Isomorphism theorems, extended Markov processes and random interlacements

Nathalie Eisenbaum (1) , Haya Kaspi

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

Several questions concerning the Gaussian free field on Z d (d ≥ 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 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≥0 with an unbounded state space and finite symmetric 0-potential densities, we construct an extended Markov process (Y t) t≥0 with a recurrent point. The generalized second Ray-Knight theorem applied to (Y t) t≥0 leads to an identity connecting the Gaussian free field associated to (X t) t≥0 to the local time process of (Y t) t≥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.

Mots clés: Markov process; excessive measure; local time; Gaussian free fields; isomorphism theorem; random interlacements; Kuznetsov process; quasi-process

Domaines: Mathématiques [math]

Nathalie Eisenbaum, Haya Kaspi. Isomorphism theorems, extended Markov processes and random interlacements. Electronic Journal of Probability, 2022, ⟨10.1214/22-EJP887⟩. ⟨hal-03897737⟩