# Category Archives: Diffusions on manifolds

## Lecture 13. The Bochner’s formula

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

## Lecture 12. The distance associated to subelliptic diffusion operators

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

## Lecture 11. Laplace-Beltrami operators on Rn

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

## Lecture 10. The heat semigroup on the circle

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

## Lecture 9. Diffusion semigroups in L^p

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

## Lecture 7. Heat kernels of subelliptic semigroups

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

## HW2. Due September 27

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