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Let $B_t$ and $W_t$ be two independent planar Brownian motions, started at uniform points in the unit square and run for unit time. Let $S$ be the intersection of their ranges, i.e., the set of points visited by both paths. Are all the connected components of  $S$ single points (almost surely on the event that $S$  is nonempty) ?

It is known (see [1]) that the range of $\{B_t\}$ is intersection-equivalent in the unit square to a dyadic fractal percolation limit set
where squares of side-length $2^{-k}$ are retained with probability $1/k$. We can deduce that $S$ is
intersection-equivalent in the unit square to a fractal percolation limit set $Q$, where squares of side-length $2^{-k}$ are retained with probability $1/k^2$. It follows from [2] that $Q$ almost surely contains connected components that are larger than points; however, intersection equivalence is not strong enough to conclude that $S$ also has this property.

 

[1] Peres, Yuval. "Intersection-equivalence of Brownian paths and certain branching processes." Communications in mathematical physics 177, no. 2 (1996): 417-434.

[2] Chayes, Jennifer T., Lincoln Chayes, and Richard Durrett. "Connectivity properties of Mandelbrot's percolation process." Probability theory and related fields 77, no. 3 (1988): 307-324.