The expected area of the filled planar Brownian loop is Pi/5
Garban, Christophe ; Ferreras, José A. Trujillo
arXiv, 0504496 / Harvested from arXiv
Text on arXiv
Let B_t be a planar Brownian loop of time duration 1 (a Brownian motion conditioned so that B_0 = B_1). We consider the compact hull obtained by filling in all the holes, i.e. the complement of the unique unbounded component of R^2\B[0,1]. We show that the expected area of this hull is Pi/5. The proof uses, perhaps not surprisingly, the Schramm Loewner Evolution (SLE). Also, using the result of Yor about the law of the index of a Brownian loop, we show that the expected areas of the regions of non-zero index n equal 1/(2 Pi n^2). As a consequence, we find that the expected area of the region of index zero inside the loop is Pi/30; this value could not be obtained directly using Yor's index description.
Publié le : 2005-04-25
Classification:  Mathematics - Probability,  Mathematical Physics 
@article{0504496,
     author = {Garban, Christophe and Ferreras, Jos\'e A. Trujillo},
     title = {The expected area of the filled planar Brownian loop is Pi/5},
     journal = {arXiv},
     volume = {2005},
     number = {0},
     year = {2005},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0504496}
}

Garban, Christophe; Ferreras, José A. Trujillo. The expected area of the filled planar Brownian loop is Pi/5. arXiv, Tome 2005 (2005) no. 0, . http://gdmltest.u-ga.fr/item/0504496/