103 research outputs found
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 -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 coupled with a rate-independent flow rule for some internal
variable . 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 and ,
where is a fixed parameter. Therefore for
and have different relaxation rates.
We address the vanishing-viscosity analysis as 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 and the one in
are involved in the jump dynamics in different ways, according to whether
, , and
On the duality between p-Modulus and probability measures
Motivated by recent developments on calculus in metric measure spaces
, we prove a general duality principle between
Fuglede's notion of -modulus for families of finite Borel measures in
and probability measures with barycenter in , with dual exponent of . We apply this general duality
principle to study null sets for families of parametric and non-parametric
curves in . 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 -Modulus (Koskela-MacManus '98, Shanmugalingam '00)
and suitable probability measures in the space of curves (Ambrosio-Gigli-Savare
'11)Comment: Minor corrections, typos fixe
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