Non-Markovian maximal couplings and a vertical reflection principle on a class of sub-Riemannian manifolds
Bibliographic record
Abstract
We develop an approach to constructing non-Markovian, non-co-adapted couplings for sub-Riemannian Brownian motions in sub-Riemannian manifolds with large symmetry groups by treating the specific cases of the three-dimensional Heisenberg group, higher-dimensional non-isotropic Heisenberg groups, SL ( 2 , R ) \operatorname {SL}(2,\mathbb {R}) and its universal cover, and SU ( 2 ) \operatorname {SU}(2) . Our primary focus is on the situation when the processes start from two points on the same vertical fiber, since in general Markovian or co-adapted couplings cannot give the sharp rate for the coupling time in this case. Non-Markovian couplings of this type on sub-Riemannian manifolds were first introduced by Banerjee-Gordina-Mariano, for the three-dimensional Heisenberg group, and were more recently extended by Bénéfice to SL ( 2 , R ) \operatorname {SL}(2,\mathbb {R}) and SU ( 2 ) \operatorname {SU}(2) , using a detailed consideration of the Brownian bridge. In contrast, our couplings are based on global isometries of the space, giving couplings that are maximal, as well as making the construction relatively simple and uniform across different manifolds. The coupled processes satisfy a reflection principle with respect to their coupling time, so that the coupling time reduces to the hitting time for one component of the Brownian motion, which is useful in explicitly bounding the tail probability of the coupling time. Further, it’s natural to use this coupling as the second stage of a two-stage coupling when considering points on different vertical fibers. We estimate the coupling time in these various situations and give applications to inequalities for the heat semigroup.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".