Related papers: Minimizing relative entropy of path measures under…
This paper is concerned with six variational problems and their mutual connections: The quadratic Monge-Kantorovich optimal transport, the Schr\"odinger problem, Brenier's relaxed model for incompressible fluids, the so-called Br\"odinger…
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 Schr\"odinger problem by…
Motivated by a problem from incompressible fluid mechanics of Brenier (JAMS 1989), and its recent entropic relaxation by Arnaudo, Cruizero, L\'eonard & Zambrini (AIHP PS 2020), we study a problem of entropic minimization on the path space…
We consider the problem of minimizing a generalized relative entropy, with respect to a reference diffusion law, over the set of path-measures with fully prescribed marginal distributions. When dealing with the actual relative entropy,…
We describe some analogy between optimal transport and the Schr\"odinger problem where the transport cost is replaced by an entropic cost with a reference path measure. A dual Kantorovich type formulation and a Benamou-Brenier type…
We propose and study a system of Schr\"odinger's problems and functional equations in probability theory. More precisely, we consider a system of variational problems of relative entropies for probability measures on a Euclidean space with…
This paper introduces a dynamic formulation of divergence-regularized optimal transport with weak targets on the path space. In our formulation, the classical relative entropy penalty is replaced by a general convex divergence, and terminal…
We study the least-energy way to reshape a probability distribution when motion is constrained to a horizontal bundle, that is, optimal transport and distribution steering in sub-Riemannian geometry, motivated by density control over…
We study the transport of Gaussian measures under the flow of the 2-dimensional defocusing Schr\"odinger equation $i \partial_t u + \Delta u = |u|^{2k} u$ posed on $\mathbb T^2$. In particular, we show that the Gaussian measures with…
In view of studying incompressible inviscid fluids, Brenier introduced in the late 80's a relaxation of a geodesic problem addressed by Arnold in 1966. Instead of inviscid fluids, the present paper is devoted to incompressible viscid…
This work studies in detail the possibility of defining a one-to-one mapping from charge densities as obtained by the time-dependent Schr\"odinger equation to external potentials. Such a mapping is provided by the Runge-Gross theorem and…
To overcome topological constraints and improve the expressiveness of normalizing flow architectures, Wu, K\"ohler and No\'e introduced stochastic normalizing flows which combine deterministic, learnable flow transformations with stochastic…
The effect of the relative entropy asymmetry is analyzed in the empirical risk minimization with relative entropy regularization (ERM-RER) problem. A novel regularization is introduced, coined Type-II regularization, that allows for…
We discuss a connection (and a proper place in this framework) of the unforced and deterministically forced Burgers equation for local velocity fields of certain flows, with probabilistic solutions of the so-called Schr\"{o}dinger…
Consider the problem of matching two independent i.i.d. samples of size $N$ from two distributions $P$ and $Q$ in $\mathbb{R}^d$. For an arbitrary continuous cost function, the optimal assignment problem looks for the matching that…
Radon-Nikodym (RN) derivative between two measures arises naturally in the affine structure of the space of probability measures with densities. Entropy, free energy, relative entropy, and entropy production as mathematical concepts…
In this paper we introduce the dynamical Schr\"odinger problem on abstract metric spaces, defined for a wide class of entropy and Fisher information functionals. Under very mild assumptions we prove a generic Gamma-convergence result…
We consider the problem to identify the most likely flow in phase space, of (inertial) particles under stochastic forcing, that is in agreement with spatial (marginal) distributions that are specified at a set of points in time. The…
We consider ergodic random magnetic Schr\"odinger operators on the metric graph $\mathbb{Z}^d$ with random potentials and random boundary conditions taking values in a finite set. We show that normalized finite volume eigenvalue counting…
We studied the Benamou--Brenier formulation of the Schr\"odinger problem, focusing on a gap between theoretical results and applications, that often involve measures with unbounded support. While the existing proof in the literature relies…