Degenerate stochastic differential equations with Hölder continuous coefficients and super-Markov chains
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Bibliographic record
Abstract
We consider the operator <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript i comma j equals 1 Overscript d Endscripts StartRoot x Subscript i Baseline x Subscript j Baseline EndRoot gamma Subscript i j Baseline left-parenthesis x right-parenthesis StartFraction partial-differential squared Over partial-differential x Subscript i Baseline partial-differential x Subscript j Baseline EndFraction plus sigma-summation Underscript i equals 1 Overscript d Endscripts b Subscript i Baseline left-parenthesis x right-parenthesis StartFraction partial-differential Over partial-differential x Subscript i Baseline EndFraction"> <mml:semantics> <mml:mrow> <mml:munderover> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>d</mml:mi> </mml:munderover> <mml:msqrt> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:msqrt> <mml:msub> <mml:mi> γ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mfrac> <mml:msup> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:mrow> </mml:mfrac> <mml:mo>+</mml:mo> <mml:munderover> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>d</mml:mi> </mml:munderover> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mfrac> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mrow> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:mfrac> </mml:mrow> <mml:annotation encoding="application/x-tex">\sum _{i,j=1}^d \sqrt {x_ix_j}\gamma _{ij}(x) \frac {\partial ^2}{\partial x_i \partial x_j}+\sum _{i=1}^d b_i(x) \frac {\partial }{\partial x_i}</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> acting on functions in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript b Superscript 2 Baseline left-parenthesis double-struck upper R Subscript plus Superscript d Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>b</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>d</mml:mi> </mml:msubsup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C_b^2(\mathbb {R}^d_+)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We prove uniqueness of the martingale problem for this degenerate operator under suitable nonnegativity and regularity conditions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma Subscript i j"> <mml:semantics> <mml:msub> <mml:mi> γ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\gamma _{ij}</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="b Subscript i"> <mml:semantics> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">b_i</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In contrast to previous work, the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b Subscript i"> <mml:semantics> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">b_i</mml:annotation> </mml:semantics> </mml:math> </inline-formula> need only be nonnegative on the boundary rather than strictly positive, at the expense of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma Subscript i j"> <mml:semantics> <mml:msub> <mml:mi> γ </mml:mi>
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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) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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 it