Polynomial asymptotic stability of damped stochastic differential equations

J. A. D. Appleby, CMDE, School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland
D. Mackey, 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. 2. (2004), pp. 1-33.

Abstract: 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.

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