Asymptotic Homology of Brownian motion on a Riemannian manifold
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914921107685376 |
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| author | Verjovsky, Alberto Vila-Freyer, Ricardo F. |
| author_facet | Verjovsky, Alberto Vila-Freyer, Ricardo F. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_13215 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Asymptotic Homology of Brownian motion on a Riemannian manifold Verjovsky, Alberto Vila-Freyer, Ricardo F. Probability Dynamical Systems 58J65, 60J65, 58J65 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. |
| title | Asymptotic Homology of Brownian motion on a Riemannian manifold |
| topic | Probability Dynamical Systems 58J65, 60J65, 58J65 |
| url | https://arxiv.org/abs/2312.13215 |