The Dual Theory Of The Generalized Birth-death Process And Its Corresponding Semigroups

In the study of theories of Markov processes, mathematicians have got a series of perfect and universal conclusions. In this paper, we mainly apply some of these conclusions to a specific q-matrix—the generalized birth-death matrix Q, and get a series of corresponding conclusions. Firstly, we describe some properties of the generalized birth-death matrix Q by its parameters, then we investigate some properties of the minimal Q-function. Secondly, we get the dual q-matrix Q^* of Q. Besides the discussion about some properties of Q^* and the minimal Q^*-function, we consider the relationship which may exist between the minimal Q-function and the minimal Q^*-function. Finally, combining the theory of semigroups of linear operators, we study two kinds of semigroups deduced by Q. In the following notation, we call the generalized birth-death matrix as Q for short, and call the dual q-matrix Q^* of Q as Q^*.In chapter two, we investigate some basic properties of Q, such as monotony, dual property and Feller-Reuter-Riley (FRR) property. We get some conclusions as follows:Proposition 2. 1. 1 Q is monotone if and only if {λ_n}_n=1^∞is decreasing.Proposition 2. 1. 2 Q is dual if and only if {λ_n}_n=1^∞is decreasing.Proposition 2. 1. 3 Q is FRR if and only if (?)γ_n = 0.We also prove that Q is zero-entrance if and only if Q is strong zero-entrance. It is wellknown that, for any q-matrix Q, it is zero-exit in l_∞if and only if it is zero-exit in l_∞⁺, so for short, we call it zero-exit. However, if a q-matrix Q is zero-entrance in l₁⁺, it is not necessarily zero-entrance in l₁, that is, Q is zero-entrance but not necessarily strong zero-entrance. As to this problem, the generalized birth-death matrix has a better conclusion. Thus we use the descriptions that Q is strong zero-entrance and zero-exit by parameters to get the descriptions that Q is monotone, dual and FRR by parameters. The following are the main results in chapter two:Proposition 2. 1. 5 Q is zero-entrance if and only if Q is strong zero-entrance.Proposition 2. 1. 7 Suppose thatμ_n＞0, n = 1, 2,…, and {γ_n} is bounded, then Q is strong zero-entrance if and only if S = Proposition 2. 1. 8 Suppose thatλ_n＞0, n = 1, 2,…, Q is zero-exit if and only if R = +∞orTheorem 2. 2. 2 The minimal Q-function is monotone if and only if {γ_n}_n=1^∞is decreasing and R = +∞.Theorem 2. 2. 3 The minimal Q-function is dual if and only if (1) {γ_n}_n=1^∞is decreasing and (2) (a) (?)γ_n = 0, S = +∞or (b) R＜+∞.Theorem 2. 2. 4 Suppose that {γ_n}_n=1^∞is decreasing, the minimal Q-function is FRR if and only if S = +∞or R＜+∞.In chapter three, we get the dual q-matrix Q^* of Q, and study some properties of Q^*. Then we find that: in some properties, such as conservative property, the conclusion of Q^* is worse than the conclusion of Q; while in some other properties, such as dual property and FRR property, the conclusion of Q^* is better than the conclusion of Q. We still find that there is intresting and regular relationship in the description of strong zero-entrance and zero-exit between Q and Q^*. The following are the main results:Proposition 3. 1. 1 Q^* is conservative if and only if (?)γ_n = 0.Proposition 3. 1. 2 Q^* is dual.Proposition 3. 1. 3 Q^* is monotone if and only if Q^* is conservative.Proposition 3. 1. 4 Q^* is FRR.Proposition 3. 1. 5 Q^* is zero-entrance if and only if Q^* is strong zero-entrance.Proposition 3. 1. 6 Suppose thatλ_n＞0, n = 1, 2,…, and {γ_n}_n=1^∞is bounded, then Q^* is strong zero-entrance if and only if R = +∞.Proposition 3. 1. 7 Suppose thatμ_n＞0, n = 1, 2,…, and{γ_n}_n=1^∞is bounded, then Q^* is zero-exit if and only if S = +∞.Obviously, if {γ_n}_n=1^∞is bounded, then Q^* is strong zero-entrance if and only if Q is zeroexit, Q^* is zero-exit if and only if Q is strong zero-entrance. Besides that, we still study some basic properties of the minimal Q^*-function:Theorem 3. 1. 10 The minimal Q^*-function is monotone if and only if (?)γ_n = 0 and S= +∞.Theorem 3. 1. 11 If (?)γ_n = 0, then the minimal Q^*-function is FRR if and only if R=+∞or S＜+∞.Finally, we find the following relationship between the minimal Q-function and the minimal Q^*-function under some conditions:Theorem 3. 1. 12 If {γ_n}_n=1^∞is decreasing, (?)γ_n = 0 and R = +∞, then the dual of the minimal Q-function is the minimal Q^*-function.In chapter four, we study the sufficient and necessary conditions under which the generalized birth-death matrix generates positive contraction C₀-semigroups on c₀ and l₁ respectively. At the same time, we study the FRR properties of the minimal Q-function by using the theory of semigroups of linear operators. The followings are the main results:Theorem 4. 1. 1 Suppose that{γ_n}_n=1^∞is decreasing sequence, then the following are equivalent:(a) Q₀ generates a positive contraction C₀-semigroup on c₀;(b) The minimal Q-function is a FRR transition function;(c) Either S = +∞or R＜+∞.Theorem 4. 1. 2 The following statements are equivalent:(a) Q₁ generates a positive contraction C₀-semigroup on l₁;(b) The operator I - Q on l_∞is injective, where Q is the operator with maximum domain on l_∞;(c) Either (i)λ_nk= 0 for some subsequence {λ_nk} of {λ_n} or (â…±)m = sup{n:λ_n = 0}＜+∞, then R₁ = +∞or R₂ = +∞. where...