$p$-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.