Convergence of stochastic flows connected with stochastic ordinary differential equations

Abstract

We obtain the limit theorem for the sequence of stochastic flows generated by the stochastic ordinary differential equations dx/dt = F$sub:n$esub:(t, x, ω)n =l, 2,.... Under some regularity and mixing conditions on the sequence F$sub:n$esub: n = l,2,..., it is shown that the associated flows converge weakly to a Brownian motion in the diffeomorphisms group and that the latter is generated by an Ito's stochastic differential equation. The strong convergence is also established: It covers the approximation theorems on stochastic flows studied by Malliavin. Ikeda Watanabe. Bismut and Shu.