Work

Sharp spectral asymptotics for nonreversible metastable diffusion processes

Year: 2020

Type: article

Abstract: Let $U_h:\mathbb R^{d}\to \mathbb R^{d}$ be a smooth vector field and consider the associated overdamped Langevin equation $$dX_t=-U_h(X_t)\,dt+\sqrt{2h}\,dB_t$$ in the low temperature regime $h\rightarrow 0$. In this work, we study the spectrum of the associated diffusion $L=-hΔ+U_h\cdot\nabla$ under the assumptions that $U_h=U_{0}+hν$, where the vector fields $U_{0}:\mathbb R^{d}\to \mathbb R^{d}$ and $ν:\mathbb R^{d}\to \mathbb R^{d}$ are independent of $h\in(0,1]$, and that the dynamics admits $e^{-\frac Vh}$ as an invariant measure for some smooth function $V:\mathbb{R}^d\rightarrow\mathbb{R}$. Assuming additionally that $V$ is a Morse function admitting $n_0$ local minima, we prove that there exists $ε>0$ such that in the limit $h\to 0$, $L$ admits exactly $n_0$ eigenvalues in the strip $\{0\leq \operatorname{Re}(z)< ε\}$, 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.

Source: Probability and Mathematical Physics

DOI: 10.2140/pmp.2020.1.3

Authors: Dorian Le Peutrec, Laurent Michel

Institutions: Université d'Orléans, Institut Denis Poisson, Université de Bordeaux, Institut de Mathématiques de Bordeaux

Language: English

Topic: Spectral Theory in Mathematical Physics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funder: Agence Nationale de la Recherche

Award: Quantitative Analysis of Metastable Processes