𝜆-continuous Markov chains. II

Abstract

Continuing the investigation in [8] we study a λ \lambda -continuous Markov operator P P . It is shown that, if P P is conservative and ergodic, P P is indeed “periodic” as is the case when the state space is discrete; there is a positive integer δ \delta , called the period of P P , such that the state space may be decomposed into δ \delta cyclically moving sets C 0 , ⋯ , C δ − 1 {C_0}, \cdots ,{C_{\delta - 1}} and, for every positive integer n , P n δ n,{P^{n\delta }} acting on each C i {C_i} alone is ergodic. It is also shown that P P maps L q ( μ ) {L_q}(\mu ) into L q ( μ ) {L_q}(\mu ) where μ \mu is the nontrivial invariant measure of P P and 1 ≦ q ≦ ∞ 1 \leqq q \leqq \infty . If μ \mu is finite and normalized then it is shown that (1) if f ∈ L ∞ ( λ ) f \in {L_\infty }(\lambda ) , then { P n δ + k f } \{ {P^{n\delta + k}}f\} converges a.e. ( λ ) (\lambda ) to g k = ∑ i = 0 δ − 1 c i + k 1 C i {g_k} = \sum \nolimits _{i = 0}^{\delta - 1} {{c_{i + k}}} {1_{{C_i}}} where c j = δ ∫ C j f d μ {c_j} = \delta {\smallint _{{C_j}}}fd\mu if 0 ≦ j ≦ δ − 1 0 \leqq j \leqq \delta - 1 and c j = c i {c_j} = {c_i} if j = m δ + i , 0 ≦ i ≦ δ − 1 j = m\delta + i,0 \leqq i \leqq \delta - 1 , (2) { P n δ + k f } \{ {P^{n\delta + k}}f\} converges in L q ( μ ) {L_q}(\mu ) to g k {g_k} if f ∈ L q ( μ ) f \in {L_q}(\mu ) , and(3) lim inf n → ∞ P n δ + k f = g k \lim {\inf _{n \to \infty }}{P^{n\delta + k}}f = {g_k} a.e. ( λ ) (\lambda ) if f ∈ L 1 ( μ ) f \in {L_1}(\mu ) and f ≧ 0 f \geqq 0 . If μ \mu is infinite, then it is shown that (1) if f ≧ 0 , f ∈ L q ( μ ) f \geqq 0,f \in {L_q}(\mu ) for some 1 ≦ q > ∞ 1 \leqq q > \infty , then lim inf n → ∞ P n f = 0 \lim {\inf _{n \to \infty }}{P^n}f = 0 a.e. ( λ ) (\lambda ) , (2) there exists a sequence { E k } \{ {E_k}\} of sets such that X = ∪ k = 1 ∞ E k X = \cup _{k = 1}^\infty {E_k} and lim n → ∞ P n δ + i 1 E k = 0 {\lim _{n \to \infty }}{P^{n\delta + i}}{1_{{E_k}}} = 0 a.e. ( λ ) (\lambda ) for i = 0 , 1 , ⋯ , δ − 1 i = 0,1, \cdots ,\delta - 1 and k = 1 , 2 , ⋯ k = 1,2, \cdots .