Hörmander densities for equations driven by measurable Markovian rough paths
Let $d, e \ge 1$ and $N \ge 2$, let $G^N(\mathbb R^d)$ be the free step-$N$ nilpotent Lie group with horizontal left-invariant generators $U_1, \dots, U_d$ and Haar measure $\lambda$, and let $a$ be a measurable map from $G^N(\mathbb R^d)$ to the symmetric $d \times d$ matrices with $$\Lambda^{-1}|\xi|^2 \le \xi^\top a(x)\,\xi \le \Lambda|\xi|^2 \qquad (x \in G^N(\mathbb R^d),\ \xi \in \mathbb R^d)$$ for some finite $\Lambda \ge 1$. For $x_0 \in G^N(\mathbb R^d)$ let $\mathbf X = \mathbf X^{a,x_0}$ be the Markov diffusion on $G^N(\mathbb R^d)$ started at $x_0$ and associated with the closure in $L^2(\lambda)$ of $$\mathcal E(f,h) = \int \sum_{i,j=1}^d a_{ij}(x)\,(U_i f)(x)\,(U_j h)(x)\,d\lambda(x), \qquad f, h \in C_c^\infty(G^N(\mathbb R^d)),$$ viewed as a geometric rough path (a Markovian rough path in the sense of Chevyrev and Ogrodnik). For $V_1, \dots, V_d \in C_b^\infty(\mathbb R^e; \mathbb R^e)$ and $y_0 \in \mathbb R^e$ let $Y$ solve the rough differential equation $$dY_t = \sum_{i=1}^d V_i(Y_t)\,d\mathbf X^i_t, \qquad Y_0 = y_0.$$ Chevyrev and Ogrodnik proved that $Y_t$, $t > 0$, has a density on the orbit of $y_0$ under the $V_i$ whenever the dimension of the span of the iterated brackets of the $V_i$ of length greater than $N$ is constant along that orbit; this condition implies Hörmander's condition on the orbit, but not conversely. Yang proved, for $e = d$, that $Y_t$, $0 < t \le 1$, has a smooth density under Hörmander's condition when $a$ depends only on the first level, is smooth, non-constant and bounded with all its derivatives, and the differential of its square root satisfies a uniform non-degeneracy condition.
Problem. Is it true that for all such $d$, $e$, $N$ and $a$, all $V_1, \dots, V_d \in C_b^\infty(\mathbb R^e; \mathbb R^e)$ whose iterated Lie brackets span $\mathbb R^e$ at every point, and all $x_0 \in G^N(\mathbb R^d)$ and $y_0 \in \mathbb R^e$, the law of $Y_t$ is absolutely continuous with respect to Lebesgue measure on $\mathbb R^e$ for every $t > 0$?
Source: Chevyrev and Ogrodnik (2018), Remark 3.11. Stated by AIM — Open Applied Problems for bounded smooth vector fields whose brackets span the whole space, with densities with respect to Lebesgue measure.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, Dirichlet forms, Hörmander's condition, rough paths
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{chevyrev2018,
author = {Chevyrev, Ilya and Ogrodnik, Marcel},
title = {A support and density theorem for {M}arkovian rough paths},
journal = {Electron. J. Probab.},
volume = {23},
year = {2018},
number = {56},
pages = {1--16},
doi = {10.1214/18-EJP184},
eprint = {1701.03002},
archivePrefix = {arXiv},
}
@article{yang2024,
author = {Yang, Guang},
title = {A version of {H}{\"o}rmander's theorem for {M}arkovian rough paths},
journal = {Potential Anal.},
volume = {60},
number = {1},
year = {2024},
pages = {173--195},
doi = {10.1007/s11118-022-10046-5},
eprint = {2005.09192},
archivePrefix = {arXiv},
}
@misc{aim-hormander-densities-for-equations-driven-by-measurable-markovian-rough-paths,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Hörmander densities for equations driven by measurable Markovian rough paths}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/1004-hormander-densities-for-equations-driven-by-measurable-markovian-rough-paths.md},
note = {Entry 1004 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-01156, "Hörmander densities for equations driven by measurable Markovian rough paths", clopen.solutions.