The Expected Area of the Filled Planar Brownian Loop is π/5

Let B_t,0≤t≤1 be a planar Brownian loop (a Brownian motion conditioned so that B₀=B₁). We consider the compact hull obtained by filling in all the holes, i.e. the complement of the unique unbounded component of \B[0,1]. We show that the expected area of this hull is π/5. The proof uses, perhaps not surprisingly, the Schramm Loewner Evolution (SLE). As a consequence of this result, using Yor's formula [17] for the law of the index of a Brownian loop, we find that the expected area of the region inside the loop having index zero is π/30; this value could not be obtained directly using Yor's index description.