关于随机环境中可数马氏链的位势问题
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摘要
本文是在Cogburn建立的随机环境中Markov链的数学模型的基础上,主要研究随机环境中可数Markov链的位势的几个相关问题。
首先,我们讨论了随机环境中离散Markov链的禁忌概率的基本分解公式,引入在禁忌集状态可达的概念,从而得到首先可达分布矩的一个结果。然后,给出随机调和函数,极端随机调和函数和必离集的概念,用必离集的概念研究随机调和函数、极端随机调和函数的性质,进而得到随机调和函数是常向量函数的一个充分条件。最后,我们研究了随机环境中可数Markov链的位势初步。首先引入随机位势函数的概念,给出随机位势函数的一个首要条件,进而讨论非负随机上调和函数的唯一分解性。
This text is on the basis of mathematics model of Markov chain in random environment that Cogburn sets up, the several relevant problems potential of denumerable Markov chain in random environment are researched mainly.First of all, we discuss taboo probability of discrete Markov chain decomposition formulas basically in random environment, and introduce the concept that the state can be reached under the condition of the taboo set, a result of the moments of fist entrance time distributions is given. Moreover, provide the concepts of random harmonic functions, extreme harmonic function and inevitable exit set, some basic properties of them are studied by the concepts of inevitable exit sets, and then a sufficient condition for random harmonic functions to be the constant vector functions is obtained. Finally, the potential of denumerable Markov chain in random environment is preliminary that we have studied. Introduce the concept of random regular function and random potential function at first, a necessary and sufficient condition of random potential function for denumerable Markov chain in random environment is provided, and the unique decomposition property of nonnegative random super-regular function is discussed.
首先,我们讨论了随机环境中离散Markov链的禁忌概率的基本分解公式,引入在禁忌集状态可达的概念,从而得到首先可达分布矩的一个结果。然后,给出随机调和函数,极端随机调和函数和必离集的概念,用必离集的概念研究随机调和函数、极端随机调和函数的性质,进而得到随机调和函数是常向量函数的一个充分条件。最后,我们研究了随机环境中可数Markov链的位势初步。首先引入随机位势函数的概念,给出随机位势函数的一个首要条件,进而讨论非负随机上调和函数的唯一分解性。
This text is on the basis of mathematics model of Markov chain in random environment that Cogburn sets up, the several relevant problems potential of denumerable Markov chain in random environment are researched mainly.First of all, we discuss taboo probability of discrete Markov chain decomposition formulas basically in random environment, and introduce the concept that the state can be reached under the condition of the taboo set, a result of the moments of fist entrance time distributions is given. Moreover, provide the concepts of random harmonic functions, extreme harmonic function and inevitable exit set, some basic properties of them are studied by the concepts of inevitable exit sets, and then a sufficient condition for random harmonic functions to be the constant vector functions is obtained. Finally, the potential of denumerable Markov chain in random environment is preliminary that we have studied. Introduce the concept of random regular function and random potential function at first, a necessary and sufficient condition of random potential function for denumerable Markov chain in random environment is provided, and the unique decomposition property of nonnegative random super-regular function is discussed.
引文
[1] Cogburn R, Markov chains in random environments: the case of Markovian environments[J].Ann Probab, 1980,8(5):908-916.
[2] Cogburn R, The ergodic theory of Markov chains in random environments [J].Z Wahrsch Verw Gebiete, 1984,66:109-128.
[3] Cogburn R, On direct Convergence and periodicity for transition probabilities of Markov chains in random environments[J].Ann Probab, 1990,18(2):642-654.
[4] Cogburn R, On products of random stochastic matrices[J]. Contemp math, 1986,50:199-273.
[5] Cogburn R, On the central limit theorem for Markov chains in random environment[J].1991, Ann Probab, 19:583-604.
[6] NawRotzk K, Discrete open systems Markov chains in a random environment[J]. I, J Inform Process Gybernet 1981,18:83-98.
[7] NawRotzki K, Discrete open systems or Markov chains in a random environment [J].II, J Inform Process 6ybernet 1982,17:567-599.
