Must observations be affine in every finite-dimensional Euclidean estimation algebra?
Let $n, m \ge 1$, and let $f : \mathbb{R}^n \to \mathbb{R}^n$ and $h = (h_1, \dots, h_m) : \mathbb{R}^n \to \mathbb{R}^m$ be smooth. The estimation algebra $E$ is the real Lie algebra of differential operators on $C^\infty(\mathbb{R}^n; \mathbb{R})$, under the commutator, generated by $$L_0 = \frac12 \Delta - \sum_{i=1}^n f_i \partial_i - \operatorname{div} f - \frac12 \sum_{j=1}^m h_j^2$$ and the operators of multiplication by $h_1, \dots, h_m$. Ocone proved that every function in a finite-dimensional $E$ is a polynomial of degree at most two. Chen and Yau proved that the $h_j$ have degree one when $E$ is finite-dimensional of maximal rank, that is, when the homogeneous degree-one polynomials in $E$ form a space of dimension $n$, and Jiao and Yau proved it for $n = 4$ when that dimension is $3$.
Problem. If $E$ is finite-dimensional over $\mathbb{R}$, must every $h_j$ be affine on $\mathbb{R}^n$?
Source: Mitter (1979), in the form recorded by Chen and Yau (1997); called the weak form of Mitter's conjecture by Jiao and Yau (2025), and Conjecture 4.1 (p. 770) of Jiao, Kang and Shi (2025).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- OC — Optimization and Control
- Secondary
- PR
- Keywords
- AIM Open Applied Problems, estimation algebras, Mitter conjecture, nonlinear filtering
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{mitter1979,
author = {Mitter, Sanjoy K.},
title = {On the analogy between mathematical problems of non-linear filtering and quantum physics},
journal = {Ricerche Automat.},
volume = {10},
number = {2},
year = {1979},
pages = {163--216},
}
@article{chen1997,
author = {Chen, Jie and Yau, Stephen S.-T.},
title = {Finite-dimensional filters with nonlinear drift {VII}: {M}itter conjecture and structure of $\eta$},
journal = {SIAM J. Control Optim.},
volume = {35},
number = {4},
year = {1997},
pages = {1116--1131},
doi = {10.1137/S0363012994272836},
}
@article{jiaoyau2025,
author = {Jiao, Xiaopei and Yau, Stephen Shing-Toung},
title = {Weak form {M}itter conjecture on nonmaximal rank estimation algebra: state dimension 4 and rank 3},
journal = {J. Syst. Sci. Complex.},
volume = {38},
number = {1},
year = {2025},
pages = {27--78},
doi = {10.1007/s11424-025-4138-z},
}
@article{jiaokangshi2025,
author = {Jiao, Xiaopei and Kang, Jiayi and Shi, Ji},
title = {Finite-dimensional filter with non-maximal rank},
journal = {Commun. Inf. Syst.},
volume = {25},
number = {4},
year = {2025},
pages = {757--785},
doi = {10.4310/CIS.260128101407},
}
@incollection{ocone1981,
author = {Ocone, Daniel},
title = {Finite dimensional estimation algebras in nonlinear filtering},
booktitle = {Stochastic Systems: The Mathematics of Filtering and Identification and Applications},
editor = {Hazewinkel, Michiel and Willems, Jan C.},
publisher = {Reidel},
address = {Dordrecht},
year = {1981},
pages = {629--636},
doi = {10.1007/978-94-009-8546-9_35},
}
@misc{aim-must-observations-be-affine-in-every-finite-dimensional-euclidean-estimation-algebra,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Must observations be affine in every finite-dimensional Euclidean estimation algebra?}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/885-must-observations-be-affine-in-every-finite-dimensional-euclidean-estimation-algebra.md},
note = {Entry 885 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-01119, "Must observations be affine in every finite-dimensional Euclidean estimation algebra?", clopen.solutions.