Existence and higher regularity of statistically steady states for the stochastic Coleman-Gurtin equation
Nathan E. Glatt-Holtz, Vincent R. Martinez, Hung D. Nguyen
TL;DR
The study addresses statistically steady states for the stochastic Coleman–Gurtin-type equation with memory by augmenting the system with a history variable to recover Markovianity. A memory-adapted functional framework with an exponentially decaying kernel is used to overcome loss of compactness, enabling Krylov–Bogoliubov construction of invariant measures. Under sufficiently smooth noise and dissipative polynomial nonlinearities, the authors prove existence of invariant measures and, in dimensions $d\le 3$, enhanced regularity of these measures' supports via a bootstrap control mechanism that leverages the Lyapunov structure. The results provide a rigorous link between noise regularity, memory effects, and the regularity of long-time statistical states for memory-driven SPDEs, with concentration on smoother Sobolev spaces and exponential moment bounds.
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.