Attractors and Expansion for Brownian Flows

Details Attractors and Expansion for Brownian Flows Attractors and Expansion for Brownian Flows

2009-09-21

arxiv.org/abs/0909.3768v1

Mathematics

Probability

Scientific paper

We show that a stochastic flow which is generated by a stochastic differential equation on $\R^d$ 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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