Armand Bernou is a postdoctoral researcher at Sapienza Università di Roma. His research focuses on understanding the influence of microscopic structures on macroscopic behaviors in mathematical models for physics and biology.

**Host University**: Sapienza Università di Roma, Italy

**Host research group or department**: Department of Mathematics - site

**Co-host University**: Aix Marseille University, France

**Secondment institution**:** **Université Paris Dauphine - PSL

**Advisor**: Professor Alessandra Faggionato

**Co-advisor**: Professor Pierre Mathieu

**Secondment mentor**: Dr. Yating Liu

**My research**

**Stochastic Homogenization and Linear Response**

This proposal is structured around three challenges:

1) Stochastic homogenization and hydrodynamic limit for particle systems on manifolds. The idea of stochastic homogenisation theory is that, considering a system of interacting particle on which randomness applies, for a limit taken in some well-defined sense (hydrodynamical limit), one will obtained a homogenized equation with effective coefficients describing the behaviour of the system at the limit. Many results are available for simple microscopic models in the mathematical literature e.g. random walks on point processes of R^d, simple exclusion processes on simple graphs... however in most cases the state space is simple. We aim at considering more physically relevant (and complex !) geometric settings for those problems, to prove those results in these new frameworks, and to focus on more evolved processes, for instance multiple random walks with site exclusion interaction.

2) Obtaining quantitative estimates in linear response theory. Linear response theory aims at describing the answer of a system to a small perturbation. Recent results have considered the case of a diffusion with a perturbation depending on time, and obtain explicit formulae for this linear response: the complex mobility matrix. However, this is only a qualitative result: there are no estimates on the speed of convergence towards this complex mobility matrix at the moment. Combining those qualitative insights with recent techniques on quantitative homogenization with which I am familiar, we hope to obtain new results for this question. Several problems regarding linear response theory for processes in random environment will also be considered.

3) Kinetic theory with physically relevant boundary conditions. Most results available in the mathematical theory for the study of a gas use either deterministic boundary conditions or conditions with a randomness that is too simple to describe the real physical setting. The goal of this challenge is to continue my program focused around the Cercignani-Lampis boundary condition, which is used in practice by physicists, and presents some novel difficulties from the mathematical point of view. A mix of techniques coming from probability and analysis will be used to tackle this important problem.

**Date started – Date End**

**01.11.2022 - 31.10.2024**