Uniqueness of a reflected rough equation with one scalar input
Let $d \ge 2$ and $2 < p < 3$, let $D \subset \mathbb{R}^d$ be a bounded connected domain with smooth boundary and inward unit normal $n$, let $f \in C_b^\infty(\mathbb{R}^d; \mathbb{R}^d)$ and $y_0 \in \overline{D}$. Let $x : [0,1] \to \mathbb{R}$ be a continuous path with $x_0 = 0$ and finite $p$-variation, enhanced to the rough path $\mathbf{X}_{s,t} = \bigl(x_t - x_s, \tfrac12 (x_t - x_s)^2\bigr)$. A solution of the normally reflected equation $$Y_t = y_0 + \int_0^t f(Y_s)\, d\mathbf{X}_s + K_t, \qquad Y_t \in \overline{D}, \qquad K_0 = 0, \qquad dK_t = n(Y_t)\, d|K|_t$$ is a pair $(Y, K)$ of continuous paths on $[0,1]$ such that $K$ has bounded variation, with variation measure $|K|$ supported on $\{t : Y_t \in \partial D\}$, and $Y$ has finite $p$-variation and is controlled by $x$ with derivative $f(Y)$: the remainder $Y_t - Y_s - f(Y_s)(x_t - x_s)$ has finite $p/2$-variation. The integral is the controlled rough integral, the limit of the compensated sums $\sum_{[s,t] \in \mathcal{P}} \bigl( f(Y_s)(x_t - x_s) + \tfrac12 Df(Y_s) f(Y_s)(x_t - x_s)^2 \bigr)$ over partitions $\mathcal{P}$. Aida proved existence of solutions of reflected equations driven by rough paths of finite $p$-variation, $p < 3$, under conditions on the domain. Deya, Gubinelli, Hofmanová and Tindel proved existence and uniqueness for one-dimensional equations reflected at zero. Gassiat constructed a reflected equation in a half-plane with uncountably many solutions, driven by a scalar path of finite $p$-variation for every $p > 2$ together with a second, nondecreasing input; he observes that when the vector field of a scalar-driven equation is never tangent to the boundary, a local change of variables to additive noise gives uniqueness.
Problem. Does this equation have at most one solution $(Y, K)$, for every such $d$, $p$, $D$, $f$, $y_0$ and $x$?
Source: Gassiat (2021), Remark 2.7(3), who conjectures that uniqueness holds for every one-dimensional driving signal, that is, his assertion (A)_{d,1,p} for all d and p. Stated by AIM — Open Applied Problems for bounded smooth domains, 2 < p < 3 and the canonical enhancement of a scalar path.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- CA
- Keywords
- AIM Open Applied Problems, reflected differential equations, rough differential equations
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{gassiat2021,
author = {Gassiat, Paul},
title = {Non-uniqueness for reflected rough differential equations},
journal = {Ann. Inst. Henri Poincar{\'e} Probab. Stat.},
volume = {57},
number = {3},
year = {2021},
pages = {1369--1387},
doi = {10.1214/20-AIHP1121},
eprint = {2001.11914},
archivePrefix = {arXiv},
}
@article{aida2015,
author = {Aida, Shigeki},
title = {Reflected rough differential equations},
journal = {Stochastic Process. Appl.},
volume = {125},
number = {9},
year = {2015},
pages = {3570--3595},
doi = {10.1016/j.spa.2015.03.008},
eprint = {1311.6104},
archivePrefix = {arXiv},
}
@article{deya2019,
author = {Deya, Aur{\'e}lien and Gubinelli, Massimiliano and Hofmanov{\'a}, Martina and Tindel, Samy},
title = {One-dimensional reflected rough differential equations},
journal = {Stochastic Process. Appl.},
volume = {129},
number = {9},
year = {2019},
pages = {3261--3281},
doi = {10.1016/j.spa.2018.09.007},
eprint = {1610.07481},
archivePrefix = {arXiv},
}
@misc{aim-uniqueness-of-a-reflected-rough-equation-with-one-scalar-input,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Uniqueness of a reflected rough equation with one scalar input}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/993-uniqueness-of-a-reflected-rough-equation-with-one-scalar-input.md},
note = {Entry 993 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-01167, "Uniqueness of a reflected rough equation with one scalar input", clopen.solutions.