Homeomorphic property of solutions of SDE driven by countably many Brownian motions with non-Lipschitzian coefficients

Abstract

We study m-dimensional SDE X t = x 0 + ∑ i = 1 ∞ ∫ 0 t σ i ( X s ) d W s i + ∫ 0 t b ( X s ) d s , where { W i } i ⩾ 1 is an infinite sequence of independent standard d-dimensional Brownian motions. The existence and pathwise uniqueness of strong solutions to the SDE was established recently in [Z. Liang, Stochastic differential equations driven by countably many Brownian motions with non-Lipschitzian coefficients, Preprint, 2004]. We will show that the unique strong solution produces a stochastic flow of homeomorphisms if the modulus of continuity of coefficients is less than | x − y | ( log 1 | x − y | ) ϑ , ϑ ∈ [ 0 , 1 ) with ( − 1 ) ϑ = 1 , and the coefficients are compactly supported.