On the density of the winding number of planar Brownian motion

TL;DR

This paper derives a detailed formula for the density of the winding number of planar Brownian motion, providing asymptotic expansions and corrections to classical laws like Spitzer's law.

Contribution

It introduces explicit formulas and higher-order corrections for the winding number density of planar Brownian motion, extending classical results.

Findings

Derived a formula for the winding number density

Obtained asymptotic expansions in inverse powers of log time

Provided higher-order corrections to Spitzer's law

Abstract

We obtain a formula for the density of the winding number of planar Brownian motion around the origin, and deduce from it asymptotic expansions in inverse powers of the logarithm of the squared time, explicit in the angular variable. In particular, we obtain the corrections of any order to the Spitzer's law, and also to a local limit theorem for the windings.