Category Archives: Curvature dimension inequalities

Let be a complete -dimensional Riemannian manifold and denote by its Laplace-Beltrami operator. As usual, we denote by the heat semigroup generated by . Throughout the Lecture, we will assume that the Ricci curvature of is bounded from below by … Continue reading →

In this Lecture, we will prove a first interesting consequence of the Bochner’s identity: We will prove that if, on a complete Riemannian manifold , the Ricci curvature is bounded from below, then the heat semigroup is stochastically complete, that … Continue reading →

The goal of this lecture is to prove the Bochner formula: A fundamental formula that relates the so-called Ricci curvature of the underlying Riemannian structure to the analysis of the Laplace–Beltrami operator. The Bochner’s formula is a local formula, we … Continue reading →

In this section we shall consider a smooth and complete Riemannian manifold with dimension . The Riemannian measure will be denoted by . The Laplace-Beltrami of will be denoted by . Since is assumed to be complete, as we have … Continue reading →

In this lecture we extend the previous results in the framework of smooth manifolds. The main idea to extend those results is that, similar computations may be performed in local coordinates charts and then we use a partition of unity. … Continue reading →

In order to apply the diffusion semigroup theory developed in the first lectures and construct without ambiguity the semigroup associated to the Laplace-Beltrami operator L, we need to know if L is essentially self-adjoint. Interestingly, this property of essential self-adjointness … Continue reading →

In this lecture we define Riemannian structures and corresponding Laplace–Beltrami operators. We first study Riemannian structures on Rn to avoid technicalities in the presentation of the main ideas and then, in a later lecture, will extend our results to the … Continue reading →

We now turn to a new part in this course. The first few lectures were devoted to the study of diffusion operators and the construction of associated semigroups. The goal of this new part of the course will be to … Continue reading →

In the previous lectures, we have seen that if L is an essentially self-adjoint diffusion operator with respect to a measure, then by using the spectral theorem one can define a self-adjoint strongly continuous contraction semigroup on L2 with generator … Continue reading →

In the previous lectures, we have proved that if L is a diffusion operator that is essentially self-adjoint then, by using the spectral theorem, we can define a self-adjoint strongly continuous contraction semigroup with generator L and this semigroup is … Continue reading →