ω～*-Continuous Semigroups And Applications To Continuous-Time Markov Chains

In the study of theories of Markov processes, there traditionally are 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 analytical method and using the theory of semigroups of linear operators, study the ω*-continuous semigroups and their applications to Continuous-Time Markov Chains(CTMCs).From [1], Every transition function is a positive strongly continuous semigroup of contractions on l\, but it isn't a strongly continuous semigroup on l_∞- In fact, the sufficient and necessary condition for a transition function to be a strongly continuous semigroup on l_∞ is that the q—matrix Q is an uniformly bounded q—matrix on l_∞- This is the trivial case. The q—matrix we usually deal with don't satisfy this property. So Anderson [1] think that l_∞ is too large a space on which to develop a really useful theory. Thus, when we discuss on l_∞, the strongly continuous semigroup isn't a suitable tool to study the Continuous-Time Markov Chains(CTMCs). A natural question is: Can we find a new tool on semigroups to study CTMCs?In this paper, we introduce a new class of semigroups which called ω^*-continuous semigroups. We give the definitions of ω^*-continuous semigroup and the ω^*-generator. Note that, the adjoint semigroup of a strongly continuous semigroup T(t) on X is, in fact, a ω*-continuous semigroup on X^*. Thus, in the chapter 2, we discuss the characteristics of this ω^*-continuous semigroup and its ω^*-generator.In chapter 3, we discuss the adjoint semigroups on the sequence Banach spaces(c₀,l₁). According to [2], we know that if X is a reflexive Banach space and T(t) is a strongly continuous semigroup on X with the infinitesimal generator A, then the adjoint semigroup T(t)^* is also a strongly continuous semigroup on X* and its infinitesimal generator is A^*, the ajoint operator of A. However, it doesn't hold for general spaces. We know that the c₀ spaces and l₁ spaces are not reflexive spaces, whether the ajoint semigroups on them are strongly continuous semigroups?...