On permanental processes

Abstract

Permanental processes can be viewed as a generalization of squared centered Gaussian processes. We develop in this paper two main directions. The first one analyses the connections of these processes with the local times of general Markov processes. The second deals with Bosonian point processes and the Bose-Einstein condensation. The obtained results in both directions are related and based on the notion of infinite divisibility.

Local Times for Spectrally Negative Lévy Processes

“…The matrix above can be divided into four sub-matrices, with the upper-right (m + 1) × (n − m + 1) sub-matrix equals to 0. We can then prove (3.24) using (3.16) is the corresponding reference measure; see also [3,5] and references therein.…”

Permanental vectors with nonsymmetric kernels

“…This two classes correspond respectively to vectors with the Laplace transform of a squared Gaussian vector to a positive power and to infinitely divisible permanental vectors. Infinitely divisible permanental processes are connected to local times of Markov processes thanks to Dynkin's isomorphism theorem and its extensions (see [3]). Besides, we have shown in [1] that for permanental vectors, infinite divisibility and positive correlation are equivalent properties.…”

Multivariate gamma distributions

“…Nevertheless, using his criteria, Eisenbaum and Kaspi [1] show that it suffices that R is the inverse of an M -matrix; see Remark 2.1. One is, for which matrices R do there exist random variables X satisfying (1.1)?…”