z and V at p and gives a bilinear mapping A : T P ⊕ T P ⊥ → T P . This is the shape operator of P at p. It satisfies: hAz (v), uip = hαp (v, u), zip ,

u, v ∈ Tp P

z ∈ Tp P ⊥ .

(9.12)

Gauss’s formula is P ∇Q V (p) U = αp (V (p), U (p)) + ∇V (p) U,

(9.13)

⊥ ∇Q V (p) Z = −AZ(p) (V (p)) + ∇V (p) Z,

(9.14)

and Weingarten’s is

where ∇⊥ refers to covariant differentiation using the induced connection on the normal bundle. Since these are local equations they apply equally well to manifolds P isometrically immersed in Q.

148

Chapter 9. Appendices

Finally recall that P is minimal P if and only if the second fundamental form has zero trace at each point, that is if αp (ei , ei ) = 0 using an orthonormal basis of Tp P , for each p. It is said to be totally geodesic when a geodesic starting at any point of P in a direction tangential to P does not leave P for a positive amount of time (and so never leaves P if P is closed). This holds if and only if the second fundamental form, or equivalently the shape operator, vanishes identically, see [56] page 59.

9.5

Intertwined stochastic flows

In this section we shall consider the situation of a stochastic flow lying above another, first in relation to the corresponding properties of their reproducing kernels and reproducing kernel Hilbert spaces. As a by-product we will obtain a decomposition of the generator of the one-point motion of the flow “upstairs” which may not agree with the canonical one obtained in Chapter 2. As usual p : N → M will denote a smooth map, which we shall assume to be surjective. For simplicity we treat only smooth, i.e. C ∞ , flows, with correspondingly smooth reproducing kernels. We are not making any constant rank hypothesis.

9.5.1

Intertwined reproducing kernels and Gaussian spaces of vector fields

˜ say, on N we have a Recall from Chapter 8 that associated to a stochastic flow ξ, ∗ ˜ ˜ reproducing kernel k with k(u, v) : Tu N → Tv N linear for each u, v ∈ N . In fact it ˜v , where E ˜u is the image of the one-point generator, B say, ˜ ∗ to E gives maps from E u but we will not need that extra refinement here. The kernel can be obtained from the generator B (2) on N × N of the two-point motion, see the discussion in Section ∗ (N × N ) with Tu∗ N × Tv∗ N , 9.2.3. Indeed, from that discussion, identifying T(u,v) we see that the symbol of B (2) is given by  (2) 1 2 ˜ ˜ v)a1 ) a (k(u, v)b1 ) + b2 (k(u, (a1 , a2 )σ B (b1 , b2 ) = a1 σ B (b1 ) + a2 σ B (b2 ) + 2 (9.15) for a1 , b1 ∈ Tu∗ N and a2 , b2 ∈ Tv∗ N . ˜ there is its reproducing kernel Associated to such a reproducing kernel, k, ˜ ·)(−) = ρ˜∗ : T ∗ N → H ˜ ˜ Hilbert space of smooth vector fields, H say, so that k(u, u u ˜ where ρ˜u : H → Tu N is the evaluation. There is also the Gaussian family of vector ˜ (·), uniquely determined by fields, to be denoted by W h i ˜ v)(a) = E a(W ˜ (u)) b(W ˜ (v)) , (9.16) bk(u, a ∈ Tu∗ N, b ∈ Tv∗ N. Note that for a, b as above, ˜ v)(a). ˜ ·)(a), k(v, ˜ ·)(b)i ˜ = h˜ ρ∗u (a), ρ˜∗v (b)iH˜ = bk(u, hk(u, H

(9.17)

9.5. Intertwined stochastic flows

149

Proposition 9.5.1. The following conditions on k¯ are equivalent: (i) The symbol of the two-point motion of the associated flow has (p × p)-projectible symbol. ˜ (ii) For any x ∈ M if p(u) = p(v) = x, then, as elements of H, ˜ ·)(T ∗ p(`)) = k(v, ˜ ·)(T ∗ p(`) k(u, u v

for any ` ∈ Tx∗ M.

(iii) In the notation of (ii), ˜ u)(T ∗ p(`)) + T ∗ (`)k(v, ˜ v)(T ∗ p(`)) − 2T ∗ (`)k(u, ˜ v)(T ∗ p(`)) = 0. Tu∗ (`)k(u, u v v v u ˜ (u) = Tv pW ˜ (v) almost surely. (iv) If p(u) = p(v), then Tu pW Proof. First note that (ii) and (iii) are equivalent by expanding ˜ ·)(T ∗ p(`)) − k(v, ˜ ·)(T ∗ p(`))||2˜ ||k(u, u v H and applying equation (9.17). To bring in (iv) write it as  2 ˜ (u) − Tv pW ˜ (v)) ] = 0 E[ `(Tu pW

for all ` ∈ Tx M,

then expand and use the relation (9.16) to see that it is equivalent to (iii). The equivalence of (i) with (ii) is immediate from equation (9.15) and Lemma 2.1.1.  Assuming criterion (ii) above we see that we have a unique family of linear maps k(x, y) : Tx∗ M → Ty M for x and y in M such that ˜ v)T ∗ p = k(p(u), p(v)) Tv p k(u, u

for all u, v ∈ N.

(9.18)

It is clear that k inherits the positivity conditions from k˜ that are required of a reproducing kernel: namely (a) k(y, x)∗ = k(x, y), (b) if xj , j = 1, . . . , k are in M and `j ∈ Txj M , then k X

`j k(xi , xj )`i > 0.

i,j=1

˜ , or If k is smooth we will say that the reproducing kernel k˜ , or equivalently H ˜ W , is projectible over p , or p-projectible, and that it lies over k or is p-related to k. Smoothness of k is inherited from that of k˜ when p is a submersion, as is easily seen from the local product structure of submersions. It is equivalent to the smoothness of the elements of H.

