Almost sure subexponential decay rates of scalar Ito-Volterra equations

J. A. D. Appleby, CMDE, School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland

E. J. Qualitative Theory of Diff. Equ., Proc. 7'th Coll. Qualitative Theory of Diff. Equ., No. 1. (2004), pp. 1-32.

Abstract: The paper studies the subexponential convergence of solutions of scalar It\^{o}-Volterra equations. First, we consider linear equations with an instantaneous multiplicative noise term with intensity $\sigma$. If the kernel obeys
\[
\lim_{t \rightarrow\infty} k'(t)/k(t)=0,
\]
and another nonexponential decay criterion, and the solution $X_\sigma$ tends to zero as $t\rightarrow\infty$, then
\[
\limsup_{t\rightarrow\infty}\frac{\log|X_{\sigma}(t)|}{\log(tk(t))}
=1-\Lambda(|\sigma|), \quad \text{a.s.}
\]
where the random variable $\Lambda(|\sigma|)\rightarrow0$ as $\sigma\rightarrow\infty$ a.s. We also prove a decay result for equations with a superlinear diffusion coefficient at zero. If the deterministic equation has solution which is uniformly asymptotically stable, and the kernel is subexponential, the decay rate of the stochastic problem is exactly the same as that of the underlying deterministic problem.

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