Annals of Probability

On the Spatial Asymptotic Behavior of Stochastic Flows in Euclidean Space

Peter Imkeller and Michael Scheutzow

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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.

Article information

Ann. Probab., Volume 27, Number 1 (1999), 109-129.

First available in Project Euclid: 29 May 2002

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Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 34F05: Equations and systems with randomness [See also 34K50, 60H10, 93E03]
Secondary: 60G48: Generalizations of martingales 60G17: Sample path properties

Stochastic differential equation stochastic flow spatial growth rate modulus of continuity GRR lemma semimartingale


Imkeller, Peter; Scheutzow, Michael. On the Spatial Asymptotic Behavior of Stochastic Flows in Euclidean Space. Ann. Probab. 27 (1999), no. 1, 109--129. doi:10.1214/aop/1022677255.

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