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Chapter 9. Appendices

˜ of vector fields on N lies over or is We shall say that a Hilbert space H ˜ is p-related to a Hilbert space of vector fields H on M if each element h ∈ H ˜ → H is an orthogonal p-related to some element p∗ (h) of H and the map p∗ : H projection (i.e. p∗ is an isometry on the orthogonal complement of its kernel, or ˜ equivalently its adjoint maps H isometrically to a subspace of H). Theorem 9.5.2. Let k˜ and k be smooth reproducing kernels for vector fields on N ˜ and H. Then k˜ lies and M respectively, with reproducing kernel Hilbert spaces H ˜ over k if and only if H lies over H. Proof. First suppose k˜ lies over k. Recall that the set {k(x, ·)(`) : x ∈ M, ` ∈ Tx∗ M } ˜ to be the closed linear is total in H. Define the “horizontal subspace” HH of H ˜ ·)(Tu p∗ (`)) : u ∈ N, ` ∈ T ∗ M }. Using the property (9.17) note span of {k(u, p(u) that for uj ∈ N and `j ∈ Tx∗j M for j = 1, . . . , k and xj = p(uj ) we have ||

k X

˜ j , ·)(Tu p∗ (`j ))||2˜ = k(u j H

j=1

=

k X i,j=1

k X

˜ i , uj )(Tu p∗ (`i )) Tuj p∗ (`j ){k(u i

i,j=1

`j k(xi , xj )`i = ||

X

k(xj , ·)(`j )||2H .

j

˜ ·)(Tu p∗ (`)) is p-related to k(p(u), ·)(`) for Since equation (9.18) tells us that k(u, all relevant u and `, we see that we have an isometry h 7→ p∗ (h) of HH onto ˜ ·)(Tu p∗ (`)) 7→ k(p(u), ·)(`), with h always p-related to H extending the map k(u, p∗ (h). ˜ is orthogonal to HH . Then we claim Tu p(h(u)) = 0 Suppose now that h ∈ H for all u ∈ N , i.e. h is vertical, so h is p-related to the zero vector field, and we set p∗ (h) = 0. To check this claim set p(u) = x and take ` ∈ Tx∗ M . Then ˜ ·)T ∗ (`), hi ˜ = 0 ` (Tu p(h(u))) = Tu∗ p(`)(˜ ρu (h)) = h˜ ρ∗u (Tu∗ p(`)), hiH˜ = hk(u, u H as required. ˜ lies over H with projection p∗ : H ˜ → H. If For the converse assume that H ˜ (·) and W (·) are the Gaussian vector fields corresponding to H ˜ and H it follows W ˜ (u) and W (p(u)) are equal in law for each u ∈ N . Now, for α ∈ T ∗ M that p∗ W p(u) ∗ M , take a = Tu∗ p(α) and b = Tv p∗ (β) in equation (9.16). This gives and β ∈ Tp(v)   ˜ v)(T ∗ p(α)) = Tv p∗ (β)k(u, ˜ v)(T ∗ p(α)) β Tv pk(u, u u i h ˜ (u))) β(Tv p(W ˜ (v))) = E α((Tu p(W = E [α(W (p(u)) β(W (p(v)))] = β (k(p(u), p(v))(α)) so k˜ lies over k.



9.5. Intertwined stochastic flows

151

˜ Remark 9.5.3. From the theorem, if k˜ lies over k, we have a decomposition of H L into HH (HH )⊥ where (HH )⊥ is the kernel of p∗ and from the proof we see that (HH )⊥ consists only of vertical vector fields; indeed it must contain all the ˜ On the other hand, elements of HH may take vertical vertical vector fields in H. values at some points. As an example let p : R2 → R be the projection onto the first co-ordinate. Let H be one-dimensional and generated by h for h(x) = ˜ have orthonormal base {h1 , h2 } given by x2 assigned norm equal to 1. Let H h1 (x, y) = (x2 , cos(y)) and h2 (x, y) = (0, sin(y)). Then HH = Rh1 . The situation is clarified in the next proposition. Proposition 9.5.4. Using the notation of the theorem suppose k˜ lies over k. Let Hu denote the horizontal subspace of Tu N as defined after Proposition 2.1.2, (it was not necessary to assume that the symbol of A has constant rank to define Hu ). Then for h ∈ H the “lift” (p∗ )∗ (h) ∈ HH of h has (p∗ )∗ (h)(u) ∈ Hu if h ∈ (ker ρp(u) )⊥ . If h is in the “redundant noise” subspace at p(u) , i.e. h ∈ ker ρp(u) , then (p∗ )∗ (h)(u) is vertical. Proof. Fix x ∈ M and u ∈ p−1 (x). Since in this section we are considering kernels such as k as giving maps k(x, y) : Tx∗ M → Ty M it will be convenient to take a splitting of Tx M into Ex , the image of ρx : H → Tx M , and a complementary subspace so that we can use the inner product on Ex induced from ρx to consider the mapping v 7→ v # as a map from Ex to Tx∗ M . Let K ⊥ (x) : H → H be the projection onto the kernel of ρx . Then K ⊥ (x)(h) = ∗ (ρx ) (ρx (h))# = k(x, ·)(h(x)# ) and so h is in the orthogonal complement of the redundant noise at x if and only if h(·) = k(x, ·)(h(x)# ). If so we know from the ˜ ·)(Tu p∗ (h(x)# )). In proof of Theorem 9.5.2 that the lift, (p∗ )∗ (h) of h, is just k(u, particular ˜ u)(Tu p∗ (h(x)# )). (p∗ )∗ (h)(u) = k(u, (9.19) On the other hand we can apply the discussion in Section 2.3, especially equation ˜ and X given by X(v) ˜ ˜ → Tv N and (2.22), to the p-related SDE’s X = ρ˜v : H ˜ X(y) = ρy ◦ p∗ : H → Ty M to see that the horizontal lift of h(x) is given by hu (h(x)) = ρ˜u (p∗ )∗ ρ∗x (h(x)# ) = ρ˜u (p∗ )∗ k(x, ·)(h(x)# ) ˜ u)(Tu p)∗ (h(x)# ) = k(u, = (p∗ )∗ (h)(u) by equation (9.19), as required. In the situation where h is in the redundant noise subspace at x, so h(x) = 0, take ` ∈ Tx∗ M and observe that: ˜ ·)(Tu p)∗ (`), (p∗ )∗ (h)i ˜ ` (Tu p ρ˜u ((p∗ )∗ (h))) = hk(u, H = hk(x, ·)(`), hiH = `(h(x)) = 0.



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Chapter 9. Appendices

Remark 9.5.5. Theorem 9.5.2 shows that, if we have a diffusion operator B over a diffusion operator A, then if B has a H¨ ormander form representation B=

1X L ˜j L ˜j 2 j X X

˜ 1, X ˜ 2 , . . . an orthonormal base for a possibly infinite dimensional Hilbert with X ˜ space H of smooth vector fields on N such that the corresponding reproducing kernel is projectible over p, then there is a decomposition B = A + BV of B into diffusion operators such that BV is vertical and A lies over A. Unlike the decomposition of cohesive operators given in Theorem 2.2.5 this is not canonical and depends on the choice of such a H¨ ormander form, even assuming such a projectible smooth H¨ ormander form exists. It agrees with the usual decomposition ormander form for B V which we use together B = AH + BV if we have a smooth H¨ with the lift of such a form for A to produce the required one for B. ˜ be threeExample 9.5.6. Take M = R and N = R2 with p(x, y) = x. Let H ˜ 2, h ˜ 3 given by ˜1, h dimensional with orthonormal base the vector fields h ˜ 1 (x, y) = (sin x, y), h

