research
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
and .
and Section VII of 
. 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
. 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
.