Quasi-stationary distribution for the Langevin process in cylindrical domains. II: Overdamped limit. (English) Zbl 1491.35017

Summary: Consider the Langevin process, described by a vector (positions and momenta) in \({\mathbb{R}^d}\times{\mathbb{R}^d} \). Let \(\mathcal{O}\) be a \(({\mathcal{C}^2}\) open bounded and connected set of \({\mathbb{R}^d} \). Recent works showed the existence of a unique quasi-stationary distribution (QSD) of the Langevin process on the domain \(D:=\mathcal{O}\times{\mathbb{R}^d}\). In this article, we study the overdamped limit of this QSD, i.e. when the friction coefficient goes to infinity. In particular, we show that the marginal law in position of the overdamped limit is the QSD of the overdamped Langevin process on the domain \(\mathcal{O}\). For Part I, see [T. Lelièvre, Stochastic Processes Appl. 144, 173–201 (2022; Zbl 1481.35294)].


35B25 Singular perturbations in context of PDEs
35R60 PDEs with randomness, stochastic partial differential equations
47B07 Linear operators defined by compactness properties
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics


Zbl 1481.35294
Full Text: DOI arXiv


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