On the Density of the Winding Number of Planar Brownian Motion

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.