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Stochastic differential equations and stochastic flows of diffeomorphisms. (English) Zbl 0554.60066
Ecole d’été de probabilités de Saint-Flour XII - 1982, Lect. Notes Math. 1097, 143-303 (1984).
[For the entire collection see Zbl 0541.00006.]
This monograph will be useful not only to established researchers, but also to advanced graduate students looking for a clear, concise introduction to stochastic differential equations. Most of the material has appeared before in work by Kunita, Bismut, Elworthy, Watanabe and others, but its presentation here is more comprehensible because it is presented as a whole.
Chapter I establishes the theory of stochastic integration with respect to continuous semimartingales, and a generalized Itô formula for the composition of continuous semimartingales. Chapter II proves the existence of a stochastic flow of homeomorphisms (resp. \(C^ k\) diffeomorphisms) for a system of stochastic differential equations in \(R^ n\) driven by Lipschitz (resp. \(C^{k,\alpha}\), where \(\alpha >0)\) vector fields; the proofs involve \(L^ p\) estimates and Kolmogorov’s continuity criterion for random fields. Chapter III treats the backwards flow, the action of the stochastic flow on tensor fields, and explicit representation of solutions in terms of the Lie algebra generated by the driving vector fields. Both Itô and Stratonovich s.d.e. are treated throughout, and all relevant formulae are given.
A minor criticism is that stochastic differential equations on manifolds are not treated in an intrinsic way, but only in local coordinates. Two important recent references which give an alternative treatment of much of this material are K. D. Elworthy’s book ”Stochastic differential equations on manifolds” (1982; Zbl 0514.58001), and P. Baxendale, ”Brownian motions in the diffeomorphism group. I”, Compos. Math. 53, 19- 50 (1984). However the present article is the most readable exposition of s.d.e.
Reviewer: R.Darling

60H20 Stochastic integral equations
60J60 Diffusion processes
58J65 Diffusion processes and stochastic analysis on manifolds
60H05 Stochastic integrals