Spatial asymptotic behavior of homeomorphic global flows for non-Lipschitz SDEs

Let x→ϕs,t(x) be a -valued stochastic homeomorphic flow produced by non-Lipschitz stochastic differential equation , where W=(W1,W2,…) is an infinite sequence of independent standard Brownian motions. We first give some estimates of modulus of continuity of {ϕs,t(⋅)}, then prove that the flow ϕs,t(x), when x nears infinity, grows slower than for some constant c>0 and integrable random variable Z via lemma of Garsia–Rodemich–Rumsey Lemma (abbreviated as GRR Lemma) improved by Arnold and Imkeller [L. Arnold, P. Imkeller, Stratonovich calculus with spatial parameters and anticipative problems in multiplicative ergodic theory, Stochastic Process. Appl. 62 (1996) 19–54] and moment estimates for one- and two-point motions.