From Brownian motion with a local time drift to Feller’s branching diffusion with logistic growth. (English) Zbl 1245.60079

Pardoux, Etienne; Wakolbinger, Anton

Electron. Commun. Probab. 16, 720-731 (2011).

Summary: 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 [V. Le and the authors, “Trees under attack: a Ray-Knight representation of Feller’s branching diffusion with logistic growth”, Probab. Theory Relat. Fields, to appear (2012; doi:10.1007/s00440-011-0408-x)] 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 J. R. Norris, L. C. G. Rogers and D. Williams [Probab. Theory Relat. Fields 74, 271–287 (1987; Zbl 0611.60052)].

MSC:

60J70	Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60J55	Local time and additive functionals
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

Ray-Knight representation; local time; Feller branching with logistic growth; Brownian motion; local time drift; Girsanov transform

Citations:

Zbl 0611.60052