**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.

Communicated by L. Hatvani.
| Received on 2003-10-06 Appeared on 2004-08-31 |

**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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