Some Large-Deviation Theorems for Branching Diffusions

Abstract

A branching diffusion process is studied when its diffusivity decreases to 0 at the rate of $\varepsilon \ll 1$ and its branching/transmutation intensity increases at the rate of $\varepsilon^{-1}$. We derive the action functionals which describe some large deviations of the processes as $\varepsilon$ tends to 0. The branching diffusion processes are closely related to systems of semilinear parabolic differential equations.