Table of Contents
Introduction vii
1 Diffusion Operators 1
1.1 Representations of Diffusion Operators 1
1.2 The Associated First-Order Operator 4
1.3 Diffusion Operators Along a Distribution 5
1.4 Lifts of Diffusion Operators 7
1.5 Notes 10
2 Decomposition of Diffusion Operators 11
2.1 The Horizontal Lift Map 11
2.2 Lifts of Cohesive Operators and The Decomposition Theorem 17
2.3 The Lift Map for SDEs and Decomposition of Noise 23
2.3.1 Decomposition of Stratonovich SDE's 24
2.3.2 Decomposition of the noise and Itô SDE's 25
2.4 Diffusion Operators with Projectible Symbols 26
2.5 Horizontal lifts of paths and completeness of semi-connections 28
2.6 Topological Implications 30
2.7 Notes 31
3 Equivariant Diffusions on Principal Bundles 33
3.1 Invariant Semi-connections on Principal Bundles 34
3.2 Decompositions of Equivariant Operators 36
3.3 Derivative Flows and Adjoint Connections 41
3.4 Associated Vector Bundles and Generalised Weitzenböck Formulae 46
3.5 Notes 58
4 Projectible Diffusion Processes and Markovian Filtering 61
4.1 Integration of predictable processes 62
4.2 Horizontality and filtrations 66
4.3 Intertwined diffusion processes 66
4.4 A family of Markovian kernels 70
4.5 The filtering equation 71
4.6 Approximations 73
4.7 Krylov-Veretennikov Expansion 74
4.8 Conditional Laws 75
4.9 An SPDE example 79
4.10 Equivariant case: skew-product decomposition 81
4.11 Conditional expectations of induced processes on vector bundles 83
4.12 Notes 85
5 Filtering with non-Markovian Observations 87
5.1 Signals with Projectible Symbol 88
5.2 Innovations and innovations processes 91
5.3 Classical Filtering 94
5.4 Example: Another SPDE 95
5.5 Notes 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 Spaces 115
7.1 Riemannian Submersions 115
7.2 Riemannian Symmetric Spaces 116
7.3 Notes 119
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 135
9.1 Girsanov-Maruyama-Cameron-Martin Theorem 135
9.2 Stochastic differential equations for degenerate diffusions 139
9.3 Semi-martingales and Γ -martingales along a Subbundle 145
9.4 Second fundamental forms and shape operators 147
9.5 Intertwined stochastic flows 148
Bibliography 159
Index 167
