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.

中文翻译：

---

Galton-Watson 过程的惩罚

我们将 Roynette、Vallois、Yor 引入的布朗运动惩罚技术应用到 Galton-Watson 过程中，惩罚函数的形式为 $P (x)s^x$，其中 P 是 p 次多项式，s $\in$ [0, 1]。我们证明了通过这种方法获得的极限鞅在大多数情况下都是经典的，除了在 s = 1（或 s $\rightarrow$ 1）的超临界情况下，我们获得了新的鞅。如果我们用这个鞅改变概率测度，我们得到一个多类型 Galton-Watson 树，它有 p 个可区分的无限刺。