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

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)

Zbl 0611.60052
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