{"ID":22973658,"CreatedAt":"2026-09-17T02:46:45.862280177Z","UpdatedAt":"2026-09-17T02:46:45.862280177Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.18855","arxiv_id":"2609.18855","title":"$p$-roughness of paths and invariance of $p$-th variation","abstract":"We introduce an intrinsic notion of p-roughness for continuous paths,for p\u003e1, 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.","short_abstract":"We introduce an intrinsic notion of p-roughness for continuous paths,for p\u003e1, 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...","url_abs":"https://arxiv.org/abs/2609.18855","url_pdf":"https://arxiv.org/pdf/2609.18855v1","authors":"[\"Rama Cont\"]","published":"2026-09-16T15:57:58Z","proceeding":"math.PR","tasks":"[\"math.PR\",\"math.CA\"]","methods":"[]","has_code":false}