Dirichlet’s and Thomson’s Principles for Non-selfadjoint Elliptic Operators with Application to Non-reversible Metastable Diffusion Processes

Abstract

We present two variational formulae for the capacity in the context of non-selfadjoint elliptic operators. The minimizers of these variational problems are expressed as solutions of boundary-value elliptic equations. We use these principles to provide a sharp estimate for the transition times between two different wells for non-reversible diffusion processes. This estimate permits to describe the metastable behavior of the system.When Ω = D \ B, where B ⊂ D ⊂ R d are smooth domains, and f = 1, 0 on B, D c , respectively, the minimal energy is called the capacity. In electrostatics, it corresponds to the total electric charge on the conductor ∂B held at unit potential and grounded at D c . It is denoted by cap D (B) and can be represented, by the divergence theorem, aswhere h is the harmonic function which solves (1.1), n B is the outward normal vector to ∂B, and σ the surface measure at ∂B.where o (1) → 0 as vanishes. This estimate appears in articles published in the 60's. A rigorous proof was first obtained by Bovier, Eckhoff, Gayrard, and Klein [8] with arguments from potential theory, and right after by Helffer, Klein and Nier [14] through Witten Laplacian METASTABILITY IN NON-REVERSIBLE DIFFUSION PROCESSES 3 analysis. We refer to Berglund [6] and Bouchet and Reygner [7] for an historical overview and further references.Recentlty, Bouchet and Reygner [7] extended the Eyring-Kramers formula to the non-reversible setting. They showed that in this context the negative eigenvalue −λ has to be replaced by the unique negative eigenvalue of (Hess U ) (σ) M.We present below a rigorous proof of this result, based on the variational formulae obtained for the capacity in the first part of the article, and on the approach developed by Bovier, Eckhoff, Gayrard, and Klein [8] in the reversible case. This estimate permits to describe the metastable behavior of the diffusion X t in the small noise limit. Analogous results have been derived for random walks in a potential field in [17,18]. Notation and ResultsWe start by introducing the main assumptions. We frequently refer to [12] for results on elliptic equations and to [11,22] for results on diffusions. 2.1. A Dirichlet's and a Thomson's principle. Fix d ≥ 2, and denote by C k (R d ), 0 ≤ k ≤ ∞, the space of real functions on R d whose partial derivatives up to order k are continuous. Let M m,n , 1 ≤ m, n ≤ d, be functions in C 2 (R d ) for which there exists a finite constant C 0 such that d m,n=1 4 C. LANDIM, M. MARIANI, I. SEO Denote by B(r) ⊂ R d , r > 0, the open ball of radius r centered at the origin, and by ∂B(r) its boundary. We assume that lim n→∞ inf z ∈B(n)

Metastability of Nonreversible Random Walks in a Potential Field and the Eyring‐Kramers Transition Rate Formula

“…We investigate in [17] the metastability behavior of such diffusions. We investigate in [17] the metastability behavior of such diffusions.…”

“…In this context, the matrix M , introduced above (2.10), is the Jacobian of the drift b D A rF .x/ at . We investigate in [17] the metastability behavior of such diffusions.…”

Metastable Markov chains

“…Theorem 14.2 appeared in Gaudillière and Landim [67], and Theorem 14.3 is due to Slowik [118]. Similar Dirichlet and Thomson principles are available in the context of diffusions processes, [90,85]. Remark 14.5.…”

“…We refer to [91,92,93,90] for more details. This model is at the origin of the study of metastability from a dynamical point of view.…”

Exit Event from a Metastable State and Eyring-Kramers Law for the Overdamped Langevin Dynamics

“…Notice that the results presented above do not provide any information concerning the average time it takes for the process (1) to go from the global minimum of f to a local minimum of f when h → 0. One also refers to [44] for generalization of [6,7] for a class of non reversible processes when f has two local minima, and to [11-13, 37, 55] for related results. Remark 6.…”