Dimensional contraction via Markov transportation distance

It is now well known that curvature conditions \`a la Bakry-Emery are equivalent to contraction properties of the heat semigroup with respect to the classical quadratic Wasserstein distance. However, this curvature condition may include a dimensional correction which up to now had not induced any strenghtening of this contraction. We first consider the simplest example of the Euclidean heat semigroup, and prove that indeed it is so. To consider the case of a general Markov semigroup, we introduce a new distance between probability measures, based on the semigroup, and adapted to it. We prove that this Markov transportation distance satisfies the same properties for a general Markov semigroup as the Wasserstein distance does in the specific case of the Euclidean heat semigroup, namely dimensional contraction properties and Evolutional variational inequalities

Calculus, heat flow and curvature-dimension bounds in metric measure spaces

The theory of curvature-dimension bounds for nonsmooth spaces has several motivations: the study of functional and geometric inequalities in structures which arc very far from being Euclidean, therefore with new non-Riemannian tools, the description of the \u201cclosure\u201d of classes of Riemannian manifolds under suitable geometric constraints, the stability of analytic and geometric properties of spaces (e.g. to prove rigidity results). Even though these goals may occasionally be in conflict, in the last few years we have seen spectacular developments in all these directions, and my text is meant both as a survey and as an introduction to this quickly developing research field

Heat Kernels, Stochastic Processes and Functional Inequalities

The general topic of the 2013 workshop Heat kernels, stochastic processes and functional inequalities was the study of linear and non-linear diffusions in geometric environments: finite and infinite-dimensional manifolds, metric spaces, fractals and graphs, including random environments. The workshop brought together leading researchers from analysis, probability and geometry and provided a unique opportunity for interaction of established and young scientists from these areas. Unifying themes were heat kernel analysis, mass transport problems and related functional inequalities such as Poincar´e, Sobolev, logarithmic Sobolev, Bakry-Emery, Otto-Villani and Talagrand inequalities. These concepts were at the heart of Perelman’s proof of Poincar´e’s conjecture, as well as of the development of the Otto calculus, and the synthetic Ricci bounds of Lott-Sturm-Villani. The workshop provided participants with an opportunity to discuss how these techniques can be used to approach problems in optimal transport for non-local operators, subelliptic operators in finite and infinite dimensions, analysis on singular spaces, as well as random walks in random media

Generalized Bakry-\'Emery curvature condition and equivalent entropic inequalities in groups

We study a generalization of the Bakry-\'Emery pointwise gradient estimate for the heat semigroup and its equivalence with some entropic inequalities along the heat flow and Wasserstein geodesics for metric-measure spaces with a suitable group structure. Our main result applies to Carnot groups of any step and to the $\mathbb{SU}(2)$ group.Comment: 76 page