Long time behavior of killed Feynman-Kac semigroups with singular Schr{\"o}dinger potentials

Abstract

In this work, we investigate the compactness and the long time behavior of killed Feynman-Kac semigroups of various processes arising from statistical physics with very general singular Schr{\"o}dinger potentials. The processes we consider cover a large class of processes used in statistical physics, with strong links with quantum mechanics and (local or not) Schr{\"o}dinger operators (including e.g. fractional Laplacians). For instance we consider solutions to elliptic differential equations, L{\'e}vy processes, the kinetic Langevin process with locally Lipschitz gradient fields, and systems of interacting L{\'e}vy particles. Our analysis relies on a Perron-Frobenius type theorem derived in a previous work [A. Guillin, B. Nectoux, L. Wu, 2020 J. Eur. Math. Soc.] for Feller kernels and on the tools introduced in [L. Wu, 2004, Probab. Theory Relat. Fields] to compute bounds on the essential spectral radius of a bounded nonnegative kernel.