Acta mathematica scientia,Series B ›› 2002, Vol. 22 ›› Issue (2): 254-260.

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THE ASYMPTOTIC PROPERTIES OF WEIGHTED MARKOV OPERATORS

 DING Yi-Ming   

  1. Wuhan Institute of Physics and Mathematics, The Chinese Academy of Sciences, Wuhan 430071, China
    Department of System Science, Beijing Normal University, Beijing 100875, China
  • Online:2002-04-15 Published:2002-04-15
  • Supported by:

    Research is partially supported by the NSFC (60174048)

Abstract:

Let (X,, μ) be a −finite measure space, P : L1 ! L1 be a Markov operator,and Qt =P1 n=0 qn(t)Pn, where {qn(t)} be a sequence satisfying:
i) qn(t)  0 and P1 n=0 qn(t) = 1 for all t > 0; ii) lim t!1 (q0(t) +P1 n=1 |qn(t) − qn−1(t)|) = 0. f 2 L1, it is proved that Qt(f) convergent strongly to a fixed point of P as t ! 0 if and only if {Qt(f)}t>0 is precompact. Qt(f) is convergent if and only if the ergodic mean operator An(f) is convergent, and they have the same limit. If P is a double stochastic operator then lim t!1 Qtf = E(f|0) for all f 2 L1, where 0 is the invariant -algebra of P. Some related results are also given.

Key words: weighted Markov operator, weakly precompact, double stochastic operator,support, invariant

CLC Number: 

  • 28D04
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