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

A model for the Furstenberg boundary is the hyperbolic disc $D=\$. The classical Poisson formula for a bounded harmonic function on the disc has the form
:$f(z) = \frac\int_0^ \hat(e^) P(z,e^)\, d\theta$
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 . Then the Poisson formula has the form

: $F(g) = \int_\hat(gz) \, dm(z)$

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

In general, let ''G'' be a semi-simple Lie group and μ a probability measure on ''G'' that is absolutely continuous. A function ''f'' on ''G'' is μ-harmonic if it satisfies the mean value property with respect to the measure μ:

:$f(g) = \int_G f(gg') \, d\mu(g')$

There is then a compact space Π, with a ''G'' action and measure ''ν'', such that any bounded harmonic function on ''G'' is given by

:$f(g) = \int_\Pi \hat(gp) \, d\nu(p)$

for some bounded function $\hat$ on Π.

The space Π and measure ''ν'' depend on the measure μ (and so, what precisely constitutes a harmonic function). However, it turns out that although there are many possibilities for the measure ν (which always depends genuinely on μ), there are only a finite number of spaces Π (up to isomorphism): these are homogeneous spaces of ''G'' that are quotients of ''G'' by some parabolic subgroup, which can be described completely in terms of root data and a given Iwasawa decomposition. Moreover, there is a maximal such space, with quotient maps going down to all of the other spaces, that is called the Furstenberg boundary.

References

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* {{citation, first=Harry, last=Furstenberg, title=Boundary theory and stochastic processes on homogeneous spaces, publisher=AMS, year=1973, pages=193–232, editor=Calvin Moore, journal=Proceedings of Symposia in Pure Mathematics, volume=26, doi=10.1090/pspum/026/0352328, isbn=9780821814260

Potential theory