Timeline for Fokker-Planck: equivalence between linear spectral gap and nonlinear displacement convexity?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| S May 24, 2022 at 8:23 | history | suggested | ViktorStein | CC BY-SA 4.0 |
typos fixed
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| May 24, 2022 at 7:04 | review | Suggested edits | |||
| S May 24, 2022 at 8:23 | |||||
| Apr 28, 2022 at 19:02 | vote | accept | leo monsaingeon | ||
| Apr 28, 2022 at 16:51 | comment | added | Tobsn | Can not add much to the elaborate answer of @AndréSchlichting below, except for one standard refernce that you may consult in order to find more details, also on André's arguments in the beginning: Bakry et al, Analysis and Geometry of Markov Diffusion Operators | |
| Apr 28, 2022 at 13:22 | history | edited | JHM | CC BY-SA 4.0 |
minor misspelling
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| Apr 28, 2022 at 13:10 | answer | added | André Schlichting | timeline score: 7 | |
| Apr 28, 2022 at 4:29 | comment | added | leo monsaingeon | @Tobsn: thanks for your input. Allow me a couple questions: 1) I don't see how my above assumption is weaker than yours? The $\lambda$-convexity also leads to contractivity (although I didn't discuss this here, but formally they are also equivalent). And 2) why should this lead to the same spectral gap? Actually this is precisely my question! and I am not aware that for Fokker-Planck the first eigenvalue (0 left aside) is the lowest eigenvalue of the Hessian of the potential. Is this your claim? (I am not saying it isn't, just that I am not aware of it.) Do you have a reference to suggest? | |
| Apr 27, 2022 at 14:19 | comment | added | Tobsn | At least on full space, exponential Wasserstein contraction of trajectories i.e. $W(\mu_{t},\mu_{t}')\le e^{-\lambda t}W(\mu_{0},\mu_{0}')$ (which is of course stronger than your assumption above) is actually equivalent to $\lambda$-strong convexity of the potential $V$ and should then also lead to the same spectral gap. | |
| Apr 27, 2022 at 13:12 | history | edited | leo monsaingeon | CC BY-SA 4.0 |
minus sign missing
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| Apr 27, 2022 at 13:03 | history | asked | leo monsaingeon | CC BY-SA 4.0 |