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Author Archives: Fabrice Baudoin
HW5. MA3160, Due 10/07/2021
Exercise 1. Three balls are randomly chosen with replacement from an urn containing 2 blue, 3 red, and 2 yellow balls. Let denote the number of red balls chosen. What are the possible values of ? What is the density … Continue reading
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H-type structures
In two works in collaboration with Erlend Grong, Gianmarco Molino and Luca Rizzi, we introduced the notion of H-type structure: H-type foliations Comparison theorems on H-type sub-Riemannian manifolds Consider a triple where is a manifold, a constant rank sub-bundle of … Continue reading
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Lecture notes: An Introduction to Dirichlet Spaces
Peter Gustav Lejeune Dirichlet An introduction to Dirichlet spaces: Lecture notes The outline of those (unpolished) lecture notes is the following: Chapter 1: Semigroups Chapter 2: Markovian semigroups and Dirichlet forms Chapter 3: Dirichlet spaces with Gaussian or sub-Gaussian heat … Continue reading
Lecture 8. Sobolev inequalities on Dirichlet spaces: The Varopoulos approach
In this lecture, we study Sobolev inequalities on Dirichlet spaces. The approach we develop is related to Hardy-Littlewood-Sobolev theory The link between the Hardy-Littlewood-Sobolev theory and heat kernel upper bounds is due to Varopoulos, but the proof I give below I … Continue reading
Lecture 7. Diffusion operators as Markovian operators
In this section, we consider a diffusion operator where and are continuous functions on and for every , the matrix is a symmetric and non negative matrix. Our goal is to prove that if is essentially self-adjoint, then the semigroup … Continue reading
Lecture 6. The Lp theory of Markovian semigroups
Our goal, in this lecture, is to define, for , on . This may be done in a natural way by using the Riesz-Thorin interpolation theorem that we recall below. Theorem: [Riesz-Thorin interpolation theorem] Let , and . Define by … Continue reading
Lecture 5. Dirichlet forms
As in the previous lecture, let be a good measurable space equipped with a -finite measure . Definition: A function on is called a normal contraction of the function if for almost every , Definition: Let be a densely defined … Continue reading
Lecture 4. Markovian semigroups
Let be a measurable space. We say that is a good measurable space if there is a countable family generating and if every finite measure on can be decomposed as where is the projection of on the first coordinate and … Continue reading
Lecture 3. Diffusion operators
In this lecture, we illustrate some of the concepts introduced earlier in the context of diffusion operators in . Throughout the lecture, we consider a second order differential operator that can be written where and are continuous functions on and … Continue reading
Lecture 2. Semigroups on Hilbert spaces: The golden triangle
Let be a Hilbert space and let be a densely defined operator on a domain . We have the following basic definitions. The operator is said to be symmetric if for , The operator is said to be non negative … Continue reading
Research and Lecture notes 