On the spatial asymptotic behavior of stochastic flows in Euclidean space

We study asymptotic growth rates of stochastic flows on Rd and their deriva tives with respect to the spatial parameter under Lipschitz conditions on t he 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 \phi(0t)(x) - phi(0t)(y)\(p) less than or equal to \x - y\p exp(cp(2) ) for all p greater than or equal to 1. 0 less than or equal to t less than or equal to T In a second step we deduce the growth rates from an integrated version of t hese moment inequalities, using the continuity lemma of Garsia, Rodemich an d Rumsey. We provide two examples to show that our results are sharp.