Penalization of Galton-Watson processes (Q2309599)

This paper studies penalization of Galton-Watson processes. The authors apply the penalization technique of Roynette-Vallois-Yor [\textit{B. Roynette} et al., Stud. Sci. Math. Hung. 43, No. 3, 295--360 (2006; Zbl 1121.60004)] for Brownian motion to Galton-Watson processes with a penalizing function of the form \(P(x)s^x\), where \(P\) is a polynomial of degree \(p\) and \(s\in [0, 1]\). The authors prove that the limiting martingales obtianed via this method are usually the classical limiting martingales, except in the super-critical case for \(s=1\) (or \(s\uparrow 1\)), where a new limiting martingale appears. When this limiting martingale is used to make a martingale change of measure, a multi-type Galton-Watson tree with \(p\) distinguished spines is obtained.