JournalsjstVol. 7, No. 4pp. 939–984

Small noise spectral gap asymptotics for a large system of nonlinear diffusions

  • Giacomo Di Gesù

    Technische Universität Wien, Austria
  • Dorian Le Peutrec

    Université Paris-Sud, Orsay, France
Small noise spectral gap asymptotics for a large system of nonlinear diffusions cover
Download PDF

A subscription is required to access this article.

Abstract

We study the L2L^2 spectral gap of a large system of strongly coupled diffusions on unbounded state space and subject to a double-well potential. This system can be seen as a spatially discrete approximation of the stochastic Allen–Cahn equation on the one-dimensional torus. We prove upper and lower bounds for the leading term of the spectral gap in the small temperature regime with uniform control in the system size. The upper bound is given by an Eyring–Kramers-type formula. The lower bound is proven to hold also for the logarithmic Sobolev constant. We establish a sufficient condition for the asymptotic optimality of the upper bound and show that this condition is fullled under suitable assumptions on the growth of the system size. Our results can be reformulated in terms of a semiclassical Witten Laplacian in large dimension.

Cite this article

Giacomo Di Gesù, Dorian Le Peutrec, Small noise spectral gap asymptotics for a large system of nonlinear diffusions. J. Spectr. Theory 7 (2017), no. 4, pp. 939–984

DOI 10.4171/JST/182