Let $B$ be a standard Brownian motion, and $A$ a process of finite variation on compacts almost surely, not necessarily adapted to the Brownian filtration.


**Question:** Denoting by $\mathcal L$ the Lebesgue measure, is it true that $$\mathcal L(\{t \, | \, B\_t = A\_t \}) = 0$$


almost surely?



Intersection of Brownian motion and finite variation process
Fix a finite interval $I$.It suffices to show that almost surely,
$$\mathcal L(\{t \in I \, | \, B\_t = A\_t \}) = 0 \,.$$
Brownian motion restricted to $\{t \in I \, | \, B\_t = A\_t \}$ has bounded variation, so a positive answer is implied by the following stronger result, a special case of Theorem 1.3 of [1]:


**Theorem[1]** Let $\{B(t): t\in [0,1]\}$ be a standard Brownian motion.
Then, almost surely, for all $S\subset [0,1]$,
if $B|\_{S}$ is of bounded variation, then $\overline{\dim}\_M S\leq 1/2$.


Here $\overline{\dim}\_M$ is the upper Minkowski dimension (a.k.a. upper box dimension.)


[1] Angel, Omer, Richárd Balka, András Máthé, and Yuval Peres. "Restrictions of Hölder continuous functions." Transactions of the American Mathematical Society 370, no. 6 (2018): 4223-4247.
<https://www.ams.org/journals/tran/2018-370-06/S0002-9947-2018-07126-4/S0002-9947-2018-07126-4.pdf>
<http://wrap.warwick.ac.uk/84253/7/WRAP-restrictions-continuous-functions-Peres-2018.pdf>


