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Mathematics - Probability Theory and Stochastic Processes | On the Geometry of Diffusion Operators and Stochastic Flows

On the Geometry of Diffusion Operators and Stochastic Flows

Series: Lecture Notes in Mathematics, Vol. 1720

Elworthy, K.D., Le Jan, Y., Li, Xue-Mei

1999, V, 105 p.

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  • About this book

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.

Content Level » Research

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

Related subjects » Analysis - Geometry & Topology - Probability Theory and Stochastic Processes

Table of contents 

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.

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