Tunnel effect for semiclassical random walks

Abstract

We study a semiclassical random walk with respect to a probability measure with a finite number $n_0$ of wells. We show that the associated operator has exactly $n_0$ eigenvalues exponentially close to $1$ (in the semiclassical sense), and that the others are $\mathcal{O}\left(h\right)$ away from $1$. We also give an asymptotic of these small eigenvalues. The key ingredient in our approach is a general factorization result of pseudodifferential operators, which allows us to use recent results on the Witten Laplacian.