Brownian charges around loops (Q1912576)

Let \((X^n_t)\) be a Poisson sequence of independent Brownian motions in \(\mathbb{R}^d\), \(d\geq 3\); let \({\mathcal L}\) be a compact oriented submanifold of \(\mathbb{R}^d\), of dimension \(d-2\) and volume \(l\); let \(\Phi_t\) be the sum of the windings of \((X^n_s\), \(0\leq s\leq t)\) around \({\mathcal L}\); then \(\Phi_t/t\) converges in law towards a Cauchy variable of parameter \(l/2\). A similar result is valid when the winding is replaced by the integral of a harmonic 1-form in \(\mathbb{R}^2 \setminus {\mathcal L}\).