Sharp spectral asymptotics for nonreversible metastable diffusion processes

Abstract

Let U h : ‫ޒ‬ d → ‫ޒ‬ d be a smooth vector field and consider the associated overdamped Langevin equation d X t = −U h (X t)dt + √ 2hd B t in the low temperature regime h → 0. In this work, we study the spectrum of the associated diffusion L = −h + U h • ∇ under the assumptions that U h = U 0 + hν, where the vector fields U 0 : ‫ޒ‬ d → ‫ޒ‬ d and ν : ‫ޒ‬ d → ‫ޒ‬ d are independent of h ∈ (0, 1], and that the dynamics admits e − V h d x as an invariant measure for some smooth function V : ‫ޒ‬ d → ‫.ޒ‬ Assuming additionally that V is a Morse function admitting n 0 local minima, we prove that there exists > 0 such that in the limit h → 0, L admits exactly n 0 eigenvalues in the strip {0 ≤ Re(z) < }, 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. 12 2. General spectral estimates 17 2A. Proof of Proposition 1.1 17 2B. Spectral analysis near the origin 20 3. Geometric preparation 25 4. Spectral analysis in the case of Morse functions 29 4A. Construction of accurate quasimodes 29 4B. Quasimodal estimates 33 4C. Proof of Theorem 1.9 37 4D. Proof of Theorem 1.11 42 Appendix: Some results in linear algebra 45 Acknowledgements. 52 References 52

Metastable diffusions with degenerate drifts

“…In particular, this assumption implies that f uniquely attains its global minimum at m ∈ U (0) . This assumption is a generalization of [22, Hypothesis 5.1] (see also [25]). By smooth perturbation of the function f , one sees that it is generically satisfied.…”

“…On the other hand, using Lemma 5.3 and proceeding in the same way as in the proofs of [5, Proposition 5.1(i)] and [25,Lemma 4.7] respectively, we also have for any j, k ∈ {1, . .…”

“…In this part, we construct Gaussian quasimodes near the separating saddle manifolds. Such constructions go back to [6,8,25]. Given a separating saddle manifold Γ ∈ U…”

Untitled

“…This allowed them in particular to justify the formal result of Bouchet and Reygner for a particular class of systems whose invariant measure is known explicitly. See also [Le Peutrec and Michel 2020] for semiclassical results on nonreversible diffusions with known invariant measure. In a different direction, a reactive path theory for multidimensional, nonreversible diffusions was developed by Lu and Nolen [2015], based on ideas by E and Vanden-Eijnden [2006].…”

Energy Landscape and Metastability of Stochastic Ising and Potts Models on Three-dimensional Lattices Without External Fields

“…Metastability is a ubiquitous phenomenon that arises when a stochastic system possesses several locally stable sets; it is observed in a wide class of models, such as in the small random perturbations of dynamical systems (e.g., [12,13,20,30,34,[37][38][39]48]), interacting particle systems consisting of sticky particles (e.g., [5,7,22,23,28,29,47,49]) and spin systems in the low-temperature regime (e.g., [1,6,[8][9][10][11]14,15,19,26,27,32,36,42,44,45]). Numerous important works are not listed here, and we refer to the references of the monographs [9,46], which provide a comprehensive introduction to this sizable topic.…”