Change of measures for Markov chains and the \(L\log L\) theorem for branching processes (Q1567212)

Let \(\{X_n\}\) be a Markov chain on a possibly uncountable state space and assume that \(E_xv(X_1)= \rho v(x)\) for some function \(v\geq 0\) and some \(\rho>0\). Then \(W_n=v (X_n)/ \rho^n v(X_0)\) is a mean 1 non-negative martingale and a multiplicative functional, and therefore defines a new set \(\widetilde P_x\) of Markov probabilities by \(\widetilde P_z(A)= E_x[W_n;A]\) for \(A\in \sigma (X_0,\dots,X_n)\). This fact is used to exhibit conditions for \(W=\lim W_n\) to be non-zero in the setting of branching processes with a general type space, and it is shown how classical \(L\log L\) conditions originating from \textit{H. Kesten} and \textit{B. P. Stigum} [Ann. Math. Stat. 37, 1211-1223 (1966; Zbl 0203.17401) and ibid. 37, 1463-1481 (1966; Zbl 0203.17402)] come out in this way. The paper may be seen as a continuation of \textit{R. Lyons}, \textit{R. Pemantle} and \textit{Y. Peres} [Ann. Probab. 23, No. 3, 1125-1138 (1995; Zbl 0840.60077)].