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.