Stability of estimates for fundamental solutions under Feynman-Kac perturbations for symmetric Markov processes

TL;DR

This paper establishes necessary and sufficient conditions for the stability of fundamental solutions under Feynman-Kac perturbations in symmetric Markov processes, extending known results to broader classes of measures.

Contribution

It provides a comprehensive criterion linking heat kernel estimate stability to Feynman-Kac semigroup fundamental solutions, generalizing previous results.

Findings

Necessary and sufficient condition for stability of fundamental solutions

Weak type global estimates under Kato class conditions

Extension of stability results to broader measure classes

Abstract

In this paper, when a given symmetric Markov process X satisfies the stability of global heat kernel two-sided (upper) estimates by Markov perturbations, we give a necessary and sufficient condition on the stability of global two-sided (upper) estimates for fundamental solution of Feynman-Kac semigroup of X. As a corollary, under the same assumptions, a weak type global two-sided (upper) estimates holds for the fundamental solution of Feynman-Kac semigroup with (extended) Kato class conditions for measures. This generalizes all known results on the stability of global integral kernel estimates by symmetric Feynman-Kac perturbations with Kato class conditions in the framework of symmetric Markov processes.