Properties Of Population Processes With Geometric Catastrophe And Their Dual Transition Functions

In the study of thcories of Markov processes, there traditionally are two methods: the probabilistic method and the analytical method. The probable method is straightforward, vivid and distinct in expression, and the probability meaning is clear. However, the analitical 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 achivements 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 special properties of population processes with geometric catastrophe and their dual transition functions.From [1], Every transition function is a positive strongly continuous semigroup of contractions on l₁, 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. A natural question is: How can we obtain the good properties of population processes with geometric catastrophe on l₁ and l_∞?In this paper, we introduce two operators Q_∞and Q₀ generated by geometric catastrophe q-matrices on l_∞and l₁ respectively. The domain D(Q_∞) is not dense in l_∞by [4]. Q_∞does not generatc any strongly continuous semigroups on l_∞. However, we will show that Q_∞generates an integrated semigroup and Q₀ generates a continuous positive contraction semigroup. Thus, in the chapter 2, we have the following results:Proposition 2. 2. 1. The geometric catastrophe q-matrix Q is monotone if and only if q≤1/2. Proposition 2. 2. 2. The geometric catastrophe q-matrix Q is always FRR.Proposition 2. 2. 3. Q is zero-exit, and thus regular.To discuss the FRR property, we also need the zero-entrance property. It, remains open whether the strong zero-entrance is equivalent to zero-entrance for a, general q-matrix. [12] has proved the equivalence for downwardly skip free q-matrix. For upwardly skip flee q-matrix: we have the following proposition:Proposition 2. 2. 4. Q is strong zero-entrance if and only if Q is zero-entrance.Proposition 2. 2. 5. Geometric catastrophe q-matrix Q is zero-entrance and thus strong zero-entrance.Theorem 2. 2. 1 For the geometric catastrophe q-matrix Q. we have(â…°) Q_∞generates a positive contraction integrated semigroup T(t) = (R_ij(t)) on l_∞;(â…±) Q₀ generates a continuous positive contraction semigroup P(t) = (P_ij(t)) on c₀, P(t) is just the minimal Q-function and is an FRR transition function;(â…²) T(t) and P(t) are stochastically monotone if and only if q≤1/2;(â…³) c₀ is an invariant subspace of the integrated semigroup T(t).Duality is an important tool in Markov processes, in particular, in continuous- time markov chains (CTMC) and interacting particle systems. In chapter 3, we will discuss the dual matrices Q of geometric catastrophe q-matrices and the condition of recurrenc and the probability of extinction of Q. In the reference [17]. Brockwell has obained beautiful results of the probability of extinction for upwardly skip free processes. For the dual processes of geometric catastrophe Q-processes, can we have the similar wonderful results? In chapter 3, we have the following theorems:Theorem 3. 2. 1 Geometric catastrophe Q-function is recurrent if and only if p＞b/a.Theorem 3. 2. 2 Forθ= 0, {x_i(0), i≥0} is the unique bounded solution of the following system andwhereTheorem 3. 2. 3 Letα= lim_i-xα_i(0). Then...