Markov uniqueness of the Bismut-gradient operator on based path space
Let $M$ be a smooth compact connected Riemannian manifold without boundary, of dimension at least two, and fix $x_0 \in M$ and $T > 0$. Let $C_{x_0}M$ be the space of continuous paths $\sigma : [0,T] \to M$ with $\sigma(0) = x_0$ (the endpoint is free), with Brownian motion measure $\mu$ for the generator $\frac12\Delta_M$. With $/\!/_t : T_{x_0}M \to T_{\sigma(t)}M$ stochastic Levi-Civita parallel translation, the Bismut tangent space at $\sigma$ is $$\mathcal H_\sigma = \{ v_t = /\!/_t\, h_t : h \in H^1([0,T]; T_{x_0}M),\ h_0 = 0 \},$$ with the damped inner product $$\langle v, w \rangle_{\mathcal H_\sigma} = \int_0^T \Bigl\langle \frac{Dv_t}{dt} + \frac12 \operatorname{Ric}^\#_{\sigma(t)} v_t,\ \frac{Dw_t}{dt} + \frac12 \operatorname{Ric}^\#_{\sigma(t)} w_t \Bigr\rangle\,dt.$$ Let $\mathrm{Cyl}$ be the smooth cylindrical functions $F(\sigma) = f(\sigma(t_1), \dots, \sigma(t_k))$, $f \in C^\infty(M^k)$, let $d : D^{2,1} \subset L^2(\mu) \to L^2\Gamma\mathcal H^*$ be the closure of the restrictions of their differentials to the $\mathcal H_\sigma$, with Hilbert space adjoint $d^*$, and let $L_0 = -d^*d$ on $\mathrm{Cyl}$. Elworthy and Li proved that Markov uniqueness of $L_0$ is equivalent to the equality of $D^{2,1}$ with the space $W^{2,1}$ of weakly differentiable functions on $C_{x_0}M$, and that it would follow if, for an Itô map $\mathcal I$ of a stochastic differential equation whose solutions are Brownian motions on $M$ and whose induced connection is the Levi-Civita connection, the conditional expectation given $\mathcal I$ of every function in $D^{2,1}$ on classical Wiener space were again in $D^{2,1}$; they also identified a gap in an earlier proof of this last property.
Problem. Is it true that for every such $M$, $x_0$ and $T$ there is exactly one symmetric, strongly continuous, sub-Markovian contraction semigroup $(P_t)$ on $L^2(\mu)$ whose generator extends $L_0$, where sub-Markovian means that $0 \le f \le 1$ implies $0 \le P_t f \le 1$?
Source: Elworthy and Li (2006), "Intertwining and the Markov uniqueness problem on path spaces" and their ICM 2006 address, Section 2.2; the definition is that of Elworthy and Li (2007), Section 6.2. Stated by AIM — Open Applied Problems for Brownian motion measure and the Levi-Civita connection on manifolds of dimension at least two.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- DG, FA
- Keywords
- AIM Open Applied Problems, Malliavin calculus, Markov uniqueness, path spaces
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@inproceedings{elworthy2006icm,
author = {Elworthy, K. David and Li, Xue-Mei},
title = {Geometric stochastic analysis on path spaces},
booktitle = {International {C}ongress of {M}athematicians. {V}ol. {III}},
publisher = {Eur. Math. Soc.},
address = {Z{\"u}rich},
year = {2006},
pages = {575--594},
doi = {10.4171/022-3/30},
eprint = {1911.09764},
archivePrefix = {arXiv},
}
@article{elworthy2007,
author = {Elworthy, K. D. and Li, Xue-Mei},
title = {It{\^o} maps and analysis on path spaces},
journal = {Math. Z.},
volume = {257},
number = {3},
year = {2007},
pages = {643--706},
doi = {10.1007/s00209-007-0145-0},
eprint = {math/0605089},
archivePrefix = {arXiv},
}
@incollection{elworthy2006intertwining,
author = {Elworthy, K. David and Li, Xue-Mei},
title = {Intertwining and the {M}arkov uniqueness problem on path spaces},
booktitle = {Stochastic Partial Differential Equations and Applications---{VII}},
series = {Lect. Notes Pure Appl. Math.},
volume = {245},
publisher = {Chapman \& Hall/CRC},
address = {Boca Raton, FL},
year = {2006},
pages = {89--95},
doi = {10.1201/9781420028720.ch9},
eprint = {1911.10969},
archivePrefix = {arXiv},
}
@misc{aim-markov-uniqueness-of-the-bismut-gradient-operator-on-based-path-space,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Markov uniqueness of the Bismut-gradient operator on based path space}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/1005-markov-uniqueness-of-the-bismut-gradient-operator-on-based-path-space.md},
note = {Entry 1005 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-01158, "Markov uniqueness of the Bismut-gradient operator on based path space", clopen.solutions.