On Representation of Excessive Measures for S-Subordinated Markov Processes

#### M. Hmissi, C. N. Jmailli

2020, v.26, Issue 5, 901-914

ABSTRACT

Let $X$ and $Y$ be two transient Markov processes defined on a common state space such that $Y$ is $\Su$-subordinated to $X$, i.e.\ each $X$-excessive function is $Y$-excessive. Based on a classical representation of excessive measures and a projection result established recently, we prove that, for each $Y$-excessive $m$, there exists a unique $X$-entrance law $(\nu_t)$ such that
$$m P_t \ = \ m_i \ + \ \nu_t V , \qquad t>0$$
where $(P_t)$ is the transition function of $X$, $m_i$ is the $Y$-invariant part of $m$ and $V$ is the potential kernel of $Y$.
Moreover, some applications of this result are derived.

Keywords: Markov process, transition function, resolvent, excessive function, excessive measure, energy functional, $\Su$-subordination, entrance law, Levy process