Timeline for Brownian motion on Metric spaces
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 22, 2014 at 1:51 | answer | added | Fabrice Baudoin | timeline score: 7 | |
| Aug 10, 2013 at 11:47 | answer | added | Hatem | timeline score: 0 | |
| Jul 4, 2013 at 12:28 | vote | accept | Matthias Ludewig | ||
| Jul 4, 2013 at 9:42 | review | Close votes | |||
| Jul 4, 2013 at 15:58 | |||||
| Jul 4, 2013 at 9:38 | comment | added | Benoît Kloeckner | To give an example, one can use Brownian motion on symmetric spaces to define their Poisson boundary. If one wants an analogue for discrete groups, one should simply use a random walk. If instead one wants an analogue for sub-Riemannian geometry, then the rescaling of random walks is probably a good approach, but then one has to choose a volume. | |
| Jul 4, 2013 at 9:35 | answer | added | Benoît Kloeckner | timeline score: 2 | |
| Jul 4, 2013 at 9:32 | comment | added | Benoît Kloeckner | @Kofi: the Brownian motion is defined on $\mathbb{R}^n$. You want to generalize it. Obviously one cannot generalize all properties, in all context. So, without telling which properties you really want, and in which context, it is not possible to answer (or, there are too many answers). Are you interested in Heisenberg groups? In random walk on hyperbolic groups? On Brownian motion on fractals? We can't tell until you give more info. | |
| Jul 4, 2013 at 9:20 | comment | added | Did | For the Riemannian case, see An Introduction to the Analysis of Paths on a Riemannian Manifold by Daniel Stroock, which essentially develops @Benoît's idea. Generalizing this to every metric space is problematic, for example when every ball of radius $\lt1$ is reduced to a point (which might be a reason to ask that you provide more context)... | |
| Jul 4, 2013 at 8:58 | comment | added | Matthias Ludewig | I feel you are being a bit mysterious here. The definition of Brownian motions on $\mathbb{R}^n$ depends not on my purpose, but on metric spaces it does? | |
| Jul 4, 2013 at 8:57 | history | edited | Matthias Ludewig | CC BY-SA 3.0 |
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| Jul 4, 2013 at 6:51 | comment | added | Benoît Kloeckner | @Kofi: yes, I think that the approach I suggest gives the usual Brownian motion on manifold. But again, you should precise to which kind of metric space you want to apply Brownian motion, and for what purpose. There is plenty of literature, but to give pointers one need more info. As it is, your question probably does not have a good answer, so that I might be inclined to vote to close. | |
| Jul 3, 2013 at 21:00 | comment | added | Did | A distance distributed by a normal distribution? Usually one is unhappy with negative distances... | |
| Jul 3, 2013 at 20:35 | comment | added | Gerald Edgar | There is something studied under the name "Brownian motion on a fractal" ... but generally this in done in settings far less general than a "metric space" ... proba.jussieu.fr/bulletin/Bmfrac.pdf | |
| Jul 3, 2013 at 18:39 | comment | added | Matthias Ludewig | Well, I would like a definition that reduces to the case of usual Brownian motion on a Riemannian manifold, if the metric space happens to have this structure. Your definition does the trick, I would say, doesn't it? | |
| Jul 3, 2013 at 15:15 | comment | added | Benoît Kloeckner | If you have a metric measure space, you can define a random walk by jumping uniformly in a ball of radius $\varepsilon$, then make $\varepsilon\to0$ while rescaling time to get a continuous stochastic process. But then you certainly need hypotheses to have convergence to a nice analogue of Brownian motion. Without more motivation about what you want from it, it is difficult to give a reasonable answer. | |
| Jul 3, 2013 at 14:46 | history | asked | Matthias Ludewig | CC BY-SA 3.0 |