Markov processes with identical hitting probabilities

Abstract:

Let $(X(t),{P^x})$ and $(Y(t),{Q^x})$ be transient Hunt processes on a state space $E$ satisfying the hypothesis of absolute continuity (Meyer’s hypothesis (L)). Let $T(K)$ be the first entrance time into a set $K$, and assume ${P^x}(T(K) < \infty ) = {Q^x}(T(K) < \infty )$ for all compact sets $K \subseteq E$. There exists a strictly increasing continuous additive functional of $X(t),A(t)$, so that if $T(t) = {\text {inf}}\{s:A(s) > t\}$, then $(X(T(t)),{P^x})$ and $(Y(t),{Q^x})$ have the same joint distributions. An analogous result is stated if $X$ and $Y$ are right processes (with an additional hypothesis). These theorems generalize the Blumenthal-Getoor-McKean Theorem and have interpretations in terms of potential theory.