Asymptotic Constancy and Stability in Nonautonomous Stochastic Differential Equations

Abstract

This paper considers the asymptotic behaviour of a scalar non-autonomous stochastic differential equation which has zero drift, and whose diffusion term is a product of a function of time and space dependent function, and which has zero as a unique equilibrium solution. We classify the pathwise limiting behaviour of solutions; solution either tends to a non-trivial, non-equilibrium and random limit, or the solution hits zero in finite time. In the first case, the exact rate of decay can always be computed. These results can be inferred from the square integrability of the time dependent factor, and the asymptotic behaviour of the corresponding autonomous stochastic equation, where the time dependent multiplier is unity.