The Geometry of Filtering

Frontiers in Mathematics Advisory Editorial Board Leonid Bunimovich (Georgia Institute of Technology, Atlanta) Benoît ...

Author: K. David Elworthy | Yves Le Jan | Xue-Mei Li

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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 Itô 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 Weitzenböck 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 Weitzenböck 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 Hörmander 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 Hörmander form. Consider now a diffusion operator not given in Hörmander 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 Hörmander 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 Hörmander forms composed of intertwined vector fields. It is shown that the Hörmander 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 Hörmander 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 Hörmander 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


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


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


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


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 carré 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)


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 σB (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


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).


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 Hörmander 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


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 Hörmander 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


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 Itô 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 Hörmander’s condition if for some (and hence all) Hörmander 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


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 Hörmander ˜ 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 ù : Rm → Rm ker ù = ker X(p(u)) and the diagram

˜ ∗ X(u)

Tu∗ N

-

6 (Tu p)∗

Tx∗ M

˜ Rm 6

˜ X(u) -Tu N

Tu p

ù

-

? -Tx M

Rm

X(x)∗

X(x)

˜ ◦ ù . 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)ù Y (p(u)). Proof. The larger square commutes by Lemma 2.1.1. For the rest we need to construct ù . It suffices to define ù 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 ù is defined with ker ù = 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 ù that the right-hand side also commutes. The uniqueness of ù 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 ù (e) to be X(u) x From now on assume that X(x) has rank independent of x ∈ M . This ensures that ù 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


The horizontal lift of X(x), which can be used to construct a Hörmander 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 ù = 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 )∗ ù = 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 Itô 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


˜ 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


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


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 Hörmander 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


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).


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 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 Weitzenböck 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


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 .


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


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 Weitzenböck 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.


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 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 Weitzenböck 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

120

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


121

122

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

124


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


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 ) ,


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


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 Hörmander 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.


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


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 Itô’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


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

The Geometry of Filtering

The geometry of filtering

Fundamentals of stochastic filtering

Filtering Theory

Fundamentals of Stochastic Filtering

Optimal Filtering

Optimal Filtering

Adaptive Filtering

Optimal Filtering

Filtering Theory

Modern geometry. The geometry of surfaces

The Geometry of Schemes

The Geometry of Schemes

The geometry of submanifolds

The Geometry of Time

The Geometry of Opinion

The Geometry of Schemes

The Geometry of Time

Geometry of the Quintic

Geometry of the quintic

The Geometry of Supermanifolds

Geometry of the Quintic

The geometry of numbers

The Geometry of Schemes

The geometry of syzygies

The foundations of Geometry

The elements of geometry

The Geometry of Submanifolds

Performance and implementation aspects of nonlinear filtering

Kalman filtering theory

QRD-RLS adaptive filtering

The Geometry of Filtering