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Aymeric Baradat

Publications and source records attributed to Aymeric Baradat.

12 recordsLinked to original sources
arXiv2026

Multiphasic formulation of Vlasov equations and applications

This work is a mathematical study of the multiphasic formulation of the Vlasov equation, which consists in recasting this collisionless kinetic equation as a system of coupled pressureless Euler equations. This framework allows to consider solutions that are only measure-valued in the velocity variable and is thus relevant to tackle physical problems where rough velocity distributions, such as Dirac masses, naturally arise. We specifically address the case of nonlinear Vlasov equations where the force field is one derivative more regular than some moments in velocity of the solution. Under this key assumption, a unified theory of local well-posedness at finite regularity (typically in Sobolev spaces) is developed from scratch for the multiphasic formulation, yielding corresponding results for the Cauchy problem of the associated Vlasov equation at low regularity. We thoroughly apply this abstract theory to two classes of equations, namely Vlasov-Poisson type systems, and Vlasov-Navier-Stokes type systems. In addition to the justification of the monokinetic limit, it also leads to specific applications for each class of equations, allowing possibly rough velocity distributions. For Vlasov-Poisson type systems, we justify the semiclassical limit from Hartree equations, and we describe the nonlinear instability of homogeneous equilibria. For Vlasov-Navier-Stokes type systems, we establish nonlinear asymptotic stability near monokinetic profiles. New results allowing a separation between the regularity in space and in velocity are proven, and along the way, we also provide new proofs of known results and generalize them to the case of rough solutions. Finally, the flexibility of the framework is exploited to obtain extensions to more sophisticated systems such as the Vlasov-Poisson equation for ions, or the Vlasov equation coupled with the compressible Navier-Stokes system.

math.AP↗
arXiv2026

A convergence rate for the entropic JKO scheme

The so-called JKO scheme, named after Jordan, Kinderlehrer and Otto, provides a variational way to construct discrete time approximations of certain partial differential equations (PDEs) appearing as gradient flows in the space of probability measures equipped with the Wasserstein metric. The method consists of an implicit Euler scheme, which can be implemented numerically. Yet, in practice, evaluating the Wasserstein distance can be numerically expensive. To address this problem, a common strategy introduced by Peyré in 2015 and which has been shown to produce faster computations, is to replace the Wasserstein distance with its entropic regularization, also known as the Schrödinger cost. In 2026, the first author, Hraivoronska and Santambrogio, proved that if the regularization parameter $\varepsilon$ is proportional to the time step $τ$, that is, $\varepsilon = ατ$ for some $α> 0$, then as $τ\to 0$, this change results in adding to the limiting PDE the additional linear diffusion term $\fracα{2} Δρ$. Our goal in this article is to provide a convergence rate under convexity assumptions between the entropic JKO scheme and the solution of the initial PDE as both $α$ and $τ$ tend to zero. This will appear as a consequence of a new bound between the classical and entropic JKO schemes.

math.AP↗
arXiv2025

Using Sinkhorn in the JKO scheme adds linear diffusion

The JKO scheme is a time-discrete scheme of implicit Euler type that allows to construct weak solutions of evolution PDEs which have a Wasserstein gradient structure. The purpose of this work is to study the effect of replacing the classical quadratic optimal transport problem by the Schrödinger problem (\emph{a.k.a.}\ the entropic regularization of optimal transport, efficiently computed by the Sinkhorn algorithm) at each step of this scheme. We find that if $ε$ is the regularization parameter of the Schrödinger problem, and $τ$ is the time step parameter, considering the limit $τ,ε\to 0$ with $\fracετ \to α\in \mathbb{R}_+$ results in adding the term $\fracα{2} Δρ$ on the right-hand side of the limiting PDE. In the case $α= 0$ we improve a previous result by Carlier, Duval, Peyr{é} and Schmitzer (2017).

math.AP↗
arXiv2023

Convergence of the Sinkhorn algorithm when the Schrödinger problem has no solution

The Sinkhorn algorithm is the most popular method for solving the entropy minimization problem called the Schrödinger problem: in the non-degenerate cases, the latter admits a unique solution towards which the algorithm converges linearly. Here, motivated by recent applications of the Schrödinger problem with respect to structured stochastic processes (such as increasing ones), we study the Sinkhorn algorithm in degenerate cases where it might happen that no solution exist at all. We show that in this case, the algorithm ultimately alternates between two limit points. Moreover, these limit points can be used to compute the solution of a relaxed version of the Schrödinger problem, which appears as the $Γ$-limit of a problem where the marginal constraints are replaced by asymptotically large marginal penalizations, exactly in the spirit of the so-called unbalanced optimal transport. Finally, our work focuses on the support of the solution of the relaxed problem, giving its typical shape and designing a procedure to compute it quickly. We showcase promising numerical applications related to a model used in cell biology.

math.OC↗
arXiv2021

Regularized unbalanced optimal transport as entropy minimization with respect to branching Brownian motion

