Timeline for What is an analogous version of the Ornstein–Uhlenbeck process on Riemannian manifolds?
Current License: CC BY-SA 4.0
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| Feb 28, 2023 at 18:43 | vote | accept | Student | ||
| Feb 15, 2023 at 5:07 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jan 16, 2023 at 4:05 | answer | added | Thomas Kojar | timeline score: 1 | |
| Jan 5, 2023 at 5:20 | comment | added | Nate Eldredge | @Student: I would say the natural analogue of the Gaussian is the heat kernel. Note that by Radon-Nikodym, saying $\mu = e^{-f} d\mathrm{vol}$ is equivalent to saying "smooth measure" and just adding the condition "strictly positive density". Any two such measures are mutually absolutely continuous. | |
| Jan 5, 2023 at 4:48 | comment | added | Student | @NateEldredge thank you for your comments. This makes sense, so I guess then the issue is to figure out a natural version of the gaussian measure on a curved space. Perhaps measures of the form $\mu = e^{-f} d\text{vol}_g$ might be more appropriate? | |
| Jan 5, 2023 at 3:14 | comment | added | Nate Eldredge | Also, if $\mu$ is the Riemannian volume measure, then the resulting $X_t$ is Brownian motion on $(M,g)$. So Ornstein-Uhlenbeck and its generalizations are, in some sense, the Brownian motion for some "weighting" of $(M,g)$. | |
| Jan 5, 2023 at 3:10 | comment | added | Nate Eldredge | Are you familiar with Dirichlet forms? That's how I would extend this, fix a smooth measure $\mu$ on $(M,g)$, and consider the Dirichlet form on $L^2(M, \mu)$ defined formally by $\mathcal{E}(f) = \int_M |\nabla f|^2\,d\mu$. It induces a continuous Markov process $X_t$ symmetric with respect to $\mu$. On $\mathbb{R}^n$ with $\mu$ being Gaussian measure, the resulting $X_t$ is the Ornstein-Uhlenbeck process. | |
| Jan 5, 2023 at 2:25 | history | edited | YCor | CC BY-SA 4.0 |
edited title
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| Jan 5, 2023 at 2:16 | history | edited | YCor | CC BY-SA 4.0 |
formatting, added tag
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| Jan 5, 2023 at 0:53 | comment | added | R W | How do you define the OU process on a general manifold? | |
| Jan 5, 2023 at 0:48 | comment | added | Student | @RW thanks for your comment, but will that operator be the generator of the OU process on the manifold? | |
| Jan 5, 2023 at 0:20 | comment | added | R W | Just take the negative gradient field of the distance function with respect to a fixed reference point. | |
| Jan 4, 2023 at 23:57 | history | asked | Student | CC BY-SA 4.0 |