Limit theorems for stochastic differential equations and stochastic flows of diffeomorphisms

Abstract

The approximation theories and limit theorems related to stochastic differential equations have been studied by many authors from various motivations. The purpose of this report is to look at the problems from the points of the views of the stochastic flow of diffeomorphisms, and present a unified method for a large class of problems.

In Section 1, we survey three limit theorems related to the diffusion processes and stochastic flows. The first is the approximation of the stochastic differential equation developed by Wong-Zakai [27], Ikeda-Nakao-Yamato [7], Ikeda-Watanabe [8], Malliavin [20], Bismut [2], Shu [25] etc. Here, the Brownian motions defining the stochastic differential equation are approximated by sequences of processes with piecewise smooth paths. Polygonal approximations and the approximations by the mollifiers are widely used. The second is the limit theorem for suitable stochastic ordinary differential equations with the small parameter ε, studied by Khasminskii [12], Papanicolaou-Kohler [22], Borodin [4], Kesten-Papanicolaou [9] etc. Under various conditions, they proved that, after a suitable change of the scale of the time, the solutions converge weakly to a diffusion process as ε → 0. The third is the limit theorem studied by Papanicolaou-Stroock-Varadhan [23], concerning the driving processes and driven processes.

In order to discuss these limit theorems rigorously in a unified method, the recent results on stochastic differential equations and stochastic flows are needed. We discuss these facts in Section 2 following partly to Le Jan [18], Le Jan-Watanabe [19] and Fujiwara-Kunita [5].

In Section 3, we shall formulate the limit theorems and state three theorems. The first (Theorem 3.1) is a rather abstract theorem. Assumptions are stated in the language of the conditional expectations and martingales. Then we consider the two special cases. Theorem 3.2 discusses the case of the mixing property and Theorem 3.3 deals with the case of the ergodic property. Then we check how the limit theorems stated in Section 1 are derived from these theorems. In Section 4, we apply these theorems to limits for stochastic partial differential equations.