Abstracts

The evolution of a set by deformation of its boundary along mean curvature flow is involved in many physical phenomena. We are interested here in this evolution, to which we add a noise that acts uniformly on its boundary, and a renormalization term. It is shown that this evolution can be coupled to a Brownian motion which remains within the set, and which at all times is uniformly distributed inside the set. This is the phenomenon of duality and intertwining established by Diaconis and Fill in the context of Markov chains in finite state spaces. Different couplings are proposed, some of which involving the local time of the Brownian motion, either on the skeleton of the set or on its boundary. These couplings also differ by more or less strong correlation between the Brownian motion inside and the vibrating boundary. When the set is a symmetric real interval, the boundary evolves as a Bessel process of dimension 3, and we recover the 2M-X Pitman theorem as a special case, with one particular coupling. This is a common work with Koléhè Coulibaly (Institut Elie Cartan de Lorraine (IECL), Nancy), and Laurent Miclo (Institut de Mathématiques de Toulouse (IMT), Toulouse).

I will introduce a generalization of the Schrödinger problem where the Brownian motion is replaced by the branching Brownian motion. Just as the Schrödinger problem admits an interpretation in terms of regularized optimal transport, our branching Schrödinger problem admits an interpretation in terms of regularized and unbalanced optimal transport, meaning that the total mass is allowed to vary along the transport. In this presentation, I will insist on the link between the competitors of both problems, and hence answer the question "How to build a competitor of the tranport problem out of a competitor of the entropic problem, and vice versa?". This goes through a characterization of all the laws having finite entropy w.r.t. the branching Brownian motion.

We revisit the Einstein equations in vacuum as optimality equations of a kind of generalized quadratic matrix-valued optimal transport problem à la Benamou-Brenier.
ref: https://hal.archives-ouvertes.fr/hal-03311171

Multi-marginal optimal transport arises in various contexts in economics, many-electron physics and fluid dynamics. Entropic optimal transport is an efficient and popular method to approximate such problems with the Sinkhorn algorithm. In this talk I will give two results concerning entropic multi-marginal optimal transport: well-posedness of the Schrödinger system (joint with Maxime Laborde) and linear convergence of the Sinkhorn algorithm.

This talk is devoted to the study of a class of Hamilton-Jacobi equations on the space of probability measures that arises naturally in connection with the Schrödinger problem for interacting particle systems. After presenting the equations and their geometrical interpretation, I will move on to illustrate the main ideas behind a general strategy for to prove uniqueness of viscosity solutions, i.e. the comparison principle. Joint work with D. Tonon (U. Padova) and R. Kraaij (TU Delft).

In this talk, I will introduce and motivate the equilibrium transport problem, which is a generalization of the optimal transport problem, and I will focus on its entropic regularization. I will show the existence and uniqueness of a solution to a generalization of the Schrodinger-Bernstein system, and I will provide an algorithm to compute it. Based on joint work with Eugene Choo, Charles Liang, and Simon Weber.

In non-imaging optics, we try to optimize the trajectory of the light from a source to a target without trying to form an image of the source on the target. Some non-imaging optic problems can be translated into optimal transport problems in a semi-discrete setting, meaning that the source is a continuous domain and the target is a finite set of points. Other problems of non-imaging optics can sometimes be rewritten into a slightly more global form than optimal transport, which we call Generated Jacobian Equations. During this presentation we will focus on these Generated Jacobian Equations, also in a semi-discrete setting. We will begin by introducing them using a non-imaging optic problem, and make the link with optimal transport. We will then present an algorithm to solve Generated Jacobian Equations, which was adapted from an existing algorithm to solve optimal transport problems. And finally we will detail the main lines of the proof of convergence of this algorithm.

The classical geometric mechanics, including the symmetries, the Lagrangian and Hamiltonian mechanics, and the Hamilton-Jacobi theory, are based on classical geometric structures such as jets, symplectic structures and contact structures. In this paper, we will continue the forgotten framework of the second-order (or stochastic) differential geometry developed originally by L. Schwartz and P.-A. Meyer, to construct the second-order (or stochastic) counterparts of those classical structures. Based on these second-order structures, we then study the symmetries of SDEs, the stochastic Lagrangian and Hamiltonian mechanics as well as their connections with the Hamilton-Jacobi-Bellman equations. The latter suggests an alternative description to the Schrödinger's problem which is equivalent to the viewpoint of optimal transport.

