Isomorphism theorems, extended Markov processes and random interlacements

Abstract

Several questions concerning the Gaussian free field on ${\mathbb{Z}}^d$ ($d\ge 3)$ are solved thanks to a Dynkin-type isomorphism theorem established by Sznitman [29]. This isomorphism theorem relates the Gaussian free field to random interlacements and has the same spirit as the generalized second Ray-Knight theorem [11]. We show here that this isomorphism theorem is actually the generalized second Ray-Knight theorem written for a Markov process which is an extension of the continuous time simple random walk on ${\mathbb{Z}}^d$. As a result, the occupation times of random interlacements are the local time processes of this extended Markov process. More generally, for any given transient Markov process ${(X_t)}_{t\ge 0}$ with an unbounded state space and finite symmetric 0-potential densities, we construct an extended Markov process ${(Y_t)}_{t\ge 0}$ with a recurrent point. The generalized second Ray-Knight theorem applied to ${(Y_t)}_{t\ge 0}$ leads to an identity connecting the Gaussian free field associated to ${(X_t)}_{t\ge 0}$ to the local time process of ${(Y_t)}_{t\ge 0}$. Besides symmetry is not required from a transient Markov process to admit an extended Markov process with a recurrent point. Given a transient Markov process, we explore the connections between its associated Kuznetsov processes, its quasi-processes, its extended Markov process and its random interlacements.