Author Archives: Fabrice Baudoin

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 →

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: … Continue reading →

The goal of this lecture is to prove that if a diffusion operator L is elliptic, then the semigroup it generates admits a smooth kernel. As a consequence, the semigroup generated by an elliptic diffusion operator is regularizing in the … Continue reading →

In this lecture, we consider a diffusion operator L which is essentially self-adjoint. Its Friedrichs extension is still denoted by L. The fact that we are now dealing with a non negative self-adjoint operator allows us to use spectral theory … Continue reading →

The goal of the next few lectures will be to introduce the semigroup generated by a diffusion operator. This semigroup will play pervasive role throughout these lectures and is the main tool associated to the curvature dimension inequalities. The construction … Continue reading →

In this first lecture we introduce the main characters of this course: The diffusion operators. Definition: A differential operator on , is called a diffusion operator if it can be written where and are continuous functions on and if for … Continue reading →

Next Fall, I will teach a graduate course on curvature dimension inequalities, and, as usual, the Lectures will be posted on this blog. The theory of curvature dimension inequalities and of their applications to the geometric analysis of manifolds is, … Continue reading →