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

Let f : R d $\rightarrow$ R be a smooth function and (Xt) t$\ge$0 be the stochastic process solution to the overdamped Langevin dynamics dXt = ----f (Xt)dt + \sqrt h dBt. Let $\Omega$ $\subset$ R d be a smooth bounded domain and assume that f | $\Omega$ is a double-well potential with degenerate barriers. In this work, we study in the small temperature regime, i.e. when h $\rightarrow$ 0 + , the asymptotic repartition of the quasi-stationary distribution of (Xt) t$\ge$0 in $\Omega$ within the two wells of f | $\Omega$. We show that this distribution generically concentrates in precisely one well of f | $\Omega$ when h $\rightarrow$ 0 + but can nevertheless concentrate in both wells when f | $\Omega$ 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 $\rightarrow$ 0 + of the first exit point distribution from $\Omega$ of (Xt) t$\ge$0 when X0 is distributed according to the quasi-stationary distribution. 1 Setting and results 1.1 Quasi-stationary distribution and purpose of this work Let (X t) t$\ge$0 be the stochastic process solution to the overdamped Langevin dynamics in R d : dX t = ----f (X t)dt + \sqrt h dB t , (1) where f : R d $\rightarrow$ R is the potential (chosen C $\infty$ in all this work), h > 0 is the temperature and (B t) t$\ge$0 is a standard d-dimensional Brownian motion. Let $\Omega$ be a C $\infty$ bounded open and connected subset of R d and introduce $\tau$ $\Omega$ = inf{t $\ge$ 0 | X t / $\in$ $\Omega$} the first exit time from $\Omega$. A quasi-stationary distribution for the process (1) on $\Omega$ is a probability measure $\mu$ h on $\Omega$ such that, when X 0 $\sim$ $\mu$ h , it holds for any time t > 0 and any Borel set A $\subset$ $\Omega$, P(X t $\in$ A | t < $\tau$ $\Omega$) = $\mu$ h (A).