### Abstract

We show that nonlinear harmonic measures on trees lack many desirable properties of set functions encountered in classical analysis. Let F be an averaging operator on R^{K} and W^{F} be the F-harmonic measure on a K-regular forward branching tree. Unless F is the usual average, W ^{F} is not a Choquet capacity; union of sets of W^{F} measure zero can have positive W^{F} measure when F is permutation invariant; and there exist sets of full W^{P} measure having "small" dimension. Let A be a monotone operator on R^{K}, then A -harmonic functions on trees need not obey the strong maximum principle unless the ratio of the ellipticity constants is close to 1.

Original language | English (US) |
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Pages (from-to) | 279-302 |

Number of pages | 24 |

Journal | Annales Academiae Scientiarum Fennicae Mathematica |

Volume | 28 |

Issue number | 2 |

State | Published - Nov 14 2003 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Annales Academiae Scientiarum Fennicae Mathematica*,

*28*(2), 279-302.