 
Summary: STABILISATION OF VOLTERRA EQUATIONS BY NOISE
JOHN A. D. APPLEBY AND AOIFE FLYNN
Abstract. The paper is concerned with the relationship between the stability
of the It^oVolterra equation
dX(t) = af(X(t)) +
t
0
k(t  s)g(X(s)) ds dt + X(t) dB(t)
and its deterministic counterpart (where = 0). It is assumed that k is
bounded by a negative exponential with exponent µ. Under mild continuity
and growth hypotheses on f, g, we show that there exists an open set in (µ, a)
parameter space in which the deterministic problem is unstable, while for an
interval of values of , all solutions of the stochastic problem tend to zero
exponentially fast, almost surely. In addition, for the scalar linear problem
when k is a negative exponential, and the deterministic problem is stable,
there is an interval of values of for which the stochastic problem is stable.
1. Introduction
This paper aims to contribute to research on the question of the stabilisation or
destabilisation of a deterministic dynamical system (differential equation, partial
differential equation or functional differential equation) by a noise perturbation,
