QuAMProcs

The mathematical analysis of metastable processes started 75 years ago with the seminal works of Kramers on Fokker-Planck equation. Although the original motivation of Kramers was to "elucidate some points in the theory of the velocity of chemical reactions", it turns out that Kramers' law is observed to hold in many scientific fields: molecular biology (molecular dynamics), economics (modelization of financial bubbles), climate modeling, etc. Moreover, several widely used efficient numerical methods are justified by the mathematical description of this phenomenon. Recently, the theory has witnessed some spectacular progress thanks to the insight of new tools coming from Spectral and Partial Differential Equations theory. Semiclassical methods together with spectral analysis of Witten Laplacian gave very precise results on reversible processes. From a theoretical point of view, the semiclassical approach allowed to prove a complete asymptotic expansion of the small eigenvalues of Witten Laplacian in various situations (global problems, boundary problems, degenerate diffusions, etc.). The interest in the analysis of boundary problems was rejuvenated by recent works establishing links between the Dirichlet problem on a bounded domain and the analysis of exit event of the domain. These results open numerous perspectives of applications. Recent progress also occurred on the analysis of irreversible processes. Very recently members of the project obtained first results on overdamped Langevin equation in irreversible context showing that Eyring-Kramers law governs the dynamics. Concerning the full (inertial) Langevin equation, a breakthrough was obtained on the analysis of Kramer-Fokker-Planck equation, using various technics : microlocal analysis, complex analysis, super-symmetry. A series of works involving two members of the consortium allowed to prove new properties of the Kramer-Fokker-Planck equation : hypoelliptic estimates, resolvent estimates, asymptotic limits of « small » eigenvalues. In the end, these works constitute the first proof that Eyring-Kramers law governs the long time evolution of the full Langevin equation. The above progresses pave the way for several research tracks motivating our project. First of all, we aim at generalizing the recent results on overdamped Langevin equations to degenerate situations (several type of « degenerate » situations exist, some of them having been studied recently by members of the project). The recent results on boundary problems should also be refined and extended in order to treat the case when the domain under consideration is the basin of attraction of the determinist dynamics (which is the most interesting case in practice). Altough this problem raises serious difficulties, it seems to be accessible with recently developed mathematical tools. Concerning irreversible processes, the variety of open problems is considerable. To quote a few, non-reversible overdamped Langevin equations (with or without boundary conditions), the Kramers-Fokker-Planck equation on bounded domains, the case of non local collision kernels, etc. Questions related to Metropolis algorithms will also be investigated. To summarize, the aim of this project is to further develop the mathematical analysis of metastable phenomena, by mixing viewpoint from probability and partial differential equation theory. The efforts of the team will be concentrated on the challenging cases of non-reversible processes and non-linear dynamics. A particular attention will also be paid to computational aspects.