The Annals of Probability

Fatou theorem of p-harmonic functions on trees

Robert Kaufman and Jang-Mei Wu

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We study bounded $p$-harmonic functions $u$ defined on a directed tree $T$ with branching order $\kappa(1<p<\infty$ \and $\kappa=2,3,\ldots)$. Denote by $BV(u)$ the set of paths on which $u$ has finite variation and $\mathscr{F}(u)$ the set of paths on which $u$ has a finite limit. Then the infimum of dim $BV(u)$ and the infimum of dim $\mathscr{F}(u)$ are equal over all bounded-harmonic functions on $T$ (with $p$ and $\kappa$ fixed); the infimum $d(\kappa, p)$ is attained and is strictly between 0 and 1 expect when $p = 2$ or $\kappa = 2$.

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Ann. Probab., Volume 28, Number 3 (2000), 1138-1148.

First available in Project Euclid: 18 April 2002

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Zentralblatt MATH identifier

Primary: 31C20: Discrete potential theory and numerical methods 31C45: Other generalizations (nonlinear potential theory, etc.)
Secondary: 31A20: Boundary behavior (theorems of Fatou type, etc.) 60G42: Martingales with discrete parameter

Fatou Theorem trees $p$-harmonic functions dimension entropy


Kaufman, Robert; Wu, Jang-Mei. Fatou theorem of p -harmonic functions on trees. Ann. Probab. 28 (2000), no. 3, 1138--1148. doi:10.1214/aop/1019160328.

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