Vol. 1, No. 1, 2020

Download this article
Download this article For screen
For printing
Recent Issues
Volume 5, Issue 2
Volume 5, Issue 1
Volume 4, Issue 4
Volume 4, Issue 3
Volume 4, Issue 2
Volume 4, Issue 1
Volume 3, Issue 4
Volume 3, Issue 3
Volume 3, Issue 2
Volume 3, Issue 1
Volume 2, Issue 4
Volume 2, Issue 3
Volume 2, Issue 2
Volume 2, Issue 1
Volume 1, Issue 1
The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
ISSN (electronic): 2690-1005
ISSN (print): 2690-0998
Author Index
To Appear
Other MSP Journals
Sharp spectral asymptotics for nonreversible metastable diffusion processes

Dorian Le Peutrec and Laurent Michel

Vol. 1 (2020), No. 1, 3–53

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.

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