˜ 2 (x, y) = (cos x, 1), h

˜ 3 (x, y) = (0, 1). h

Then H has orthonormal base {h1 , h2 } where h1 (x) = sin x and h2 (x) = cos x. (So ˜ which only take vertical values are the scalar A = 12 ∆.) The only elements of H 3 H ˜ ˜ 1 and h ˜ 2 and these are the “lifts” of multiples of h . Therefore H is spanned by h 1 2 h and h respectively. Note that at (0, 1) this lift of h1 is vertical. Also, at (x, y), A=

9.5.2

2 1 ∂2 1 ∂2 2 ∂ (1 + y . + ) + (cos x + y sin x) 2 ∂x2 2 ∂y 2 ∂x∂y

Intertwined stochastic flows that induce Levi-Civita connections

Consider a Riemannian submersion p : N → M . Is it possible to construct a smooth stochastic flow ξ˜. of Brownian motions on N that lies over a stochastic flow on M and such that the connection determined by ξ˜. , in the sense of Section 8.2, is the Levi-Civita connection? We shall show that this is so if and only if the submersion has totally geodesic fibres and the horizontal distribution on N determined by the Riemannian structures is integrable, at least when N is compact. This is essentially “local triviality” of the Riemannian structure of the submersion, see [83], and the discussion in the Notes on Chapter 7, Section 7.3. In general, given p : N → M we will say that a, possibly local, stochastic flow ξ˜. on M lies over such a flow ξ. on M , or that ξ˜. and ξ. are intertwined by p if: (i) They are defined on the same probability space {Ω, F, P}.

9.5. Intertwined stochastic flows

153

(ii) The lifetimes ζ : M × Ω → (0, ∞] and ζ˜ : N × Ω → (0, ∞] of ξ. and ξ˜. almost surely satisfy ˜ ω) 6 ζ(p(y), ω) ζ(y, for all y ∈ N. (iii) Almost surely, for all y ∈ N we have   p ξ˜t (y, ω) = ξt (p(y), ω)

˜ ω). t < ζ(y,

Recall that a local stochastic flow on manifold M determines a reproducing kernel k and a drift vector field, A say, and up to law it is determined by them as the solution flow of the SDE on M : dxt = ρxt ◦ dWt + A(xt ) dt,

(9.20)

for a Wiener process {Wt }t > 0 of vector fields determined up to law by the reproducing kernel Hilbert space H where, as usual, ρx denotes the evaluation map, evaluated at the point x of M . Proposition 9.5.7. Let ξ˜. be a (possibly local) smooth stochastic flow on N with ˜ Let ξ. , k, and A be corresponding objects on M . reproducing kernel k˜ and drift A. Then ξ˜. lies over ξ. , or lies over a flow with the same law as ξ. , if and only if k˜ and k and also A˜ and A are p-related. Proof. Two flows with the same laws have the same k and H. Intertwining of the flows implies that this is true of their two-point motions and so of the symbols of the generators of these motions, as in Lemma 2.1.1. Since those symbols determine the reproducing kernels by equation (9.15) above, it follows that the reproducing kernels are p-related. By looking at the generators of the one-point motions we see that intertwining also implies the drifts are p-related. ˜ → H be Conversely if the kernels and the drifts are p-related let p∗ : H the projection of the reproducing kernel Hilbert spaces induced by p as given by ˜ t }t is the Wiener process such that ξ˜. is the solution flow of Theorem 9.5.2. If {W the SDE ˜ t + A˜ dt, dut = ρ˜ut ◦ dW then the p-related SDE ˜ t + A dt dxt = ρxt p∗ ◦ dW has flow with the same law as ξ. , since p∗ W. is a Wiener process with law given  by H. But this flow is intertwined with ξ˜. . To simplify the exposition we will consider intertwined stochastic differential equations ˜ t ) ◦ dBt + A(y ˜ t ) dt, dyt = X(y dxt = X(xt ) ◦ dBt + A(xt ) dt

(9.21) (9.22)

154

Chapter 9. Appendices

˜ : N × Rm → T N and X : M × Rm → T M are p-related as are the so that X vector fields A˜ and A. By Proposition 9.5.7 above this will allow us to cover the case of intertwined stochastic flows: we can easily extend what follows to allow for infinite dimensional noise. ˜ is non-degenerate and determines the given RieWe shall suppose that X mannian structure on N , as then will X for M . Recall from Section 3.3 that the ˘ determined by such an X, the LW connection of the SDE in the connection ∇ terminology of [36], is just the projection of the trivial connection on the product bundle M × Rn onto T M . The covariant derivative of a vector field V is given by ˘ u = X(x)d[z 7→ Y (z)(V (z))](u) ∇V

u ∈ Tx M,

where Y is the Rm -valued one-form defined by Yx = X(x)∗ = [X(x)|ker X(x)⊥ ]−1 . This can be written in terms of the version k # of the reproducing kernel used in Chapter 8 as ˘ u = d[z 7→ k # (x, z)(V (z))](u) ∇V

u ∈ Tx M,

(9.23)

#

remembering that k (x, y) = X(y)Yx . In particular two different SDE’s may have the same reproducing kernel but then their LW connections will be the same. See Theorem 8.1.3 and Remark 8.2.2, or for more details of what follows [36]. This connection has the defining condition ˘ u (X(−)(e)) = 0 ∇

u ∈ Tx M.

(9.24)

M

Moreover it is the Levi-Civita connection ∇ if and only if the exterior derivative, written d1 Y : ∧2 T M → Rm , of Y treated as an Rm -valued one-form satisfies X(x) (d1 Y )x = 0

x ∈ M.

(9.25)

An X determining the Levi-Civita connection can be obtained via Nash’s embedding theorem. In fact any metric connection can be obtained from some X by a theorem of Narasimhan & Ramanan [80] with [36], or see [89]. By a trivial extension of any X : M × Rm → T M , as above, we mean some X 0 : M × Rm+q → T M for some q, again linear on the fibres and smooth, together with a fixed orthogonal projection π of Rm+q onto Rm , or a closed subspace of Rm , such that X 0 (x) = X(x) ◦ π. The induced connection does not change after taking a trivial extension since this does not change the reproducing kernel k . Consider the SDE ˜ t ) ◦ dWt + A(y ˜ t ) dt, dyt = X(y dxt = X(xt ) ◦ dBt + A(xt ) dt,

(9.26) (9.27)

˜ where W. is now a Brownian motion on Rm+q and B. is one on Rm with X(y) : m+q m → Ty N and X(x) : R → Tx M . We shall say these are weakly p-related if R ˜ is p-related to a trivial extension of X and the vector fields A˜ and A are the X p-related. Because of Proposition 9.5.7 this is essentially equivalent to their flows being intertwined:

9.5. Intertwined stochastic flows

155

Proposition 9.5.8. The SDE above are weakly p-related if and only if their reproducing kernels k˜ and k are p-related in the sense of equation (9.18). Proof. Recall that the reproducing kernels are related to the SDE by k(x, y) = ˜ X(y)X(x)∗ : Tx∗ M → Ty M and similarly for k. If the SDE’s are p-related their reproducing kernels are p-related, and so the same holds if the SDE’s are weakly p-related. ˜ and H denote For the converse assume that the kernels are p-related. Let H their reproducing kernel Hilbert spaces. By Theorem 9.5.2 the canonical SDE’s ˜ → T N and ρ· : H → T N are weakly p-related (extending our definition to ρ˜· : H ˜ be defined cover the infinite dimensional case and using p∗ ). Let X˜ : Rm+q → H m by e 7→ x ˜(·)(e) with X : R → H defined similarly. For u ∈ N and x = p(u) we have the commutative diagram: X˜

Rm+q

-

ρ˜u

˜ H

-Tu N

p∗

Rm

-

? H

Tu p

? -Tx M

X ρx ∗ ˜ → R by π = X p∗ X . This is an orthonormal projection onto Define π : R a subspace of Rm because X˜ , X and p∗ are all surjective orthogonal projections. Moreover m+q

m

X(x)π(e) = X(x)X ∗ p∗ X˜ (e) = ρx X X ∗ p∗ X˜ (e) ρu X˜ (e) = ρx p∗ X˜ (e) = Tu p˜ ˜ = Tu pX(u)(e) as required.



Theorem 9.5.9. Consider a Riemannian submersion p : N → M with a smooth X : M × Rm → T M as above, inducing the Riemannian metric on M . Then there ˘˜ is ˜ on N which is weakly p-related to X and whose induced connection ∇ exists X ˘ the Levi-Civita connection if and only if ∇, the connection induced by X, is the Levi-Civita connection of M , the submersion has totally geodesic fibres, and the horizontal subspace of p : N → M is integrable. ˜ We can replace X by a trivial extenProof. First suppose there exists such an X. ˜ are p-related. At each y ∈ N we sion if necessary and so assume that X and X can decompose Rm into ⊥ ˜ y [Rm ] [Rm ] ⊕ Qy [Rm ] ⊕ K Rm = Kp(y)

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Chapter 9. Appendices

where, as usual Kx⊥ is the orthogonal projection of Rm onto the orthogonal com˜ ˜ y is the projection onto the kernel of X(y), plement of the kernel of X(x), while K ⊥ ˜ and Qy = I − Kp(y) − Ky . Fixing y and setting p(y) = x, take an orthonormal base {e1 , . . . , em } for Rm with e1 , . . . , en in Kx⊥ [Rm ], and en+1 , . . . , en+p in the image of Qy . Here n = dim M and n + p = dim N . Set X j = X(−)ej and let X j,H denote its horizontal lift. We have, for any i, j, j,H i,H − ∇N [X i,H , X j,H ](y) = ∇N X i,H (y) X X j,H (y) X   ˜ = X(y) d(Y˜ X j,H )(y)(X i,H ) − d(Y˜ X i,H )(y)(X j,H ) .

Now by formula (2.22), for u ∈ N : ˜ X i,H (u) = X(u)Y (p(u))X i (p(u)) giving ˜ ⊥ (u)Y (p(u))X i (p(u)) = Y (p(u))X i (p(u)). Y˜u X i,H (u) = K Thus, using the formula for the exterior derivative recalled in equation (9.29) below:
˜ [X i,H , X j,H ](y) = X(y) d(Y X j )(p(y))(X i (p(y)) − d(Y X i )(p(y))(X j (p(y))
˜ = X(y) 2d1 Yx (X i (x), X j (x)) + Yx ([X i , X j ](x)) 1 ˜ Yx (X i (x), X j (x))) + hy ([X i , X j ](x)). = X(y)(2d

Let PuH and PuV denote the projections of Tu N onto its horizontal and vertical subspaces for any u ∈ N . From [83] we know that hy ([X i , X j ](x)) = P H ([X i,H , X j,H ](y)). Consequently we have 1 ˜ PyV ([X i,H , X j,H ](y)) = X(y)(2d Yx (X i (x), X j (x))).

˜ are p-related, In particular since X and X 1 ˜ Yx (X i (x), X j (x))) = X(x)(2d1 Yx (X i (x), X j (x))). 0 = T pX(y)(2d

Thus X(x)d1 Yx = 0 for all x ∈ M and so X induces the Levi-Civita connection. To prove integrability of the horizontal subspace it is convenient to quote O’Neill again. In [83] he defines a tensor A, which we will write as AO to distinguish it from shape operators, which gives a bilinear skew-symmetric map (a, b) 7→ AO ab: T N ⊕ T N → T N such that, if U and V are horizontal vector fields, then V N AO U (u) V (u) = P ∇U (u) V =

1 V P ([U, V ](u)). 2 u

It will therefore suffice to show that AO u v = 0 for all horizontal tangent vectors u, v at y.

9.5. Intertwined stochastic flows

157

For this take U = X i,H and V = X j,H . Then, using equation (9.24) and O’Neill’s result that covariant differentiation  commutes with horizontal lifts, [83], N j,H M j = 0 if j 6 n. But X j,H (y) = 0 if = hy ∇X i (x) X we see that ∇X i,H (y) X j,H j > n. Thus since AO is a tensor we have AO (y) = 0 for all i, j, proving X i,H (y) X O that A vanishes on horizontal vectors. To see that p has totally geodesic fibres first note that, as our U and V vary through all i and j, so P V ∇N U (y) V determines all values of the second fundamental form of the leaf through y of the horizontal foliation. It therefore vanishes and so correspondingly does the shape operator of the leaf, see Section 9.4. This shape operator is given by the horizontal component of the covariant derivative of each vertical vector, Z say, in a horizontal direction. Thus P H ∇N U Z = 0 for all horizontal vector fields U . If also U is basic, i.e. the horizontal lift of a vector field on M , then as observed by O’Neill the bracket [Z, U ] is vertical and so ∇N Z U must be vertical. ˜ j vanishes In particular this holds for U = X j,H . Now take 1 6 j 6 n . Then ∇N X j,H j V ˜j N ˜ at Y by equation (9.24). However X = X − P X . Therefore ∇Z(y) X j,H = ˜ j and so if αV denotes the second fundamental form of the fibre −∇N P V X y

Z(y)

p−1 (x) at y we have ˜ j (y)) = P H ∇N P V X ˜ j = 0. αyV (Z(y), P V X Z(y) ˜ j = 0 by equation (9.24), Further, if n + 1 6 j 6 n + p we have ∇N Z(y) X V V ˜j j ˜ is vertical for such j. However so that αy (Z(y), P X (y)) = 0 since each X 1 n+p ˜ ˜ (y) span Ty N , so this implies that αyV = 0. Thus the fibres are X (y), . . . , X totally geodesic. For the converse suppose that X determines the Levi-Civita connection of M and that the horizontal distribution is integrable. Let X H : N × Rm → T N be the horizontal lift of X. Choose some X : N × Rq → T N which induces the Riemannian metric for N and its Levi-Civita connection. Set X V = P V X . This will induce a connection ∇V on the vertical tangent bundle V T N which restricts to the Levi-Civita connection of the fibres. Moreover if v ∈ Ty N and B is a vertical vector field we see immediately that V P V ∇N v B = ∇v B.

