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

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.


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
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