The asymptotic properties of weighted Markov operators (Q698908)

Let \((X.\Sigma,u)\) be a \(\sigma\)-finite measure space, and \(D=\{f\in L^1: f\geq 0,\|f\|=1\}\). An operator \(P: L^1\to L^1\) is called Markov operator if \[ Pf\geq 0 \quad\text{for } f\geq 0, \qquad \|Pf\|=\|f\|\quad \text{for } f\geq 0. \] If \(f\in D\) is a fixed point of \(P\), \(f\) is called a stationary density. Let \(\{q_n(t)\}\) be a sequence of nonnegative real functions from \(\mathbb{R}^+\to[0,1]\) such that \[ \sum_{n=0}^\infty q_n(t)=1\quad \text{for all } t>0, \] \[ \lim_{t\to\infty}(q_0(t)+\sum_{n=1}^\infty|q_n(t)-q_{n-1}(t)|)=0. \] The weighted Markov operator \(\{Q_t\}\) is defined by \(Q_tf=\sum_{n=0}^\infty q_n(t)P^nf\). There exist examples of weighted Markov operators: (1) \(q_n(m)={1\over n}\) whenever \(0\leq m<n\) and \(q_n(m)=0\) for \(m\geq n\). The weighted Markov operator is \(A_nf={1\over n}\sum_{k=0}^{n-1}P^kf\). (2) \(q_n(t)=e^{-t}{t^n\over n!}\). The weighted Markov operator plays an important role in understanding the asymptotic behavior of solutions to the linear Boltzmann equation. (3) \(q_n(t)=\varepsilon(1-\varepsilon)^n\), where \(\varepsilon\in(0,1)\), which is used to consider the stability of discrete nonsingular system under randomly applied stochastic perturbations. In the article under review, the limit behavior of \(\{Q_t\}\) is studied. Theorem 2.3. If one of the two sequences \(\{A_nf\}\) and \(\{Q_tf\}\) is weakly precompact, then both limits coincide. Let \(\operatorname {supp}f=\{x: g(x)\neq 0\}\) which is defined up to measure zero. A stationary density \(g\) has maximal support if \(u(\operatorname {supp}g- \operatorname {supp}f)=0\) for every stationary density \(f\), and the maximal support is denoted by \(G(P)\). Theorem 4.3. Let \(g\) be a stationary density with maximal support \(G(P)\). Consider the \(\sigma\)-algebra \(\Sigma_*=\{A\in\Sigma: P(1_A\cdot g)/g=1_A\}\). If \(f\in L^1\) and \(\operatorname {supp}f\subset G(P)\), then \[ \lim_{n\to\infty}A_nf=g\cdot E(f/g|\Sigma_*). \]