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My publications, activities and projects are featured at the bottom of this page.

I'm mainly interested in probability theory and more specifically in the large deviations of large systems of evolving random particles. My concern is not about interaction between particles, by the way most of my papers deal with non-interacting particles, but about the structure of the thermodynamic limits of some evolving random particle systems subject to marginal constraints.  Although non-interacting, these particle systems give rise to truely interesting limit behaviors, very close in spirit to optimal transport.

Sanov's theorem is the prototype of the large deviation results that I invoke. Since it crucially involves a relative entropy, I was drawn to solve entropy minimization problems under infinite dimensional constraints. To this purpose, in the early 2000's, I developed an approach for solving convex minimization problems in infinite dimension, see  PDF  and  PDF .

In the early 30's, Erwin Schrödinger addressed the following statistical physics problem, see  PDF  and Section VII of  Mathdoc . Let a large particle system, in contact with some heat bath, be in equilibrium at the initial time t=0. Now, suppose that a little later, at time t=1 (say), you observe it far away from equilibrium. Although highly unlikely, this spontaneous deviation from equilibrium may happen with a positive probability. Therefore, one may wonder, conditionally on the occurrence of this very rare event, what the most likely evolution of the whole particle system during the time interval [0,1] is. As was shown by Schrödinger himself and later formalized by Hans Föllmer in his Saint-Flour lecture notes, the answer to this question leads to a relative entropy minimization problem on a space of path measures subject to prescribed initial and final marginal measures.

The beauty of Schrödinger's problem is that it is a random version of the renowned Monge-Kantorovich optimal transport problem. It was addressed ten years before the celebrated 1942 paper by Leonid Kantorovich (also a Nobel laureate, precisely for this contribution). The solution of Schrödinger's problem is an interpolation between the initial and final prescribed marginals in the  space of all probabilty measures on the configuration space. It almost looks like a geodesic and indeed, it is a viscous regularization of McCann's displacement interpolation. As such, it might be used to perform rigorous analytic calculations in some settings where  displacement interpolations would only provide us with (reliable but) informal guesses. In particular, by considering random walks on graphs rather than the Brownian motion in Schrödinger's original formulation, we obtain a natural notion of displacement interpolation on a discrete graph PDF  arxiv-icon . This opens the way to developing an analogue in the discrete setting of the Lott-Sturm-Villani (LSV) theory of a synthetic notion of Ricci curvature , see Cédric Villani's textbook. Indeed, the LSV theory relies in an essential manner upon the notion of displacement interpolation on geodesic spaces (excluding de facto discrete spaces).

For more details about Schrödinger's problem and the above research program, one may have a look at my recent survey paper  PDF  arxiv .