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We study a class of semi-linear differential Volterra equations withpolynomial-type potentials that incorporates the effects of memory while beingsubjected to random perturbations via an additive Gaussian noise. We show thatfor a broad class of non-linear potentials and sufficiently regular noise thesystem always admits invariant probability measures, defined on the extendedphase space, that possess higher regularity properties dictated by thestructure of the nonlinearities in the equation. Furthermore, we investigatethe singular limit as the memory kernel collapses to a Dirac function.Specifically, provided sufficiently many directions in the phase space arestochastically forced, we show that there is a unique stationary measure towhich the system converges, in a suitable Wasserstein distance, at exponentialrates independent of the decay of the memory kernel. We then prove theconvergence of the statistically steady states to the unique invariantprobability of the classical stochastic reaction-diffusion equation in thedesired singular limit. As a consequence, we establish the validity of thesmall memory approximation for solutions on the infinite time horizon$[0,\infty)$.

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