Minicourses

Mathav Murugan - Stability and characterization of the Harnack inequalities and heat kernel estimates

 

Which properties of space are stable under perturbations? We survey known results, relevant open problems, and some methods developed to answer this question for potential theoretic properties such as recurrence/transience, Harnack inequalities, and heat kernel estimates. There will be a special focus on the recent progress on the stability of the elliptic Harnack inequality and an improved understanding of the relationship between elliptic and parabolic Harnack inequalities. 

 

Research seminars

Alexander Grigor'yan - Heat kernels on fractals and walk dimension

We discuss elements of Analysis on Ahlfors-regular metric spaces, in particular, on fractals, based on the notion of the heat kernel. Such spaces are characterized by two parameters: the Hausdorff dimension and the walk dimension, where the latter determines the space/time scaling for a diffusion process. We present various approaches to the notion of the walk dimension, in particular, an approach via Besov function spaces. We also discuss heat kernel bounds for diffusion and jump processes.  

Batu Güneysu - A new notion of subharmonicity for locally integrable functions on locally smoothing spaces

Given a strongly local Dirichlet space, I will explain a new notion of subharmonicity for locally integrable functions in terms of the heat kernel, which is called 'local shift defectivity', and which turns out to be equivalent to distributional subharmonicity in the   Riemannian case. Some regularity properties of these functions on a new class of strongly local Dirichlet, so called 'locally smoothing spaces', are then presented. These spaces include all Riemannian manifolds (without any curvature assumptions), finite dimensional RCD spaces, Carnot groups, and Sierpinski gaskets. As a byproduct of this   regularity theory, one obtains in this general framework a proof of a conjecture by Braverman, Milatovic, Shubin on the positivity of distributional 1-superharmonic L^2-functions on complete Riemannian manifolds. This is joint work with Stefano Pigola, Giona Veronello and Peter Stollmann.

 

Mathav Murugan - Heat kernel for reflected diffusion and extension property on uniform domains

In this talk, I report recent progress on heat kernel estimates for reflected diffusion on uniform domains where the underlying space admits sub-Gaussian heat kernel bounds. A key novelty of our work is the use of an extension operator that extends functions from the domain of the Dirichlet form for the reflected diffusion to that of the diffusion in the ambient space. 

 

Neil O'Connell - Interacting diffusion on positive definite matrices

We consider systems of Brownian particles in the space of positive definite matrices, which evolve independently apart from some simple interactions. We give examples of such processes which have an integrable structure. These are related to K-Bessel functions of matrix argument and multivariate generalisations of these functions. The latter are eigenfunctions of a particular quantisation of the non-Abelian Toda lattice.

 

Effie Papageorgiou - Asymptotic behaviour of solutions to the heat equation on noncompact symmetric spaces

In Euclidean space, the Central Limit Theorem of probability represented in the PDE setting can be described as follows: starting with absolutely integrable initial data, the solution to the heat equation converges, as time goes to infinity, to the mass of the initial data times the heat kernel, in all Lp norms, albeit at different rates. Analogous heat asymptotics may or may not hold on Riemannian manifolds. Our aim is to discuss noncompact symmetric spaces, generalizing earlier results of J.L. Vàzquez on real hyperbolic spaces. More precisely, we discuss the heat equation related to the Laplace-Beltrami operator and to the distinguished Laplacian. In the first case, if the data is bi-K-invariant, the convergence is true, but may fail otherwise. In the second case, we observe phenomena more similar to the Euclidean setting. Joint work with J.-Ph. Anker (Université d’Orléans) and H.-W. Zhang (Ghent University).

 

Philipp Sürig - Gaussian upper bounds for subsolutions of Leibenson's equation on Riemannian manifolds

We consider solutions of the differential inequality ∂_t u ≤ Δ_p u^1/(p-1), on complete Riemannian manifolds, where p>1. We prove that non-negative bounded weak solutions to this inequality have a pointwise Gaussian upper bound.

 

Jeannette Woerner - Limit theorems of the empirical measure of multidimensional Dunkl processes

In the talk we will derive limit theorems for the empirical measure of N-dimensional Dunkl processes when N tends to infinity. We consider both freezing and high temperature limits, which means that either the influence of the involved Brownian motion or the correlations beween components is removed. In the freezing case the limiting law involves semicircle and Marchenko Pastur distributions from free probability, whereas in the high temperature case the limiting law is the law of a one-dimensional Dunkl processes. Comparing both limiting laws provides some interesting insight in the structure.