Random attractors for stochastic evolution equations driven by fractional Brownian motion

TL;DR

This paper proves the existence of a random attractor for stochastic evolution equations driven by fractional Brownian motion with Hurst parameter in (1/2,1), using direct methods and stopping times for estimates.

Contribution

It introduces a direct approach to establish random attractors for fractional Brownian motion driven equations without cohomology transformation.

Findings

Existence of a pullback attractor for non-autonomous systems with Hölder continuous noise.

Existence and uniqueness of a random attractor under small noise conditions.

Development of a priori estimates using stopping times for stochastic equations.

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 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 $H\in (1/2,1)$. Under a smallness condition for that noise we will show the existence and uniqueness of a random attractor for the stochastic evolution equation.