Attractors and Expansion for Brownian Flows

Georgi Dimitroff (Fraunhofer ITWM)
Michael Scheutzow (Technische Universit├Ąt Berlin)

Abstract


We show that a stochastic flow which is generated by a stochastic differential equation on $\mathbb{R}^d$ with bounded volatility has a random attractor provided that the drift component in the direction towards the origin is larger than a certain strictly positive constant $\beta$ outside a large ball. Using a similar approach, we provide a lower bound for the linear growth rate of the inner radius of the image of a large ball under a stochastic flow in case the drift component in the direction away from the origin is larger than a certain strictly positive constant $\beta$ outside a large ball. To prove the main result we use chaining techniques in order to control the growth of the diameter of subsets of the state space under the flow.

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Pages: 1193-1213

Publication Date: July 3, 2011

DOI: 10.1214/EJP.v16-894

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