The short memory limit for long time statistics in a stochastic Coleman-Gurtin model of heat conduction

TL;DR

This paper investigates the long-term statistical behavior of a stochastic heat conduction model with memory effects, showing convergence to classical models as memory effects diminish, and establishing conditions for unique invariant measures.

Contribution

It demonstrates the existence and uniqueness of invariant measures for the stochastic memory model and proves the validity of short memory approximations in the zero-memory limit.

Findings

Existence of a unique invariant measure under certain stochastic forcing conditions.

Exponential convergence to the invariant measure independent of memory decay rate.

Convergence of the memory-affected system to the classical reaction-diffusion model as memory vanishes.

Abstract

We consider 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. Our main study is the long time statistics of the system in the singular regime as the memory kernel collapses to a Dirac function. Specifically, we show that provided that sufficiently many directions in the phase space are stochastically forced, there is a unique invariant probability measure to which the system converges, with respect to a suitable Wasserstein-type topology, and at an exponential rate which is independent of the decay rate of the memory kernel. We then prove the convergence of this unique statistically steady state to the unique invariant probability measure of the classical stochastic reaction-diffusion equation in the zero-memory limit. Consequently, we establish the global-in-time validity of the short memory approximation.