Poissonian ensembles of loops of one-dimensional diffusions
Résumé
We study the analogue of Poissonian ensembles of Markov loops ("loop soups") in the setting of one-dimensional diffusions. First we give a detailed description of the corresponding intensity measure. Then we identify the law of the occupation field of the Poissonian ensembles of loops. Finally we explain how to sample this Poissonian ensembles using two-dimensional Markov processes. We also state and prove a Vervaat-like relation between bridges conditioned by the value of their minimum and excursion that holds for all the diffusion we consider and not just for the Brownian motion.
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