Smooth perturbations at the same high irregularity exponents
Fix an integer $d \ge 1$ and parameters $\rho > 1/2$, $1/2 < \gamma < 1$ and $0 < \delta < 1$. For a continuous path $w : [0,1] \to \mathbb{R}^d$ let $$\|\Phi^w\|_{\gamma,\rho} = \sup_{\xi \in \mathbb{R}^d \setminus \{0\},\ 0 \le s < t \le 1} |\xi|^\rho (t - s)^{-\gamma} \left| \int_s^t e^{i \xi \cdot w(r)}\, dr \right|;$$ $w$ is $(\gamma, \rho)$-irregular if this is finite, and $\rho$-irregular if it is $(\gamma, \rho)$-irregular for some $\gamma > 1/2$. A path is smooth on $[0,1]$ if it extends smoothly to a neighbourhood. Catellier and Gubinelli proved that if $w$ is $\rho$-irregular and $\varphi \in C^\beta$ with $1/2 \le \beta < 1$, then $w + \varphi$ is $(\rho - 1/(2\beta))$-irregular, and that a fractional Brownian path of Hurst index $H$ plus an adapted Hölder continuous process is almost surely $\rho$-irregular for every $\rho < 1/(2H)$. Galeati and Gubinelli proved that if $w$ is strongly $\rho$-irregular, a stronger condition that also controls polynomial phases, then $w + \varphi$ is $\tilde\rho$-irregular for every $\tilde\rho < \rho$ and every smooth $\varphi$.
Problem. If $w \in C^\delta([0,1]; \mathbb{R}^d)$ and $\|\Phi^w\|_{\gamma,\rho} < \infty$, must $\|\Phi^{w + \varphi}\|_{\gamma,\rho} < \infty$ for every $\varphi \in C^\infty([0,1]; \mathbb{R}^d)$?
Source: Catellier and Gubinelli (2016), Section 1, who conjecture that w + φ is (ρ, γ)-irregular whenever w is and φ is smooth; recalled by Galeati and Gubinelli (2024), Section 5.4. Stated by AIM — Open Applied Problems for δ-Hölder paths with ρ > 1/2 and 1/2 < γ < 1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- CA
- Keywords
- AIM Open Applied Problems, irregular paths, occupation measures, regularization by noise
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{catellier2016,
author = {Catellier, R{\'e}mi and Gubinelli, Massimiliano},
title = {Averaging along irregular curves and regularisation of {ODE}s},
journal = {Stochastic Process. Appl.},
volume = {126},
number = {8},
year = {2016},
pages = {2323--2366},
doi = {10.1016/j.spa.2016.02.002},
eprint = {1205.1735},
archivePrefix = {arXiv},
}
@article{galeati2024,
author = {Galeati, Lucio and Gubinelli, Massimiliano},
title = {Prevalence of {$\rho$}-irregularity and related properties},
journal = {Ann. Inst. Henri Poincar{\'e} Probab. Stat.},
volume = {60},
number = {4},
year = {2024},
pages = {2415--2467},
doi = {10.1214/23-AIHP1399},
eprint = {2004.00872},
archivePrefix = {arXiv},
}
@misc{aim-smooth-perturbations-at-the-same-high-irregularity-exponents,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Smooth perturbations at the same high irregularity exponents}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/998-smooth-perturbations-at-the-same-high-irregularity-exponents.md},
note = {Entry 998 at commit 4f5b165, 2026-11-04},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01163, "Smooth perturbations at the same high irregularity exponents", clopen.solutions.