Almost sure finiteness for the total occupation time of an $(d,\alpha,\beta)$-superprocess
Why this work is in the frame
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Bibliographic record
Abstract
For $0<\alpha\leq 2$ and $0<\beta\leq 1$ let $X$ be the $(d,\alpha,\beta)$-superprocess, i.e. the superprocess with $\alpha$-stable spatial movement in $R^d$ and $(1+\beta)$-stable branching. Given that the initial measure $X_0$ is Lebesgue on $R^d$, Iscoe conjectured in [7] that the total occupational time $\int_0^\infty X_t(B)dt$ is a.s. finite if and only if $d\beta < \alpha$, where $B$ denotes any bounded Borel set in $R^d$ with non-empty interior.<br /> <br /> In this note we give a partial answer to Iscoe's conjecture by showing that $\int_0^\infty X_t(B)dt<\infty$ a.s. if $2d\beta < \alpha$ and, on the other hand, $\int_0^\infty X_t(B)dt=\infty$ a.s. if $d\beta > \alpha$.<br /> <br /> For $2d\beta< \alpha$, our result can also imply the a.s. finiteness of the total occupation time (over any bounded Borel set) and the a.s. local extinction for the empirical measure process of the $(d,\alpha,\beta)$-branching particle system with Lebesgue initial intensity measure.
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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 | 0.002 | 0.004 |
| 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 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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