Some large-deviation theorems for branching diffusions. (English) Zbl 0759.60024

Lee, Tzong-Yow

Ann. Probab. 20, No. 3, 1288-1309 (1992).

A branching diffusion process is studied when its diffusivity decreases to 0 at the rate of \(\varepsilon<<1\) and its branching-transmutation intensity increases at the rate of \(\varepsilon^{-1}\). The author derives the action functionals which describe some large deviations of the processes as \(\varepsilon\to 0\). In particular, the asymptotic probability as \(\varepsilon\to 0\) that the sample tree contains a branch close to a given function \(\varphi(t)\), and the asymptotic probability that the sample tree contains a 2-branch close to given functions \((\varphi_ 1(t),\varphi_ 2(t))\) are obtained.

Reviewer: A.D.Borisenko (Kiev)

MSC:

60F10	Large deviations
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60J60	Diffusion processes
35B25	Singular perturbations in context of PDEs
35K55	Nonlinear parabolic equations

Keywords:

large-deviation theorems; branching diffusion process; action functionals