Cooling down stochastic differential equations: almost sure convergence

TL;DR

This paper establishes conditions under which stochastic differential equations with decreasing noise levels almost surely converge, using Lyapunov functions and Lojasiewicz properties, with optimality demonstrated by counterexamples.

Contribution

It provides new almost sure convergence results for SDEs with diminishing noise, extending previous work with optimal conditions and Lyapunov function techniques.

Findings

Almost sure convergence of Lyapunov function values

Gradient convergence to zero under specified decay rates

Optimality of decay rate conditions demonstrated by counterexamples

Abstract

We consider almost sure convergence of the SDE $dX_t=\alpha_t d t + \beta_t d W_t$ under the existence of a $C^2$-Lyapunov function $F:\mathbb R^d \to \mathbb R$. More explicitly, we show that on the event that the process stays local we have almost sure convergence in the Lyapunov function $(F(X_t))$ as well as $\nabla F(X_t)\to 0$, if $|\beta_t|=\mathcal O( t^{-\beta})$ for a $\beta>1/2$. If, additionally, one assumes that $F$ is a Lojasiewicz function, we get almost sure convergence of the process itself, given that $|\beta_t|=\mathcal O(t^{-\beta})$ for a $\beta>1$. The assumptions are shown to be optimal in the sense that there is a divergent counterexample where $|\beta_t|$ is of order $t^{-1}$.