Asymptotics for scaled Kramers–Smoluchowski equations in several dimensions with general potentials

Abstract

In this paper, we generalize the results of Evans and Tabrizian [3], by deriving asymptotics for the time-rescaled Kramers-Smoluchowski equations, in the case of a general non-symmetric potential function with multiple wells. The asymptotic limit is described by a system of reaction-diffusion equations whose coefficients are determined by the Kramers constants at the saddle points of the potential function and the Hessians of the potential function at global minima.2010 Mathematics Subject Classification. 35K15, 35K57, 35J20.

Metastability of one-dimensional, non-reversible diffusions with periodic boundary conditions

“…Proposition 9.1). This sort of analysis has been carried out in [6,18] for reversible diffusions based on ideas from PDEs.…”

“…By Lemma 7.2, since S E (τ ) is a stopping time for the filtration {F t } and since X(S E (t)) belongs to E for all t, 7.6 Uniqueness of limit points. The proof of the uniqueness of limit points of the sequence Q θ relies on a PDE approach to metastability [6,18].…”

“…We present in this article an entirely new approach inspired by results in partial differential equations obtained by Evans, Tabrizian and Seo, Tabrizian [6,18]. Here is the idea.…”

Metastable Markov chains

“…. Properties (A1), (A2) have been proved in [57,117] for elliptic operators on R d of the form L N f = e N V ∇ · (e −N V a∇f ) and in [94] for one-dimensional diffusions with periodic boundary conditions. It is an open problem to prove these conditions in the context of interacting particle systems, say for condensing zero-range processes.…”

Variational Formulae for the Capacity Induced by Second-Order Elliptic Differential Operators

“…Properties (P1), (P2) have been proved in [13,35] for elliptic operators on R d of the form L ǫ f = e V /ǫ ∇ · (e −V /ǫ a∇f ) and in [29] for one-dimensional diffusions with periodic boundary conditions. It is an open problem to prove these conditions in the context of interacting particle systems.…”