Polynomial Recurrence for SDEs with a Gradient-Type Drift, Revisited

Abstract

In this paper, polynomial recurrence bounds for a class of stochastic differential equations with a rotational symmetric gradient type drift and an additive Wiener process are established, as well as certain a priori moment inequalities for solutions. The key feature of this paper is that the approach does not use Lyapunov functions because it is not clear how to construct them. The method based on Dynkin’s (nonrandom) chain of equations is applied instead. Another key feature is that the asymptotic conditions on the potential near infinity are assumed as inequalities—which allows for more flexibility compared to a single limit at infinity, making it less restrictive.