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In the first part of this work, we consider second order supersymmetricdifferential operators in the semiclassical limit, including theKramers-Fokker-Planck operator, such that the exponent of the associatedMaxwellian $\phi$ is a Morse function with two local minima and one saddlepoint. Under suitable additional assumptions of dynamical nature, we establishthe 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 thefunction $\phi$ has precisely one local minimum and one saddle point. We alsodiscuss further examples of supersymmetric operators, including the WittenLaplacian and the infinitesimal generator for the time evolution of a chain ofclassical anharmonic oscillators.

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