26/02/2025, 16:00 — 17:00 — Online
Luis Fredes, Université de Bordeaux
Almost upper directed Markov chains on trees
A transition matrix U on a tree T is said to be almost upper directed if the allowed steps are from a node to its parent or its descendants. In this talk, as a warm-up, I will start with the case where the tree is N and I will characterise the recurrence, positive recurrence, and invariant distribution of these transition matrices. I will then present the theorems for general trees and a technique that allows one to compute an invariant distribution at a given vertex without requiring knowledge of the full invariant measure. These results encompass the case of birth and death processes (BDPs), which possess almost upper directed transition matrices. Their properties were studied in the 1950s by Karlin and McGregor, whose approach relies on deep connections between the theory of BDPs, the spectral properties of their transition matrices, the moment problem, and the theory of orthogonal polynomials. Our approach is mainly combinatorial and uses elementary algebraic methods. This talk is based on two joint works with J.-F. Marckert.
