Some Formulas for the Energy Functional of a Markov Process

In this paper we shall establish two formulas relating the energy functional of a Markov process to that of a subprocess. Let X be a right Markov process and M an exact multiplicative functional of X. Writing (X, M) for the corresponding subprocess, let L and LM denote the energy functional of X and (X, M) respectively. Suppose that M doesn’t vanish on [0,ζ[, and define an additive functional A by dA _t = -dM _t /M _t- . Then given an X-excessive measure ξ and an X-excessive function u we have (1.1)$$ {L^M}\left( {\xi, u} \right) = L\left( {\xi, u} \right) + {v^{\xi }}(u) $$ where υξ is the Revuz measure of A relative to X and ξ. Formula (1.1) appears as (3.27) in [GSt] in the special case M _t = e-qt.