[8] Chung K L, Markov chains with stationary transition probabilities[M]. 2nd ed 1967,Springer, New York.
[9] Orey S, Markov chains with stochastically stationary transition probabilities[J]. Ann Probab ,1991,19(3):907-928.
[10] Blumenthal R M, Getoor R K, Markov processes and potential theory[M]. New York, Academic Press, 1968.
[11] Kemeny J G, Snell J L, Knapp A W, Denumerable Markov Chains[M].New York, Springer-verlag, 19?6.
[12] Kemeny J G, Snell J L, Potentials for Denumerable Markov chains[J]. J Math Anal and appl, 1961(b)3:196-260.
[13] Dynkin E B, Markov Processes (volume Ⅱ)[M].New York, Springerverlag, 1965.
[14] 胡迪鹤,关于某些随机阵的调和函数[J].数学学报,1979,22:276-190.
[15] 胡迪鹤,从p-m链到随机环境中的马尔可夫链[J].数学年刊,2004,25:1-14.
[16] Hu Dihe, The existence and moments of Canonical branching chain in random environment[J].Math Acta Sci, 2004,24B(3):499-506.
[17] Hu Dihe, The relations among various Markov chains[J]. Wuhan University J of Natural sciences, Vol6. No 3,2001:643-648.
[18] 胡迪鹤,随机环境中的生灭过程的基本概念及存在性[J].武汉大学学报(理学版),2003:1-5.
[19] 胡迪鹤,随机环境中的Markov过程的构造及等价定理[J].中国科学,A辑,2004,34(3):268-282.
[20] 胡迪鹤,随机过程理论(基础:理论-应用)[M].武汉,武汉大学出版社,2000.
[21] 胡迪鹤,可数状态的马尔可夫过程论[M].武汉,武汉大学出版社,1983.
[22] 胡迪鹤,杨广宇,李永奎,随机转移矩阵的随机调和函数[J].武汉大学学报(理学版),2005,51(1):1-6.
[23] Hu Dihe, The classification and period of states for Markov chain in random environment[J].Math Acta Sic, 2005,25B(1):23-29.
[24] 李永奎,胡迪鹤,关于随机环境中可数马氏链的位势[J].(待发).
[25] 李永奎,胡迪鹤,随机环境中离散马氏链的禁忌概率[J].(待发).
[26] 肖争艳,胡迪鹤,绕积马氏链的状态分类[J].数学物理学报,2003,23A(3):306-313.
[27] Nawrotzki K, Finite Markov chains in stationary random environment[J]. Ann probab, 1982,10:1041-1046.
[28] 李应求,关于马氏环境中的马氏链的几点注记[J],数学进展,1999,28(4):358-360.
[29] 李应求,双无限环境中马氏链的存在和不可约性[J].数学物理学报,2001,21A(4):439-442.
[30] Ofer Zeitouni, Random walks in random environments[J]. ICM2002, Vol, Ⅲ, 117-127.
[31] Freed Solomon, Random walks in a random environment[J]. Ann Probab, 1975, Vol3, No1:1-31.
[32] Firas Rassoul Agha, The point of view of the particle on the law of large numbers for random walks in a mixing random environment[J]. Ann Probab, vol31, No3:1441-1463.
[33] David Freedman, Markov chains[M].New York:Springer Verkag, 1983.
[34] Yuri, Kifer, Limit theorems for random transformations and processes in random environment[J].Trans Ameri Math Soci, Volue350, 1998: 1481-1518.
[35] Wilhelm stannat, A remark on the CLT for a random walk in random environment[J].Probab Theory Relat Fields, 2004,130:377-387.
[36] 万成高,马氏环境中马氏链的强大数定理[J].应用概率统计,2003,19(2):155-160.
[2] Cogburn R, The ergodic theory of Markov chains in random environments [J].Z Wahrsch Verw Gebiete, 1984,66:109-128.
[3] Cogburn R, On direct Convergence and periodicity for transition probabilities of Markov chains in random environments[J].Ann Probab, 1990,18(2):642-654.
[4] Cogburn R, On products of random stochastic matrices[J]. Contemp math, 1986,50:199-273.
