Timeline for Find strictly subharmonic function that vanishes at infinity
Current License: CC BY-SA 4.0
12 events
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| Feb 22, 2021 at 14:33 | comment | added | MikeG | @WillieWong Thanks, I have found that also. Actually I prefer that in Rmk 0.2 in this doc which uses less advanced results. I am very very interested in MaoWao's comment about the relationship between recurrence of BM and bdd subharmonic function being constant. Do you have any idea about this? | |
| Feb 22, 2021 at 13:37 | comment | added | Willie Wong | A proof of the Liouville theorem for subharmonic functions on the plane is given in this MSE post by way of the Hadamard Three Circle Theorem. | |
| Feb 22, 2021 at 13:34 | history | became hot network question | |||
| Feb 22, 2021 at 6:51 | vote | accept | MikeG | ||
| Feb 22, 2021 at 6:51 | comment | added | MikeG | @MaoWao: where can I find materials discussing this issue? | |
| Feb 22, 2021 at 6:39 | comment | added | MaoWao | The Brownian motion on $\mathbb{R}^2$ is recurrent, which implies that every bounded subharmonic function is constant. | |
| Feb 22, 2021 at 6:31 | answer | added | Willie Wong | timeline score: 5 | |
| Feb 22, 2021 at 6:18 | comment | added | Willie Wong | In dimension $n \geq 3$, a non-explicit way: choose your favourite strictly positive Schwartz class function $\chi$ (say, the Gaussian). Its Fourier transform is also Schwartz. So $\hat{\chi}(\xi) / |\xi|^2$ is in $L^1$ and of rapid decay, and you can apply the Fourier inversion formula to get $\triangle^{-1} \chi$ which will give you a smooth example. Alternatively just take an arbitrary positive Schwartz function and convolve with $|x|^{2-n}$. In $\mathbb{R}^2$ I am pretty sure this is ruled out by a version of Liouville's theorem. | |
| Feb 22, 2021 at 5:44 | comment | added | MikeG | Thank you!@Daniele | |
| Feb 22, 2021 at 5:41 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Minor Math Jaxing (proper bracket scaling and $exp\to\exp$
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| Feb 22, 2021 at 5:33 | review | First posts | |||
| Feb 22, 2021 at 6:47 | |||||
| Feb 22, 2021 at 5:32 | history | asked | MikeG | CC BY-SA 4.0 |