A method for finding the fixed vector of a stochastic operator (Q2342023)

The problem is to compute a stationary probability vector \(x\), i.e., to solve the linear system \((I-P)x=0\) of size \(n\), where \(P\) is a stochastic matrix. The idea is to solve the equivalent system \((I-P+\Lambda)x=s\) instead, where \(s=[s_1,\dots,s_n]^T\) is a stochastic vector and \(\Lambda=s \ell^T\) with \(\ell=[\lambda_1,\dots,\lambda_n]^T\). The system is solved by iteration and conditions are given for \(s\) and the choice of \(\ell\) to make \(I-P+\Lambda\) contractive, hence for \(x_{m+1}=s+(P-\Lambda)x_m\) to converge geometrically to a solution (unique up to normalization). A slightly adapted but similar result is obtained in two infinite-dimensional cases, where \(P\) is a stochastic operator in \(\ell_1\) or a stochastic integral operator in \(L_1\) (on a measurable set \(G\subseteq\mathbb{R}^n\)).