Nonlinear harmonic measures on trees

Robert Kaufman, José G. Llorente, Jang-Mei Gloria Wu

Research output: Contribution to journalArticle

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 RK and WF 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 WF measure zero can have positive WF measure when F is permutation invariant; and there exist sets of full WP measure having "small" dimension. Let A be a monotone operator on RK, 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 languageEnglish (US)
Pages (from-to)279-302
Number of pages24
JournalAnnales Academiae Scientiarum Fennicae Mathematica
Volume28
Issue number2
StatePublished - Nov 14 2003

ASJC Scopus subject areas

  • Mathematics(all)

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  • Cite this

    Kaufman, R., Llorente, J. G., & Wu, J-M. G. (2003). Nonlinear harmonic measures on trees. Annales Academiae Scientiarum Fennicae Mathematica, 28(2), 279-302.