On the Geometry of Diffusion Operators and Stochastic Flows

1. Front Matter

   Pages I-2

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2. Introduction

   • K. D. Elworthy, J. Le Jan, Xue-Mei Li

   Pages 3-6

3. Construction of connections

   • K. David Elworthy, Yves Le Jan, Xue-Mei Li

   Pages 7-29

4. The infinitesimal generators and associated operators

   • K. David Elworthy, Yves Le Jan, Xue-Mei Li

   Pages 30-56

5. Decomposition of noise and filtering

   • K. David Elworthy, Yves Le Jan, Xue-Mei Li

   Pages 57-75

6. Application: Analysis on spaces of paths

   • K. David Elworthy, Yves Le Jan, Xue-Mei Li

   Pages 76-86

7. Stability of stochastic dynamical systems

   • K. David Elworthy, Yves Le Jan, Xue-Mei Li

   Pages 87-94

8. Appendices

   • K. David Elworthy, Yves Le Jan, Xue-Mei Li

   Pages 95-110

9. Back Matter

   Pages 111-116

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Stochastic differential equations, and Hoermander form representations of diffusion operators, can determine a linear connection associated to the underlying (sub)-Riemannian structure. This is systematically described, together with its invariants, and then exploited to discuss qualitative properties of stochastic flows, and analysis on path spaces of compact manifolds with diffusion measures. This should be useful to stochastic analysts, especially those with interests in stochastic flows, infinite dimensional analysis, or geometric analysis, and also to researchers in sub-Riemannian geometry. A basic background in differential geometry is assumed, but the construction of the connections is very direct and itself gives an intuitive and concrete introduction. Knowledge of stochastic analysis is also assumed for later chapters.

• Measure
• Riemannian geometry
• Stochastic differential equations
• connections
• differential forms
• differential geometry
• manifold
• path space
• sub-Riemannian geometry

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