Let (X, ∑, μ) be a σ-finite measure space, P : LI → L1 be a Markov operator, and Qt = ∑n=0 ∞ qn(t)Pn, where {qn(t)} be a sequence satisfying:i) qn(t) ≥ 0 and ∑n=0 ∞ qn(t)=1 for all t >0;ii)lim (q0(t) + ∑n...Let (X, ∑, μ) be a σ-finite measure space, P : LI → L1 be a Markov operator, and Qt = ∑n=0 ∞ qn(t)Pn, where {qn(t)} be a sequence satisfying:i) qn(t) ≥ 0 and ∑n=0 ∞ qn(t)=1 for all t >0;ii)lim (q0(t) + ∑n=1 ∞ |qn(t) -qn-1(t)|) = 0.t→∞f ∈ 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 Qtf = E(f|∑0) for all f ∈ L1, where ∑0 is the invariant σ-algebra ofP. Some related results are also given.展开更多
Let X be a compact metric space and C（X） be the space of all continuous functions on X. In this article, the authors consider the Markov operator T ： C（X）N C（X）N defined by for any f = （f1,f2,… ,fN）, where ...Let X be a compact metric space and C（X） be the space of all continuous functions on X. In this article, the authors consider the Markov operator T ： C（X）N C（X）N defined by for any f = （f1,f2,… ,fN）, where （pij） is a N x N transition probability matrix and {wij } is an family of continuous transformations on X. The authors study the uniqueness, ergodicity and unidimensionality of T＊-invariant measures where T＊ is the adjoint operator of T.展开更多
基金Research is partially supported by the NSFC (60174048)
文摘Let (X, ∑, μ) be a σ-finite measure space, P : LI → L1 be a Markov operator, and Qt = ∑n=0 ∞ qn(t)Pn, where {qn(t)} be a sequence satisfying:i) qn(t) ≥ 0 and ∑n=0 ∞ qn(t)=1 for all t >0;ii)lim (q0(t) + ∑n=1 ∞ |qn(t) -qn-1(t)|) = 0.t→∞f ∈ 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 Qtf = E(f|∑0) for all f ∈ L1, where ∑0 is the invariant σ-algebra ofP. Some related results are also given.
文摘Let X be a compact metric space and C（X） be the space of all continuous functions on X. In this article, the authors consider the Markov operator T ： C（X）N C（X）N defined by for any f = （f1,f2,… ,fN）, where （pij） is a N x N transition probability matrix and {wij } is an family of continuous transformations on X. The authors study the uniqueness, ergodicity and unidimensionality of T＊-invariant measures where T＊ is the adjoint operator of T.
