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

In this paper, when a given symmetric Markov process X satisfies the stability of global heat kernel two-sided (upper) estimates by Markov perturbations (See Definition 1.2), 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.