Repartition of the quasi-stationary distribution and first exit point density for a double-well potential

Let f : R d → R be a smooth function and (Xt) t≥0 be the stochastic process solution to the overdamped Langevin dynamics dXt = −−f (Xt)dt + √ h dBt. Let Ω ⊂ R d be a smooth bounded domain and assume that f | Ω is a double-well 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. 1 Setting and results 1.1 Quasi-stationary distribution and purpose of this work Let (X t) t≥0 be the stochastic process solution to the overdamped Langevin dynamics in R d : dX t = −−f (X t)dt + √ h dB t , (1) where f : R d → R is the potential (chosen C ∞ in all this work), h > 0 is the temperature and (B t) t≥0 is a standard d-dimensional Brownian motion. Let Ω be a C ∞ bounded open and connected subset of R d and introduce τ Ω = inf{t ≥ 0 | X t / ∈ Ω} the first exit time from Ω. A quasi-stationary distribution for the process (1) on Ω is a probability measure µ h on Ω such that, when X 0 ∼ µ h , it holds for any time t > 0 and any Borel set A ⊂ Ω, P(X t ∈ A | t < τ Ω) = µ h (A).