Additive functionals and excursions of Kuznetsov processes (Q2576004)

Let \(B\) be a continuous additive functional for a standard process \((X_t)_{t\in R}\), and let \((Y_t)_{t\in R}\) be a stationary Kuznetsov process with the same semigroup of transitions. The purpose of the paper is to give, without duality hypothesis, the conditional law \(P^{x,y}\) of the excursion straddling an arbitrary random time, given the initial state \(x\) and the final state \(y\), as regular probabilities in terms of the predictable exit measures for some closed random and homogeneous subset \(M\) of \(R_+\), and also for a regenerative system consisting of the closure of the set of times the regular points of an arbitrary continuous additive functional are visited. Also, the conditional laws of pairs of excursions for a Markov process with random birth and death \((Y_t)_{t\in R}\) having the same semigroup as \(X\) are given. In this respect, an ``additive functional'' for \((Y_t)_{t\in R}\) is defined and the result concerning probability measures \(P^{x,y}\) is extended to \(Y\).