Polynomial asymptotic stability of damped stochastic differential equations.

Appleby, John A.D. and Mackey, Dana
(2004)
Polynomial asymptotic stability of damped stochastic differential equations.
In: The 7th Colloquium on the Qualitative Theory of Differential Equations, 14 - 18 July, 2003, Szeged, Hungary.

The paper studies the polynomial convergence of solutions
of a scalar nonlinear Itˆo stochastic differential equation
dX(t) = −f(X(t)) dt + (t) dB(t)
where it is known, a priori, that limt!1 X(t) = 0, a.s. The intensity
of the stochastic perturbation is a deterministic, continuous
and square integrable function, which tends to zero more
quickly than a polynomially decaying function. The function f
obeys limx!0 sgn(x)f(x)/|x| = a, for some > 1, and a > 0. We
study two asymptotic regimes: when tends to zero sufficiently
quickly the polynomial decay rate of solutions is the same as for the
deterministic equation (when 0). When decays more slowly,
a weaker almost sure polynomial upper bound on the decay rate
of solutions is established. Results which establish the necessity
for to decay polynomially in order to guarantee the almost sure
polynomial decay of solutions are also proven.