Asymptotic Homology of Brownian motion on a Riemannian manifold

We prove, using the celebrated result by Spitzer about winding of planar Brownian motion, and the existence of harmonic morphisms $f:M\to {\mathbb{S}}^1$ representing cohomology classes in ${\text{H}}^1(M,\mathbb{Z})$, that there is a stochastic process $H_t:\mathcal{C}(M)\to \text{Hom}({\text{H}}^1(M;\mathbb{R}),\mathbb{R})\simeq {\text{H}}_1(M;\mathbb{R})$ ($t\in [0,\mathrm{\infty })$), where $\mathcal{C}(M)=\{\alpha :[0,\mathrm{\infty })\to M:\alpha \text{is continuous}\}$, which has a multivariate Cauchy distribution i.e. such that for each nontrivial cohomology class $[\omega ]\in {\text{H}}^1(M;\mathbb{R}),\mathbb{R})$, represented by a closed 1-form $\omega$, in the de Rham cohomology, the process $A_t^{\omega }:\mathcal{C}(M)\to \mathbb{R}$ ($t\in [0,\mathrm{\infty })$) with $A_t^{\omega }(B)=H_t(B)([\omega ]),B\in \mathcal{C}(M)$ converges in distribution, with respect to Wiener measure on $\mathcal{C}(M)$, to a Cauchy’s distribution, with parameter 1. The process describes the ``homological winding" of the Brownian paths in $M$, thus it can be regarded as a generalization of Spitzer result. The last section discusses the asymptotic behavior of holonomy along Brownian paths.