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

We study a semiclassical random walk with respect to a probability measure with a finite number n0 of wells. We show that the associated operator has exactly n0 eigenvalues exponentially close to 1 (in the semiclassical sense), and that the others 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.

analysis of PDEs, probability, spectral theory
Mathematical Subject Classification 2010
Primary: 35S05, 35P15, 47A10, 60J05
Received: 22 January 2014
Revised: 9 January 2015
Accepted: 9 February 2015
Published: 10 May 2015
Jean-François Bony
Institut Mathématiques de Bordeaux
Université de Bordeaux
351, cours de la Libération
33405 Talence
Frédéric Hérau
Laboratoire de Mathématiques Jean Leray
Université de Nantes
2, rue de la Houssinière
44322 Nantes
Laurent Michel
Laboratoire J. A. Dieudonné
Université de Nice
Parc Valrose
06000 Nice