# zbMATH — the first resource for mathematics

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.

##### 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
Full Text: