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Consider a Brownian bridge of length $r$ on the plane. What is the expected (non-signed) smallest area of the disc spanned by the loop? By "non-signed" I mean that if a loop goes around a unit square 8 times in the clockwise direction and then 3 times in the counterclockwise direction, then the area is 15. A similar problem was discussed here, for example. But under the definion used there, the area of the loop described above would be 5.

An easier question could be: Is the expected area $O(r)$?

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  • $\begingroup$ I'm not quite sure that I understand what you mean by "unsigned area"... but the unsigned area does not seem to depend continuously on the loop: does that not bother you? Another possible problem: are you sure that the unsigned are isn't always infinity? My guess would be that the contribution coming from "little loops" whose size is between $\epsilon$ and $2\epsilon$ is $\mathcal O(1)$. $\endgroup$
    – André Henriques
    May 14, 2013 at 14:42
  • $\begingroup$ @André: Since I am not a specialist in Brownian motion, my goal is just to understand the situation. It could be that the expected area is $\infty$, but then I would like to see the proof of that. It is also possible that the question does not make sense as formulated and the "expected area" is not defined. That would be also a good answer to my question. $\endgroup$
    – user6976
    May 14, 2013 at 15:15

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If I understand your definition of area correctly, the Brownian bridge ends up having infinite filling area because of small regions with large winding number -- a result of Yor shows that the expected area of the region with winding number $k$ is $\sim 1/k^2$. This contributes $\sim 1/k$ to the area of a filling, so the expected area is infinite.

On the other hand, there are asymptotic bounds on the growth of the area of the random walk bridge. I wrote a paper with Bruno Schapira which shows that the expected homological filling area of a random walk bridge of length $n$ grows like $n \log \log n$. (The homological filling area of a curve is the area of the smallest surface of arbitrary genus whose boundary is the curve -- in this case, it's the integral of the absolute value of the winding number over the plane.)

Roughly, we estimated the homological filling area of the random walk bridge by estimating the area of the region with winding number $k$ for all $k$. A rescaled random walk stays close enough to a Brownian motion that the region with winding number $>k$ has expected area $\sim 1/k$ when $k\le \log n$. When $k>\log n$, the area drops off quickly, so the homological filling area grows like $\log \log n$.

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This is not exactly your definition of area, but there is a physics paper where authors compute the area of points with given winding number (except winding 0).

Winding of planar Brownian curves A Comtet, J Desbois, S Ouvry - Journal of Physics A: …, 1990

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