The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 2

Abstract

We consider the first exit point distribution from a bounded domain Ω of the stochastic process (X t ) t≥0 solution to the overdamped Langevin dynamicsstarting from the quasi-stationary distribution in Ω. In the small temperature regime (h → 0) and under rather general assumptions on f (in particular, f may have several critical points in Ω), it is proven that the support of the distribution of the first exit point concentrates on some points realizing the minimum of f on ∂Ω. Some estimates on the relative likelihood of these points are provided. The proof relies on tools from semi-classical analysis.r é s u m é Dans ce travail, nous étudions la distribution du point de sortie d'un domaine borné Ω pour le processus stochastique (X t ) t≥0 solution de la dynamique de Langevin suramortieinitialement distribué suivant la distribution quasi-stationnaire dans Ω. Dans la limite basse température h → 0 et sous des hypothèses générales sur la fonction f (f pouvant notamment avoir plusieurs points critiques dans Ω), nous montrons que la distribution du lieu de sortie se concentre sur certains points réalisant le minimum de f sur ∂Ω. Nous calculons aussi les probabilités relatives de sortir autour de chacun de ces points. Nos preuves reposent sur des outils issus de l'analyse semi-classique.

Eyring-Kramers exit rates for the overdamped Langevin dynamics: the case with saddle points on the boundary

“…, z n0 }. Moreover, Equations ( 35) and (36) extend the results of [33,Theorem 2.1] and [21,22,51,52,62,75] to the case when f has critical points on ∂Ω. Let us however acknowledge that even if the techniques mentioned above seem inherently limited to h-log limits, some of them are robust enough to apply to non reversible elliptic processes, or quasilinear parabolic equations (see [19-22, 33, 47, 48]), whereas we only consider reversible dynamics here.…”

“…Using the procedure of step 2 of the proof of [62,Proposition 14] and since the domain of attraction A({f < min ∂Ω f }) of {f < min ∂Ω f } for the dynamics (12) (see [62,Section 1.2.2] for the definition of A({f < min ∂Ω f })) is equal to Ω (by item 1 of (Ω-f )), (35) and (36) extends to all x ∈ K, K a compact set of Ω. It remains to prove (38).…”

Repartition of the Quasi-stationary Distribution and First Exit Point Density for a Double-Well Potential

“…41-42 there). Notice lastly that (21) shows that some tunneling effect of order √ h appears nevertheless when ∂C 1 ∩ ∂C 2 = ∅ (see indeed (19)), contrary to the case ∂C 1 ∩ ∂C 2 = ∅ when α 1 (h) and α 2 (h) do not have the same asymptotic expansion, see (18). As expected, when ∂C 1 ∩ ∂C 2 = ∅, the independence between the two wells in this case is hence generically weaker.…”

“…, z 1,n1 } = ∂C 1 ∩ ∂Ω with an explicit repartition given by (25). Adapting the proof of [6, Proposition 11] (see also [18]) by using ( 20) and ( 21), one can also show that when X 0 = x ∈ C 1 , the law of X τΩ concentrates when h → 0 on {z 1,1 , . .…”

“…We show in particular that ν h generically concentrates in one well (see Theorem 1 below) but can also concentrate in both wells when the function f is (nearly) even (see Theorems 2 and 3 below). According to the analysis led in [7] (see also the preprint [6] which concatenates the results of [7] and of [18]), the second phenomenon can only appear when the potential function f admits degenerate deepest barriers. It is particularly unstable (see Remark 4 below) and arises from a strong tunneling effect between the wells.…”