# Poissons ensemble of loops of one-dimensional diffusions

@article{Lupu2013PoissonsEO, title={Poissons ensemble of loops of one-dimensional diffusions}, author={Titus Lupu}, journal={arXiv: Probability}, year={2013}, volume={157}, pages={1-162} }

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…

## 18 Citations

From loop clusters of parameter 1/2 to the Gaussian free field

- Mathematics
- 2014

We consider a transient symmetric Markov jump process on a network and the associated Poisson ensemble of loops ("loop soup") of parameter 1/2. We construct a coupling between the Poisson ensemble of…

From loop clusters and random interlacement to the free field

- Mathematics
- 2014

It was shown by Le Jan that the occupation field of a Poisson ensemble of Markov loops ("loop soup") of parameter one-half associated to a transient symmetric Markov jump process on a network is half…

Markov loops, free field and Eulerian networks

- Mathematics
- 2015

We investigate the relations between the Poissonnian loop ensemble arising in the construction of random spanning trees, the free field, and random Eulerian networks. 1 The loop ensemble and the free…

Loop cluster on discrete circles

- Mathematics
- 2015

The loop clusters of a Poissonian ensemble of Markov loops on a finite or countable graph have been studied in \cite{Markovian-loop-clusters-on-graphs}. In the present article, we study the loop…

Loop percolation on discrete half-plane

- Mathematics
- 2014

We consider the random walk loop soup on the discrete half-plane and study the percolation problem, i.e. the existence of an infinite cluster of loops. We show that the critical value of the…

Cluster explorations of the loop soup on a metric graph related to the Gaussian free field

- Mathematics
- 2020

We consider the loop soup at intensity ${1\over 2}$ conditioned on having local time $0$ on a set of vertices with positive occupation field in their vicinities. We give a relation between this loop…

The Vervaat transform of Brownian bridges and Brownian motion

- Mathematics
- 2015

For a continuous function f 2 C([0;1]), define the Vervaat transform V (f)(t) := f( (f) +t mod 1) +f(1)1ft+ (f) 1g f( (f)), where (f) corresponds to the first time at which the minimum of f is…

Cluster explorations of the loop soup on a metric graph related to the Gaussian free field

- Mathematics
- 2020

We consider the loop soup at intensity 12 conditioned on having local time 0 on a set of vertices with positive occupation field in their vicinities. We give a relation between this loop soup and the…

On Clusters of Brownian Loops in d Dimensions

- Mathematics
- 2020

We discuss random geometric structures obtained by percolation of Brownian loops, in relation to the Gaussian Free Field, and how their existence and properties depend on the dimension of the ambient…

Squared Bessel processes of positive and negative dimension embedded in Brownian local times

- Mathematics
- 2018

The Ray--Knight theorems show that the local time processes of various path fragments derived from a one-dimensional Brownian motion $B$ are squared Bessel processes of dimensions $0$, $2$, and $4$.…

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