The Brenier problem is a variational approach of Euler equations whose solutions are path measures, minimising the kinetic energy. Using an appropriate notion of energy, this approach can be extended to different fluid evolution models. In particular, the Nelson (or mean velocity) kinetic energy links Navier-Stokes equations to the Brenier-Schrödinger problem.
This talk will introduce the Brenier-Shrödinger problem on compact manifolds with boundary and explain the link with viscous fluid evolution, especially concerning the impermeability condition. We will also give existence results in compact manifolds with boundary such as rectangles or regular triangles.

Motivated by entropic optimal transport, we investigate the parabolic equation $( \partial_t + \mathsf{b} \cdot \nabla + \Delta _{ \mathsf{a}}/2+V)g=0$ with a nonnegative final boundary condition. It is well-known that the viscosity solution $g$ of this PDE is represented by the Feynman-Kac formula when the drift $\mathsf{b}$, the diffusion matrix $\mathsf{a}$ and the scalar potential $V$ are regular enough and not growing too fast. Extending this result to a setting where $\mathsf{b}$ and $V$ are not assumed to be regular and locally bounded requires to introduce a new trajectorial notion of solution to this PDE based on semimartingale extension of Markov generators. As a by-product, we characterize the drift of Schrödinger bridges when $V$ belongs to some Kato class. Our probabilistic approach relies on stochastic derivatives, semimartingales, Girsanov's theorem, and the Hamilton-Jacobi-Bellman equation satisfied by $\log g$. Preprint: arXiv:2104.09171.

First of all, we briefly discuss Schrödinger’s and Nelson’s problems and the relation between them. Nelson’s problem can be considered a continuum limit of Schrödinger’s problem. The zero-noise limit of Schrödinger’s problem is Monge-Kantorovich's problem with a quadratic cost. (We do not discuss the recent rapid development of Schrödinger’s problems.) Then we discuss their generalizations as stochastic optimal transport that is stochastic optimal control with fixed marginals. Time permitting, we discuss our recent results where the superposition principle for diffusion processes plays a crucial role.

In this talk I will firstly review standard Multi-Marginal Optimal Transport (a number N of marginals is fixed) focusing, in particular, on the applications in Quantum Mechanics (in this case the marginals are all the same and represent the electron density). I will then extend the Optimal Transportation problem to the grand canonical setting: only the expected number of marginals/electrons is now given (i.e. we can now dene a OT problem with a fractional number of marginals). I will compare these two problems and show how they behave dierently despite considering the same cost functions. Existence of minimisers, Monge solution, duality, entropic formulation and numerics will be discussed.

We present a pair of adjoint optimal control problems characterizing a class of time-symmetric stochastic processes defined on random time intervals. The associated PDEs are of free-boundary type. The particularity of our approach is that it involves two adjoint optimal stopping times adapted to a pair of filtrations, the traditional increasing one and another, decreasing. They are the keys of the time symmetry of the construction, which can be regarded as a generalization of "Schrödinger's problem" (1931-32) to space-time domains. The relation with the notion of "Hidden diffusions" is also described.

When considering a diffusion, the well-known Hörmander condition for the process to admit a smooth density is that the iterated Lie brackets of the noise-inducing vector fields generate the tangent space at every point. Moreover, very often we know that the typical displacement in the direction Y is of order √tᵏ if ze need k brackets to generate Y. However, in the case where the drift is needed to generate the tangent space, little is known in general about the typical dispacements. In this talk, I will present a couple of such examples where we can describe the small time limit rather precisely.

Optimal transport (OT) has recently gained lot of interest in machine learning. It is a natural tool to compare in a geometrically faithful way probability distributions. It finds applications in both supervised learning (using geometric loss functions) and unsupervised learning (to perform generative model fitting). OT is however plagued by the curse of dimensionality, since it might require a number of samples which grows exponentially with the dimension. In this talk, I will explain how to leverage entropic regularization methods to define computationally efficient loss functions, approximating OT with a better sample complexity. More information and references can be found on the website of our book "Computational Optimal Transport" https://optimaltransport.github.io/

We derive Wasserstein distance bounds between the probability distributions of stochastic integrals and diffusion processes with jumps, based on the integrands appearing in their stochastic integral representations. Our approach uses classical and forward-backward stochastic calculus arguments and allows us to consider a large class of target distributions constructed using Brownian stochastic integrals and point processes. This talk is based on a joint work with Jean-Christophe Breton (Rennes).

