Estimates for Nonlinear Harmonic Measures on Trees

TL;DR

This paper provides estimates for nonlinear harmonic measures on directed trees, relating the solution's value at the root to the size of a set with initial data, using an averaging operator.

Contribution

It introduces new estimates for nonlinear harmonic measures on trees, connecting the solution at the root to the initial data set size via an averaging operator.

Findings

Derived bounds for solutions based on set size

Established estimates for nonlinear harmonic measures on trees

Connected initial data to solution values through averaging operators

Abstract

In this paper we give some estimates for nonlinear harmonic measures on trees. In particular, we estimate in terms of the size of a set $D$ the value at the origin of the solution to $ u(x)=F((x,0),\dots,(x,m-1))$ for every $x\in\mathbb{T}_m, $ a directed tree with $m$ branches with initial datum $f+\chi_D$. Here $F$ is an averaging operator on $\mathbb{R}^m$, $x$ is a vertex of a directed tree $\mathbb{T}_m$ with regular $m$-branching and $(x,i)$ denotes a successor of that vertex for $0\le i\le m-1$.