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Let $d = 2$. Do we have that with $P_x$—probability $1$, for every $T> 0$ the complement $W[0, T]^c$ of the Brownian path up to time $T$ has infinitely many connected components?

I had seen this post, and I was wondering if this result is also covered in the 1954 Dvoretsky/Erdos/Kakutani paper. If not, can somebody provide me a reference/supply an answer?

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Yes. Just observe that

(1) on any fixed time interval the Brownian path intersects itself with positive probability (easy to see);

(2) but the above implies that on any time interval the Brownian path intersects itself with probability 1 (divide that interval into many sub-intervals and use the Markov property);

(3) so, with probability 1 in any neighbourhood of a fixed point on the Brownian path there will be at least one connected component of the complement, and this implies the claim.

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    $\begingroup$ Why is (1) "easy to see"? $\endgroup$
    – Liviu Nicolaescu
    Commented Oct 25, 2015 at 12:27
  • $\begingroup$ By conformal invariance, this probability must be the same for all intervals (consider the time interval $[0,c]$; then the mapping $z\mapsto z/\sqrt{c}$ sends the trajectory on $[0,c]$ to trajectory on $[0,1]$). I think it's evident that the Brownian trajectory on the time interval $[0,1]$ intersects itself with positive probability. $\endgroup$
    – Serguei Popov
    Commented Oct 25, 2015 at 17:25
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    $\begingroup$ In dimensions $\geq 4$ a.s. a Brownian path will not have self-intersections according to Dvoretsky, Erdos and Kakutani. The fact that the dimension is $\leq 2$ must enter the argument. I don't see where you used the information about dimension in your argument. $\endgroup$
    – Liviu Nicolaescu
    Commented Oct 25, 2015 at 18:46
  • $\begingroup$ Step 3 should have an argument, too, to rule out something like a space-filling curve that intersects itself a lot but which has empty complement. $\endgroup$
    – Douglas Zare
    Commented Oct 25, 2015 at 19:01
  • $\begingroup$ A proof of the existence of self-intersections in dimensions $\leq 3$ can be found in Sec. 9.3 of Morters and Peress' book research.microsoft.com/en-us/um/people/peres/brbook.pdf $\endgroup$
    – Liviu Nicolaescu
    Commented Oct 25, 2015 at 19:02

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