#### Vol. 8, No. 2, 2015

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Tunnel effect for semiclassical random walks

### Jean-François Bony, Frédéric Hérau and Laurent Michel

Vol. 8 (2015), No. 2, 289–332
##### 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 $\mathsc{O}\left(h\right)$ 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.

##### Keywords
analysis of PDEs, probability, spectral theory
##### Mathematical Subject Classification 2010
Primary: 35S05, 35P15, 47A10, 60J05