Branching Brownian motion with decay of mass and the non-local\n Fisher-KPP equation
Bibliographic record
Abstract
In this work we study a non-local version of the Fisher-KPP equation,\n\\begin{equation*} \\begin{cases} \\frac{\\partial u}{\\partial\nt}=\\tfrac{1}{2}\\Delta u +u (1- \\phi \\ast u), \\quad t>0, \\quad x\\in \\mathbb{R},\nu(0,x)=u_0(x), \\quad x\\in \\mathbb{R} \\end{cases} \\end{equation*} and its\nrelation to $\\textit{branching Brownian motion with decay of mass}$ as\nintroduced by Addario-Berry and Penington (2017), i.e. a particle system\nconsisting of a standard branching Brownian motion (BBM) with a competitive\ninteraction between nearby particles. Particles in the BBM with decay of mass\nhave a position in $\\mathbb{R}$ and a mass, branch at rate 1 into two daughter\nparticles of the same mass and position, and move independently as Brownian\nmotions. Particles lose mass at a rate proportional to the mass in a\nneighbourhood around them (as measured by the function $\\phi$).\n We obtain two types of results. First, we study the behaviour of solutions to\nthe partial differential equation above. We show that, under suitable\nconditions on $\\phi$ and $u_0$, the solutions converge to 1 behind the front\nand are globally bounded, improving recent results of Hamel and Ryzhik\n(arxiv:arXiv:1307.3001). Second, we show that the hydrodynamic limit of the BBM\nwith decay of mass is the solution of the non-local Fisher-KPP equation. We\nthen harness this to obtain several new results concerning the behaviour of the\nparticle system.\n
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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 | 0.002 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.003 | 0.003 |
| Insufficient payload (model declined to judge) | 0.010 | 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".