128 citations · 197 across the 16 of their papers we have counts for
27 papers
Synthetic approaches to Ricci flows
Matthias Erbar, Marco Flaim, Eric Hupp +3
We review different notions of synthetic Ricci flow that apply to time-dependent families of metric measure spaces and which are based on properties of the heat flow, ideas from op…
Synthetic notions of Ricci flow for metric measure spaces
Matthias Erbar, Zhenhao Li, Timo Schultz
We develop different synthetic notions of Ricci flow in the setting of time-dependent metric measure spaces based on ideas from optimal transport. They are formulated in terms of d…
Optimal transport of stationary point processes: Metric structure, gradient flow and convexity of the specific entropy
Matthias Erbar, Martin Huesmann, Jonas Jalowy +1
We develop a theory of optimal transport for stationary random measures with a focus on stationary point processes and construct a family of distances on the set of stationary rand…
Approximation of Splines in Wasserstein Spaces
Jorge Justiniano, Martin Rumpf, Matthias Erbar
This paper investigates a time discrete variational model for splines in Wasserstein spaces to interpolate probability measures. Cubic splines in Euclidean space are known to minim…
Covariance-modulated optimal transport and gradient flows
Martin Burger, Matthias Erbar, Franca Hoffmann +2
We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the distribution. Such transport metrics…
Gradient flow formulation of diffusion equations in the Wasserstein space over a metric graph
Matthias Erbar, Dominik Forkert, Jan Maas +1
This paper contains two contributions in the study of optimal transport on metric graphs. Firstly, we prove a Benamou-Brenier formula for the Wasserstein distance, which establishe…
Tamed spaces -- Dirichlet spaces with distribution-valued Ricci bounds
Matthias Erbar, Chiara Rigoni, Karl-Theodor Sturm +1
We develop the theory of tamed spaces which are Dirichlet spaces with distribution-valued lower bounds on the Ricci curvature and investigate these from an Eulerian point of view.…
Entropic curvature and convergence to equilibrium for mean-field dynamics on discrete spaces
Matthias Erbar, Max Fathi, André Schlichting
We consider non-linear evolution equations arising from mean-field limits of particle systems on discrete spaces. We investigate a notion of curvature bounds for these dynamics bas…
The one-dimensional log-gas free energy has a unique minimiser
Matthias Erbar, Martin Huesmann, Thomas Leblé
We prove that, at every positive temperature, the infinite-volume free energy of the one dimensional log-gas, or beta-ensemble, has a unique minimiser, which is the Sine-beta proce…
On the geometry of geodesics in discrete optimal transport
Matthias Erbar, Jan Maas, Melchior Wirth
We consider the space of probability measures on a discrete set , endowed with a dynamical optimal transport metric. Given two probability measures supported in a subset $Y \sub…
Super Ricci flows for weighted graphs
Matthias Erbar, Eva Kopfer
We present a notion of super Ricci flow for time-dependent finite weighted graphs. A challenging feature is that these flows typically encounter singularities where the underlying…
Rigidity of cones with bounded Ricci curvature
Matthias Erbar, Karl-Theodor Sturm
We show that the only metric measure space with the structure of an -cone and with two-sided synthetic Ricci bounds is the Euclidean space for integer. T…
Computation of Optimal Transport on Discrete Metric Measure Spaces
Matthias Erbar, Martin Rumpf, Bernhard Schmitzer +1
In this paper we investigate the numerical approximation of an analogue of the Wasserstein distance for optimal transport on graphs that is defined via a discrete modification of t…
Poincaré, modified logarithmic Sobolev and isoperimetric inequalities for Markov chains with non-negative Ricci curvature
Matthias Erbar, Max Fathi
We study functional inequalities for Markov chains on discrete spaces with entropic Ricci curvature bounded from below. Our main results are that when curvature is non-negative, bu…
A gradient flow approach to the Boltzmann equation
Matthias Erbar
We show that the spatially homogeneous Boltzmann equation evolves as the gradient flow of the entropy with respect to a suitable geometry on the space of probability measures which…
Smoothing and non-smoothing via a flow tangent to the Ricci flow
Matthias Erbar, Nicolas Juillet
We study a transformation of metric measure spaces introduced by Gigli and Mantegazza consisting in replacing the original distance with the length distance induced by the transpor…
Ricci curvature bounds for weakly interacting Markov chains
Matthias Erbar, Christopher Henderson, Georg Menz +1
We establish a general perturbative method to prove entropic Ricci curvature bounds for interacting stochastic particle systems. We apply this method to obtain curvature bounds in…
Gradient flow structure for McKean-Vlasov equations on discrete spaces
Matthias Erbar, Max Fathi, Vaios Laschos +1
In this work, we show that a family of non-linear mean-field equations on discrete spaces can be viewed as a gradient flow of a natural free energy functional with respect to a cer…
Optimal transport, Cheeger energies and contractivity of dynamic transport distances in extended spaces
Luigi Ambrosio, Matthias Erbar, Giuseppe Savaré
We introduce the setting of extended metric-topological measure spaces as a general "Wiener-like" framework for optimal transport problems and nonsmooth metric analysis in infinite…
From large deviations to Wasserstein gradient flows in multiple dimensions
Matthias Erbar, Jan Maas, Michiel Renger
We study the large deviation rate functional for the empirical distribution of independent Brownian particles with drift. In one dimension, it has been shown by Adams, Dirr, Peleti…
Discrete Ricci curvature bounds for Bernoulli-Laplace and random transposition models
Matthias Erbar, Jan Maas, Prasad Tetali
We calculate a Ricci curvature lower bound for some classical examples of random walks, namely, a chain on a slice of the n-dimensional discrete cube (the so-called Bernoulli-Lapla…
Curvature bounds for configuration spaces
Matthias Erbar, Martin Huesmann
We show that the configuration space over a manifold M inherits many curvature properties of the manifold. For instance, we show that a lower Ricci curvature bound on M implies for…
On the Equivalence of the Entropic Curvature-Dimension Condition and Bochner's Inequality on Metric Measure Spaces
Matthias Erbar, Kazumasa Kuwada, Karl-Theodor Sturm
We prove the equivalence of the curvature-dimension bounds of Lott-Sturm-Villani (via entropy and optimal transport) and of Bakry--Émery (via energy and Γ_2$-calculus) in complete…
Gradient flow structures for discrete porous medium equations
Matthias Erbar, Jan Maas
We consider discrete porous medium equations of the form \partial_t ρ_t = Δϕ(ρ_t), where Δis the generator of a reversible continuous time Markov chain on a finite set X, and ϕis a…