Penalization of Galton-Watson processes

TL;DR

This paper extends penalization techniques to Galton-Watson processes, revealing new martingales in super-critical cases and linking them to multi-type trees with infinite spines.

Contribution

It introduces a novel penalization approach for Galton-Watson processes, producing new martingales in super-critical regimes and connecting to multi-type branching structures.

Findings

Most limiting martingales are classical except in super-critical case s=1.

New martingales are obtained for s approaching 1 in super-critical case.

Change of measure yields multi-type Galton-Watson trees with infinite spines.

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 $P (x)s^x$ where P is a polynomial of degree p and s $\in$ [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 $\rightarrow$ 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.