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Let $W$ be a standard $d$-dimensional Brownian motion with $W_0 = 0$ almost surely.

Fix a constant $\lambda > 0$ and timeframe $T > 0$, and consider the event

$$ E_T := \{|B_s| \geq \lambda s\ \text{ for all } 0 \leq s \leq T\}.$$

Question: Can we describe the distribution of $W$ on $C[0, T]$ conditional on $E_T$?

Note that $|B_s| \sim \sqrt s$ with high probability, so this linear growth is in fact a rare event on large time frames.

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    $\begingroup$ 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? $\endgroup$
    – Martin Hairer
    Commented Oct 22 at 6:38
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    $\begingroup$ @David Why 3D and what do you mean exactly by Bessel process with drift? It’s certainly ‘something like that’… $\endgroup$
    – Martin Hairer
    Commented Oct 22 at 18:22
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    $\begingroup$ @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... $\endgroup$
    – Martin Hairer
    Commented Oct 22 at 21:08
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    $\begingroup$ @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). $\endgroup$
    – Mateusz Kwaśnicki
    Commented Oct 22 at 21:47
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    $\begingroup$ @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". $\endgroup$
    – Thomas Kojar
    Commented Oct 27 at 1:25

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