Feynman-Kac Formulas for Polaron Models & the Ultraviolet Problem

Feynman-Kac formulas provide probabilistic representations of the semigroup generated by a selfadjoint operator, i.e., the solution of the corresponding heat equation. We first review this concept in the traditional setting for Schrödinger-type operators. Then, we discuss how to derive Feynman-Kac formulas for a class of models with operator-valued potentials, describing a non-relativistic particle linearly coupled to a bosonic quantum field. This class contains ultraviolet regularized versions of well-known models like the spin boson model, the Fröhlich polaron and the Nelson model.

We then discuss how to remove the ultraviolet regularization, by an application of stochastic calculus. If time allows, we present some recent applications of the so-obtained formulas. The talk is based on joint works with Daniel Fröhlich, Fumio Hiroshima and Oliver Matte.

This initiative is part of the "PhD Lectures" activity of the project "Departments of Excellence 2023-2027" of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to PhD students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.