Dominique Bakry

Harnack inequalities on a manifold with positive or negative Ricci curvature

Several new Harnack estimates for positive solutions of the heat equation on a complete Riemannian manifold with Ricci curvature bounded below by a positive (or a negative) constant are established. These estimates are sharp both for small time, for large time and for large distance, and lead to new estimates for the heat kernel of a manifold with Ricci curvature bounded below

Characterization of Markov semigroups on ℝ Associated to Some Families of Orthogonal Polynomials

We give a characterization of the eigenvalues of Markov operators which admit an orthogonal polynomial basis as eigenfunctions, in the Hermite and the Laguerre cases, as well as for the sequences of orthogonal polynomials associated to some probability measures on ℕ. In the Hermite case, we also give a description of the path of the associated Markov processes, as well as a geometric interpretation.

Weighted Nash Inequalities

Nash or Sobolev inequalities are known to be equivalent to ultracontractive properties of Markov semigroups, hence to uniform bounds on their kernel densities. In this work we present a simple and extremely general method, based on weighted Nash inequalities, to obtain non-uniform bounds on the kernel densities. Such bounds imply a control on the trace or the Hilbert-Schmidt norm of the heat kernels. We illustrate the method on the heat kernel on

Dimension dependent hypercontractivity for Gaussian kernels

\dR

On Harnack estimates for positive solutions of the heat equation on a complete manifold

naturally associated with the measure with density

Symmetric Diffusions with Polynomial Eigenvectors

C_a\exp(-|x|^a)

Weak Sobolev Inequalities

, with

Dyson Processes Associated with Associative Algebras: The Clifford Case

1

Symmetric Markov Diffusion Operators

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 Lévy semigroup are also investigated. Curvature-dimension criteria are the main tool in the analysis.

h -Transforms and Orthogonal Polynomials

Abstract We establish several new Harnack estimates for the nonnegative solutions of the heat equation on a complete Riemannian manifold with Ricci curvature bounded by a positive or negative constant. This extends to symmetric diffusions whose generator satisfies a “curvature-dimension” inequality.