[5] Cogburn R, On the central limit theorem for Markov chains in random environment[J].1991, Ann Probab, 19:583-604.
[6] NawRotzk K, Discrete open systems Markov chains in a random environment[J]. I, J Inform Process Gybernet 1981,18:83-98.
[7] NawRotzki K, Discrete open systems or Markov chains in a random environment [J].II, J Inform Process 6ybernet 1982,17:567-599.
[8] Chung K L, Markov chains with stationary transition probabilities[M]. 2nd ed 1967,Springer, New York.
[9] Orey S, Markov chains with stochastically stationary transition probabilities[J]. Ann Probab ,1991,19(3):907-928.
[10] Blumenthal R M, Getoor R K, Markov processes and potential theory[M]. New York, Academic Press, 1968.
[11] Kemeny J G, Snell J L, Knapp A W, Denumerable Markov Chains[M].New York, Springer-verlag, 19?6.
[12] Kemeny J G, Snell J L, Potentials for Denumerable Markov chains[J]. J Math Anal and appl, 1961(b)3:196-260.
[13] Dynkin E B, Markov Processes (volume Ⅱ)[M].New York, Springerverlag, 1965.
[14] 胡迪鹤,关于某些随机阵的调和函数[J].数学学报,1979,22:276-190.
[15] 胡迪鹤,从p-m链到随机环境中的马尔可夫链[J].数学年刊,2004,25:1-14.
[16] Hu Dihe, The existence and moments of Canonical branching chain in random environment[J].Math Acta Sci, 2004,24B(3):499-506.
[17] Hu Dihe, The relations among various Markov chains[J]. Wuhan University J of Natural sciences, Vol6. No 3,2001:643-648.
[18] 胡迪鹤,随机环境中的生灭过程的基本概念及存在性[J].武汉大学学报(理学版),2003:1-5.
[19] 胡迪鹤,随机环境中的Markov过程的构造及等价定理[J].中国科学,A辑,2004,34(3):268-282.
[20] 胡迪鹤,随机过程理论(基础:理论-应用)[M].武汉,武汉大学出版社,2000.
[21] 胡迪鹤,可数状态的马尔可夫过程论[M].武汉,武汉大学出版社,1983.
[22] 胡迪鹤,杨广宇,李永奎,随机转移矩阵的随机调和函数[J].武汉大学学报(理学版),2005,51(1):1-6.
[23] Hu Dihe, The classification and period of states for Markov chain in random environment[J].Math Acta Sic, 2005,25B(1):23-29.
[24] 李永奎,胡迪鹤,关于随机环境中可数马氏链的位势[J].(待发).
[25] 李永奎,胡迪鹤,随机环境中离散马氏链的禁忌概率[J].(待发).
[26] 肖争艳,胡迪鹤,绕积马氏链的状态分类[J].数学物理学报,2003,23A(3):306-313.
[27] Nawrotzki K, Finite Markov chains in stationary random environment[J]. Ann probab, 1982,10:1041-1046.
[28] 李应求,关于马氏环境中的马氏链的几点注记[J],数学进展,1999,28(4):358-360.
[29] 李应求,双无限环境中马氏链的存在和不可约性[J].数学物理学报,2001,21A(4):439-442.
[30] Ofer Zeitouni, Random walks in random environments[J]. ICM2002, Vol, Ⅲ, 117-127.
[31] Freed Solomon, Random walks in a random environment[J]. Ann Probab, 1975, Vol3, No1:1-31.
[32] Firas Rassoul Agha, The point of view of the particle on the law of large numbers for random walks in a mixing random environment[J]. Ann Probab, vol31, No3:1441-1463.
[33] David Freedman, Markov chains[M].New York:Springer Verkag, 1983.
[34] Yuri, Kifer, Limit theorems for random transformations and processes in random environment[J].Trans Ameri Math Soci, Volue350, 1998: 1481-1518.
[35] Wilhelm stannat, A remark on the CLT for a random walk in random environment[J].Probab Theory Relat Fields, 2004,130:377-387.
[36] 万成高,马氏环境中马氏链的强大数定理[J].应用概率统计,2003,19(2):155-160.