# Soutenance de thèse: Francesco Mori

*Extreme value statistics of stochastic processes : from Brownian motion to active particles*

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### Francesco Mori

Rare extreme events tend to play a major role in a wide range of contexts, from finance to climate. Hence, understanding their statistical properties is a relevant task, which opens the way to many applications. In this thesis, we investigate the extremal properties of several stochastic processes, including Brownian motion (BM), active particles, and BM with resetting. In the first part, we investigate the times at which extrema of one-dimensional stochastic processes occur. In particular, in the case of a BM of fixed duration, we compute the probability distribution of the time between the global maximum and the global minimum. Moreover, we derive the distribution of the time of the maximum for stationary stochastic processes, both at equilibrium and out-of-equilibrium. This analysis leads to the formulation of a simple criterion to detect nonequilibrium fluctuations in steady states. In the second part, we focus on the run-and-tumble particle (RTP) model. We compute exactly the survival probability for a single RTP in d dimensions, showing that this quantity is completely universal, i.e., independent of d and the speed fluctuations of the particle. We extend this universality to other observables (time of the maximum and records) and generalized RTP models. Moreover, we also investigate the position distribution of a single RTP at late times. We show that, under certain conditions, a condensation transition can be observed in the large-deviation regime where the particle is far from its starting position. Finally, we introduce a new technique, analog to the Hamilton-Jacobi-Bellman equation, to optimally control a dynamical system through stochastic resetting.

Jury :

S. Majumdar (directeur de thèse)

D. Dean (rapporteur)

M. Marsili (rapporteur)

M. Evans

J. Krug

C. Monthus

G. Schehr

R. Voiturier

**Date/Time : ** 10/06/2022 - *14:00 - 17:30*

**Location : ** Petit amphi, bâtiment Pascal n° 530

#### Upcoming seminars

- Séminaire du LPTMS : Stéphane Ouvry (LPTMS) - 9 May 2023 - 11:00
- Séminaire du LPTMS : Denis Bernard (ENS Paris) - 16 May 2023 - 11:00
- Physcis-Biology interface seminar : Guillaume Sarfati - 2 Jun 2023 - 11:00
- Séminaire du LPTMS : Laure Dumaz (DMA, ENS Paris) - 6 Jun 2023 - 11:00
- Séminaire de Physique Statistique du LPTMS : Nina Javerzat (SISSA) - 8 Jun 2023 - 11:00
- Séminaire du LPTMS : Amit Ghosal (IISER, Kolkata) - 20 Jun 2023 - 11:00
- Séminaire de Physique Statistique du LPTMS : Gérard Ben Arous (New York Univ.) - 22 Jun 2023 - 11:00
- Séminaire du LPTMS : Olesia Dogonasheva (ENS Paris and Moscow University) - 27 Jun 2023 - 11:00
- Soutenance de thèse : Lara Koehler - 10 Jul 2023 - 14:00
- Soutenance de thèse Lorenzo Rosso - 18 Jul 2023 - 10:00
- Séminaire du LPTMS : David Martin (Chicago University) - 5 Sep 2023 - 11:00
- Soutenance de thèse : Ana Flack - 25 Sep 2023 - 16:00
- Soutenance de thèse Benjamin De Bruyne - 16 Oct 2023 - 14:30