Summary
Let P = T* be a conservative Markov operator on L ∞(X, ∑, m), and let h(x) = lim sup P 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)
∥Tn(I-T)∥u∥1→ 0 for u ε L 1(E0),
-
(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 ∥Pt(x,·)-Pt+α(x,·)∥=0 a.e. on E 0,
-
(b)
for a.e. α ε ℝ, lim ∥Pt(x,·) -P t+α(x,·)∥ =2 a.e. on E 1.
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Research done during a sabbatical visit at the Ohio State University
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Lin, M. On the “zero-two” law for conservative Markov processes. Z. Wahrscheinlichkeitstheorie verw Gebiete 61, 513–525 (1982). https://doi.org/10.1007/BF00531621
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DOI: https://doi.org/10.1007/BF00531621