Timeline for Distribution of Brownian motion conditional on linear growth
Current License: CC BY-SA 4.0
23 events
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| Oct 27 at 1:49 | comment | added | Nate River | Oh nice, so it has been looked at before. Will check it out, thanks! @ThomasKojar | |
| Oct 27 at 1:25 | comment | added | Thomas Kojar | @NateRiver I might be wrong, but after doing Brownian scaling, this seems to be related to a Brownian meander e.g. see here arxiv.org/pdf/1710.02350 "SOME RESULTS ON THE BROWNIAN MEANDER WITH DRIFT" where $v=\lambda$ and take $x_{0}>0$ so that we just have $$\min_{s} B_{s}>\lambda.$$ For general techniques on conditioning on open sets, you can also see here "Brownian motion conditioned to stay in a cone". | |
| Oct 22 at 21:47 | comment | added | Mateusz Kwaśnicki | @MartinHairer: That's not really relevant for the question, but, somewhat counterintuitively, the drift term has no “$-1$” in it. I never really trust my own calculations, so I double checked it in Williams's Path Decomposition…, doi.org/10.1112/plms/s3-28.4.738 (see page 744). | |
| Oct 22 at 21:08 | comment | added | Martin Hairer | @MateuszKwaśnicki I think the drift should be $\lambda(\coth(\lambda Z_t)-1)$ rather than $\lambda\coth(\lambda Z_t)$ in $d=1$ but otherwise I agree. My point was just that while that process is indeed something like "Bessel-3 with drift" it definitely isn't any of the naïve guesses for what this actually means... | |
| Oct 22 at 20:13 | comment | added | Mateusz Kwaśnicki | …when $d \geqslant 2$: the radial part $|B_t|$ can be then described as follows. Let $Y_t$ be the $d$-dimensional Bessel process $Y_t$, consider $Y_t - \lambda t$, and condition this process to never go below $0$. Call this conditional process $Z_t$. Then $|B_t| = Z_t + \lambda t$. But $Y_t - \lambda t$ is no longer time-homogeneous when $d \geqslant 2$, and so I am skeptical about explicit formulae. (In both comments I mean the limiting case as $T \to \infty$, should have mentioned this earlier.) | |
| Oct 22 at 20:10 | comment | added | Mateusz Kwaśnicki | @MartinHairer: If I were to guess what David was willing to say is that the radial part of the conditioned process will behave roughly as $Z_t + \lambda t$, where $Z_t$ is the 3-d Bessel process. This is a reasonable approximation as long as $t$ is not too close to $0$ and $|B_t|$ is not too far away from the boundary $\lambda t$. In dimension $d = 1$, by a straightforward calculation, $|B_t| = Z_t + \lambda t$, where $dZ_t = dW_t + \lambda \coth(\lambda Z_t) dt$, and $\lambda \coth(\lambda x) \sim \frac{1}{x}$ as $x \to 0^+$. That said, I do not think there is an equally simple description… | |
| Oct 22 at 18:22 | comment | added | Martin Hairer | @David Why 3D and what do you mean exactly by Bessel process with drift? It’s certainly ‘something like that’… | |
| Oct 22 at 9:42 | comment | added | Nate River | @David Very interesting. I don’t mind the simplifying assumption $|W_0| = \varepsilon$. | |
| Oct 22 at 8:30 | comment | added | David | Your conditioned process should be a 3d Bessel process with drift $\lambda$. (Assuming you'd start with $W_0 >0$.) | |
| Oct 22 at 7:28 | comment | added | Nate River | Indeed, some sort of characterisation. (am aware that this is somewhat open ended) @RobertWegner | |
| Oct 22 at 7:25 | comment | added | Robert Wegner | @Nate River oh I see you don't just want to construct it, you want to understand and calculate with it, maybe something like the elegant construction of the Brownian bridge. | |
| Oct 22 at 7:16 | comment | added | Nate River | Right, this makes sense from the theoretical viewpoint, but i don’t see any way to describe the resulting probability measure on the preimage of $1$ just from the disintegration machinery. @RobertWegner | |
| Oct 22 at 7:15 | comment | added | Robert Wegner | So I thought (using Wikipedia page notatoon) consider $Y$ to be Wiener space and $X$ to be $\mathbb{R}$, both as polish spaces. Your events indicator function $\pi$ is hopefully measurable and "disintegrates" Wiener space into the preimage of 0 and preimage of 1. On both of these sets (one having full measure and one zero, I think), you now obtain new measures such that "Fubini" holds, i.e. the law of total probability in your case. | |
| Oct 22 at 7:11 | comment | added | Nate River | @RobertWegner Sorry, which variable/sigma-algebra do we disintegrate with respect to? | |
| Oct 22 at 7:07 | comment | added | Robert Wegner | @Nate River Does the disintegration theorem not yield the desired conditional probability measure concentrated on the subset $E_T$ of Wiener space? Unless I am misunderstanding something it should be an exercise of checking the conditions, and at a quick glance it looks good to me. | |
| Oct 22 at 6:57 | history | edited | Nate River | CC BY-SA 4.0 |
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| Oct 22 at 6:56 | comment | added | Nate River | @ClaudeChaunier Ah yes, I am interested in the distribution on $C[0, T]$. | |
| Oct 22 at 6:56 | comment | added | Nate River | @MartinHairer Wait, I did not expect $E_0$ to have positive probability - in that case I can remove the $\varepsilon$ entirely. The SDE characterisation on $[0, T]$ sounds interesting indeed. | |
| Oct 22 at 6:39 | comment | added | Claude Chaunier | @NateRiver, I guess you mean the limiting distribution at time $T$. Not later on when it's fully Brownian. | |
| Oct 22 at 6:38 | comment | added | Martin Hairer | The event $E_0$ has positive probability (for any fixed $T$ and $\lambda$), so it will just converge to $W$, conditioned on $E_0$. With a bit more effort one can derive a time-inhomogeneous SDE for it (with probably a reasonably nice explicit form if you send $T\to \infty$), is that what you're looking for? | |
| Oct 22 at 4:19 | history | edited | Nate River | CC BY-SA 4.0 |
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| Oct 22 at 3:25 | history | edited | Nate River | CC BY-SA 4.0 |
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| Oct 22 at 3:17 | history | asked | Nate River | CC BY-SA 4.0 |