Timeline for Ergodicity of the solution to some SDE
Current License: CC BY-SA 4.0
9 events
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| Jul 12, 2023 at 19:57 | history | edited | Fawen90 | CC BY-SA 4.0 |
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| Jul 12, 2023 at 7:00 | comment | added | Fawen90 | @ThomasKojar We don't need to set $\bar x$ to be the initial condition in general. We seek such a point $\bar x$ to ensure the above inequality, and we wonder whether such a point exists or not | |
| Jul 11, 2023 at 20:20 | comment | added | Thomas Kojar | even in the case of $\bar{x}$ being the initial condition this doesn't seem possible. For example, suppose we set $T=\tau_{r}-\theta$, for $\tau_{r}$ the exit time from ball $B_{\bar{x}}(r)$. Then the moment we ask $T>0 \Rightarrow \tau_{r}>\theta$ for deterministic $\theta>0$, we get a reduced probability $P[\tau_{r}>\theta]<1$. | |
| Jul 11, 2023 at 18:05 | comment | added | Fawen90 | $\bar x$ is the point that we seek to satisfy this desired equality | |
| Jul 11, 2023 at 17:43 | comment | added | Thomas Kojar | is $\bar{x}$ the initial condition or a generic point different from initial condition? I took it to mean a generic point. If so, then the probability of visiting a ball around it is strictly less than one if the process have a drift. | |
| Jul 11, 2023 at 17:36 | comment | added | Fawen90 | @ThomasKojar I don't think recurrence is related to my desired result... Even $X_\infty$ is infinite, while it is possible to stay in some open ball for a while when $t$ is not that large... | |
| Jul 11, 2023 at 16:29 | comment | added | Thomas Kojar | as soon a process has a drift, then we lose recurrence eg. see here math.stackexchange.com/questions/4570678/… for the probability of hitting a value. | |
| Jul 11, 2023 at 15:10 | history | edited | Fawen90 | CC BY-SA 4.0 |
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| Jul 11, 2023 at 7:46 | history | asked | Fawen90 | CC BY-SA 4.0 |