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

We study m-dimensional SDE Xt=x0+∑i=1∞∫0tσi(Xs)dWsi+∫0tb(Xs)ds, where {Wi}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|(log1|x−y|)ϑ, ϑ∈[0,1) with (−1)ϑ=1, and the coefficients are compactly supported.