路径的$p$-粗糙性与$p$次变差的不变性
$p$-roughness of paths and invariance of $p$-th variation
- Mathematical Institute, University of Oxford（牛津大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出路径的$p$-粗糙性概念，通过离散$p$-能量的均匀收敛刻画，证明其等价于粗粒化误差的介观抵消，并蕴含$p$次变差的不变性，适用于布朗运动与分数布朗运动，为路径微积分提供划分稳健基础。
AI中文摘要：
我们为连续路径引入了一个内蕴的$p$-粗糙性概念，其中$p>1$，该概念通过离散$p$-能量在所有足够细的均匀网格上的均匀收敛来定义。我们证明，这一自平均性质等价于离散$p$-能量的粗粒化误差的介观抵消，从而给出一个可在单一均匀多分辨率上验证的刻画。$p$-粗糙性细化了有限$p$次变差性质，并蕴含了$p$次变差在一类划分序列上的不变性。我们证明布朗运动几乎必然为2-粗糙的，而具有Hurst参数$H$的分数布朗运动几乎必然为$1/H$-粗糙的。我们还基于Faber-Schauder系数推导了$p$-粗糙性的判据。这些结果给出了高阶路径微积分和能量占据测度的划分稳健表述。最后，我们将粗粒化解释为$p$-能量的重整化流，并证明$p$-粗糙性类在临界时间-振幅缩放下是稳定的，其中线性$p$-能量轮廓为不动点。
英文摘要：
We introduce an intrinsic notion of p-roughness for continuous paths,for p>1, defined by the uniform convergence of discrete p-energies over all shifted sufficiently fine uniform grids. We prove that this self-averaging property is equivalent to mesoscopic cancellation of the coarse-graining error for discrete p-energy, yielding a characterization that can be verified on a single uniform multiresolution. $p$-roughness refines the finite p-th variation property and implies the invariance of p-th variation across a class of partition sequences. We prove that Brownian motion is almost surely 2-rough and that fractional Brownian motion with Hurst parameter $H$ is almost surely $1/H$-rough. We also derive criteria for p-roughness based on Faber-Schauder coefficients. These results yield partition-robust formulations of higher-order pathwise calculus and energy occupation measures. Finally, we interpret coarse-graining as a renormalization flow for the p-energy and show that the p-roughness class is stable under critical time-amplitude scaling, with linear p-energy profiles as fixed points.
