Timeline for Poisson Furstenberg Boundary of topological groups, reference request
Current License: CC BY-SA 3.0
10 events
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Mar 10, 2017 at 16:52 | comment | added | Uri Bader | Colin, I shall add: contributing to the confusion is the fact that for a semisimple Lie group $G$ with a minimal parabolic $P$: 1) the Furstenberg Boundary is $G$-homeomorphic to $G/P$ and 2) for any reasonable $\mu$ the Furstenberg-Poisson boundary could be realized as well on $G/P$, that is it is isomorphic to $(G/P,\nu)$ as a $G$-Lebesgue space for some measure $\nu$ (which depends on $\mu$). | |
Mar 10, 2017 at 16:34 | comment | added | Colin Reid | Thanks, that clears things up. I think I got confused by the fact that both concepts appear in the same paper of Furstenberg, and they are closely related in that context but maybe not in general. I also found a few references to a 'Furstenberg boundary' of a random walk, but perhaps this is something else again. | |
Mar 10, 2017 at 16:23 | history | edited | Colin Reid | CC BY-SA 3.0 |
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Mar 10, 2017 at 16:03 | comment | added | Uri Bader | Both objects were defined by Furstenberg. The second one was originally given (by Furstenberg) the name "Poisson Boundary", and this is also how it is called in Furman's survey, but as I mentioned, it is now called mainly the "Furstenberg-Possion Boundary". Admittedly, confusing terminology, but too late to be changed. | |
Mar 10, 2017 at 16:01 | comment | added | Uri Bader | 2) given a locally compact second countable $G$ and a probability measure $\mu$ on $G$, the unique $G$-measured space $(B,\nu)$ where $\nu$ satisfies $\mu*\nu=\nu$ and for which the "Poisson Transform" $L^\infty(B)\to H^{\infty}(G)$ taking $f$ to $g\mapsto \int fdg\nu$ is an isometry onto the space of $\mu$-harmonic functions on $G$, is called "The Furstenberg-Poisson Boundary of $(G,\mu)$". | |
Mar 10, 2017 at 15:58 | comment | added | Uri Bader | @YCor is definitely right. People tend to agree nowadays on the following terminology: 1) given a locally compact group $G$, its maximal strongly proximal minimal compact action is termed "The Furstenberg Boundary of $G$" and | |
Mar 10, 2017 at 15:20 | comment | added | YCor | If the is "Poisson" in the name, it unambiguously refers to the probabilitsic guy. Actually these used to be called "Poisson boundary", although they were coined by Furstenberg (Poisson worked much before Lebesgue initiated measure theory!). So it tends to be called "Poisson-Furstenberg boundary" now. On the other hand, it turns out that Furstenberg did other achievements, and his name is thus also associated to this universal strongly proximal space... | |
Mar 10, 2017 at 14:54 | comment | added | Colin Reid | The terminology is a bit confusing as the word 'boundary' is overloaded. Poisson boundaries are usually defined for random walks, and then it is sensitive to what probability measure you use. Perhaps what I have described should be called just the Furstenberg boundary, and then for a Poisson-Furstenberg boundary, you need to specify a probability measure, as you say. I will edit the answer. | |
Mar 10, 2017 at 13:49 | comment | added | YCor | I'm sceptical about "It looks like this universal boundary is what is usually meant by the Poisson-Furstenberg boundary of a locally compact group"; as far as I know these are distinct objects. Look e.g. at Erschler's survey on Poisson-Furstenberg boundaries (mathunion.org/ICM/ICM2010.2/Main/icm2010.2.0681.0704.pdf) or Hartman's (math.northwestern.edu/~hartman/pdfs/FPB-course.pdf) | |
Mar 10, 2017 at 10:47 | history | answered | Colin Reid | CC BY-SA 3.0 |