## Electronic Journal of Probability

### Attractors and Expansion for Brownian Flows

#### 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 $\beta$<em></em> 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 $\beta$<em></em> outside a large ball. To prove the main result we use <em>chaining techniques</em> in order to control the growth of the diameter of subsets of the state space under the flow.

#### Article information

Source
Electron. J. Probab., Volume 16 (2011), paper no. 42, 1193-1213.

Dates
Accepted: 3 July 2011
First available in Project Euclid: 1 June 2016

https://projecteuclid.org/euclid.ejp/1464820211

Digital Object Identifier
doi:10.1214/EJP.v16-894

Mathematical Reviews number (MathSciNet)
MR2827455

Zentralblatt MATH identifier
1221.37109

Rights

#### Citation

Dimitroff, Georgi; Scheutzow, Michael. Attractors and Expansion for Brownian Flows. Electron. J. Probab. 16 (2011), paper no. 42, 1193--1213. doi:10.1214/EJP.v16-894. https://projecteuclid.org/euclid.ejp/1464820211

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