On the regularity of the extinction probability of a branching process in varying and random environments

摘要

We consider a supercritical branching process in time-dependent environment xi. We assume that the offspring distributions depend regularly (C-k or real-analytically) on real parameters lambda. We show that the extinction probability q lambda(xi), given the environment xi 'inherits' this regularity whenever the offspring distributions satisfy a condition of contraction-type. Our proof makes use of the Poincare metric on the complex unit disc and a real-analytic implicit function theorem.