Frequently dense harmonic functions and universal martingales on trees
Evgeny Vasilevich Abakumov, Vassili Nestoridis, Massimo A. Picardello · Proceedings of the American Mathematical Society · 2021
On a large class of infinite trees T T , we prove the existence of harmonic functions h h , with respect to suitable transient transition operators P P , that satisfy the following universal property: h h is the Poisson transform of a martingale on the end-point boundary Ω \Omega of T T (equipped with the harmonic measure induced by P P ) such that, for every measurable function f f on Ω \Omega , it contains a subsequence converging to f f in measure. Moreover, the martingale visits every open set of measurable functions with positive lower density.