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 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 | 0.000 |
| 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".