On the Feynman--Kac semigroup for some Markov processes

TL;DR

This paper investigates the Feynman--Kac semigroup for certain Markov processes, establishing the existence and bounds of the associated transition density under specific conditions, with illustrative examples.

Contribution

It proves the existence of the density for the Feynman--Kac semigroup and derives explicit upper and lower bounds under transition density assumptions.

Findings

Existence of the density $p_t^A(x,y)$ for the Feynman--Kac semigroup.

Explicit upper and lower bounds for the density.

Application to specific examples of Markov processes.

Abstract

For a (non-symmetric) strong Markov process $X$, consider the Feynman--Kac semigroup \[T_t^Af(x):=\mathbb {E}^x\bigl[e^{A_t}f(X_t)\bigr],\quad x\in {\mathbb {R}^n}, t>0,\] where $A$ is a continuous additive functional of $X$ associated with some signed measure. Under the assumption that $X$ admits a transition probability density that possesses upper and lower bounds of certain type, we show that the kernel corresponding to $T_t^A$ possesses the density $p_t^A(x,y)$ with respect to the Lebesgue measure and construct upper and lower bounds for $p_t^A(x,y)$. Some examples are provided.