(9.28)

˜ ˜ : N × Rm × Rq → T N by X(u)(e, f ) = X H (u)(e) + X V (u)(f ). We Define X ˜ induces the Levi-Civita connection. It is clearly p-related to X. claim that X ˜ 1 Y˜ = 0. For this, first note that To prove the claim we will show that Xd Y˜y (v) = (p∗ Y )y (v) + YyV (P V v)

v ∈ Ty N

where YyV : V Ty N → Rq is the adjoint of X V (y). Observe that d1 p∗ Y = p∗ d1 Y
∗ 1 ˜ d Yy ) = hy X(x)d1 Y (Ty p−, Ty p−) = 0 because X induces the Leviand X(y)(p Civita connection. Thus ˜ 1 Y˜ = Xd ˜ 1 (Y V P V ). Xd

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Chapter 9. Appendices

Recall that for vector fields A and B on N and a, possibly vector-valued, one-form φ such as Y V P V , 2d1 (φ)(A(y), B(y)) = d (φ(B(−))) (A(y)) − d (φ(A(−))) (B(y)) − φy ([A, B](y)). (9.29) We can conclude by considering three cases: (i) A and B vertical. Then d1 (Y V P V )(A(y), B(y)) = d1 (Y V |p−1 (x) )(A(y), B(y)) 1 ˜ and so X(y)d (Y V P V )(A(y), B(y)) = 0 because X V induces the Levi-Civita connection on each fibre of p.

(ii) A and B horizontal. Then d1 (Y V P V )(A(y), B(y)) = −(YyV P V )([A, B](y)) = 0 because the horizontal distribution is assumed to be integrable. (iii) A horizontal and B vertical. Then
2d1 (Y V P V )(A(y), B(y)) = d Y V (B(−)) (A(y)) − YyV P V ([A, B](y)) so that   1 N ˜ (Y V P V )(A(y), B(y)) = ∇VA(y) B − P V ∇N 2X(y)d A(y) B − ∇B(y) A = P V ∇N B(y) A p−1 (x)

−1

p (x) using equation (9.28). However P V ∇N the B(y) A = AA(y) (B(y)) for A shape operator of the fibre of p through y, and this vanishes because the fibres are totally geodesic. 

Remark 9.5.10. The conditions on the submersion required to have such commuting SDE’s are very strong. See the discussion in the Notes on Chapter 7, Section 7.3. In fact for N complete and M simply connected, N is just the product of M with a Riemannian manifold and p is the projection: using this the sufficiency assertion given above becomes trivial. However the direct proof given here seems more illuminating. Remark 9.5.11. As described briefly in Remark 7.2.4, and proved in [36], for a Riemannian symmetric space M with natural projection p : K → M = K/G, the right-invariant SDE on the group K lies over an SDE on M which induces the Levi-Civita connection. The right-invariant SDE induces the flat right-invariant connection on K, with adjoint the flat left-invariant connection. Theorem 9.5.9 shows that in general there can be no G-invariant SDE on K which induces the Levi-Civita connection on K.

Bibliography [1] M. Arnaudon and S. Paycha. Factorisation of semi-martingales on principal fibre bundles and the Faddeev-Popov procedure in gauge theories. Stochastics Stochastics Rep., 53(1-2):81–107, 1995. [2] Alan Bain and Dan Crisan. Fundamentals of stochastic filtering, volume 60 of Stochastic Modelling and Applied Probability. Springer, New York, 2009. ´ [3] D. Bakry and Michel Emery. Diffusions hypercontractives. In S´eminaire de probabilit´es, XIX, 1983/84, volume 1123 of Lecture Notes in Math., pages 177–206. Springer, Berlin, 1985. [4] F. Baudoin. Conditioning and initial enlargement of filtration on a Riemannian manifold. Ann. Probab., 32(3A):2286–2303, 2004. [5] Fabrice Baudoin. An introduction to the geometry of stochastic flows. Imperial College Press, London, 2004. [6] P. Baxendale. Gaussian measures on function spaces. Amer. J. Math., 98(4):891–952, 1976. [7] P. Baxendale. Brownian motions in the diffeomorphism groups I. Compositio Math., 53:19–50, 1984. [8] Ya. I. Belopolskaya and Yu. L. Dalecky. Stochastic equations and differential geometry, volume 30 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1990. Translated from the Russian. [9] L. B´erard-Bergery and J.-P. Bourguignon. Laplacians and Riemannian submersions with totally geodesic fibres. Illinois J. Math., 26(2):181–200, 1982. [10] E. Berger, R. Bryant, and P. Griffiths. The Gauss equations and rigidity of isometric embeddings. Duke Math. J., 50(3):803–892, 1983. [11] Jean-Michel Bismut. Large deviations and the Malliavin calculus, volume 45 of Progress in Mathematics. Birkh¨ auser Boston Inc., Boston, MA, 1984. [12] Hermann Boerner. Representations of groups. With special consideration for the needs of modern physics. Translated from the German by P. G.

K.D. Elworthy et al., The Geometry of Filtering, Frontiers in Mathematics, DOI 10.1007/978-3-0346-0176-4, © Springer Basel AG 2010

159

160

Bibliography Murphy in cooperation with J. Mayer-Kalkschmidt and P. Carr. Second English edition. North-Holland Publishing Co., Amsterdam, 1970.

