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

Abstract

Let f : R d → R be a smooth function and (Xt) t≥0 be the stochastic process solution to the overdamped Langevin dynamicsLet Ω ⊂ R d be a smooth bounded domain and assume that f | Ω is a doublewell potential with degenerate barriers. In this work, we study in the small temperature regime, i.e. when h → 0 + , the asymptotic repartition of the quasi-stationary distribution of (Xt) t≥0 in Ω within the two wells of f | Ω . We show that this distribution generically concentrates in precisely one well of f | Ω when h → 0 + but can nevertheless concentrate in both wells when f | Ω admits sufficient symmetries. This phenomenon corresponds to the so-called tunneling effect in semiclassical analysis. We also investigate in this setting the asymptotic behaviour when h → 0 + of the first exit point distribution from Ω of (Xt) t≥0 when X0 is distributed according to the quasi-stationary distribution.

The exit from a metastable state: Concentration of the exit point distribution on the low energy saddle points, part 1

“…It is difficult to measure the quality of the approximation of u h by projecting an ansatz on Span(u h ), since the error is related to the ratio of λ h over λ 2,h (see Lemma 28). Moreover, when (A1) is not satisfied, it is difficult to predict in which well ν h concentrates when it does, as explained in [31]. This is again due to the fact that this prediction relies on a very accurate comparison between λ h and λ 2,h .…”

“…In the symmetric case depicted in Figure 3 the quasi-stationary distribution ν h concentrates in the two wells (z 1 , z) and (z, z 2 ) (see [31]): i.e. for any a 1 < b 1 such that (a 1 , b 1 ) ⊂ (z 1 , z) and x 1 ∈ (a 1 , b 1 ), and a 2 < b 2 such that (a 2 , b 2 ) ⊂ (z, z 2 ) and…”

“…However, it is proved in [31], that this equal repartition of ν h when h → 0 is unstable with respect to perturbations. Indeed, changing a little bit the value of the determinant of the hessian matrix at x 1 or x 2 , or the normal derivative at z 1 or z 2 of the symmetric potential f depicted in Figure 3 (while keeping the fact that (A1) is not satisfied) makes ν h concentrates in the limit h → 0 in only one of the two wells (z 1 , z) or (z, z 2 ), and [P1] and [P2] then also hold.…”

“…Remark 12. The main goal of[31] is to study the repartition of ν h when h → 0 in the double-well case (where (A1) does not hold). In particular, in this case, it is shown that the asymptotic behaviour when h → 0 of ν h which generically happens is the following: ν h concentrates in only one of the two wells in the limit h → 0, and [P1] and [P2] hold.When (A1) is not satisfied, the analysis of the repartition of ν h is tricky.…”

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

“…The asymptotic estimate (45) is a consequence of the fact that f is an even function (see [23,Section 1]). The asymptotic estimates (46) and (47) are proved exactly as Proposition 5, see [23,Section 1]. Let us also mention that Proposition 6 is a consequence of the results [47].…”