Convergence rates of the solution of a Volterra-type stochastic differential equations to a non-equilibrium limit

Abstract

This paper concerns the asymptotic behaviour of solutions of functional differential equations with unbounded delay to non-equilibrium limits. The underlying deterministic equation is presumed to be a linear Volterra integro-differential equation whose solution tends to a non-trivial limit. We show when the noise perturbation is bounded by a non-autonomous linear functional with a square integrable noise intensity, solutions tend to a non-equilibrium and non-trivial limit almost surely and in mean-square. Exact almost sure convergence rates to this limit are determined in the case when the decay of the kernel in the drift term is characterised by a class of weight functions.