Brownian Loops, Singular Homology, and Asymptotic Cycles
Abstract
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 Ht∈ H1(M;R). For every linear choice of smooth closed representatives of H1(M;R), the corresponding Stratonovich homology differs from Ht by a uniformly bounded term, almost surely and uniformly in time. The classes Ht 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 Ht/t→0. For elliptic diffusions, the large-deviation principle of Galkin--Mariani passes to HT/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.
Related
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BibTeX
@misc{verjovsky2026brownian,
title = {{Brownian Loops, Singular Homology, and Asymptotic Cycles}},
author = {Alberto Verjovsky and Ricardo F. Vila-Freyer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27851},
eprint = {2609.27851},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27851},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Brownian Loops, Singular Homology, and Asymptotic Cycles
AU - Alberto Verjovsky
AU - Ricardo F. Vila-Freyer
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27851
UR - https://arxiv.org/abs/2609.27851
AB - 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∈ 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→0. 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.
ER -