p-roughness of paths and invariance of p-th variation
Abstract
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.
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BibTeX
@misc{cont2026proughness,
title = {{$p$-roughness of paths and invariance of $p$-th variation}},
author = {Rama Cont},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18855},
eprint = {2609.18855},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18855},
}
RIS
TY - UNPB
TI - p-roughness of paths and invariance of p-th variation
AU - Rama Cont
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18855
UR - https://arxiv.org/abs/2609.18855
AB - 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.
ER -