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

Abstract

We consider three magnetic relativistic Schrödinger 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-Itô 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 Lévy processes $X$.