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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn057, 48 pages, doi:10.1093/imrn/rnn057 published on June 11, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

Tunnel Effect for Kramers–Fokker–Planck Type Operators: Return to Equilibrium and Applications

Frédéric Hérau1, Michael Hitrik2 and Johannes Sjöstrand3

1 Laboratoire de Mathématiques, Université de Reims, Moulin de la Housse B.P. 1039 51687 Reims cedex 2, France and UMR 6056 CNRS
2 Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA
3 CMLS Ecole Polytechnique, 91128 Palaiseau cedex, France and UMR 7640 CNRS

Correspondence: Correspondence to be sent to: hitrik{at}math.ucla.edu

In the first part of this work, we consider second-order supersymmetric differential operators in the semiclassical limit, including the Kramers–Fokker–Planck operator, such that the exponent of the associated Maxwellian {phi} is a Morse function with two local minima and one saddle point. Under suitable additional assumptions of dynamical nature, we establish the {long time} convergence to the equilibrium for the associated heat semigroup, with the rate given by the first nonvanishing, exponentially small, eigenvalue. In the second part of the paper, we consider the case when the function {phi} has precisely one local minimum and one saddle point. We also discuss further examples of supersymmetric operators, including the Witten Laplacian and the infinitesimal generator for the time evolution of a chain of classical anharmonic oscillators.


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