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Let x → ϕ s , t ( x ) be a R d -valued stochastic homeomorphic flow produced by non-Lipschitz stochastic differential equation ϕ s , t ( x ) = x + ∫ s t σ ( ϕ s , u ( x ) ) ⋅ d W u + ∫ s t b ( ϕ s , u ( x ) ) d u , where W = ( W 1 , W 2 , … ) is an infinite sequence of independent standard Brownian motions. We first give some estimates of modulus of continuity of { ϕ s , t ( ⋅ ) } , then prove that the flow ϕ s , t ( x ) , when x nears infinity, grows slower than Z exp { c ln ln | x | } 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.