[13] Jean-Michel Bony, Fabrizio Broglia, Ferruccio Colombini, and Ludovico Pernazza. Nonnegative functions as squares or sums of squares. J. Funct. Anal., 232(1):137–147, 2006. [14] Raymond Brummelhuis. A counterexample to the Fefferman-Phong inequality for systems. C. R. Acad. Sci. Paris S´er. I Math., 310(3):95–98, 1990. [15] Z. Brze´zniak and K. D. Elworthy. Stochastic differential equations on Banach manifolds. Methods Funct. Anal. Topology, 6(1):43–84, 2000. [16] A. Carverhill. Conditioning a lifted stochastic system in a product space. Ann. Probab., 16(4):1840–1853, 1988. [17] A. P. Carverhill and K. D. Elworthy. Flows of stochastic dynamical systems: the functional analytic approach. Z. Wahrsch. Verw. Gebiete, 65(2):245–267, 1983. [18] Man Duen Choi and Tsit Yuen Lam. Extremal positive semidefinite forms. Math. Ann., 231(1):1–18, 1977/78. [19] Yvonne Choquet-Bruhat, C´ecile DeWitt-Morette, and Margaret DillardBleick. Analysis, manifolds and physics. North-Holland Publishing Co., Amsterdam, second edition, 1982. [20] Michael Cranston and Yves Le Jan. Asymptotic curvature for stochastic dynamical systems. In Stochastic dynamics (Bremen, 1997), pages 327–338. Springer, New York, 1999. [21] Dan Crisan, Michael Kouritzin, and Jie Xiong. Nonlinear filtering with signal dependent observation noise. Electron. J. Probab., 14:no. 63, 1863– 1883, 2009. [22] H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon. Schr¨ odinger operators with application to quantum mechanics and global geometry. Texts and Monographs in Physics. Springer-Verlag, Berlin, study edition, 1987. [23] R. W. R. Darling. Approximating Ito integrals of differential forms and geodesic deviation. Z. Wahrsch. Verw. Gebiete, 65(4):563–572, 1984. [24] M. H. A. Davis and M. P. Spathopoulos. Pathwise nonlinear filtering for nondegenerate diffusions with noise correlation. SIAM J. Control Optim., 25(2):260–278, 1987. [25] B. K. Driver. A Cameron-Martin type quasi-invariance theorem for Brownian motion on a compact Riemannian manifold. J. Functional Analysis, 100:272–377, 1992. [26] T. E. Duncan. Some filtering results in Riemann manifolds. Information and Control, 35(3):182–195, 1977.

Bibliography

161

[27] A. Eberle. Uniqueness and non-uniqueness of semigroups generated by singular diffusion operators, volume 1718 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1999. [28] D. G. Ebin and J. Marsden. Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math., pages 102–163, 1970. [29] Robert J. Elliott and Michael Kohlmann. Integration by parts, homogeneous chaos expansions and smooth densities. Ann. Probab., 17(1):194–207, 1989. [30] K. D. Elworthy. Stochastic Differential Equations on Manifolds. LMS Lecture Notes Series 70, Cambridge University Press, 1982. [31] K. D. Elworthy. Geometric aspects of diffusions on manifolds. In P. L. Hennequin, editor, Ecole d’Et´e de Probabilit´es de Saint-Flour XV-XVII, 1985-1987. Lecture Notes in Mathematics 1362, volume 1362, pages 276– 425. Springer-Verlag, 1988. [32] K. D. Elworthy. Stochastic flows on Riemannian manifolds. In Diffusion processes and related problems in analysis, Vol. II (Charlotte, NC, 1990), volume 27 of Progr. Probab., pages 37–72. Birkh¨auser Boston, Boston, MA, 1992. [33] K. D. Elworthy and W. S. Kendall. Factorization of harmonic maps and Brownian motions. In From local times to global geometry, control and physics (Coventry, 1984/85), Pitman Res. Notes Math. Ser., 150, pages 75–83. Longman Sci. Tech., Harlow, 1986. [34] K. D. Elworthy, Y. Le Jan, and Xue-Mei Li. Equivariant diffusions on principal bundles. In Stochastic analysis and related topics in Kyoto, volume 41 of Adv. Stud. Pure Math., pages 31–47. Math. Soc. Japan, Tokyo, 2004. [35] K. D. Elworthy, Y. Le Jan, and Xue-Mei Li. Concerning the geometry of stochastic differential equations and stochastic flows. In ’New Trends in stochastic Analysis’, Proc. Taniguchi Symposium, Sept. 1994, Charingworth, ed. K. D. Elworthy and S. Kusuoka, I. Shigekawa. World Scientific Press, 1996. [36] K. D. Elworthy, Y. Le Jan, and Xue-Mei Li. On the geometry of diffusion operators and stochastic flows, Lecture Notes in Mathematics 1720. Springer, 1999. [37] K. D. Elworthy and S. Rosenberg. Homotopy and homology vanishing theorems and the stability of stochastic flows. Geom. Funct. Anal., 6(1):51–78, 1996. [38] K. D. Elworthy and M. Yor. Conditional expectations for derivatives of certain stochastic flows. In J. Az´ema, P.A. Meyer, and M. Yor, editors, Sem. de Prob. XXVII. Lecture Notes in Maths. 1557, pages 159–172. SpringerVerlag, 1993.

162

Bibliography

[39] K. David Elworthy. The space of stochastic differential equations. In Stochastic analysis and applications, volume 2 of Abel Symp., pages 327– 337. Springer, Berlin, 2007. ´ [40] M. Emery. Stochastic calculus in manifolds. Universitext. Springer-Verlag, Berlin, 1989. With an appendix by P.-A. Meyer. [41] A. Estrade, M. Pontier, and P. Florchinger. Filtrage avec observation discontinue sur une vari´et´e. Existence d’une densit´e r´eguli`ere. Stochastics Stochastics Rep., 56(1-2):33–51, 1996. [42] Maria Falcitelli, Stere Ianus, and Anna Maria Pastore. Riemannian submersions and related topics. World Scientific Publishing Co. Inc., River Edge, NJ, 2004. [43] M. Freidlin. Functional integration and partial differential equations, volume 109 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1985. [44] Kenrˆo Furutani. On a differentiable map commuting with an elliptic pseudodifferential operator. J. Math. Kyoto Univ., 24:197–203, 1984. [45] Zhong Ge. Betti numbers, characteristic classes and sub-Riemannian geometry. Illinois J. Math., 36(3):372–403, 1992. [46] Peter B. Gilkey, John V. Leahy, and Jeonghyeong Park. Spectral geometry, Riemannian submersions, and the Gromov-Lawson conjecture. Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca Raton, FL, 1999. [47] Yuri E. Gliklikh. Ordinary and stochastic differential geometry as a tool for mathematical physics, volume 374 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1996. With an appendix by the author and T. J. Zastawniak. [48] S. I. Goldberg and T. Ishihara. Riemannian submersions commuting with the Laplacian. J. Differential Geom., 13(1):139–144, 1978. [49] Mikhael Gromov. Carnot-Carath´eodory spaces seen from within. In Sub-Riemannian geometry, volume 144 of Progr. Math., pages 79–323. Birkh¨auser, Basel, 1996. [50] Istv´an Gy¨ ongy. Stochastic partial differential equations on manifolds. II. Nonlinear filtering. Potential Anal., 6(1):39–56, 1997. [51] Robert Hermann. A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle. Proc. Amer. Math. Soc., 11:236–242, 1960. [52] J. E. Humphreys. Introduction to Lie algebras and representation theory, volume 9 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1978. Second printing, revised.

