SPDEs with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi mathvariant="bold-italic">α</mml:mi></mml:mrow></mml:math>-Stable Lévy Noise: A Random Field Approach
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Bibliographic record
Abstract
This paper is dedicated to the study of a nonlinear SPDE on a bounded domain in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math>, with zero initial conditions and Dirichlet boundary, driven by an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:math>-stable Lévy noise <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:mi>Z</mml:mi></mml:mrow></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mi>α</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn mathvariant="normal">0,2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mi>α</mml:mi><mml:mo>≠</mml:mo><mml:mn fontstyle="italic">1</mml:mn></mml:math>, and possibly nonsymmetric tails. To give a meaning to the concept of solution, we develop a theory of stochastic integration with respect to this noise. The idea is to first solve the equation with “truncated” noise (obtained by removing from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mrow><mml:mi>Z</mml:mi></mml:mrow></mml:math> the jumps which exceed a fixed value <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mml:mrow><mml:mi>K</mml:mi></mml:mrow></mml:math>), yielding a solution <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M9"><mml:mrow><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>K</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>, and then show that the solutions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M10"><mml:msub><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>L</mml:mi><mml:mo>></mml:mo><mml:mi>K</mml:mi></mml:math> coincide on the event <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M11"><mml:mi>t</mml:mi><mml:mo>≤</mml:mo><mml:msub><mml:mrow><mml:mi>τ</mml:mi></mml:mrow><mml:mrow><mml:mi>K</mml:mi></mml:mrow></mml:msub></mml:math>, for some stopping times <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M12"><mml:mrow><mml:msub><mml:mrow><mml:mi>τ</mml:mi></mml:mrow><mml:mrow><mml:mi>K</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> converging to infinity. A similar idea was used in the setting of Hilbert-space valued processes. A major step is to show that the stochastic integral with respect to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M13"><mml:mrow><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>K</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> satisfies a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M14"><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:math>th moment inequality. This inequality plays the same role as the Burkholder-Davis-Gundy inequality in the theory of integration with respect to continuous martingales.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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