Pathwise mild solutions for superlinear stochastic evolution equations and their attractors

TL;DR

This paper develops a framework for analyzing superlinear stochastic evolution equations using pathwise mild solutions, establishing global well-posedness and the existence of random attractors in infinite-dimensional spaces.

Contribution

It introduces a novel approach to handle superlinear stochastic PDEs with time-dependent generators and constructs random attractors under new conditions.

Findings

Construction of an infinite-dimensional stationary Ornstein-Uhlenbeck process

Proof of global well-posedness for the class of equations studied

Existence of a random attractor for reaction-diffusion equations with random generators

Abstract

We investigate stochastic parabolic evolution equations with time-dependent random generators and locally Lipschitz continuous drift terms. Using pathwise mild solutions, we construct an infinite-dimensional stationary Ornstein-Uhlenbeck type process, which is shown to be tempered in suitable function spaces. This property, together with a bootstrapping argument based on the regularizing effect of parabolic evolution families, is then applied to prove the global well-posedness and the existence of a random attractor for reaction-diffusion equations with random non-autonomous generators and nonlinearities satisfying certain growth and dissipativity assumptions.