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An introduction to the geometry of stochastic flows. (English) Zbl 1085.60002

London: Imperial College Press (ISBN 1-86094-481-7/hbk; 978-1-86094-726-1/ebook). x, 140 p. (2004).
The monograph deals with the local geometry of stochastic flows on Euclidean space which are induced by a stochastic differential equation (sde) driven by a finite number of Brownian motions. The cornerstone of the monograph is the so-called Chen-Strichartz formula which allows a representation of the action of the flow on smooth functions as the exponential of an infinite sum of Lie brackets of the generating vector fields with iterated Stratonovich integrals of the driving Brownian motions as coefficients. After a formal derivation of the formula in the first chapter, the author deals with the representation of the solution of an sde as a smooth map of the initial condition and the lift of the driving Brownian motion in a Carnot group. Several cases (commutative, 2- and \(N\)-step nilpotent and hypoelliptic) are discussed in detail emphasizing the induced sub-Riemannian geometry. Appendices on basic stochastic calculus and vector fields, Lie groups and Lie algebras help the non-experts in those fields to better understand the monograph.

MSC:

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H07 Stochastic calculus of variations and the Malliavin calculus
53C17 Sub-Riemannian geometry
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