Abstracts

We will present a short review of the use of entropic optimal transport and its associated Sinkhorn Algorithm in the context of the multi-marginal time discretisation and resolution of control problems governed by the Fokker Planck or the Euler equation.

Einstein's equations in vacuum can be recovered from a variational principle strikingly similar to the one needed to get the isothermal Euler equations with a correspondance between the cosmological constant and the speed of sound. This requires the introduction of a suitable phase space, just as in kinetic theory, to express the Einstein equations as a kind of generalized, matrix-valued and kinetic, version of the isothermal Euler equations.

Entropic optimal transport (which has a long history as will certainly be recalled at this conference in honour of Jean-Claude) has received a lot of attention in the last decade in connection with efficient approximation algorithms (à la Sinkhorn). In this talk, I will address the speed of convergence under quite general assumptions on the transport cost (which do not imply that optimal plans are given by maps or are even unique). This will be done by proving separately a limsup and liminf bounds and show that they match. The limsup will be achieved by block approximation and a refinement of Alexandrov’s theorem, the liminf will be based on Minty’s trick. This is a recent joint work with Paul Pegon and Luca Tamanini.

Semiconcavity is a central and powerful notion in control theory. However, when studying certain properties of optimisers such as their long time behaviour, one is faced with the task of showing that the value function enjoys some kind of uniform in time semiconvexity property. Such notion appears to be much less stable than semiconcavity, especially when the control problems at hand fails to have a convex structure. In this talk, we show how to obtain semiconvexity estimates for non-convex stochastic control problems by constructing and analysing the contractive effect of coupling by reflection on the adjoint process of the forward-backward SDE systems stemming from the stochastic maximum principle. Applications to Feynman-Kac semigroups and functional inequalities will be given at the end of the talk.

We present some gradient estimates for the Schrӧdinger potentials, which we refer to as “correctors’ estimates“ in view of the control interpretation of the Schrӧdinger potentials, and we show how these bounds can lead to new quantitative stability estimates for the Schrӧdinger problem as well as how they can be leveraged in order to prove in the small-time limit that the gradients of the Schrӧdinger potentials converge to the Brenier map.
Based on an ongoing work with: A. Chiarini, G. Conforti and L. Tamanini

In her PhD thesis (Rouen, 2021), Laurène Valade has studied the isovector algebras of certain types of partial differential equations. Her results apply in particular to the equation $u_t = a(q) u_{qq} + b(q) u_q + c(q) u$ with a, b, c smooth and $a \neq 0$. For b = 0, this includes the (possibly backward-)heat equation with time–independent potential. Therefore we have common ground with the papers of Lescot-Zambrini (2004, 2008) and Lescot-Quintard-Zambrini (2015). Interesting connections appear to Rosencrans’ classical works (1976,1977)

Hopefully so!
In this talk -- based on Masoero, Raimondo and Antunes, Physica D (2015) -- we describe how it is defined and which properties it should have.

We obtain existence results for the solution of nonlocal semilinear parabolic PDEs on Rd with polynomial gradient nonlinearities using a tree-based probabilistic representation driven by a subordinated Levy process. This probabilistic representation applies to the solution of the equation itself, as well as to its partial derivatives by associating one of d marks to the initial tree branch. Partial derivatives are dealt with by integration by parts and subordination of Brownian motion, and fractional semilinear elliptic PDEs on open balls are also included in this approach by using alpha-stable processes.
Joint work with Guillaume Penent.

In this talk we first recall when and how the Asakura-Oosawa model of colloids appear in the scientific litterature. The depletion force between large spherical particules indeed is a marginal effect of the bath/emulsion of infinitely many small particles in which the large ones are living. We will compute explicitely the (hidden) depletion pair potential via the Gibbs formalism and verify that the associated Boltzmann measure is reversible for an appropriate gradient dynamics. At the end, the question of packing phenomena for emulsion with very high density will be considered.

We show that the ground state transform of the Schrödinger operator for a molecule with Coulomb interaction leads to a metric measure space with non-smooth generalized Ricci curvature bounded by a Kato function. We extend Bismut's formula for the gradient of the heat kernel to this situation and derive uniform Lipschitz bounds. Joint work with Bat Güneysu (Chemnitz).

We show how to break the curse of dimension for the estimation of optimal transport distance between two smooth distributions for the Euclidean squared distance. The approach relies on essentially one tool: represent inequality constraints in the dual formulation of OT by equality constraints with a sum of squares in reproducing kernel Hilbert space. By showing this representation is tight in the variational formulation, one can then leverage smoothness to break the curse. (*) However, the constants associated with the algorithm a priori scale exponentially with the dimension.

TBA

Reciprocal processes are stochastic processes that the current state only depends on the nearest past and future. In this talk, large deviations for some classes of reciprocal processes over a finite time interval will be discussed. These reciprocal processes are assumed to have fixed values for initial positions. We study large deviations for two types of terminal positions: (i) with fixed values, and (ii) with densities. It turns out that two different methods can be employed for these two types, and we will discuss possible connections of these two methods.

In this talk, I will introduce quantum correlations, one of the key aspects of quantum information theory [1], starting with the seminal example of Bell nonlocality [2]. I will review Bell’s theorem, which shows how quantum mechanics cannot be efficiently simulated classically. The discussion will be somewhat of a mix of physics and computer science. Generalizations of Bell nonlocality and other types of quantum correlations will be explored. As a conclusion, experimental research and applications of quantum correlations will be briefly discussed.

[1] Wilde, Mark M. Quantum information theory. Cambridge University Press, 2013.
[2] Bell, John S. "On the Einstein Podolsky Rosen paradox." Physics Physique Fizika 1.3 (1964): 195.