Summary: STOCHASTIC VOLTERRA DIFFERENTIAL EQUATIONS IN
WEIGHTED SPACES
JOHN A. D. APPLEBY AND MARKUS RIEDLE
Abstract. In the following paper, we provide a stochastic analogue to work
of Shea and Wainger by showing that when the measure and state-independent
diffusion coefficient of a linear It^o­Volterra equation are in appropriate Lp­
weighted spaces, the solution lies in a weighted Lp­space in both an almost
sure and moment sense.
1. Introduction
This paper examines the asymptotic stability and decay rates, in various modes of
stochastic convergence, of solutions of stochastically perturbed Volterra equations
to the equilibrium solution of a related unperturbed deterministic Volterra equation.
For deterministic equations the phenomenon of asymptotic stability has been shown
to be distinct from that of exponential stability. These phenomena were shown to
coincide in linear Volterra integrodifferential equations by Murakami [28, 29] if and
only if the kernel lies in an exponentially weighted L1
­space. On the other hand,
non­exponential rates of decay in spaces of integrable functions with general weights
have been considered by Gelfand et al. [11], Shea and Wainger [30] and Jordan and
Wheeler [17]. An account of this research is summarised in Grippenberg et al. [12].

Source: Applebaum, David - Department of Probability and Statistics, University of Sheffield
Dublin City University, School of Mathematical Sciences
Küchler, Uwe - Institut für Mathematik, Humboldt-Universität zu Berlin

Collections: Mathematics