From Step Reinforced Random Walks to Noise Reinforced Lévy Processes and Excursion Theory for Markov Processes Indexed by Lévy Trees

Abstract

The content of this thesis, written under the supervision of Jean Bertoin and Armand Riera from September 2020 to June 2023 at the University of Zürich, is composed by two independent and non-related parts. The first one is titled From step reinforced random walks to noise reinforced Lévy processes and falls into the broader setting of reinforcement of stochastic processes. It is composed by the two works [1] and [2], the first one written in collaboration with Marco Bertenghi. The first work [1] has been published in Journal of Statistical Physics, while [2] has been published in Electronic Journal of Probability. The second part of this thesis is titled Excursion theory for Markov processes indexed by Lévy trees, and belongs to the broader framework of stochastic geometry. It is composed by the papers [3] and [4], both written in collaboration with Armand Riera. The first work [3] has been published in Probability Theory and Related Fields, while [4] is still work in progress at an advanced stage.

[1] M. Bertenghi and A. Rosales-Ortiz. "Joint invariance principles for random walks with positively and negatively reinforced steps" (2022). J. Stat. Phys. 189.

[2] A. Rosales-Ortiz. "Noise reinforced Lévy processes: Lévy-Itô decomposition and applications" (2023). Electron. J. Probab. 28. [3] A. Riera and A. Rosales-Ortiz. "The structure of the local time of Markov processes indexed by Lévy trees" (2024), Probab. Theory Related Fields, 189.

[4] A. Riera and A. Rosales-Ortiz. "Excursion theory for Markov processes indexed by Lévy trees" (work in progress).