We consider an infinite locally finite tree $T$ equipped with nearest neighbor transition coefficients, giving rise to a space of harmonic functions. We show that, except for trivial cases, the generic harmonic function on $T$ has dense range in the complex numbers. By looking at forward-only transition coefficients, we show that the generic harmonic function induces a boundary martingale that approximates in probability all measurable functions on the boundary of $T$. We also study algebraic genericity, spaceability and frequent universality of these phenomena.
Abakumov, E., Nestoridis, V., Picardello, A.m. (2017). Universal properties of harmonic functions on trees. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 445(2), 1181-1187 [10.1016/j.jmaa.2016.03.078].
Universal properties of harmonic functions on trees
PICARDELLO, ANGELO MASSIMO
2017-01-01
Abstract
We consider an infinite locally finite tree $T$ equipped with nearest neighbor transition coefficients, giving rise to a space of harmonic functions. We show that, except for trivial cases, the generic harmonic function on $T$ has dense range in the complex numbers. By looking at forward-only transition coefficients, we show that the generic harmonic function induces a boundary martingale that approximates in probability all measurable functions on the boundary of $T$. We also study algebraic genericity, spaceability and frequent universality of these phenomena.File | Dimensione | Formato | |
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