Consider a [Brownian bridge](http://en.wikipedia.org/wiki/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](http://www.eurandom.tue.nl/events/workshops/2006/YEP2006/Ferreras.pdf), 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 $O(r)$?