The expected area of the filled planar Brownian loop is Pi/5

Details The expected area of the filled planar Brownian loop is Pi/5 The expected area of the filled planar Brownian loop is Pi/5

2005-04-25

arxiv.org/abs/math/0504496v1

Mathematics

Probability

15 pages, 3 figures

Scientific paper

10.1007/s00220-006-1555-2

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.

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