Almost sure subexponential decay rates of scalar Ito-Volterra equations.

Appleby, John A.D.
(2004)
Almost sure subexponential decay rates of scalar Ito-Volterra equations.
In: 7th Colloquim on the Qualitative Theory of Differential Equations, 14-18 July, 2003, Szeged, Hungary.

The paper studies the subexponential convergence of
solutions of scalar Itˆo-Volterra equations. First, we consider linear
equations with an instantaneous multiplicative noise term with
intensity . If the kernel obeys
lim
t!1
k0(t)/k(t) = 0,
and another nonexponential decay criterion, and the solution X
tends to zero as t ! 1, then
limsup
t!1
log |X(t)|
log(tk(t))
= 1 − (||), a.s.
where the random variable (||) ! 0 as ! 1 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.

Item Type:

Conference or Workshop Item (Paper)

Event Type:

Conference

Refereed:

Yes

Uncontrolled Keywords:

almost sure exponential asymptotic stability; Itˆo-
Volterra equations;