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

We study m-dimensional SDE X-t = x(0) + Sigma(infinity)(i=1) f(0)(t) sigma(i)(X-s)dW(s)(i) + f(0)(t) b(X-s)ds, where [W-i] >= 1 is an infinite sequence of independent standard d-dimensional Brownian motions. The existence and pathwise uniqueness of strong solutions to the SIDE 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|)(nu), nu is an element of [0, 1) with (- 1)(nu) = 1, and the coefficients are compactly supported. (c) 2005 Elsevier SAS. All rights reserved.