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
In this paper,we study the eigenvalue problem of the Markov diffusion operator L^(2),and give generalized inequalities for eigenvalues of the operator L^(2)on a Markov diffusion triple.By applying these inequalities,w...In this paper,we study the eigenvalue problem of the Markov diffusion operator L^(2),and give generalized inequalities for eigenvalues of the operator L^(2)on a Markov diffusion triple.By applying these inequalities,we then get some new universal bounds for eigenvalues of a special Markov diffusion operator L^(2)on bounded domains in an Euclidean space.Moreover,our results can reveal the relationship between the(k+1)-th eigenvalue and the first k eigenvalues in a relatively straightforward manner.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.
基金Supported by the Open Research Fund of Key Laboratory of Nonlinear Analysis and Applications(Central China Normal University),Ministry of Education,P.R.China(Grant No.NAA2025ORG011)Science and Technology Plan Project of Jingmen(Grant No.2024YFZD076)+3 种基金Research Team Project of Jingchu University of Technology(Grant No.TD202006)Research Project of Jingchu University of Technology(Grant Nos.HX20240049HX20240200)the Teaching Reform Research Project of Hubei Province(Grant No.2024496)。
文摘In this paper,we study the eigenvalue problem of the Markov diffusion operator L^(2),and give generalized inequalities for eigenvalues of the operator L^(2)on a Markov diffusion triple.By applying these inequalities,we then get some new universal bounds for eigenvalues of a special Markov diffusion operator L^(2)on bounded domains in an Euclidean space.Moreover,our results can reveal the relationship between the(k+1)-th eigenvalue and the first k eigenvalues in a relatively straightforward manner.
