The Annals of Applied Probability· 2026Q1
Asymptotic stability and cut-off phenomenon for the underdamped Langevin dynamics
- 1citations
- Q1SCImago
- 2026year
Short summary
A new necessary condition guarantees convergence to a unique attractor for zero-noise Langevin dynamics, proven sharp for linear models, and enables proof of the cut-off phenomenon in the small-noise regime.
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Key points
- Introduced a necessary condition for zero-noise underdamped Langevin dynamics to converge to a unique attractor.
- Demonstrated this condition is sharp for a broad class of linear models.
- Proved the cut-off phenomenon in the small-noise regime, providing precise mixing time asymptotics.
- Developed new methods to handle the degeneracy of the non-elliptic infinitesimal generator.
AI-generated from the title and abstract; the full text is not read.
Abstract
In this article, we provide a detailed analysis of the long-time behavior of the underdamped Langevin dynamics. We first provide a necessary condition guaranteeing that the zero-noise dynamical system converges to its unique attractor. We also observed that this condition is sharp for a large class of linear models. We then prove the so-called cut-off phenomenon in the small-noise regime under this condition. This result provides the precise asymptotics of the mixing time of the process and of the distance between the distribution of the process and its stationary measure. The main difficulty of this work relies on the degeneracy of its infinitesimal generator which is not elliptic, thus requiring a new set of methods.
The authors' abstract, as published at the source. The Annals of Applied Probability, 2026 · DOI ↗
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Field: Statistics and Probability
Statistics and ProbabilityMathematics