Self-improvement of the Bakry-\'Emery condition and Wasserstein contraction of the heat flow in RCD(K,\infty) metric measure spaces

We prove that the linear heat flow in a RCD(K,\infty) metric measure space (X,d,m) satisfies a contraction property with respect to every L^p-Kantorovich-Rubinstein-Wasserstein distance. In particular, we obtain a precise estimate for the optimal W_\infty-coupling between two fundamental solutions in terms of the distance of the initial points. The result is a consequence of the equivalence between the RCD(K,\infty) lower Ricci bound and the corresponding Bakry-\'Emery condition for the canonical Cheeger-Dirichlet form in (X,d,m). The crucial tool is the extension to the non-smooth metric measure setting of the Bakry's argument, that allows to improve the commutation estimates between the Markov semigroup and the Carr\'e du Champ associated to the Dirichlet form. This extension is based on a new a priori estimate and a capacitary argument for regular and tight Dirichlet forms that are of independent interest.Comment: (v2) Minor corrections. A discussion of quasi-regular Dirichlet forms has been added (Section 2.3) to cover the case of a sigma-finite reference measure. The proof of the quasi regularity of the Cheeger energy has been added (Thm. 4.1

Lecture Notes on Gradient Flows and Optimal Transport

We present a short overview on the strongest variational formulation for gradient flows of geodesically $\lambda$-convex functionals in metric spaces, with applications to diffusion equations in Wasserstein spaces of probability measures. These notes are based on a series of lectures given by the second author for the Summer School "Optimal transportation: Theory and applications" in Grenoble during the week of June 22-26, 2009

Optimal transport in competition with reaction: the Hellinger-Kantorovich distance and geodesic curves

We discuss a new notion of distance on the space of finite and nonnegative measures which can be seen as a generalization of the well-known Kantorovich-Wasserstein distance. The new distance is based on a dynamical formulation given by an Onsager operator that is the sum of a Wasserstein diffusion part and an additional reaction part describing the generation and absorption of mass. We present a full characterization of the distance and its properties. In fact the distance can be equivalently described by an optimal transport problem on the cone space over the underlying metric space. We give a construction of geodesic curves and discuss their properties

From Poincar\'e to logarithmic Sobolev inequalities: a gradient flow approach

We use the distances introduced in a previous joint paper to exhibit the gradient flow structure of some drift-diffusion equations for a wide class of entropy functionals. Functional inequalities obtained by the comparison of the entropy with the entropy production functional reflect the contraction properties of the flow. Our approach provides a unified framework for the study of the Kolmogorov-Fokker-Planck (KFP) equation

Balanced-Viscosity solutions for multi-rate systems

Several mechanical systems are modeled by the static momentum balance for the displacement $u$ coupled with a rate-independent flow rule for some internal variable $z$. We consider a class of abstract systems of ODEs which have the same structure, albeit in a finite-dimensional setting, and regularize both the static equation and the rate-independent flow rule by adding viscous dissipation terms with coefficients $\varepsilon^\alpha$ and $\varepsilon$, where $00$ is a fixed parameter. Therefore for $\alpha \neq 1$ $u$ and $z$ have different relaxation rates. We address the vanishing-viscosity analysis as $\varepsilon \downarrow 0$ of the viscous system. We prove that, up to a subsequence, (reparameterized) viscous solutions converge to a parameterized curve yielding a Balanced Viscosity solution to the original rate-independent system, and providing an accurate description of the system behavior at jumps. We also give a reformulation of the notion of Balanced Viscosity solution in terms of a system of subdifferential inclusions, showing that the viscosity in $u$ and the one in $z$ are involved in the jump dynamics in different ways, according to whether $\alpha>1$, $\alpha=1$, and $\alpha \in (0,1)$

On the duality between p-Modulus and probability measures

Motivated by recent developments on calculus in metric measure spaces $(X,\mathsf d,\mathfrak m)$, we prove a general duality principle between Fuglede's notion of $p$-modulus for families of finite Borel measures in $(X,\mathsf d)$ and probability measures with barycenter in $L^q(X,\mathfrak m)$, with $q$ dual exponent of $p\in (1,\infty)$. We apply this general duality principle to study null sets for families of parametric and non-parametric curves in $X$. In the final part of the paper we provide a new proof, independent of optimal transportation, of the equivalence of notions of weak upper gradient based on $p$-Modulus (Koskela-MacManus '98, Shanmugalingam '00) and suitable probability measures in the space of curves (Ambrosio-Gigli-Savare '11)Comment: Minor corrections, typos fixe