Poisson ensembles of loops of one-dimensional diffusions

Abstract

We study the analogue of Poisson ensembles of Markov loops ('loop soups') in the setting of one-dimensional diffusions. We give a detailed description of the corresponding intensity measure. The properties of this measure on loops lead us to an extension of Vervaat's bridge-to-excursion transformation that relates the bridges conditioned by their minimum and the excursions of all the diffusion we consider and not just the Brownian motion. Further we describe the Poisson point process of loops, their occupation fields and explain how to sample these Poisson ensembles of loops using two-dimensional Markov processes. Finally we introduce a couple of interwoven determinantal point processes on the line which is a dual through Wilson's algorithm of Poisson ensembles of loops and study the properties of these determinantal point processes.