Random attractors via pathwise mild solutions for stochastic parabolic evolution equations

TL;DR

This paper establishes the existence of finite-dimensional exponential attractors for stochastic parabolic evolution equations in Banach spaces using pathwise mild solutions, advancing understanding of their long-term behavior.

Contribution

It introduces a novel approach to analyze stochastic PDEs with time-dependent operators by employing pathwise mild solutions to prove the existence of exponential attractors.

Findings

Existence of random exponential attractors with finite fractal dimension.

Attractors attract solutions at an exponential rate.

Framework applicable to non-stationary stochastic PDEs.

Abstract

We investigate the longtime behavior of stochastic partial differential equations (SPDEs) with differential operators that depend on time and the underlying probability space. In particular, we consider stochastic parabolic evolution problems in Banach spaces with additive noise and prove the existence of random exponential attractors. These are compact random sets of finite fractal dimension that contain the global random attractor and are attracting at an exponential rate. In order to apply the framework of random dynamical systems, we use the concept of pathwise mild solutions. This approach is essential for our setting since the stochastic evolution equation cannot be transformed into a family of PDEs with random coefficients via the stationary Ornstein-Uhlenbeck process.