| In the study of theories of Markov processes, there traditionally exist two methods: the probabilistic method and the analytical method. Recently mathematician investigate theories of Markov processes using the theory of semigroups of linear operators, and obtain many results. In this paper, we, mainly by means of analytic method and using the theory of semigroups of linear operators, study the transition functions, q-matrices, and the continuous dependence of transition functions on their q-matrices in continuous-time Markov chains (CTMCs).We know that every transition function is a positive strongly continuous semigroup of contractions on l1, but it isn't a positive strongly continuous semigroup of contractions on l. Thus Anderson [2] think that l is too large a space on which to develop a really useful theory. However, are there similar applications of the integrated semigroups which have developed recently to CTMCs. Li Y.R. [33, 36] discussed the above question, and proved that the transition function is a positive once strong-contraction integrated semigroup on l(we call Markov integrated semigroup), and established the relationship between the transition functions and the Markov integrated semigroups. A natural question is: whether does there exist a subspace of l on which any transition function is a positive strongly continuous semigroup of contractions. According to the theory of Markov integrated semigroup developed by Li Y.R.[33, 36], we obtain the following results.Theorem 4.2.2 Let P(t) = (pij(t)) be a q-function of q-matrix Q, and G(t) be the Markov integrated semigroup with generator Q. DefineC1 = {, P(t)x is continuous function of t > 0 on l}Then the transition function P(t) is a positive contraction C0 semigroup on C1, and its generator 0 is the part of in C1, i.e.,Theorem 4.2.3 Let P(t) = (pij(t)) be a q-function of q-matrix Q, and G{t) be the Markov integrated semigroup with generator . Then the following statements are equivalent:(a) is densely defined in l, i.e.,D() = l(b) q-matrix Q is uniformly bounded, i.e., (c) is a bounded operator on Theorem 4.2.4 Let P(t) = (pij(t)) be a q-function of q-matrix Q, and G(t) be the Markov integrated semigroup with generator . Then the following statements are equivalent:(a) P(t) = (pij(t)) is a Feller-Reuter-Riley transition function;(b) the part of in c0 generates a positive C0 semigroup of contractions on c0.Moreover, let A be the generator of P{t) on cq. Then A is the part of in c0, i.e., A = 0.In the theories of Markov process, the question on approximation is always of most importance. Approximation by discrete skeletons has long been the mainstay of this theory. David Williams [17] considered a kind of approximation, that of the CTMCs by the finite CTMCs, where the finite CTMCs is well understood. Anderson [2] stated that there exists a sequence of transition functions with uniformly bounded q-matrices to converge to a given minimal transition function. In this paper, we study the same question by means of theory of semigroups of linear opreraors. Many useful results are obtained. Considering our question in l1, we have:Theorem 5.1.2 Let Q = (qij) be a g-matrix, and F(t) = (fij(t)) be the minimal q-function. Let nQ be nq-matrix, and nF{t) = {nfij(t)) be the minimal nq-fiinction. Suppose(a) - Q is injective on l; that is, the equationhas no solution other than the trivial solution y = 0 for some A > 0;ThenTheorem 5.1.3 Let Q = {qij) be a q--matrix, and F(t) = (fij(t)) be the minimal q-function. Suppose(a) - Q is injective on l for some A > 0;(b)The truncated matrix nQ = (nqij) satisfies (i) if i,j < n, nqij = qij; (ii) if other, nqij can be arbitrary to ensure that nQ is a nq-matrix.Finally, let nF(t) = {nfij{t)) be the minimal nq-function. Then,Theorem 5.1.4 Let P(t) = {pij(t)) and nP{t) = (nPij{t)) be transition functions (not necessary the minimal ones), and R(X) = (rij(A))and nR = (nrij be the corresponding resolvent functions, respectively. Consider the following statements:(a) nPij(t)...
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