We consider the problem of minimizing the entropy of a law with respect to the law of a reference branching Brownian motion under density constraints at an initial and final time. We call this problem the branching Schrödinger problem by analogy with the Schrödinger problem, where the reference process is a Brownian motion. Whereas the Schrödinger problem is related to regularized (a.k.a. entropic) optimal transport, we investigate here the link of the branching Schrödinger problem with regularized unbalanced optimal transport. This link is shown at two levels. First, relying on duality arguments, the values of these two problems of calculus of variations are linked, in the sense that the value of the regularized unbalanced optimal transport (seen as a function of the initial and final measure) is the lower semi-continuous relaxation of the value of the branching Schrödinger problem. Second, we also explicit a correspondence between the competitors of these two problems, and to that end we provide a fine description of laws having a finite entropy with respect to a reference branching Brownian motion. We investigate the small noise limit, when the noise intensity of the branching Brownian motion goes to $0$: in this case we show, at the level of the optimal transport model, that there is convergence to partial optimal transport. We also provide formal arguments about why looking at the branching Brownian motion, and not at other measure-valued branching Markov processes, like superprocesses, yields the problem closest to optimal transport. Finally, we explain how this problem can be solved numerically: the dynamical formulation of regularized unbalanced optimal transport can be discretized and solved via convex optimization.

math.PR↗
arXiv2021

$Γ$-convergence for a class of action functionals induced by gradients of convex functions

Given a real function $f$, the rate function for the large deviations of the diffusion process of drift $\nabla f$ given by the Freidlin-Wentzell theorem coincides with the time integral of the energy dissipation for the gradient flow associated with $f$. This paper is concerned with the stability in the hilbertian framework of this common action functional when $f$ varies. More precisely, we show that if $(f_h)_h$ is uniformly $λ$-convex for some $λ\in \mathbb{R}$ and converges towards $f$ in the sense of Mosco convergence, then the related functionals $Γ$-converge in the strong topology of curves.

math.OC↗
arXiv2020

Monge-Ampère gravitation as a $Γ$-limit of good rate functions

Monge-Ampère gravitation is a modification of the classical Newtonian gravitation where the linear Poisson equation is replaced by the nonlinear Monge-Ampère equation. This paper is concerned with the rigorous derivation of Monge-Ampère gravitation for a finite number of particles from the stochastic model of a Brownian point cloud, in the spirit of a previous work by the third author [A double large deviation principle for Monge-Ampère gravitation, 2016]. The main step in this derivation is the $Γ-$convergence of the good rate functions corresponding to a one-parameter family of large deviation principles. Surprisingly, the derived model includes dissipative phenomena. As an illustration, we show that it leads to sticky collisions in one space dimension.

math.OC↗
arXiv2020

Minimizing relative entropy of path measures under marginal constraints

We study generalizations of the Schrödinger problem in statistical mechanics in two directions: when the density is constrained at more than two times, and when the joint law of the initial and final positions for the particles is prescribed. This is done in agreement with the so-called Brödinger problem recently introduced to regularize Brenier's variational model for incompressible fluids. We recover generalizations of the standard factorization result for the Radon-Nikodym derivative of the solution $P$ with respect to the reference measure $R$: this density can be written in terms of an additive functional on the set of constrained times. The specificity of this work is that we place ourselves in the case when $R$ is Markov (or reciprocal), and that we use Markovian methods rather than classical convex analysis arguments. In this setting, it appears that a natural assumption to be made on the reference measure $R$ is of irreducibility type.

math.PR↗
arXiv2019

Small noise limit and convexity for generalized incompressible flows, Schrödinger problems, and optimal transport

This paper is concerned with six variational problems and their mutual connections: The quadratic Monge-Kantorovich optimal transport, the Schrödinger problem, Brenier's relaxed model for incompressible fluids, the so-called Brödinger problem recently introduced by M. Arnaudon & al. [3], the multiphase Brenier model, and the multiphase Brödinger problem. All of them involve the minimization of a kinetic action and/or a relative entropy of some path measures with respect to the reversible Brownian motion. As the viscosity parameter $ν\to 0$ we establish Gamma-convergence relations between the corresponding problems, and prove the convergence of the associated pressures arising from the incompressibility constraints. We also present new results on the time-convexity of the entropy for some of the dynamical interpolations. Along the way we extend previous results by H. Lavenant [30] and J-D. Benamou & al. [10].

math.AP↗
arXiv2019

On the existence of a scalar pressure field in the Brödinger problem

This work deals with the entropic regularization of the Brenier problem for perfect incompressible fluids introduced by Arnaudon, Cruzeiro, Léonard and Zambrini. We show that as in the original setting, there exists a scalar pressure field which is interpreted as the Lagrange multiplier associated to the incompressibility constraint. The proof goes through a reformulation of the problem in PDE terms.

math.AP↗
arXiv2018

Nonlinear instability in Vlasov type equations around rough velocity profiles

In the Vlasov-Poisson equation, every configuration which is homogeneous in space provides a stationary solution. Penrose gave in 1960 a criterion for such a configuration to be linearly unstable. While this criterion makes sense in a measure-valued setting, the existing results concerning nonlinear instability always suppose some regularity with respect to the velocity variable. Here, thanks to a multiphasic reformulation of the problem, we can prove an "almost Lyapounov instability" result for the Vlasov-Poisson equation, and an ill-posedness result for the kinetic Euler equation and the Vlasov-Benney equation (two quasineutral limits of the Vlasov-Poisson equation), both around any unstable measure.

math.AP↗