2,139 research outputs found
Entropies, convexity, and functional inequalities
Our aim is to provide a short and self contained synthesis which generalise
and unify various related and unrelated works involving what we call
Phi-Sobolev functional inequalities. Such inequalities related to Phi-entropies
can be seen in particular as an inclusive interpolation between Poincare and
Gross logarithmic Sobolev inequalities. In addition to the known material,
extensions are provided and improvements are given for some aspects. Stability
by tensor products, convolution, and bounded perturbations are addressed. We
show that under simple convexity assumptions on Phi, such inequalities hold in
a lot of situations, including hyper-contractive diffusions, uniformly strictly
log-concave measures, Wiener measure (paths space of Brownian Motion on
Riemannian Manifolds) and generic Poisson space (includes paths space of some
pure jumps Levy processes and related infinitely divisible laws). Proofs are
simple and relies essentially on convexity. We end up by a short parallel
inspired by the analogy with Boltzmann-Shannon entropy appearing in Kinetic
Gases and Information Theories.Comment: Formerly "On Phi-entropies and Phi-Sobolev inequalities". Author's
www homepage: http://www.lsp.ups-tlse.fr/Chafai
Dimension dependent hypercontractivity for Gaussian kernels
We derive sharp, local and dimension dependent hypercontractive bounds on the
Markov kernel of a large class of diffusion semigroups. Unlike the dimension
free ones, they capture refined properties of Markov kernels, such as trace
estimates. They imply classical bounds on the Ornstein-Uhlenbeck semigroup and
a dimensional and refined (transportation) Talagrand inequality when applied to
the Hamilton-Jacobi equation. Hypercontractive bounds on the Ornstein-Uhlenbeck
semigroup driven by a non-diffusive L\'evy semigroup are also investigated.
Curvature-dimension criteria are the main tool in the analysis.Comment: 24 page
About the analogy between optimal transport and minimal entropy
We describe some analogy between optimal transport and the Schr\"odinger
problem where the transport cost is replaced by an entropic cost with a
reference path measure. A dual Kantorovich type formulation and a
Benamou-Brenier type representation formula of the entropic cost are derived,
as well as contraction inequalities with respect to the entropic cost. This
analogy is also illustrated with some numerical examples where the reference
path measure is given by the Brownian or the Ornstein-Uhlenbeck process. Our
point of view is measure theoretical and the relative entropy with respect to
path measures plays a prominent role
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