Brownian Windings

Given a point and a loop in the plane, one can define a relative integer which counts how many times the curve winds around the point. We will discuss how this winding function, defined for almost every points in the plane, allows to define some integrals along the loop. Then, we will investigate some properties of it when the loop is Brownian.
In particular, we will explain how to recover data such as the Lévy area of the curve and its occupation measure, based on the values of the winding of uniformly distributed points on the plane.