Timeline for how many nonparabolic ends guarantee a nonconstant harmonic function on Riemannian manifold?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Dec 25, 2012 at 9:36 | comment | added | R W | @timur - any harmonic function of $\mathbb R^1$ is affine, so that there are no non-constant bounded ones. This is also true in any dimension. | |
| Dec 25, 2012 at 9:34 | comment | added | R W | @unknown (google) - yes, this is a particular case of the Poisson formula. | |
| Dec 24, 2012 at 17:22 | comment | added | John Pardon | One should probably add that if we choose a nonparabolic end of $M$, then the function which assigns to $p\in M$ the probability of brownian motion started at $p$ escaping out the chosen end is a harmonic function. If there are at least two nonparabolic ends, then this function will be nonconstant (it approaches 1 deep in the chosen end, and approaches 0 deep in every other nonparabolic end). | |
| Dec 24, 2012 at 16:56 | comment | added | timur | I am just trying to understand the meaning of the statement. | |
| Dec 24, 2012 at 16:56 | comment | added | timur | What is an example of a nonconstant bounded harmonic function on $R^1$? | |
| Dec 23, 2012 at 15:46 | comment | added | R W | Actually it makes sense for an arbitrary Markov process (chain). The space of bounded harmonic functions (i.e., those which satisfy the mean value property with respect to the transition probabilities) is isometric to the space of bounded functions on the path space measurable with respect to the sigma-algebra of time shift invariant sets. This sigma-algebra describes the "stochastically significant" behaviour of the Markov process at infinity. The key words are "Poisson formula" (this is the above isometry; the name comes from the usual Poisson formula for the disk) and "Poisson boundary". | |
| Dec 23, 2012 at 12:53 | comment | added | Dmitri Panov | Very interesting. If you don't mind would it be possible to expand a little bit the phrase " there will be a non-trivial behaviour at infinity, i.e., a non-constant bounded harmonic function"? How can one give to this a precise mathematical sense? Is there a reference for this kind of reasoning? | |
| Dec 23, 2012 at 12:27 | history | answered | R W | CC BY-SA 3.0 |