Flow of diffeomorphisms induced by a geometric multiplicative functional. (English) Zbl 0918.60009

Lyons, Terry; Qian, Zhongmin

Probab. Theory Relat. Fields 112, No. 1, 91-119 (1998).

The paper describes a new approach for constructing a stochastic flow of diffeomorphisms. The authors prove that the unique multiplicative functional solution to a differential equation driven by a geometric multiplicative functional constitutes a flow of local diffeomorphisms. A geometric multiplicative functional is constructed using Lyons forms [see T. Lyons, Math. Res. Lett. 1, No 4, 451-464 (1994; Zbl 0835.34004)]. If the driving path is a Brownian motion (or more generally, a continuous semimartingale), the result in particular gives the answer to an open problem proposed by N. Ikeda and S. Watanabe [“Stochastic differential equations and diffusion processes” (1981; Zbl 0495.60005)].

Reviewer: V.Oganyan (Erevan)

MSC:

60D05	Geometric probability and stochastic geometry
58D25	Equations in function spaces; evolution equations

Keywords:

diffeomorphisms; multiplicative functional; Brownian motion; stochastic flow of diffeomorphisms; differential equation driven by a geometric multiplicative functional; flow of local diffeomorphisms