Asymptotic Homology of Brownian motion on a Riemannian manifold

Abstract

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.