Attractors and Expansion for Brownian Flows

TL;DR

This paper establishes conditions under which stochastic flows generated by SDEs in ^d have random attractors and provides bounds on the growth rate of images of large balls, using chaining techniques for analysis.

Contribution

It introduces new criteria for the existence of random attractors and bounds on growth rates in stochastic flows with bounded volatility.

Findings

Existence of random attractors under certain drift conditions.

Lower bounds for the growth rate of images of large balls.

Application of chaining techniques to control growth of flow diameters.

Abstract

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.