A Probabilistic Approach to the Zero-Mass Limit Problem for Three Magnetic Relativistic Schrodinger Heat Semigroups

TL;DR

This paper investigates the zero-mass limit of three magnetic relativistic Schrödinger heat semigroups, proving their convergence using path integral representations and limit theorems for semimartingales as the mass parameter approaches zero.

Contribution

It introduces a probabilistic framework to analyze the zero-mass limit for magnetic relativistic Schrödinger operators, establishing convergence via path integral and semimartingale techniques.

Findings

Proved convergence of heat semigroups as mass parameter tends to zero.

Established a limit theorem for exponentials of semimartingales.

Connected the zero-mass limit to Lévy process functionals.

Abstract

We consider three magnetic relativistic Schr\"odinger operators which correspond to the same classical symbol $\sqrt{(\xi-A(x))^2+m^2}+V(x)$ and whose heat semigroups admit the Feynman-Kac-It\^o type path integral representation $E[e^{-S^m(x,t, X)}g(x+X(t))]$. Using these representations, we prove the convergence of these heat semigroups when the mass--parameter $m$ goes to zero. Its proof reduces to the convergence of $e^{-S^m(x,t;X)}$, which yields a limit theorem for exponentials of semimartingales as functionals of L\'evy processes $X$.