Timeline for Second order brownian motion $\ddot{x}(t) = \xi(t)$
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 29, 2021 at 1:14 | vote | accept | kevinkayaks | ||
| Jun 29, 2021 at 1:56 | |||||
| Jun 21, 2021 at 17:06 | comment | added | Roger V. | @kevinkayaks indeed, an integral is a sum, whereas a sum of gaussian random variables is a gaussian random variable. | |
| Jun 21, 2021 at 17:05 | comment | added | kevinkayaks | I am curious though about the joint distribution of position and velocity. In this case from an analogous method should I start from $P(x,v,t) = \langle \delta(x-\int_0^t dt_1 \int_0^{t_1} dt_2 \xi(t_2)) \delta(v-\int_0^t dt_1 \xi(t_1))\rangle$ and take fourier transforms over both $x$ and $v$? Or am I misunderstanding how to write the joint distribution as an expectation of delta functions? | |
| Jun 21, 2021 at 17:03 | comment | added | kevinkayaks | Thanks @Roger ! This looks great. I was playing with this but could not figure it out with the additional integral in $z$. Recognizing that the entire integral is a gaussian random variable is the key. | |
| Jun 21, 2021 at 14:24 | history | answered | Roger V. | CC BY-SA 4.0 |