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On the Geometry of Diffusion Operators and Stochastic Flows : Lecture Notes in Mathematics - K.D. Elworthy

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On the Geometry of Diffusion Operators and Stochastic Flows

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

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Paperback | 17 November 1999

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132 Pages

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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.

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You Can Find This Book In

Non-Fiction Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Medicine Medicine in General Probability & Statistics Applied Mathematics Stochastics Geometry Differential & Riemannian Geometry Functional Analysis & Transforms

Introduction	p. 3
Construction of connections	p. 7
Construction of connections	p. 7
Basic Classes of Examples	p. 14
Adjoint connections, torsion skew symmetry, basic formulae	p. 18
Example: Homogeneous spaces continued	p. 26
The infinitesimal generators and associated operators	p. 30
The irrelevance of drift in dimension greater than 1	p. 30
Torsion Skew Symmetry	p. 35
The 'divergence operator' [delta]	p. 37
Hormander form generators on differential forms	p. 42
On the infinitesimal generator	p. 47
Decomposition of noise and filtering	p. 57
A canonical decomposition of the noise driving a stochastic differential equation	p. 57
Canonical decomposition of the Gaussian field W[subscript t]	p. 60
Filtering out redundant noise	p. 63
Application: Analysis on spaces of paths	p. 76
Integration by parts and Clark-Ocone formulae	p. 78
Logarithmic Sobolev Inequality	p. 82
Analysis on C[subscript id] (DiffM)	p. 83
Stability of stochastic dynamical systems	p. 87
Universal Connections as L-W connection	p. 95
Creation and Annihilation operators (notation for section 2.4)	p. 101
Basic formulae	p. 103
List of notation	p. 109
Table of Contents provided by Blackwell. All Rights Reserved.

On the Geometry of Diffusion Operators and Stochastic Flows

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