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 language||English (US)|
|Number of pages||24|
|Journal||Annales Academiae Scientiarum Fennicae Mathematica|
|State||Published - Nov 14 2003|
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