Polynomial asymptotic stability of damped stochastic differential equations

• Main Authors: John Appleby, D. Mackey
• Format: Article
• Language: English
• Published: University of Szeged 2004-08-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=189

The paper studies the polynomial convergence of solutions of a scalar nonlinear It\^{o} stochastic differential equation\[dX(t) = -f(X(t))\,dt + \sigma(t)\,dB(t)\] where it is known, {\it a priori}, that $\lim_{t\rightarrow\infty} X(t)=0$, a.s. The intensity of the stochastic perturbation $\sigma$ is a deterministic, continuous and square integrable function, which tends to zero more quickly than a polynomially decaying function. The function $f$ obeys $\lim_{x\rightarrow 0}\mbox{sgn}(x)f(x)/|x|^\beta = a$, for some $\beta>1$, and $a>0$.We study two asymptotic regimes: when $\sigma$ tends to zero sufficiently quickly the polynomial decay rate of solutions is the same as for the deterministic equation (when $\sigma\equiv0$). When $\sigma$ decays more slowly, a weaker almost sure polynomial upper bound on the decay rate of solutions is established. Results which establish the necessity for $\sigma$ to decay polynomially in order to guarantee the almost sure polynomial decay of solutions are also proven.