Bibliography

163

[53] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes , second edition. North-Holland, 1989. [54] M. Joannides and LeGland F. Nonlinear filtering with continuous time perfect observations and noninformative quadratic variation. In Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, December 1012, 1997, page pp. 16451650, 1997. [55] S. Kobayashi and K. Nomizu. Foundations of differential geometry, Vol. I. Interscience Publishers, 1969. [56] S. Kobayashi and K. Nomizu. Foundations of differential geometry, Vol. II. Interscience Publishers, 1969. [57] I. Kol´aˇr, P. W. Michor, and J. Slov´ ak. Natural operations in differential geometry. Springer-Verlag, Berlin, 1993. [58] Hiroshi Kunita. Cauchy problem for stochastic partial differential equations arising in nonlinear filtering theory. Systems Control Lett., 1(1):37–41, 1981/82. [59] Hiroshi Kunita. Stochastic flows and stochastic differential equations, volume 24 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1990. [60] S. Kusuoka. Degree theorem in certain Wiener Riemannian manifolds. Stochastic analysis, Proc. Jap.-Fr. Semin., Paris/France 1987, Lect. Notes Math. 1322, 93-108 , 1988. [61] Serge Lang. Introduction to differentiable manifolds. Universitext. SpringerVerlag, New York, second edition, 2002. [62] Joan-Andreu L´ azaro-Cam´ı and Juan-Pablo Ortega. Reduction, reconstruction, and skew-product decomposition of symmetric stochastic differential equations. Stoch. Dyn., 9(1):1–46, 2009. [63] Yves Le Jan and Olivier Raimond. Solutions statistiques fortes des ´equations diff´erentielles stochastiques. C. R. Acad. Sci. Paris S´er. I Math., 327(10):893–896, 1998. [64] Yves Le Jan and Olivier Raimond. Integration of Brownian vector fields. Ann. Probab., 30(2):826–873, 2002. [65] R´emi L´eandre. Applications of the Malliavin calculus of Bismut type without probability. WSEAS Trans. Math., 5(11):1205–1210, 2006. [66] Y. Le Jan and S. Watanabe. Stochastic flows of diffeomorphisms. In Stochastic analysis (Katata/Kyoto, 1982), North-Holland Math. Library, 32,, pages 307–332. North-Holland, Amsterdam, 1984. [67] S. Lemaire. Invariant jets of a smooth dynamical system. Bull. Soc. Math. France, 129(3):379–448, 2001.

164

Bibliography

[68] Xue-Mei Li. Stochastic differential equations on noncompact manifolds: moment stability and its topological consequences. Probab. Theory Related Fields, 100(4):417–428, 1994. [69] M. Liao. Factorization of diffusions on fibre bundles. Trans. Amer. Math. Soc., 311(2):813–827, (1989). [70] Ming Liao. Decomposition of stochastic flows and Lyapunov exponents. Probab. Theory Related Fields, 117(4):589–607, 2000. [71] Andre Lichnerowicz. Quelques th´eoremes de g´eom´etrie diff´erentielle globale. Comment. Math. Helv., 22:271–301, 1949. [72] S. V. Lototsky and B. L. Rozovskii. Wiener chaos solutions of linear stochastic evolution equations. Ann. Probab., 34(2):638–662, 2006. [73] Sergey Lototsky, Remigijus Mikulevicius, and Boris L. Rozovskii. Nonlinear filtering revisited: a spectral approach ii. In Proc. 35th IEEE Conf. on Decision and Control, volume 4 of Kobe,Japan,1996, pages 4060–4064. Omnipress, Madison,WI., 1996. [74] Sergey Lototsky and Boris L. Rozovskii. Recursive multiple Wiener integral expansion for nonlinear filtering of diffusion processes. In Stochastic processes and functional analysis (Riverside, CA, 1994), volume 186 of Lecture Notes in Pure and Appl. Math., pages 199–208. Dekker, New York, 1997. [75] Henry P. McKean. Stochastic integrals. AMS Chelsea Publishing, Providence, RI, 2005. Reprint of the 1969 edition, with errata. [76] P. W. Michor. Gauge theory for fiber bundles, volume 19 of Monographs and Textbooks in Physical Science. Lecture Notes. Bibliopolis, Naples, 1991. [77] D Mitter, S.K.and Ocone. Multiple integral expansions for nonlinear filtering. In 18th IEEE Conf. on decision and Control including Symp. on Adaptive Processes, volume 18 of 1979, pages 329–334. IEEE, 1979. [78] S.A. Molchanov. Diffusion processes and riemannian geometry. Russian Mathematical Surveys, 30:2–63, 1975. [79] Richard Montgomery. A tour of subriemannian geometries, their geodesics and applications, volume 91 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2002. [80] M. S. Narasimhan and S. Ramanan. Existence of universal connections. American J. Math., 83, 1961. [81] D Ocone. Multiple integral expansions for nonlinear filtering. Stochastics, 10(1):1–30, 1983. [82] O. A. Ole˘ınik. On linear equations of the second order with a non-negative characteristic form. Mat. Sb. (N.S.), 69 (111):111–140, 1966.

Bibliography

165

[83] Barrett O’Neill. The fundamental equations of a submersion. Michigan Math. J., 13:459–469, 1966. [84] E. Pardoux. Nonlinear filtering, prediction and smoothing. In Stochastic systems: the mathematics of filtering and identification and applications (Les Arcs, 1980), volume 78 of NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., pages 529–557. Reidel, Dordrecht, 1981. ´ [85] Etienne Pardoux. Filtrage non lin´eaire et ´equations aux d´eriv´ees partielles ´ ´ e de Probabilit´es de Saint-Flour XIX— stochastiques associ´ees. In Ecole d’Et´ 1989, volume 1464 of Lecture Notes in Math., pages 67–163. Springer, Berlin, 1991. [86] Alberto Parmeggiani. A class of counterexamples to the Fefferman-Phong inequality for systems. Comm. Partial Differential Equations, 29(9-10):1281– 1303, 2004. [87] E. J. Pauwels and L. C. G. Rogers. Skew-product decompositions of Brownian motions. In Geometry of random motion (Ithaca, N.Y., 1987), volume 73 of Contemp. Math., pages 237–262. Amer. Math. Soc., 1988. [88] Monique Pontier and Jacques Szpirglas. Filtering with observations on a Riemannian symmetric space. In Stochastic differential systems (Bad Honnef, 1985), volume 78 of Lecture Notes in Control and Inform. Sci., pages 316–329. Springer, Berlin, 1986. [89] D. Quillen. Superconnection character forms and the Cayley transform. Topology, 27(2):211–238, 1988. [90] Michael Reed and Barry Simon. Methods of modern mathematical physics. I. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, second edition, 1980. Functional analysis. [91] Daniel Revuz and Marc Yor. Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition, 1999. [92] L. C. G. Rogers and D. Williams. Diffusions, Markov processes, and martingales. Vol. 2. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000. [93] S. Rosenberg. The Laplacian on a Riemannian manifold, volume 31 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1997. An introduction to analysis on manifolds. [94] Paulo R. C. Ruffino. Decomposition of stochastic flows and rotation matrix. Stoch. Dyn., 2(1):93–107, 2002.

