ED Mathématiques et Informatique
Semiclassical hypocoercivity and Eyring-Kramers law for degenerate Fokker-Planck operators
by Loïs DELANDE (IMB - Institut de Mathématiques de Bordeaux)
The defense will take place at 15h00 - Salle de Conférence 351 cours de la Libération, Bâtiment A33 - F 33 405 TALENCE
in front of the jury composed of
- Laurent MICHEL - Professeur des universités - Université de Bordeaux - Directeur de these
- Michael HITRIK - Professor - University of California - Rapporteur
- Luc HILLAIRET - Professeur des universités - Université d'Orléans - Rapporteur
- Jean-Marc BOUCLET - Professeur des universités - Université Paul Sabatier - Examinateur
- Mouez DIMASSI - Professeur des universités - Université de Bordeaux - Examinateur
- Tony LELIEVRE - Professeur - Ecole des Ponts ParisTech - Examinateur
- Boris NECTOUX - Professeur assistant - Université Clermont Auvergne - Examinateur
In this thesis, we investigate certain Fokker-Planck operators and the Witten Laplacian in the low temperature regime, considering potentials that are not necessarily Morse. Our main focus is on the spectral behavior near zero of the associated operators, for which we aim to provide a precise characterization. Such a spectral description allows us to derive detailed insights into the long-time dynamics of the solutions, including quantitative results on return to equilibrium and metastability. We begin with the analysis of the Witten Laplacian, a selfadjoint operator. Our approach involves adapting recent quasimode constructions to our non-Morse setting. Under a generic assumption on the degenerate potential, we successfully derive the desired spectral description. Next, we turn to the Fokker-Planck operator with generalized degenerate coefficients. Here, degeneracy refers to the fact that the microlocal symbol of the operator is no longer locally quadratic. Leveraging the results obtained for the Witten Laplacian, we address the analytical challenges introduced by this degeneracy. Our strategy partly relies on resolvent estimates derived via hypocoercive techniques. As a result, we are once again able to establish an Eyring-Kramers-type formula for the spectrum of this operator.