Summary

1. Hetranso AHNI

2. Ashot ALEKSIAN

3. Romain RAVAILLE

4. Others

Hetranso AHNI

    Since January 2024, I co-supervise with Jean-François Jabir the PhD of Hetranso AHNI. The funding is provided by UJM through EDSIS. PhD subject.

    First, Hetranso is getting familiar to the Freidlin and Wentzell theory, the latter being central in the PhD.

    For few weeks, he is using his knowledge on Euler-Lagrange equations in order to optimize the different good rate functions which intervene in the large deviations theory. 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 encouraging results on the subject which give a glimpse that he will defend a good PhD before the end of October 2026.

Ashot ALEKSIAN

    I have had the great pleasure to 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 are funding together the PhD. The subject is "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 (that is to say a diffusion in which the empirical measure intervenes in the drift). About the initial objectives of the PhD as it has been given 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, he is rigorous and he understands 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

    Up to now, we (Ashot, Aline, me and 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. We have submitted the related paper (to appear in ESAIM Probability and statistics): "Self-interacting diffusions: long-time behaviour and exit-problem in the uniformly convex case".

    Moreover, Ashot ALEKSIAN has proven (alone) a large deviations principle for the self-interacting diffusion. This result and the initiatives that he has taken have allowed us to obtain the Kramers'type law for the self-interacting diffusion without uniform convexity hypothesis. This lead us (Ashot, Aline and me) to the paper (revision in Probability Theory and Related Fields): "Exit-problem for a class of non-Markov processes with path dependency".

    Also, Ashot is interested in self-stabilizing diffusions and one of his project is to extend my results on the exit-time to the case where interacting force is repulsive. We have submitted an article (minor revision 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 is working with japanese colleagues (Kentaro FUJIE and Hiroshi WAKUI). They are interested in the long-time convergence for a solution of a mollification from a nonlinear PDE with explosive repulsion.

    To sum up, despite the initial conditions of the PhD thesis have not been the best due to pandemie (confinment in particular), the advance of the thesis is conform to our expectations and even further. The defense of this excellent thesis was in November 20^th of 2023.

Romain RAVAILLE

    I have co-supervised with Olivier ALATA (50% each) the PhD thesis of Romain RAVAILLE from October 2017 to July 2021. Jean Monnet University funded that thesis. That thesis lead to two publications (one in a computer science national journal with Olivier Alata and Rémi Émonet as co-authors and one in an international journal with Daniel Adams, Gonçalo dos Reis, William Salkeld and me as co-authors).

Others

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

    I have also advise a master internship of a student from ENS Rennes in 2016.