Existence and higher regularity of statistically steady states for the stochastic Coleman-Gurtin equation

TL;DR

This paper proves the existence and higher regularity of statistically steady states for a class of stochastic Volterra equations with memory effects, overcoming compactness challenges through adapted functional spaces and control techniques.

Contribution

It introduces a novel approach using adapted functional spaces and control arguments to establish higher regularity of invariant measures in stochastic equations with memory.

Findings

Invariant probability measures exist for broad nonlinear potentials.

Memory effects require specialized functional spaces for analysis.

Higher regularity of stationary states is achieved under smooth noise assumptions.

Abstract

We study a class of semi-linear differential Volterra equations with polynomial-type potentials that incorporates the effects of memory while being subjected to random perturbations via an additive Gaussian noise. We show that for a broad class of non-linear potentials, the system always admits invariant probability measures. However, the presence of memory effects precludes access to compactness in a typical fashion. In this paper, this obstacle is overcome by introducing functional spaces adapted to the memory kernels, thereby allowing one to recover compactness. Under the assumption of sufficiently smooth noise, it is then shown that the statistically stationary states possess higher-order regularity properties dictated by the structure of the nonlinearity. This is established through a control argument that asymptotically transfers regularity onto the solution by exploiting the underlying Lyapunov structure of the system in a novel way.