Timeline for Second order brownian motion $\ddot{x}(t) = \xi(t)$
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Aug 18, 2021 at 9:10 | comment | added | kevinkayaks | Makes sense! I was misunderstanding your use of $W$ | |
| Aug 18, 2021 at 9:00 | comment | added | Kurt G. | @kevinkayaks : to your doubts in the newer comments : no we have $E[W(u)W(v)]=\min(u,v)$, which implies $E[x^2]=t^3/3\,.$ Writing $W(u)=\int_0^u\xi(s)\,ds$ where $\xi$ is white noise we get the same relations from $E[\xi(u)\xi(v)]=\delta(u-v)\,.$ | |
| Aug 18, 2021 at 8:54 | comment | added | Kurt G. | @kevinkayaks : to your question in the comment from Jun 20 : the joint distribution of position and velocity satisfies $\partial_t P=-v\,\partial_x P+\frac{1}{2}\partial_{vv}P$ as I derived in my first answer to your original question. | |
| Aug 17, 2021 at 22:03 | comment | added | kevinkayaks | Per your earlier answer here physics.stackexchange.com/questions/653279/… you simply forgot to eat up one of the integrals between writing $E(W(u)W(v))$ and $\min(u,v)$ if I am correct. | |
| Aug 17, 2021 at 21:54 | comment | added | kevinkayaks | $E[W(u)W(v)] = \delta(u-v)$ by definition, does it not ? This gives $E(x^2) = t^3/2$, not $t^3/3$. | |
| Jun 21, 2021 at 19:11 | history | edited | Kurt G. | CC BY-SA 4.0 |
modified after kevinaykas clarified what $D$ means, removed one more stray index i
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| Jun 21, 2021 at 19:02 | history | edited | Kurt G. | CC BY-SA 4.0 |
modified after kevinaykas clarified what $D$ means
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| Jun 20, 2021 at 18:37 | comment | added | kevinkayaks | Yes, 1D Kurt - $D$ is just a diffusivity. Regardless the variance is not challenging, and the velocity distribution satisifies a diffusion equation $ \partial_t P(v,t) = \frac{\partial^2}{\partial v ^2} P(v,t)$, but what equations do the position distribution or the joint distribution of position and velocity satisfy? | |
| Jun 20, 2021 at 17:02 | comment | added | Kurt G. | In that case, even simpler. We can drop the index $i$ and have a variance of $D^2t^3/3$ of the position $D\int_0^tW(s)\,ds\,.$ Same calculation. | |
| Jun 20, 2021 at 11:43 | comment | added | Quillo | I am not sure it's a D dimensional problem.. probably D is just the diffusion coefficient. | |
| Jun 20, 2021 at 11:04 | history | answered | Kurt G. | CC BY-SA 4.0 |