We precent some results concerning a first-order mean field planning problem associated to a convex Hamiltonian with quadratic growth and a monotone interaction term with polynomial growth.
Combining ideas from optimal transport, convex analysis, and renormalized solutions to the continuity equation, we exploit the variational structure of the system and we prove existence and (at least partial) uniqueness of a weak solution. A crucial step of our approach relies on a careful analysis of distributional subsolutions to Hamilton-Jacobi equations and on a measure-theoretic description of the optimality via a suitable contact-defect measure.
(In collaboration with Carlo Orrieri and Alessio Porretta)

The Schrödinger problem as "noised" optimal transport is by now a well-established interpretation. From this perspective several natural questions stem, as for instance the convergence rate as the noise parameter vanishes of many quantities: optimal value, Schrödinger bridges and potentials... As for the optimal value, after the works of Erbar-Maas-Renger and Pal a first-order Taylor expansion is available. A first aim of this talk is to improve this result in a twofold sense: from the first to the second order and from the Euclidean to the Riemannian setting (and actually far beyond). From the proof it will be clear that the statement is in fact a particular instance of a more general result. For this reason, in the second part of the talk we introduce a large class of dynamical variational problems, extending far beyond the classical Schrödinger problem, and for them we prove Γ-convergence towards the geodesic problem and a suitable generalization of the second-order Taylor expansion.
(based on joint works with G. Conforti, L. Monsaingeon and D. Vorotnikov)

In this talk we consider a class of variational shape optimisation problems for densities that repel each other at a certain distance. Typical examples are given by the Dirichlet functional and the Rayleigh functional minimised in the class of functions that attain some H^{^1} boundary conditions, subject to the constraint that the supports of different densities are at a certain fixed distance. We show a connection with solutions to variational elliptic systems with nonlocal competing interactions, investigate the optimal regularity of the solutions (and prove uniform estimates with respect to the distance parameter), prove a free-boundary condition and derive some preliminary results characterising the free boundary. The talk is based in the following works:
[1] N. Soave, H. Tavares, S. Terracini, A. Zilio, Variational problems with long-range interaction, Arch. Rational Mech. Anal.228(2018), 743–772.
[2] Nicola Soave, Hugo Tavares and Alessandro Zilio, Free boundary problems with long-range interactions: uniform Lipschitz estimates in the radius, arXiv:2106.03661.

In this talk, I will describe several approaches to introduce a quantum analogue of the optimal transport problem and related Wasserstein distances, motivated by applications in different research areas. The main focus will be on two recent ones: a "Quantum optimal transport with quantum channels" (joint with G. De Palma), and a "quantum Wasserstein distance of order 1" (joint with G. De Palma, M. Marvian and Seth Lloyd).

We will survey some results on the construction of candidate models for Brownian motion on Wasserstein space and the connection to the Dean-Kawasaki equation appearing in the theory of fluctuating hydrodynamics.

Optimal "ballistic" transport problems for matrix-valued measures naturally arise, in the spirit of Y. Brenier. Comm. Math. Phys. (2018) 364(2) 579-605, from a certain dual formulation of nonlinear evolutionary equations with a particular quadratic structure reminiscent both of the incompressible Euler equation and of the quadratic Hamilton-Jacobi equation. The examples include the ideal incompressible MHD, the template matching equation, the multidimensional Camassa-Holm (also known as the Hdiv geodesic equation), EPDiff, Euler-alpha, KdV and Zakharov-Kuznetsov equations, the equations of motion for the incompressible isotropic elastic fluid and for the damping-free Maxwell's fluid. This yields the existence of a new type of continuous and Sobolev in time generalized solutions to the initial-value problems for the above mentioned PDE. We also establish a sharp upper bound on the optimal value of the dual problem, and explore the weak-strong uniqueness issue. The approach also suggests a funny but unambiguous way how to construct pseudoentropic regularizations of many PDE in the spirit of the Schroedinger problem.