Vol. 1, No. 1, 2020

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Sharp spectral asymptotics for nonreversible metastable diffusion processes

Dorian Le Peutrec and Laurent Michel

Vol. 1 (2020), No. 1, 3–53
Abstract

Let Uh : d d be a smooth vector field and consider the associated overdamped Langevin equation

dXt = Uh(Xt)dt + 2hdBt

in the low temperature regime h 0. In this work, we study the spectrum of the associated diffusion L = hΔ + Uh under the assumptions that Uh = U0 + hν, where the vector fields

U0 : d d andν : d d

are independent of h (0,1], and that the dynamics admits eV h dx as an invariant measure for some smooth function V : d . Assuming additionally that V is a Morse function admitting n0 local minima, we prove that there exists 𝜖 > 0 such that in the limit h 0, L admits exactly n0 eigenvalues in the strip {0 Re(z) < 𝜖}, which have moreover exponentially small moduli. Under a generic assumption on the potential barriers of the Morse function V , we also prove that the asymptotic behaviors of these small eigenvalues are given by Eyring–Kramers type formulas.

Keywords
nonreversible overdamped Langevin dynamics, metastability, spectral theory, semiclassical analysis, Eyring–Kramers formulas
Mathematical Subject Classification 2010
Primary: 35P15, 35Q82, 60J60, 81Q12, 81Q20
Milestones
Received: 26 September 2019
Revised: 24 August 2020
Accepted: 15 September 2020
Published: 16 November 2020
Authors
Dorian Le Peutrec
Institut Denis Poisson
Université d’Orléans
Orléans
France
Laurent Michel
Institut Mathématiques de Bordeaux
Université de Bordeaux
Talence
France