Processes and heat kernels with symmetries (Conference with mini-courses)

Neil O'Connell - A Markov chain on reverse plane partitions

I will discuss a natural Markov chain on reverse plane partitions which is closely related to the Toda lattice.


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.  

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. 


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. V ́azquez 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 (Universit ́e d’Orl ́eans) and H.-W. Zhang (Ghent University).


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.