发表机构
Tongji University（同济大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明分数布朗运动中H=1/4是可测局域性的精确阈值，并引入精确尺度演算，区分粗糙路径的存在性与局部可恢复性。
AI 中文摘要
自Young 1936年定理以来，不规则积分一直围绕阈值1/2组织：高于该阈值时，路径决定积分；而低于该阈值时，则需要高阶数据。对于分数布朗运动，H = 1/4是典型高斯增强的阈值，尽管几何粗糙提升对每个H > 0都存在。我们证明H = 1/4反而是可测局域性的精确阈值。对于具有独立分量的d维分数布朗运动，d至少为2，若0 < H ≤ 1/4，则不存在正测度Borel集支持哪怕一个满足Chen关系的有限非对角二阶坐标，且该坐标可由路径增量逐区间可测地得到。我们不施加任何矩、Hölder、几何性、平稳性或缩放假设。若1/4 < H ≤ 1/2，则标准Hölder范围内的每个全律局部粗糙路径提升自动为平方可积，因此无需假设L2条件即可分类；对于H > 1/2，每个具有自然分级Hölder界的有限步增强唯一地是Young签名。我们还引入了经典可微性以下的精确尺度演算。匹配的二进差分以尖锐的O(ε²)误差恢复归一化导数，实现精确的局部反演和无损重构。乘法和光滑函数演算精确地转移到尺度坐标，而corona商产生精确的通用导子。重新插入Hölder振幅并使用Fourier正规排序，可为每个正输入正则性和每个较低粗糙指数产生强几何提升，并具有明确的紫外速率和稳定性。因此，粗糙提升存在于四分之一以下，但正测度域上的可测提升在该处不能是区间局域的。这些结果将存在性与局部可恢复性分开，并表明不可微性不会破坏精确的微分信息。
英文摘要
Since Young's 1936 theorem, irregular integration has been organized around the threshold 1/2: above it the path determines the integral, while below it higher-order data are needed. For fractional Brownian motion, H = 1/4 is the threshold for the canonical Gaussian enhancement, although geometric rough lifts exist for every H > 0. We prove that H = 1/4 is instead the exact threshold for measurable locality. For d-dimensional fractional Brownian motion with independent components, d at least 2, if 0 < H <= 1/4, no positive-measure Borel set supports even one finite off-diagonal second-level coordinate satisfying Chen's relation and measurable interval by interval from path increments. No moment, Holder, geometricity, stationarity, or scaling assumption is imposed. If 1/4 < H <= 1/2, every full-law local rough-path lift in the standard Holder range is automatically square-integrable and hence classified without an assumed L2 condition; for H > 1/2, every finite-step enhancement with natural graded Holder bounds is uniquely the Young signature. We also introduce an exact scale calculus below classical differentiability. Matched dyadic differences recover normalized derivatives with sharp O(epsilon^2) error, exact localized inversion, and lossless reconstruction. Multiplication and smooth functional calculus transport exactly to scale coordinates, while the corona quotient yields an exact universal derivation. Reinserting the Holder amplitude and using Fourier-normal ordering produces strong geometric lifts for every positive input regularity and every lower rough exponent, with explicit ultraviolet rates and stability. Thus rough lifts exist below one quarter, but no measurable lift on a positive-measure domain can be interval-local there. The results separate existence from local recoverability and show that nondifferentiability does not destroy exact differential information.
