Integrated Semigroups Of Linear Operators And Their Applications To Continuous-Time Markov Chains

In the study of theories of Markov processes, there traditionally exist two methods: the probable method and the analytical method. The probable method is straightforward, vivid and distinct in expression, and the probability meaning is clear. However, the analytical method is concise, lucid and lively in expression. As far as the application is concerned, many specialists such as physicists, biologists and chemists are fond of the results expressed by the probable method. However, the results expressed by the analytical method is much easier to combine the probability theory with the achievements of other branches of mathematics and apply the achievements of modern mathematics. In this paper, we use the analytical method. By using the theory of semigroups of linear operators, we study the integrated semigroups of linear operators and their applications to Continuous-Time Markov Chains (CTMCs).According to the theory of Continuous-Time Markov Chains, given a g-matrix Q, it is possible that there exist infinite transition functions and therefore there maybe exist infinite positive contraction semigroups (Co semigroups) on l1 derived from q-matrix Q. But each positive contraction semigroup has and only has one infinitesimal generator. It is natural to ask the following question: what relationship is between g-matrix Q and the infinitesimal generators of the positive contraction semigroups derived from g-matrix Q? In the first part of this paper, we discuss this question. We place restrictions on g-matrix Q, that is, we discuss the conditions on which the operators and derived from q-matrix Q are the generators of positive contraction semigroups on l1 or c0 (in order to define q-matrix Q must be a Feller- Reuter-Riley q-matrix).First, we obtain the relationships among the operators derived from q-matrix Q:Secondly, we discuss the conditions on which , Ql1 and generates positive contraction semigroups on l1 or c0. We have the following results:Theorem 3.1.1 Ql1 generates a positive contraction semigroup on l1 iff there exists A>0 such that Ql1 is injective on l1;Theorem 3.1.3 generates a positive contraction semigroup on l1 iff there exists A>0 such that is injective on l,When is well defmed,Theorem 3.2.1 generates a positive contraction semigroup on c0 iff there exists A>0 such that is injective on l1.At last, we apply the above results to the birth and death process and obtain the numeral descriptions of the above conditions. When q-matrix Q is a birth-death matrix, denotewhere 0, i = 0,1, 2, 3, DenoteWe have the following results:Theorem 4.1.1 generates a positive contraction semigroup on CQ iff either (i) = 0 for some subsequence of ; or (ii) if m = sup{n;un = 0} < +00, then 5 = ;Theorem 4.2.1 Ql1 generates a positive contraction semigroup on l1 iff either (i)= 0 for some subsequence of ; or (ii) if m = sup{n= 0} < +00, then 5 = +00;Theorem 4.2.2 generates a positive contraction semigroup on l\ iff either (i) = 0 for some subsequence of or (ii) if m = sup{n; = 0} < then R = +00.In the second part of this paper, we discuss the applications of integrated semigroups to CTMCs. It is well-known that there exist far-reaching applications of CQ semigroups to CTMCs. But are there similar applications of integrated semigroups which have been developed recently to CTMCs? Y.R. Li has discussed the question in [41]. Y.R. Li in [41] has discussed the properties of transition functions on and obtained that a generally unlimited q-matrix Q can generate a positive once integrated semigroup with contractions on Up to now, it is the first time to apply the theories of integrated semigroups to CTMCs.On the base of Li's paper [41], we first discuss the relationships between q-matrix Q and the generators of positive once integrated semigroups on derived from transition functions. Then we discuss the converse question, that is, given a positive once integrated semigroup on , on what condition, we can find a transition function which corresponds to the q-matrix Q. We have the following r...