## Abstract

Let P = T* be a conservative Markov operator on L_{∞}(X, ∑, m), and let h(x) = lim supP^{n}(I-P)f: ∥f ∥_{∞}≦1. Then h(x) is zero or two a.e. The sets E_{0}={h = 0} and E_{1} = {h = 2} are invariant, and we have: (a) ∥T^{n}(I-T)∥u∥_{1}→ 0 for u ε L_{1}(E_{0}), (b) ∥ |T^{n}(I-T) |u∥_{1}=2∥u∥ for every n, 0≦u ε L_{1}(E_{1}). If ∑ is countably generated and P is given by P(x, A), we have (a) ∥P^{n}(x,·)-P^{n+1}(x,·) ∥→ 0 a.e. on E_{0}, (b) ∥P^{n}(x,·)-P^{n+1}(x,·)∥=2 a.e. on E_{1}, for every n. A sufficient (but not necessary) condition for m(E_{1}) = 0 is that σ(P)∩|λ| = l = 1. If P_{t} is a conservative semi-group given by P_{t}(x, A) bi-measurable, there are invariant sets E_{0} and E_{1} such that: (a) ∀ α ε ℝ, lim ∥P_{t}(x,·)-P_{t+α}(x,·)∥=0 a.e. on E_{0}, (b) for a.e. α ε ℝ, lim ∥P_{t}(x,·) -P_{t+α}(x,·)∥ =2 a.e. on E_{1}.

Original language | English |
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Pages (from-to) | 513-525 |

Number of pages | 13 |

Journal | Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete |

Volume | 61 |

Issue number | 4 |

DOIs | |

State | Published - 1 Dec 1982 |

## ASJC Scopus subject areas

- Analysis
- Statistics and Probability
- Mathematics (all)