A $\lambda$-invariant measure of a sub-Markov chain is a left eigenvector of its transition matrix of eigenvalue $\lambda$. In this article, we give an explicit integral representation of the $\lambda$-invariant measures of subcritical Bienaym\'e--Galton--Watson processes killed upon extinction, i.e.\ upon hitting the origin. In particular, this characterizes all quasi-stationary distributions of these processes. Our formula extends the Kesten--Spitzer formula for the (1-)invariant measures of such a process and can be interpreted as the identification of its minimal $\lambda$-Martin entrance boundary for all $\lambda$. In the particular case of quasi-stationary distributions, we also present an equivalent characterization in terms of semi-stable subordinators. Unlike Kesten and Spitzer's arguments, our proofs are elementary and do not rely on Martin boundary theory.

翻译：亚马可夫链的 $lambda$- Watson 参数是其过渡矩阵的左半导体元值 $\ lambda$\ lambda$。 在本条中,我们明确代表了亚临界比亚基那伊姆\ e- Galton- Watson 进程在灭绝时即在撞击源头时死亡的 $lumbda$- 内变量值的变量值。 特别是, 这具有这些过程所有准静止分布的特点。 我们的公式扩展了该过程的( 1- ) 变量的 Kesten- Spitzer 公式, 并可以解释为所有\ lambda 美元 的最小的 $\ lambda$- Martin 入口边界。 在半静止分布的特定情况下, 我们还在半稳定的副协调员方面提出了等同的特征。 与 Kesten 和 Spitzer 的论据不同, 我们的证据是基本的, 并不依赖 Martin 边界理论 。