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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.

We study asymptotic growth rates of stochastic flows on $\mathbf{R}^d$ and their derivatives with respect to the spatial parameter under Lipschitz conditions on the local characteristics of the generating semimartingales. In a first step these conditions are seen to imply moment inequalities for the flow $\phi$ of the form $$E \sup_{0 \le t \le T}|\phi_{0t} (x) - \phi_{0t} (y)|^p\le|x-y|^p\exp(cp^2)\quad\text{for all $p\ge 1$.}$$ In a second step we deduce the growth rates from an integrated version of these moment inequalities, using the continuity lemma of Garsia, Rodemich and Rumsey. We provide two examples to show that our results are sharp.