Hölder continuity for parabolic SPDEs with merely bounded transport noise
On $\mathbb{T}^d$, $d \ge 2$, consider the scalar Itô equation $$du = \nabla \cdot (a \nabla u)\,dt + \sum_{k \ge 1} (b_k \cdot \nabla u)\,dW^k_t, \qquad u(0) = u_0,$$ where the Brownian motions $W^k$ are independent, $u_0 \in C^\infty(\mathbb{T}^d)$ is deterministic, and the coefficients are progressively measurable in $(t, \omega)$ and measurable in $x$, with deterministic bounds $$|a| + \Bigl(\sum_k |b_k|^2\Bigr)^{1/2} \le M, \qquad \eta^\top \Bigl(a - \tfrac12 \sum_k b_k \otimes b_k\Bigr) \eta \ge \nu |\eta|^2 \quad (\eta \in \mathbb{R}^d)$$ for some $M < \infty$ and $\nu > 0$. The solution is the variational one, in $C([0,T]; L^2) \cap L^2(0,T; H^1)$, defined by testing against smooth spatial functions, with the stochastic terms integrated in the Itô sense. Dareiotis and Gerencsér proved that it is bounded, and that it is almost surely continuous at each fixed interior point. Agresti, Sauerbrey and Veraar proved that if moreover the $b_k$ are bounded in $C^{3+\delta}(\mathbb{T}^d; \ell^2)$ uniformly in time, for some $\delta \in (0,1)$, the solution is almost surely Hölder continuous, with an exponent that may depend on $\omega$.
Problem. With no spatial regularity imposed on $a$ or $b$, does the solution have a modification such that, almost surely, for every $0 < \tau < T < \infty$ there is a $\gamma = \gamma(\omega, \tau, T) > 0$ with $$u \in C^{\gamma/2, \gamma}([\tau, T] \times \mathbb{T}^d)?$$ Here the Hölder seminorm uses the denominator $|t - r|^{\gamma/2} + \operatorname{dist}(x, y)^\gamma$, and the assertion concerns joint continuity on one event of probability one, not continuity almost surely at each separately fixed point.
Source: Agresti, Sauerbrey and Veraar (2025), §1.2, first open problem (rough transport noise). Stated by AIM — Open Applied Problems on the torus, with smooth deterministic initial data.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, De Giorgi–Nash–Moser estimates, stochastic PDEs, transport noise
References
@misc{agresti2025flow,
author = {Agresti, Antonio and Sauerbrey, Max and Veraar, Mark},
title = {A stochastic flow approach to {D}e {G}iorgi--{N}ash--{M}oser estimates for {SPDE}s with smooth transport noise},
year = {2025},
eprint = {2511.12692},
archivePrefix = {arXiv},
}
@article{dareiotis2015,
author = {Dareiotis, Konstantinos and Gerencs{\'e}r, M{\'a}t{\'e}},
title = {On the boundedness of solutions of {SPDE}s},
journal = {Stoch. Partial Differ. Equ. Anal. Comput.},
volume = {3},
number = {1},
year = {2015},
pages = {84--102},
doi = {10.1007/s40072-014-0043-5},
}
@article{dareiotis2017,
author = {Dareiotis, Konstantinos and Gerencs{\'e}r, M{\'a}t{\'e}},
title = {Local {$L_\infty$}-estimates, weak {H}arnack inequality, and stochastic continuity of solutions of {SPDE}s},
journal = {J. Differential Equations},
volume = {262},
number = {1},
year = {2017},
pages = {615--632},
doi = {10.1016/j.jde.2016.09.038},
}
@misc{aim-rough-transport-spde-holder,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Hölder continuity for parabolic SPDEs with merely bounded transport noise}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/454-rough-transport-spde-holder.md},
note = {Entry 454 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00469, "Hölder continuity for parabolic SPDEs with merely bounded transport noise", clopen.solutions.