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Higher-Order Paracontrolled Calculus for Geometric Rough Paths
Abstract
We study the rough differential equation dut/dt= F(ut) ζt, u(0) = u0 ∈ ℝd, where F : ℝd → L(ℝd, ℝd) is a sufficiently regular vector field, and the noise ζ is a distribution in 𝒞α−1(ℝ) for α∈ (1/4, 1/3). We construct solutions to this equation at lower regularity than previous work of Gubinelli-Imkeller-Perkowski and Prömel-Trabs using ideas from the high-order paracontrolled viewpoint of Bailleul-Bernicot. The key tool in our work is the development of a harmonic-analytic framework for proving a set of iterated commutator estimates which are necessary for the solution theory. We also show the procedure of enhancement when the noise ζ= dZ, where Z is a realization of fractional Brownian motion with Hurst parameter H ∈ (1/4, 1/3).
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BibTeX
@misc{hoose2026higherorder,
title = {{Higher-Order Paracontrolled Calculus for Geometric Rough Paths}},
author = {Tim Van Hoose},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17307},
eprint = {2610.17307},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17307},
}
RIS
TY - UNPB
TI - Higher-Order Paracontrolled Calculus for Geometric Rough Paths
AU - Tim Van Hoose
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.17307
UR - https://arxiv.org/abs/2610.17307
AB - We study the rough differential equation du_t/dt= F(u_t) ζ_t, u(0) = u_0 ∈ ℝ^d, where F : ℝ^d → L(ℝ^d, ℝ^d) is a sufficiently regular vector field, and the noise ζ is a distribution in 𝒞^α−1(ℝ) for α∈ (1/4, 1/3). We construct solutions to this equation at lower regularity than previous work of Gubinelli-Imkeller-Perkowski and Prömel-Trabs using ideas from the high-order paracontrolled viewpoint of Bailleul-Bernicot. The key tool in our work is the development of a harmonic-analytic framework for proving a set of iterated commutator estimates which are necessary for the solution theory. We also show the procedure of enhancement when the noise ζ= dZ, where Z is a realization of fractional Brownian motion with Hurst parameter H ∈ (1/4, 1/3).
ER -