On the "zero-two" law for conservative Markov processes

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Abstract

Let P = T* be a conservative Markov operator on L(X, ∑, m), and let h(x) = lim supPn(I-P)f: ∥f ∥≦1. Then h(x) is zero or two a.e. The sets E0={h = 0} and E1 = {h = 2} are invariant, and we have: (a) ∥Tn(I-T)∥u∥1→ 0 for u ε L1(E0), (b) ∥ |Tn(I-T) |u∥1=2∥u∥ for every n, 0≦u ε L1(E1). If ∑ is countably generated and P is given by P(x, A), we have (a) ∥Pn(x,·)-Pn+1(x,·) ∥→ 0 a.e. on E0, (b) ∥Pn(x,·)-Pn+1(x,·)∥=2 a.e. on E1, for every n. A sufficient (but not necessary) condition for m(E1) = 0 is that σ(P)∩|λ| = l = 1. If Pt is a conservative semi-group given by Pt(x, A) bi-measurable, there are invariant sets E0 and E1 such that: (a) ∀ α ε ℝ, lim ∥Pt(x,·)-Pt+α(x,·)∥=0 a.e. on E0, (b) for a.e. α ε ℝ, lim ∥Pt(x,·) -Pt+α(x,·)∥ =2 a.e. on E1.

Original languageEnglish
Pages (from-to)513-525
Number of pages13
JournalZeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
Volume61
Issue number4
DOIs
StatePublished - 1 Dec 1982

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Mathematics (all)

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