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Category Archives: Dirichlet forms at NYU
Lecture notes: A short introduction to Dirichlet spaces
Here are the lecture notes of the minicourse given at NYU Abu Dhabi.
Lecture 7. Further topics
In this lecture let X be a locally compact and complete metric space equipped with a Radon measure μ supported on X. Let (ℰ, ℱ = dom(ℰ)) be a Dirichlet form on X. We assume throughout that the heat semigroup … Continue reading
Posted in Dirichlet forms at NYU
Tagged Dirichlet forms, Hunt process, Regular Dirichlet form
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Lecture 6. Gagliardo-Nirenberg inequalities in Dirichlet spaces
Let (X,μ,ℰ,ℱ) be a Dirichlet space and {Pt}t∈[0,∞) denote the associated Markovian semigroup. Throughout the lecture, we shall assume that P_t admits a measurable heat kernel pt(x,y) satisfying, for some C > 0 and β > 0, for μ × … Continue reading
Lecture 5. Examples of Dirichlet spaces
Contents Riemannian manifolds Carnot groups Sierpinski gasket Cheeger space Riemannian manifolds Let (M,g) be an n-dimensional Riemannian manifold with Riemannian volume measure μ and Riemannian distance d. We consider the quadratic form ℰ on M, which is obtained as the … Continue reading
Posted in Dirichlet forms at NYU
Tagged Dirichlet forms, Heat kernel estimates, Sierpinski gasket
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Lecture 4. The Lp theory of semigroups and diffusion operators as generators of Dirichlet forms
Contents Lp theory of semigroups Diffusion operators as generators of Dirichlet forms The Lp theory of heat semigroups Our goal, in this section, is to define, for 1 ≤ p ≤ +∞, Pt on Lp := Lp(X,μ). This may be … Continue reading
Lecture 3. Markov semigroups and Dirichlet forms
Let (X, ℬ) be a measurable space. We say that (X, ℬ) is a good measurable space if there is a countable family generating ℬ and if every finite measure γ on (X × X, ℬ ⊗ ℬ) can be … Continue reading
Lecture 2. Quadratic forms in Hilbert spaces
Contents Quadratic forms and generators Semigroups and generators A first example: The Dirichlet energy on an open set Quadratic forms and generators Definition A quadratic form ℰ on H is a non-negative definite, symmetric bilinear form 𝒟(ℰ) × 𝒟(ℰ) → … Continue reading
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Tagged Dirichlet forms, quadratic-forms-hilbert-space
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Lecture 1: Semigroups and generators
Contents Preliminaries: Self-adjoint Operators Semigroups and generators Preliminaries: Self-adjoint Operators Let (H,⟨⋅,⋅⟩) be a Hilbert space with norm ‖f‖2=⟨f,f⟩ and let A be a H-valued densely defined operator on a domain 𝒟(A). We recall the following basic definitions. The operator … Continue reading
Dirichlet forms at NYU Abu Dhabi
On January 20, I will give a 4 hours mini course on the Dirichlet forms at the NYU campus of Abu Dhabi. Lectures will be posted on this blog and I will prepare an extended set of lecture notes. This … Continue reading
Research and Lecture notes 