We know how to compute the density of the bars in the $0$-dimensional persistence diagram for a trajectory of the standard Brownian motion with constant drift $m$, – it is given by
\[
\frac{4m^2e^{-2m\Delta}(1+e^{-2m\Delta})}{(1-e^{-2m\Delta})^3},
\]
But what about other drifts?
Luckily, we have the theory of scale functions for general Brownian motions (i.e., the processes on the real line satisfying the strong Markov property, with trajectories that are continuous a.s.).
Namely, if we have the process conditioned to stay within an interval $[l,u]$, and to exit with certainty from one of the ends, then one can show (see e.g., Revuz-Yor, VII.3.2) that there exists a continuous, strictly increasing scale function $s$ such that for $x\in [b,d], l<b<d<r$, the probability for the process starting at $x$ to leave the interval $[b,d]$ through $b$ is given by
\[
\frac{s(x)-s(b)}{s(d)-s(b)}.
\]
If we know that the process is eventually exiting through $d$, then the total number of {\em windings around the interval $[b,d]$}, starting anywhere below $b$ is distributed geometrically, with the probability of success
\[
p(b,d)=\frac{s(d)-s(b)}{s(u)-s(b)}.
\]
Hence, the expected number of such crossings, and, consequently, the expected content of the quadrant $Q_{b,d}={(b’,d’): l<b'<b<d<d'<r }$ is given by $1/p(b,d)$, and the density of the point process $\mathbf{PH}_0$ at $(b,d)$ is given by
\[
\partial_b\partial_dp(b,d)^{-1} (\mbox{here } \partial_b=\partial/\partial b \mbox{ etc.})
\]
These standard facts become relevant if one wants to understand the point process of the $0$-dim bars obstructing cancellation of a coupled pair $(b,d)$. This happens when $b<d$ are critical values of a (local) minimum and a (local) maximum which form a bar in the persistent diagram. Then the obstacles (introduced in ELMS and ELMSZ) are the bars dominated by $(b,d)$ in the depth poset, ibid.
Rather than define these notions, we refer to the figure below: all the small bars are obstacles, and we want to understand how many of them are there.
Here the cancellation of the bar $(l,h)$ is obstructed by several bars, in each of the three segments (green, black, red) shown. There are two obstructing bars on the green segment, three on the black one (one of them is $(b,d)$), one on the red.
When the trajectory is a realization of a Brownian motion, obviously, countably many. However, the trajectories of the Brownian motions are necessarily tame, and so the question makes sense. One only needs to understand the nature of the segments of the trajectory $B_t$ adjacent to the minima and maxima points.
Here we have a remarkable statement, the Williams decomposition. It says that each of the three segments (green, black and red in the sketch above) is distributed, essentially, as the $3$-dimensional Bessel process (Brownian motion conditioned to stay above $l$, starting at $l$) (i.e., $\mathrm{Bes}_0(3)+l$ and stopped when it reaches $u$. This is true as stated for the middle segment, and with some reorientations of space and time for the other two segments as well.
The scale function for $\mathrm{Bes}_0(3)$ is well-known: it is $s(x)=-1/x$.
Putting all together, we can compute the density of the obstacle bars. For example, for $l=0, u=1$ it is given by
\[
\mu=\frac{b(1-b)+d(1-d)}{(d-b)^3}.
\]
The contour plot on the left shows the level sets of the log of the density $\mu$ for $u=0, l=1$.
The $\Delta^{-3}$ behavior near the diagonal is, of course not surprising.