Penalization of Galton–Watson processes

Abstract

We apply the penalization technique introduced by Roynette, Vallois, Yor for Brownian motion to Galton–Watson processes with a penalizing function of the form P(x)sx where P is a polynomial of degree p and s∈[0,1]. We prove that the limiting martingales obtained by this method are most of the time classical ones, except in the super-critical case for s=1 (or s→1) where we obtain new martingales. If we make a change of probability measure with this martingale, we obtain a multi-type Galton–Watson tree with p distinguished infinite spines.