Markov processes with identical hitting probabilities

Abstract

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