Research interests

My research work is focusing on the behaviour of constrained measure-valued dynamics, with a special interest in their stability and approximation properties. I have been working on mean-field theory, large deviations, optimal transport, stochastic control, viscosity solutions and filtering theory. Some of my motivations are listed below.

• Estimation and filtering. When modelling a dynamical phenomenon, a discrepancy is often measured between the model prediction and the targeted quantity; it is then natural to look for an improved version of the model that takes into account the measured error. A first example is the filtering problem, which computes the conditional distribution of a system in real-time given a noisy measurement on it. When it comes to high-dimensional systems, famous examples are given by statistical mechanics and data analysis, like the Gibbs principle and the Schrödinger bridge problem. In all physical experiments, measurements are affected by uncertainties of different natures; thus, the question of stability comes along with the estimation problem.

• Sampling and optimal transport. A fundamental question in probability and statistics is to sample from probability measures that minimise a given cost functional, possibly under additional constraints. A prominent example is the optimal transport problem, which searches for an optimal measure on a product space with imposed marginal laws. Sampling from measures can be achieved by constructing appropriate dynamics with the desired large time behaviour, preferably satisfying the same constraints as the target. A famous example is given by the diffusion models in machine learning, which turns out to be closely connected to Schrödinger bridges. The question of stability and long-time convergence for the sampling dynamics is once again crucial.

Contact

E-mail: louis-pierre.chaintron-at-epfl.ch