arXiv subjects

Ricardo F. Vila-Freyer

Publications and source records attributed to Ricardo F. Vila-Freyer.

2 recordsLinked to original sources

arXiv2026

Brownian Loops, Singular Homology, and Asymptotic Cycles

Let $M$ be a closed connected Riemannian manifold. A continuous semimartingale segment can be closed by a Borel family of paths of uniformly bounded length, producing a singular homology class $H_t\in H_1(M;\R)$. For every linear choice of smooth closed representatives of $H^1(M;\R)$, the corresponding Stratonovich homology differs from $H_t$ by a uniformly bounded term, almost surely and uniformly in time. The classes $H_t$ are additive under time shift up to a uniformly bounded error. For Brownian motion this comparison yields the Gaussian central limit theorem with covariance given by the normalized Hodge inner product, a functional central limit theorem for the polygonal interpolation of the closed homology classes, and the almost sure limit $H_t/t\to0$. For elliptic diffusions, the large-deviation principle of Galkin--Mariani passes to $H_T/T$ with the same rate function. The construction uses Schwartzman's closing procedure and is independent, at these asymptotic scales, of the chosen bounded closing family.

math.PR↗

arXiv2024

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 $H_t:{\mathcal C}(M)\to{\text{Hom}(\text{H}^1(M;\mathbb R), \mathbb R)}\simeq{\text{H}_1(M;\mathbb R)}$ ($t\in[0,\infty)$), where ${\mathcal C}(M)= \{ α:[0, \infty) \to M :α\,\, \text{is continuous} \}$, which has a multivariate Cauchy distribution i.e. such that for each nontrivial cohomology class $[ω]\in{\text{H}^1(M;\mathbb R), \mathbb R)}$, represented by a closed 1-form $ω$, in the de Rham cohomology, the process $A^ω_t:{\mathcal C}(M)\to\mathbb R\,$ ($t\in[0,\infty)$) with $A^ω_t(B)=H_t(B)([ω]),\, B\in{\mathcal C}(M)$ 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.

math.PR↗