 
Summary: Nonexponential Stability of Scalar Stochastic
Volterra Equations
John A. D. Appleby 1
School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland.
David W. Reynolds 2
School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland.
Abstract
We study convergence rates to zero of solutions of the scalar equation
dX(t) = f(X(t)) +
t
0
k(t  s)g(X(s)) ds dt + h(X(t)) dB(t),
where f, g, h are globally Lipschitz, xg(x) > 0 for nonzero x, and k is continuous,
integrable, positive and limt k(t  s)/k(t) = 1, for s > 0. Then
lim sup
t
X(t)
k(t)
= a.e. on A,
for nontrivial solutions satisfying limt X(t) = 0 on A, a set of positive probability.
