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, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper L left-parenthesis 2 comma double-struck upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>SL</mml:mi> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {SL}(2,\mathbb {R})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and its universal cover, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper U left-parenthesis 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>SU</mml:mi> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {SU}(2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper L left-parenthesis 2 comma double-struck upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>SL</mml:mi> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {SL}(2,\mathbb {R})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper U left-parenthesis 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>SU</mml:mi> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {SU}(2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , 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 distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".