Random attractors for stochastic evolution equations driven by fractional Brownian motion
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 . 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 H\"older continuous driving function. In a second part, we shall consider the random setup: stochastic equations having as driving process a fractional Brownian motion with . Under a smallness condition for that noise we will show the existence and uniqueness of a random attractor for the stochastic evolution equation.
Cite
@article{arxiv.1307.6743,
title = {Random attractors for stochastic evolution equations driven by fractional Brownian motion},
author = {H. Gao and M. J. Garrido-Atienza and B. Schmalfuss},
journal= {arXiv preprint arXiv:1307.6743},
year = {2013}
}