Julian Tugaut

Associate Professor
Hors Classe
Jean Monnet University

Summary

Since January 2024, I co-supervise with Jean-François Jabir (50% each) the PhD of Hetranso AHNI. The funding is provided by a doctoral allocation from UJM through doctoral school EDSIS (École Doctorale Sciences Ingénierie Santé). PhD subject.

First, Hetranso has acquired some knowledges and intuitions about the Freidlin and Wentzell theory, the latter being central in the PhD. He has also read pioneering papers dealing with the flocking models in population dynamics.

For few weeks, he is using his knowledge on Euler-Lagrange equations and on the Pontryagin principle in order to optimize the different good rate functions intervening in the large deviations theory for the Langevin stochastic processes. One of the first aim is to study the exit cost of one particle in a kinetic mean-field system of interacting particles without assuming that the initial law is a Dirac measure. Also, Hetranso proceeds to numerical simulations on Python to validate his first intuitions.

Hetranso has first hopeful results on the subject which give a glimpse that he will defend a good even a very good PhD before the end of October 2026.

I have had the great pleasure to co-supervise with Aline KURTZMANN (50% each) the PhD thesis of Ashot ALEKSIAN from September 2020 to November 2023. The ANR project METANOLIN and University Jean Monnet have funded together the PhD. The subject was "Metastability for self-interacting diffusions". The initial objective was to establish Kramers'type law results for the first exit-time of a self-interacting diffusion (namely a diffusion in which the empirical measure intervenes in the drift). About the initial objectives of the PhD as it was stated in 2020: PhD subject.

Ashot ALEKSIAN has a lot of qualities like autonomy and ability to take initiative. Moreover, he got familiar very fast with the subject (self-interacting diffusions and Freidlin and Wentzell theory). Furthermore, during the three years of the PhD fellowship, he was rigorous and he has understood a lot of mathematical tools. Last but not the least: he is humanly as great as scientifically. If you have any question on this remarkable student, I would be delighted to answer if you write me at the address: julian (dot) tugaut (at) univ-st-etienne (dot) fr

In one year, Ashot and his advisors (Aline and me) plus Pierre Del Moral have succeeded to establish a Kramers'type law for self-interacting diffusions in the case where both confining potential and interacting potential are convex. This leads to the paper (published in ESAIM Probability and statistics): "Self-interacting diffusions: long-time behaviour and exit-problem in the uniformly convex case".

Then, Ashot ALEKSIAN has also proven (alone) a large deviations principle for the self-interacting diffusion. This result and the initiatives that he has taken yield to a proof of the Kramers'type law for the self-interacting diffusion without uniform convexity hypothesis leading us (Ashot, Aline and me) to the submitted paper: "Exit-problem for a class of non-Markov processes with path dependency".

Furthermore, Ashot was interested in self-stabilizing diffusions and one of his project was the extension of my results on the exit-time to the case of possibly repulsive interacting force. Together, we have published an article (in Electronic Journal of Probability) on this subject: "Measure-dependent non-linear diffusions with superlinear drifts: asymptotic behaviour of the first exit-times".

Finally, Ashot has also submitted a paper with an advisor of his postdoc. In this work, they provide a general study of the time inhomogeneous (with respect to the time) processes.

To sum up, despite the initial conditions of the PhD thesis have not been ideal due to pandemie (confinment in particular), the thesis has been conform to our expectations and even further. The defense of this excellent thesis was the Monday November 20^th of 2023. Since then, Ashot is postdoctoral researcher at Toulouse School of Economics.

From October 2017 to July 2021, I have co-supervised with Olivier ALATA (50% each) the PhD thesis of Romain RAVAILLE.

The funding was from a doctoral allocation from UJM through the doctoral school EDSIS (École Doctorale Sciences Ingénierie Santé). Initially, the PhD has been dealing with the non stationary Gaussian Processes and their applications in images and videos. However, during the second visit of William Salkeld at Université Jean Monnet for two weeks, our student has taken another way: the reflected nonlinear stochastic processes.

That thesis leads to two publications: one in a computer science national journal with Olivier Alata and Rémi Émonet as co-authors (on the subject of non stationary Gaussian Processes) and one in Stochastic Processes and their Applications with Daniel Adams, Gonçalo dos Reis, William Salkeld and me as co-authors (about reflected McKean-Vlasov diffusions).

The Defense of that PhD was the Wednesday 7^th of July in 2021. Since then, Romain is working for private compagnies.

I will supervise a master internship of a student at ISFA on June and July 2025. He will work on the existence of nonlinear stochastic processes in the general dimensionnal case without assuming neither Lipschit hypotheses nor the rotational invariance of the interacting potential.

From February 2023 to June 2023, I have supervised the master internship of a student from Amiens. He has worked on large deviations principles for Langevin processes.

I have also supervised a master internship of a student from ENS Rennes in 2016. He worked on the Mogulskii theorem.