No 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.

Let X be a symmetric stable process of index α ∈ (1, 2] and let L x t denote the local time at time t and position x.We call V (t) the most visited site of X up to time t. We prove the transience of V , that is, lim t→∞ |V (t)| = ∞ almost surely. An estimate is given concerning the rate of escape of V . The result extends a well-known theorem of Bass and Griffin for Brownian motion. Our approach is based upon an extension of the Ray-Knight theorem for symmetric Markov processes, and relates stable local times to fractional Brownian motion and further to the winding problem for planar Brownian motion.,

We show that fractional Brownian motions with index in (0, 1] satisfy a remarkable property: their squares are infinitely divisible. We also prove that a large class of Gaussian processes are sharing this property. This property then allows the construction of two-parameters families of processes having the additivity property of the squared Bessel processes.

We show that, up to multiplication by constants, a Gaussian process has an infinitely divisible square if and only if its covariance is the Green function of a transient Markov process.