Electronic Journal of Probability

Attractors and Expansion for Brownian Flows

Georgi Dimitroff and Michael Scheutzow

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37H10: Generation, random and stochastic difference and differential equations [See also 34F05, 34K50, 60H10, 60H15]
Secondary: 60G90 60H10: Stochastic ordinary differential equations [See also 34F05]

Stochastic flow stochastic differential equation attractor chaining

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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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