Summary: STOCHASTIC VOLTERRA DIFFERENTIAL EQUATIONS IN
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.
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. , Shea and Wainger  and Jordan and
Wheeler . An account of this research is summarised in Grippenberg et al. .