80 research outputs found
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
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
Metric measure spaces with Riemannian Ricci curvature bounded from below
In this paper we introduce a synthetic notion of Riemannian Ricci bounds from
below for metric measure spaces (X,d,m) which is stable under measured
Gromov-Hausdorff convergence and rules out Finsler geometries. It can be given
in terms of an enforcement of the Lott, Sturm and Villani geodesic convexity
condition for the entropy coupled with the linearity of the heat flow. Besides
stability, it enjoys the same tensorization, global-to-local and
local-to-global properties. In these spaces, that we call RCD(K,\infty) spaces,
we prove that the heat flow (which can be equivalently characterized either as
the flow associated to the Dirichlet form, or as the Wasserstein gradient flow
of the entropy) satisfies Wasserstein contraction estimates and several
regularity properties, in particular Bakry-Emery estimates and the L^\infty-Lip
Feller regularization. We also prove that the distance induced by the Dirichlet
form coincides with d, that the local energy measure has density given by the
square of Cheeger's relaxed slope and, as a consequence, that the underlying
Brownian motion has continuous paths. All these results are obtained
independently of Poincar\'e and doubling assumptions on the metric measure
structure and therefore apply also to spaces which are not locally compact, as
the infinite-dimensional onesComment: (v2) Minor typos, proof of Proposition 2.3, proof of Theorem 4.8:
corrected. Proof of Theorem 6.2: corrected and simplified, thanks to the new
Lemma 2.8. Lemma 3.6 and 4.6 (of v1) removed, since no more neede
- …