Frequently dense harmonic functions and universal martingales on trees

Abakumov, Evgeny; Nestoridis, Vassili; Picardello, Massimo

doi:10.1090/proc/15355

Frequently dense harmonic functions and universal martingales on trees
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by Evgeny Abakumov, Vassili Nestoridis and Massimo A. Picardello PDF

Proc. Amer. Math. Soc. 149 (2021), 1905-1918 Request permission

Abstract:

On a large class of infinite trees $T$, we prove the existence of harmonic functions $h$, with respect to suitable transient transition operators $P$, that satisfy the following universal property: $h$ is the Poisson transform of a martingale on the end-point boundary $\Omega$ of $T$ (equipped with the harmonic measure induced by $P$) such that, for every measurable function $f$ on $\Omega$, it contains a subsequence converging to $f$ in measure. Moreover, the martingale visits every open set of measurable functions with positive lower density.

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