## 60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]

### Refine

#### Document Type

- Article (2)

#### Language

- English (2)

#### Has Fulltext

- yes (2)

#### Is part of the Bibliography

- no (2)

#### Keywords

- Brownian motion (1)
- Feller branching with logistic growth (1)
- Girsanov transform (1)
- Ray-Knight representation (1)
- alpha-stable branching (1)
- coalescent (1)
- genealogy (1)
- local time (1)
- local time drift (1)
- lookdown construction (1)

#### Institute

- Mathematik (2)

From Brownian motion with a local time drift to Feller's branching diffusion with logistic growth
(2011)

We give a new proof for a Ray-Knight representation of Feller's branching diffusion with logistic growth in terms of the local times of a reflected Brownian motion H with a drift that is affine linear in the local time accumulated by H
at its current level. In Le et al. (2011) such a representation was obtained by an approximation through Harris paths that code the genealogies of particle systems. The present proof is purely in terms of stochastic analysis, and is inspired by previous work of Norris, Rogers and Williams (1988).

We determine that the continuous-state branching processes for which the genealogy, suitably time-changed, can be described by an autonomous Markov process are precisely those arising from $\alpha$-stable branching mechanisms. The random ancestral partition is then a time-changed $\Lambda$-coalescent, where $\Lambda$ is the Beta-distribution with parameters $2-\alpha$ and $\alpha$, and the time change is given by $Z^{1-\alpha}$, where $Z$ is the total population size. For $\alpha = 2$ (Feller's branching diffusion) and $\Lambda = \delta_0$ (Kingman's coalescent), this is in the spirit of (a non-spatial version of) Perkins' Disintegration Theorem. For $\alpha =1$ and $\Lambda$ the uniform distribution on $[0,1]$, this is the duality discovered by Bertoin & Le Gall (2000) between the norming of Neveu's continuous state branching process and the Bolthausen-Sznitman coalescent.
We present two approaches: one, exploiting the `modified lookdown construction', draws heavily on Donnelly & Kurtz (1999); the other is based on direct calculations with generators.