Talk:Poisson boundary

Initial comments

The article as created is probably not so good. Reading only the first section and the parts about discrete random walks should be OK but the more technical parts are a bit bare-bones and maybe not precise enough (I included them so that the article is encyclopedic enough but I don't feel competent enough to edit them much more for the moment). In particular I think the article should include a more comprehensive and detailed discussion on the Poisson and Martin boundaries for Brownian motion on symmetric spaces. jraimbau (talk) 16:26, 30 August 2016 (UTC)[reply]

Nilpotent groups

I don't think that the Martin boundary, even the minimal Martin boundary is trivial for nilpotent groups. The poisson boundary always is, but you need the random walk to be centered to have triviality of the Martin boundary. Actually, minimal harmonic functions on abelian groups are exactly multiplicative functions. When the random walk on Z^d is non-centered, Ney and Spitzer showed that there are a lot of such functions (it is topologically a sphere, and it is also the whole Martin boundary, which is thus minimal). M. Dussaule, 12.06.17, 11:50

I added the provision that the random walk be symmetric for triviality to hold. jraimbau (talk) 11:24, 12 June 2017 (UTC)[reply]

I edited some things in the article. First, since the whole family of Martin boundaries is mentionned for the Laplacian on manifolds, I added the whole family in the context of a finitely generated group.

Second, I made a more precise statement about nilpotent groups. When it is centered, the Martin boundary is trivial (not only the minimal one and actually, if the minimal Martin boundary is trivial, then the whole Martin boundary is trivial, since every harmonic function is an integral of Martin kernels over the minimal boundary). When it is non-centered, we don't know.

Maybe we should refer to the case of an abelian group: Ney and Spitzer proved that when the random walk is non-centered, the Martin boundary is a sphere of the appropriate dimension : Peter Ney and Frank Spitzer. “The Martin boundary for random walk”. In: Transactions of the American Mathematical Society 11 (1966), pp. 116–132.

Finally, I stated that the Martin boundary of a finite range random walk on a hyperbolic group is the Gromov boundary. It was proved by Ancona: lano Ancona. “Positive harmonic functions and hyperbolicity”. In: Potential theory-surveys and problems . Lecture notes in mathematics, Springer, 1988, pp. 1–23.

I actually don't know how to include new references in the bibliography, but if someone reads this, he/she should either tell me how to do so or include Ney and Spitzer and Ancona.

M. Dussaule — Preceding unsigned comment added by 2A01:CB05:838D:5400:E0F4:C1B9:31D2:B1AC (talk) 17:30, 8 October 2017 (UTC)[reply]