Stabilization of Volterra Equations by Noise

This paper aims to contribute to research on the question of the stabilization or destabilization of a deterministic dynamical system (differential equation, partial differential equation, or functional differential equation) by a noise perturbation, and in particular, perturbations which transform the differential equation (or FDE) to one of Itô-type. Mao has written an interesting paper [9] devoted to the study of stabilization and destabilization of nonlinear finite-dimensional differential equations, which has been extended to examine the stabilization of partial differential equations by Caraballo et al. [6]. The asymptotic behaviour of linear functional differential equations with bounded delay has been studied by Mohammed and Scheutzow [11], wherein it is shown that time delays in the diffusion coefficient can destablilize a linear functional differential equation. The stabilization of nonlinear finite-dimensional functional differential equations with (sufficiently small) bounded delay has been covered by the author in [3], and the destabilization of even-dimensional equations in [2]. In the latter paper, however, the delay can be unbounded, so Volterra equations can be destabilized, as a special case. For scalar, linear, convolution Volterra equations with positive and integrable kernel, Appleby has shown in [1] that the corresponding family of Itô-Volterra equations with a diffusion term of the form σx is almost surely asymptotically stable, provided the deterministic problem is uniformly asymptotically stable, so the addition of noise is not destabilizing. To the authors’ knowledge, the issue of stabilization of a Volterra equation by noise perturbations of Itô-type has not, to date, been studied. The question is of interest in applications in