Resolvent Decomposition Theorems and Their Application in Denumerable Markov Processes with Instantaneous States
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Resolvent Decomposition Theorems and Their Application in Denumerable Markov Processes with Instantaneous States Anyue Chen1,2 Received: 4 January 2019 © The Author(s) 2019
Abstract The basic aim of this paper is to provide a fundamental tool, the resolvent decomposition theorem, in the construction theory of denumerable Markov processes. We present a detailed analytic proof of this extremely useful tool and explain its clear probabilistic interpretation. We then apply this tool to investigate the basic problems of existence and uniqueness criteria for denumerable Markov processes with instantaneous states to which few results have been obtained even until now. Although the complete answers regarding these existence and uniqueness criteria will be given in a subsequent paper, we shall, in this paper, present part solutions of these very important problems that are closely linked with the subtle Williams S and N conditions. Keywords Denumerable Markov processes · Transition functions · Resolvents · Taboo probabilities · Existence · Uniqueness Mathematics Subject Classification (2010) Primary 60J27; Secondary 60J35
1 Introduction The basic aim of this paper is to provide a fundamental tool, the resolvent decomposition theorem, in the construction theory of continuous time Markov chains (CTMC). This extremely useful theorem has a very clear probabilistic interpretation. It is just the Laplace transform version of first-entrance–last-exit decomposition law. Let {X t ; t ≥ 0} be a homogeneous continuous time Markov chain defined on the countable state space E = {e1 , e2 , e3 , . . .} and let P(t) = { pi j (t); i, j ∈ E, t ≥ 0} be
B
Anyue Chen [email protected]; [email protected]
1
Department of Mathematics, Southern University of Science and Technology, Xueyuan Boulevard 1088, Shenzhen 518055, China
2
Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, UK
123
Journal of Theoretical Probability
its transition function. Then, in matrix form, this family of real matrix functions P(t) satisfies P(t) ≥ 0 P(t)1 ≤ 1 P(t + s) = P(t)P(s)
(t ≥ 0) (t ≥ 0)
(1.1) (1.2)
(t ≥ 0, s ≥ 0)
(1.3)
and lim P(t) = P(0) = I
t→0+
(1.4)
where I denotes the identity matrix on E × E and 1 denotes the column vector on E whose components are all 1. In the following, 1 will always denote the column vector with an appropriate dimension (either finite or infinite) whose components are all 1. If the equality holds in (1.2) for all t ≥ 0, then the transition function P(t) is called honest. One of the fundamental results in the theory of CTMC is that if P(t) satisfies (1.1)–(1.4), then the following limit exists lim
t→0+
P(t) − I =Q t
(1.5)
where the matrix Q = {qi j ; i, j ∈ E} satisfies 0 ≤ qi j < +∞ −∞ ≤ qii ≤ 0 and
qi j ≤ − qii
(i = j) (∀ i ∈ E) (∀ i ∈ E).
(1.6) (1.7)
(1.8)
i= j
In the following, we shall always denote qi = −qii (i ∈ E). Note that in (1.7), qii may not be finite. If qi = −qii < +∞, then the state i ∈ E is called stable, while if qi = +∞ then the stat
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