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.

151 pages

References in corpus (8)

• From loop clusters and random interlacement to the free field
• Convergence of the two-dimensional random walk loop soup clusters to CLE
• Decomposition of Brownian loop-soup clusters
• The first passage sets of the 2D Gaussian free field: convergence and isomorphisms
• Excursions Above the Minimum for Diffusions
• Loop cluster on the discrete circle
• Markovian loop soups: permanental processes and isomorphism theorems
• Markovian loop clusters on graphs

Cited by in corpus (6)

• The Vervaat transform of Brownian bridges and Brownian motion
• Jump stochastic differential equations for the characterisation of the Bragg peak in proton beam radiotherapy
• Markov loops, complex free field and Eulerian circuits
• Crossing exponent in the Brownian loop soup
• The spans in Brownian motion
• Markovian loop soups: permanental processes and isomorphism theorems