Category Archives: Diffusions on manifolds

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 therefore only need to prove it … Continue reading →

In this lecture we prove that most of the results that were proven for Laplace-Beltrami operators may actually be generalized to any locally subelliptic operator. Let be a locally subelliptic diffusion operator defined on . For every smooth functions , … 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 manifold case. … Continue reading →

In the next few lectures, we will show that the diffusion semigroups theory we developed may actually be extended without difficulties to a manifold setting. As a motivation, we start with a very simple example. We first study the heat … 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 this Lecture, we use the local regularity theory of subelliptic operators, to prove the existence of heat kernels. Proposition: Let be a locally  subelliptic diffusion operator with smooth coefficients that is essentially self-adjoint with respect to a measure . Denote … Continue reading →

Exercise: Show that if is the Laplace operator on , then for ,   Exercise: Let be an essentially self-adjoint diffusion operator on . Show that if the constant function and if , then   Exercise: Let be an essentially self-adjoint diffusion operator … Continue reading →

For geometric purposes, it is often very useful to use the language of vector fields to study diffusion operators. Let be a non-empty open set. A smooth vector field   on is a smooth map We will often regard a … Continue reading →

In this lecture, we show that the diffusion semigroup that was constructed in the previous lectures appears as the solution of a parabolic Cauchy problem. Under an ellipticity and completeness assumption, it is moreover the unique square integrable solution. Proposition: Let be an … Continue reading →

Exercise 1. Let be a linear operator such that: is a local operator, that is if on a neighborhood of then ; If has a global maximum at with then . Show that for and , where , and are continuous … Continue reading →