Tunnel effect for semiclassical random walks

Jean-François Bony (1) , Frédéric Hérau (2) , Laurent Michel (3)

1 IMB - Institut de Mathématiques de Bordeaux
2 LMJL - Laboratoire de Mathématiques Jean Leray
3 LJAD - Laboratoire Jean Alexandre Dieudonné

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.

Mots clés: Semiclassical analysis; Spectral theory; Probability

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]; Théorie spectrale [math.SP]

Jean-François Bony, Frédéric Hérau, Laurent Michel. Tunnel effect for semiclassical random walks. Analysis & PDE, 2015, 8 (2), pp.289-332. ⟨10.2140/apde.2015.8.289⟩. ⟨hal-00927089⟩