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

Abstract

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.