  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 $H_t:{\mathcal C}(M)\to{\text{Hom}(\text{H}^1(M;\mathbb R),
\mathbb R)}\simeq{\text{H}_1(M;\mathbb R)}$ ($t\in[0,\infty)$), where
${\mathcal C}(M)= \{ \alpha:[0, \infty) \to M :\alpha \,\, \text{is continuous}
\}$, 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
$A^\omega_t:{\mathcal C}(M)\to\mathbb R\,$ ($t\in[0,\infty)$) with
$A^\omega_t(B)=H_t(B)([\omega]),\, B\in{\mathcal C}(M)$ 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.
