Timeline for Use stochastic process to express solution to Laplace equation in the whole space
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 7, 2021 at 2:02 | comment | added | Jacob Lu | Many thanks for your suggestions! They are really helpful! | |
| Feb 3, 2021 at 21:00 | comment | added | Mateusz Kwaśnicki | If there is a Markov process $X_t$ with infinitesimal generator $L = b(x) \cdot \nabla + \delta$, then its potential operator $U$, defined by $U f(x) = \mathbb E^x \int_0^\infty f(X_t) dt$, is roughly the inverse of $-L$. Detailed statements require extra care, and this belongs to the field of probabilistic potential theory. Off the top of my head, I do not have a good reference, but, especially if you are willing to read more about it, you can try Dynkin's two-volume book Markov processes. Although very old, it is still a very good read. (Of course, there are more excellent resources.) | |
| Feb 3, 2021 at 19:44 | comment | added | Jacob Lu | Thanks a lot! What if we consider a slightly more general equation $b(x)\cdot\nabla u + \Delta u = f$ with a suitable vector field $b(x)$? Can we get a similar conclusion in this situation? Do we need more structure condition for $b$ to make the argument true (for example, $b(x)$ is a divergence free vector field). | |
| Feb 3, 2021 at 19:39 | vote | accept | Jacob Lu | ||
| Feb 3, 2021 at 9:00 | history | answered | Mateusz Kwaśnicki | CC BY-SA 4.0 |