Feynman-Kac formula for Levy processes and semiclassical (Euclidean) momentum representation

TL;DR

This paper extends the Feynman-Kac formula to Levy processes, analyzing the semiclassical limit of quantum systems with Levy-type potentials using large deviation techniques, and providing asymptotic results in both configuration and momentum spaces.

Contribution

It introduces a Feynman-Kac formula for Levy processes and explores the semiclassical limit of quantum systems with Levy potentials, including detailed asymptotics.

Findings

Limiting behavior matches new aspects of semiclassical quantum mechanics.

Provides asymptotic formulas for Levy process contributions.

Demonstrates the formula with non-trivial Levy process examples.

Abstract

We prove a version of the Feynman-Kac formula for Levy processes and integro-differential operators, with application to the momentum representation of suitable quantum (Euclidean) systems whose Hamiltonians involve L\'{e}vy-type potentials. Large deviation techniques are used to obtain the limiting behavior of the systems as the Planck constant approaches zero. It turns out that the limiting behavior coincides with fresh aspects of the semiclassical limit of (Euclidean) quantum mechanics. Non-trivial examples of Levy processes are considered as illustrations and precise asymptotics are given for the terms in both configuration and momentum representations.