布朗环、奇异同调与渐近环
Brownian Loops, Singular Homology, and Asymptotic Cycles
- Instituto de Matemáticas Universidad Nacional Autónoma de México（墨西哥国立自治大学数学研究所）
- CIMAT（墨西哥数学中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对闭黎曼流形上的半鞅，通过有界闭合过程构造奇异同调类，证明其渐近性质，并推导布朗运动的高斯中心极限定理及椭圆扩散的大偏差原理。
AI中文摘要：
设 $M$ 为闭连通黎曼流形。一个连续半鞅段可以通过一族长度一致有界的路径进行闭合，从而产生一个奇异同调类 $H_t\in H_1(M;\R)$。对于 $H^1(M;\R)$ 的每个光滑闭代表元的线性选择，相应的 Stratonovich 同调与 $H_t$ 相差一个一致有界的项，几乎必然且关于时间一致成立。类 $H_t$ 在时间平移下具有可加性，误差一致有界。对于布朗运动，该比较给出了以归一化 Hodge 内积为协方差的高斯中心极限定理、闭合同调类多边形插值的泛函中心极限定理，以及几乎必然极限 $H_t/t\to0$。对于椭圆扩散，Galkin--Mariani 的大偏差原理以相同的速率函数传递到 $H_T/T$。该构造使用了 Schwartzman 闭合过程，并且在这些渐近尺度下，与所选的有界闭合族无关。
英文摘要：
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.
