Cooling down stochastic differential equations: Almost sure convergence

Steffen Dereich and Sebastian Kassing

Stochastic Processes and their Applications, 2022, vol. 152, issue C, 289-311

Abstract: We consider almost sure convergence of the SDE dXt=αtdt+βtdWt under the existence of a C2-Lyapunov function F:Rd→R. More explicitly, we show that on the event that the process stays local we have almost sure convergence in the Lyapunov function (F(Xt))t≥0 as well as ∇F(Xt)→0, if |βt|=O(t−β) for a β>1/2. If, additionally, one assumes that F is a Łojasiewicz-function, we get almost sure convergence of the process itself, given that |βt|=O(t−β) for a β>1. The assumptions are shown to be optimal in the sense that there is a divergent counterexample where |βt| is of order t−1.

Keywords: Stochastic gradient flow; Brownian particle; Cooling down; Łojasiewicz-inequality; Almost sure convergence (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1016/j.spa.2022.06.020

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