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Stochastic differential equations and stochastic flows of homeomorphisms. (English) Zbl 0555.60039
Stochastic analysis and applications, Adv. Probab. Relat. Top. 7, 269-291 (1984).
[For the entire collection see Zbl 0541.00008.]
1. A Stratonovich stochastic differential equation (s.d.e.) on a non- compact connected manifold, driven by $$C^ 2$$ vector fields, generates a stochastic flow of homeomorphisms, if and only if a certain adjoint equation has a strongly complete solution; an s.d.e. for the backwards flow is also given.
2. Given a square integrable stochastic flow of homeomorphisms on $${\mathbb{R}}^ d$$, adapted to the filtration of an m-dimensional Brownian motion, and with a mild condition on the expected value of each one-point motion, there exists a collection of m vector fields for which the corresponding Itô s.d.e. generates the given stochastic flow. [A related result was proved by P. Baxendale, Compos. Math. 53, 19-50 (1984; Zbl 0547.58041)].
3. A similar representation result is obtained, replacing $${\mathbb{R}}^ d$$ by a non-compact manifold M, and Itô s.d.e. by Stratonovich s.d.e.. The formulas and techniques in this paper will be useful to workers in this area.