On the Density of the Winding Number of Planar Brownian Motion

Abstract

We obtain a formula for the density $$f(\theta , t)$$ of the winding number of a planar Brownian motion $$Z_t$$ around the origin. From this formula, we deduce an expansion for $$f(\log (\sqrt{t})\,\theta ,\,t)$$ in inverse powers of $$\log \sqrt{t}$$ and $$(1+\theta ^2)^{1/2}$$ which in particular yields the corrections of any order to Spitzer’s asymptotic law (in Spitzer, Trans. Am. Math. Soc. 87:187–197, 1958). We also obtain an expansion for $$f(\theta ,t)$$ in inverse powers of $$\log \sqrt{t}$$ , which yields precise asymptotics as $$t \rightarrow \infty $$ for a local limit theorem for the windings.