Jump diffusions and Feynman-Kac transforms

A generic strong Markov process​ may have both the continuous (diffusive) part and the purely discontinuous (jumping) part, and its infinitesimal generator is a mixture of local and non-local operators.

Suppose a strong Markov process  $X$ has a jointly continuous transition density function with two-sided estimates that contain both the Gaussian and the power law components, we will show that under certain Kato class conditions, the heat kernel of a non-local Feynman-Kac semigroup of $X$ has similar two-sided estimates, but with a set of possible different constants.