Elworthy K.D., Le Jan Y., Xue-Mei L. On the Geometry of Diffusion Operators and Stochastic Flows

Berlin: Springer-Verlag, 1999. — 116 p.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.Introduction
Construction of connections
The infinitesimal generators and associated operators
Decomposition of noise and filtering
Application: Analysis on spaces of paths
Stability of stochastic dynamical systems
Appendices