Furstenberg boundary
In potential theory, a discipline within applied mathematics, the Furstenberg boundary is a notion of boundary associated with a group. It is named for Harry Furstenberg, who introduced it in a series of papers beginning in 1963 (in the case of semisimple Lie groups). The Furstenberg boundary, roughly speaking, is a universal moduli space for the Poisson integral, expressing a harmonic function on a group in terms of its boundary values.
Motivation[edit]
A model for the Furstenberg boundary is the hyperbolic disc . The classical Poisson formula for a bounded harmonic function on the disc has the form
where P is the Poisson kernel. Any function f on the disc determines a function on the group of Möbius transformations of the disc by setting F(g) = f(g(0)). Then the Poisson formula has the form
where m is the Haar measure on the boundary. This function is then harmonic in the sense that it satisfies the mean-value property with respect to a measure on the Möbius group induced from the usual Lebesgue measure of the disc, suitably normalized. The association of a bounded harmonic function to an (essentially) bounded function on the boundary is one-to-one.
Construction for semi-simple groups[edit]
In general, let G be a semi-simple Lie group and
There is then a compact space
for some bounded function on
The space
References[edit]
- Borel, Armand; Ji, Lizhen, Compactifications of symmetric and locally symmetric spaces (PDF)
- Furstenberg, Harry (1963), "A Poisson Formula for Semi-Simple Lie Groups", Annals of Mathematics, 77 (2): 335–386, doi:10.2307/1970220, JSTOR 1970220
- Furstenberg, Harry (1973), Calvin Moore (ed.), "Boundary theory and stochastic processes on homogeneous spaces", Proceedings of Symposia in Pure Mathematics, 26, AMS: 193–232, doi:10.1090/pspum/026/0352328, ISBN 9780821814260