Tunnel effect for Kramers-Fokker-Planck type operators: return to equilibrium and applications

Frédéric Hérau (1) , Michael Hitrik (2) , Johannes Sjöstrand (3)

1 LMR - Laboratoire de Mathématiques de Reims
2 Department of Mathematics [UCLA]
3 CMLS - Centre de Mathématiques Laurent Schwartz

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 non-vanishing, 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.

Mots clés: return to equilibrium; heat semigroup; eigenvalue splitting; supersymmetry; Witten Laplacian; Kramers; Fokker-Planck; semiclassical limit

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

Frédéric Hérau, Michael Hitrik, Johannes Sjöstrand. Tunnel effect for Kramers-Fokker-Planck type operators: return to equilibrium and applications. 2008. ⟨hal-00214252⟩