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

TL;DR

This paper studies the long-term behavior and compactness of killed Feynman-Kac semigroups associated with various processes in statistical physics, especially those involving singular Schr{"o}dinger potentials.

Contribution

It extends analysis to a broad class of processes linked to quantum mechanics and Schr{"o}dinger operators, using advanced spectral radius bounds and Perron-Frobenius theorems.

Findings

Established conditions for compactness of semigroups.

Derived bounds on the essential spectral radius.

Applied results to processes like Lévy and Langevin.

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.