Large-Time Analysis of the Langevin Dynamics for Energies Fulfilling Polyak-Łojasiewicz Conditions

2025/10/28 by Fornasier, Massimo, Sun, Lukang, Ward, Rachel
#Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2510.24925

Abstract

In this work, we take a step towards understanding overdamped Langevin dynamics for the minimization of a general class of objective functions L. We establish well-posedness and regularity of the law ρ_t of the process through novel a priori estimates, and, very importantly, we characterize the large-time behavior of ρ_t under truly minimal assumptions on L. In the case of integrable Gibbs density, the law converges to the normalized Gibbs measure. In the non-integrable case, we prove that the law diffuses. The rate of convergence is O(1/t). Under a Polyak-Lojasiewicz (PL) condition on L, we also derive sharp exponential contractivity results toward the set of global minimizers. Combining these results we provide the first systematic convergence analysis of Langevin dynamics under PL conditions in non-integrable Gibbs settings: a first phase of exponential in time contraction toward the set of minimizers and then a large-time exploration over it with rate O(1/t).

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