We consider the first exit point distribution from a bounded domain Ω of the stochastic process (Xt)t≥0 solution to the overdamped Langevin dynamics dXt=−∇f(Xt)dt+hdBt starting from the quasi-stationary distribution in Ω. In the small temperature regime (h→0) and under rather general assumptions on f (in particular, f may have several critical points in Ω), it is proven that the support of the distribution of the first exit point concentrates on some points realizing the minimum of f on ∂Ω. Some estimates on the relative likelihood of these points are provided. The proof relies on tools from semi-classical analysis.
The exit from a metastable state: Concentration of the exit point distribution on the low energy saddle points, part 1 / Di Gesù, G. F.; Lelievre, T.; Le Peutrec, D.; Nectoux, B.. - In: JOURNAL DE MATHÉMATIQUES PURES ET APPLIQUÉES. - ISSN 0021-7824. - 138:(2020), pp. 242-306. [10.1016/j.matpur.2019.06.003]
The exit from a metastable state: Concentration of the exit point distribution on the low energy saddle points, part 1
Di Gesù G. F.;
2020
Abstract
We consider the first exit point distribution from a bounded domain Ω of the stochastic process (Xt)t≥0 solution to the overdamped Langevin dynamics dXt=−∇f(Xt)dt+hdBt starting from the quasi-stationary distribution in Ω. In the small temperature regime (h→0) and under rather general assumptions on f (in particular, f may have several critical points in Ω), it is proven that the support of the distribution of the first exit point concentrates on some points realizing the minimum of f on ∂Ω. Some estimates on the relative likelihood of these points are provided. The proof relies on tools from semi-classical analysis.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.