• Corpus ID: 229349054

The dynamical Schr\"odinger problem in abstract metric spaces

  title={The dynamical Schr\"odinger problem in abstract metric spaces},
  author={L{\'e}onard Monsaingeon and Luca Tamanini and Dmitry Vorotnikov},
In this paper we introduce the dynamical Schrödinger 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 towards the geodesic problem as the noise parameter ε ↓ 0. We also investigate the connection with geodesic convexity of the driving entropy, and study the dependence of the entropic cost on the parameter ε. Some examples and applications are discussed. MSC [2020… 

Figures from this paper

Stability of a class of action functionals depending on convex functions
<p style='text-indent:20px;'>We study the stability of a class of action functionals induced by gradients of convex functions with respect to Mosco convergence, under mild assumptions on the
Viscosity solutions of Hamilton-Jacobi equation in $RCD(K,\infty)$ spaces and applications to large deviations
The aim of this paper is twofold. In the setting of RCD(K,∞) metric measure spaces, we derive uniform gradient and Laplacian contraction estimates along solutions of the viscous approximation of the
Convex functions defined on metric spaces are pulled back to subharmonic ones by harmonic maps
If u : Ω ⊂ Rd → X is a harmonic map valued in a metric space X and E : X→ R is a convex function, in the sense that it generates an EVI0-gradient flow, we prove that the pullback E ◦ u : Ω → R is
Hamilton--Jacobi equations for controlled gradient flows: the comparison principle
Motivated by recent developments in the fields of large deviations for interacting particle system and mean field control, we establish a comparison principle for the Hamilton–Jacobi equation
Schrödinger encounters Fisher and Rao: a survey
This short note reviews the dynamical Schrödinger problem on the non-commutative Fisher-Rao space of positive semi-definite matrixvalued measures, and discusses in particular connections with Gaussian optimal transport, entropy, and quantum Fisher information.


A Convexity Principle for Interacting Gases
A new set of inequalities is introduced, based on a novel but natural interpolation between Borel probability measures on R d . Using these estimates in lieu of convexity or rearrangement
The mean field Schrödinger problem: ergodic behavior, entropy estimates and functional inequalities
We study the mean field Schrödinger problem (MFSP), that is the problem of finding the most likely evolution of a cloud of interacting Brownian particles conditionally on the observation of their
Orlicz Space Regularization of Continuous Optimal Transport Problems
In this work we analyze regularized optimal transport problems in the so-called Kantorovich form, i.e. given two Radon measures on two compact sets, the aim is to find a transport plan, which is
Optimal Transport losses and Sinkhorn algorithm with general convex regularization
A new class of convex-regularizedOptimal Transport losses is introduced, which generalizes the classical Entropy-regularization of Optimal Transport and Sinkhorn divergences, and a generalized Sinkinghorn algorithm is proposed, which unifies many regularizations and numerical methods previously appeared in the literature.
Eulerian Calculus for the Contraction in the Wasserstein Distance
This proof is based on the insight that the porous medium equation does not increase the size of infinitesimal perturbations along gradient flow trajectories and on an Eulerian formulation for the Wasserstein distance using smooth curves.
A formula for the time derivative of the entropic cost and applications
Second order differentiation formula on $RCD^*(K,N)$ spaces
Aim of this paper is to prove the second order differentiation formula for $H^{2,2}$ functions along geodesics in $RCD^*(K,N)$ spaces with $N < \infty$. This formula is new even in the context of
Optimal Transport
The globalization theorem for the Curvature-Dimension condition
The Lott–Sturm–Villani Curvature-Dimension condition provides a synthetic notion for a metric-measure space to have Ricci-curvature bounded from below and dimension bounded from above. We prove that
An Optimal Transport Approach for the Schrödinger Bridge Problem and Convergence of Sinkhorn Algorithm
This paper exploit the equivalence between the Schrodinger Bridge problem and the entropy penalized optimal transport in order to find a different approach to the duality, in the spirit of optimal