Homogeneous Denumerable Markov Processes

I Construction Theory of Sample Functions of Homogeneous Denumerable Markov Processes.- I The First Construction Theorem.- § 1.1 Introduction.- § 1.2 Definition of transformation gn.- § 1.3 Convergence of the sequence X(n)(?) (n?1).- § 1.4 Further properties of X(n)(?) (n?1).- § 1.5 The first construction theorem.- II The Second Construction Theorem.- § 2.1 Introduction.- § 2.2 The mapping Tmn.- § 2.3 The mapping Wn.- § 2.4 Constructing auxiliary functions.- § 2.5 The second construction theorem.- § 2.6 Summary.- § 2.7 Two notes.- II Theory of Minimal Nonnegative Solutions for Systems of Nonnegative Linear Equations.- III General Theory.- § 3.1 Introduction.- § 3.2 Definition of a system of nonnegative linear equations and definition, existence and uniqueness of its minimal nonnegative solution.- § 3.3 Comparison theorem and linear combination theorem.- § 3.4 Localization theorem.- § 3.5 Connecting property of the minimal nonnegative solution.- § 3.6 Limit theorem.- § 3.7 Matrix representation.- § 3.8 Dual theorem.- IV Calculation.- § 4.1 Some lemmas.- § 4.2 Reduction of the problems.- § 4.3 Ordinary systems of strictly nonhomogeneous equations with dimension n.- V Systems of 1-Bounded Equations.- § 5.1 Introduction.- § 5.2 First-type leading-outside systems of equations.- § 5.3 First-type consistent systems of equations.- § 5.4 Tailed random systems of strictly nonhomogeneous equations.- § 5.5 Regular systems of equations.- § 5.6 Pseudo-normal systems of equations.- § 5.7 Pseudo-normal systems of equations of finite dimension.- § 5.8 Second-type regular systems of equations.- III Homogeneous Denumerable Markov Chains.- VI General Theory.- § 6.1 Introduction.- § 6.2 Transition probabilities.- § 6.3 Distribution and moments of the first passage time.- § 6.4 Distribution and moments of the first passage time of a homogeneous finite Markov chain.- § 6.5 Distribution and moments of the times of passage.- § 6.6 Criteria for recurrence.- § 6.7 Distribution and moments of additive functionals.- § 6.8 Derived Markov chains and criteria for atomic almost closed sets.- VII Martin Exit Boundary Theory.- § 7.1 Introduction.- § 7.2 Decomposition for Markov chains.- § 7.3 Limit behaviour of excessive functions.- § 7.4 Green functions and Martin kernels.- § 7.5 h-chains.- § 7.6 Limit theorem for Martin kernels.- § 7.7 Martin boundaries.- § 7.8 Distribution of x?.- § 7.9 Martin expressions of excessive functions.- § 7.10 Exit space.- § 7.11 Uniqueness theorem.- § 7.12 Minimal excessive functions.- § 7.13 Terminal random variables.- § 7.14 Criteria for potentials and excessive functions, Riesz decomposition.- § 7.15 Criteria for minimal harmonic functions, minimal potentials and minimal excessive functions.- § 7.16 Atomic exit spaces and nonatomic exit spaces.- § 7.17 Blackwell decomposition of the state space.- VIII Martin Entrance Boundary Theory.- § 8.1 Introduction.- § 8.2 The first group of lemmas.- § 8.3 Properties of finite excessive measures.- § 8.4 The second group of lemmas.- § 8.5 Entrance boundary.- § 8.6 Entrance space and the expressions of excessive measures.- IV Homogeneous Denumerable Markov Processes.- IX Minimal Q-Processes.- § 9.1 Introduction.- § 9.2 Transition probabilities.- § 9.3 Distribution and moments of the first passage time.- § 9.4 Criterion for the positive recurrence.- § 9.5 Distribution and moments of integral-type functionals.- § 9.6 Distribution and moments of integral-type functionals on pseudo-translatable sets.- § 9.7 Extensions of the results in § 9.3.- X Q-Processes of Order One.- § 10.1 Introduction.- § 10.2 Transition probabilities.- § 10.3 Distribution and moments of the first passage time.- XI Arbitrary Q-Processes.- § 11.1 Strengthening of the first construction theorem.- § 11.2 Transition probability.- § 11.3 Decomposition theorems for excessive measures and excessive functions.- V Construction Theory of Homogeneous Denumerable Markov Processes.- XII Criteria for the Uniqueness of Q-Processes.- § 12.1 Introduction.- § 12.2 Lemmas.- § 12.3 Proof of the main theorem.- § 12.4 The case of diagonal type.- § 12.5 The bounded case.- § 12.6 The case when E is finite.- § 12.7 The case of a branch Q-matrix.- § 12.8 Another criterion and the finite and nonconservative case.- § 12.9 Independence of the two conditions in Theorem 12.1.1.- § 12.10 Probability interpretation of Condition (i) in Theorem 12.1.1.- XIII Construction of Q-Processes.- § 13.1 Construction theorem.- § 13.2 Specifications of all the Q-processes.- § 13.3 Expression of $$\left\{ {Q,\,{\Pi _{{{\left( {\partial X} \right)}_{e,}}\,x\,E}}} \right\}$$-processes.- § 13.4 Discussion.- XIV Qualitative Theory.- § 14.1 Introduction.- § 14.2 Statement of results.- § 14.3 Reduction of the construction problem of B-type Q-processes, Doob processes.- § 14.4 Reduction of the construction problem of B?F-type Q-processes.- § 14.5 Proofs of Theorems 14.2.1–14.2.3.- § 14.6 Proof and examples of applications of Theorem 14.2.4.- § 14.7 Proofs of Theorems 14.2.5–14.2.10.