Cooling down stochastic differential equations: almost sure convergence

Abstract

We consider almost sure convergence of the SDE $d{X}_t={\alpha }_{t}dt+{\beta }_{t}d{W}_t$ 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 $t^{-1}$.

References in corpus (2)

• Mathematics of Deep Learning
• The complex chemical Langevin equation

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