166

Bibliography

[95] D. Stroock and S. R. S. Varadhan. On degenerate elliptic-parabolic operators of second order and their associated diffusions. Comm. Pure Appl. Math., 25:651–713, 1972. [96] D. W. Stroock. Lectures on stochastic analysis: diffusion theory, volume 6 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1987. [97] D. W. Stroock and S. R. S. Varadhan. Multidimensional diffusion processes, volume 233 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1979. [98] H. J. Sussmann. Orbits of families of vector fields and integrability of distributions. Trans. Amer. Math. Soc., 180:171–188, 1973. [99] Kazuaki Taira. Diffusion processes and partial differential equations. Academic Press Inc., Boston, MA, 1988. [100] Daniel Tataru. On the Fefferman-Phong inequality and related problems. Comm. Partial Differential Equations, 27(11-12):2101–2138, 2002. [101] J. C. Taylor. Skew products, regular conditional probabilities and stochastic differential equation: a technical remark. In Sminaire ´ de probabilit´es, XXVI, Lecture Notes in Mathematics 1526, ed. J. Az´ema, P. A. Meyer and M. Yor, pages 113–126, 1992. [102] B Tsirelson. Filtrations of random processes in the light of classification theory. (i). a topological zero-one law. arXiv:math.PR/0107121, 2001. [103] A. Ju. Veretennikov and N. V. Krylov. Explicit formulae for the solutions of stochastic equations. Mat. Sb. (N.S.), 100(142)(2):266–284, 336, 1976. [104] Jaak Vilms. Totally geodesic maps. J. Differential Geometry, 4:73–79, 1970. [105] Bill Watson. Manifold maps commuting with the Laplacian. J. Differential Geometry, 8:85–94, 1973. [106] Bill Watson. δ-commuting mappings and Betti numbers. Tˆ ohoku Math. J. (2), 27(2):135–152, 1975.

Index (∇), 143 GLM , 42 M + , 61 CM + , 61 D0 (x), 30 D0 (x), 30 Γ-martingale, 93, 146 k # (x, y), 122 p-related, 19 reproducing kernels, 149 reproducing kernel Hilbert space, 150 SDE, 24 adjoint semi-connection, 41 along S diffusion operator, 5 semi-martingale, 145 basic diffusion, 82 sigma-algebra, 82 co-differential, 120 co-differentials, 32 cohesive, 17 descends, 26 conditional law, 75 connection, 2 on natural bundles, 131 adjoint, 124 canonical, 118 complete, 29 determined by stochastic flows, 118

Levi-Civita, 45, 118 LW, 41, 45, 118, 124, 154 partial, 13 semi-, 13 stochastically complete, 105 strongly stochastically complete, 106 coupling, 142 covariant derivative for LW connection, 154 from semi-connection, 35 curvature form, 36 operator, 43, 58 parallel, 113 Ricci, 43, 46, 58 Weitzenb¨ock, 47, 52, 54, 58, 85, 119 damped parallel translation, 85 derivative flow, 45 development anti-, 146 stochastic, 145 diffeomorphism space Ds , 121 differential form pull-back, 31 diffusion equivariant, 102 measure, 61 operator, 1 diffusion operators G-invariant, 58

168 distribution, 5 regular, 6 vertical, 6 Duncan-Mortensen-Zakai equation, 89 exponential martingale, 135 and explosion, 135 flow derivative, 44, 45, 48, 83, 113 gradient, 45 holonomy, 30 of isometries, 48 of SDE, 44 on differential forms, 57 stochastic, 121, 125 frame bundle, 42 orthonormal, 117 r-th order, 131 Gauss’s formula, 147 Gradient Brownian SDE, 45 H¨ormander condition, 109, 111, 113 form, x, 20, 23, 24, 41, 51, 55, 101, 111, 128, 144 representation, 3 Hamiltonian systems stochastic, 58 Heisenberg group, 5, 20, 21, 39, 48 Hermann’s theorem, 103, 111, 112, 120 Hodge-Kodaira Laplacian, 119 Hodge-Kodaira-Laplacian eigenforms, 120 holonomy group, 36 flow, 30 group, 36 stochastic, 105 holonomy invariant at x0 , 108

Index diffusion operator, 108 stochastically, 108 horizontal, 9 lift of A, 18 horizontal lift map, 11, 15 of one-point motion, 127 horizontal subspace of a semi-connection, 13 innovations process, 91 intertwined diffusions, 66 flows, 152 Itˆo equation, 76, 143 Kallianpur-Striebel formula, 88 Krylov-Veretennikov formula, 74, 85 Laplacian fractional powers, 120 Hodge-Kodaira, 119 Lie differentiation, 3 lies over a flow, 152 LW connection, 41 covariant derivative, 154 Malliavin covariance operator deterministic, 111 minimal fibres, 115, 116, 119 minimal submanifold, 148 mixed type SDE, 76, 144 natural bundle, 131 observations process, 87 noise-free, 85, 99 operator curvature, 43 diffusion, 1 semi-elliptic, 1 shape, 147 symbol, 1

Index operators pseudo-differential, 120 vertical, 119 parallel translation stochastic, 105 damped, 85 predictable S ∗ -valued process, 63 principal semi-connection on diffeomorphism bundle, 122 projectible, 116 reproducing kernel, 149 projectible, operator, 11 pull-back, 31 redundant noise, 25, 151 representation linear isotropy, 117 reproducing kernel, 122, 126 Hilbert space, 122, 126 of SDE, 128 projectible, 149 Riemannian submersion, 115 SDE Itˆo, 143 mixed type, 76, 144 second fundamental form, 45, 147 semi-connection adjoint, 124 complete, 29 covariant derivative, 35 induced by reproducing kernel, 123 invariant, 34 linear, 35 on diffeomorphism bundle, 122, 127 one-form, 35, 39 principal, 34 semi-elliptic, 1 semi-groups P V , P H , 105

169 commuting, 104, 106 shape operator, 45, 147 skew-product decomposition of flows, 128 skew-product decomposition, 58, 81, 83 spans E, 122 stochastic flow, 142 jets of, 132 skew-product decomposition, 128, 129 stochastic partial differential equation, 95 submanifold minimal, 148 totally geodesic, 148 submersion, 8 Riemannian, 101, 115 symbol, 1 projectible, 14 symmetric space, 116 irreducible, 119 locally, 114 totally geodesic, 120, 148 fibres, 101, 102, 120, 152 vector field basic, 101 vertical, 9 vertical part, no, 28 Weingarten’s formula, 147 Weitzenb¨ock curvature, 85 Wiener chaos expansion, 75