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arXiv · 1307.6743

Random attractors for stochastic evolution equations driven by fractional Brownian motion

H. Gao·M. J. Garrido-Atienza·B. Schmalfuss
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Abstract

The main goal of this article is to prove the existence of a random attractor for a stochastic evolution equation driven by a fractional Brownian motion with $H\in (1/2,1)$. We would like to emphasize that we do not use the usual cohomology method, consisting of transforming the stochastic equation into a random one, but we deal directly with the stochastic equation. In particular, in order to get adequate a priori estimates of the solution needed for the existence of an absorbing ball, we will introduce stopping times to control the size of the noise. In a first part of this article we shall obtain the existence of a pullback attractor for the non-autonomous dynamical system generated by the pathwise mild solution of an nonlinear infinite-dimensional evolution equation with non--trivial Hölder continuous driving function. In a second part, we shall consider the random setup: stochastic equations having as driving process a fractional Brownian motion with $H\in (1/2,1)$. Under a smallness condition for that noise we will show the existence and uniqueness of a random attractor for the stochastic evolution equation.

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H. Gao, M. J. Garrido-Atienza, B. Schmalfuss. 2013-07-25. Random attractors for stochastic evolution equations driven by fractional Brownian motion. https://arxiv.org/abs/1307.6743

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