Tunnel effect for semiclassical random walk

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 exponentially close to 1 eigenvalues (in the semiclassical sense), and that the other are O(h) 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.