Topological dimensions of random attractors for stochastic partial differential equations with delay

TL;DR

This paper estimates the Hausdorff and fractal dimensions of random attractors for stochastic PDEs with delay, transforming the equations into an auxiliary Hilbert space to analyze their dynamical properties.

Contribution

It introduces a novel method to estimate the dimensions of random attractors for delayed stochastic PDEs using a transformation into an auxiliary Hilbert space.

Findings

Established the existence of random attractors for the class of stochastic PDEs with delay.

Provided upper bounds for the Hausdorff and fractal dimensions of these attractors.

Proved the differentiability properties of the associated random dynamical system.

Abstract

The aim of this paper is to obtain an estimation of Hausdorff as well as fractal dimensions of random attractors for a class of stochastic partial differential equations with delay. The stochastic equation is first transformed into a delayed random partial differential equation by means of a random conjugation, which is then recast into an auxiliary Hilbert space. For the obtained equation, it is firstly proved that it generates a random dynamical system (RDS) in the auxiliary Hilbert space. Then it is shown that the equation possesses random attractors by a uniform estimate of the solution and the asymptotic compactness of the generated RDS. After establishing the variational equation in the auxiliary Hilbert space and the $\mathbb{P}$ almost surely differentiable properties of the RDS, an upper estimate of both Hausdorff and fractal dimensions of the random attractors are obtained.