Orbital Properties of Regular Chain


The strong Markov process had been obtained by Ray-Knight compacting; its orbit natures are discussed; the significance probability of kolmogorov forward and backward equations are explained.

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Zhang, K. , Du, H. , Meng, H. and Ba, M. (2014) Orbital Properties of Regular Chain. Applied Mathematics, 5, 3311-3317. doi: 10.4236/am.2014.521308.

1. Introduction

General Markov chain only has locally strong Markov property, which is the main obstruction to solve the pro- blem of Markov chain constructing [1] [2] . The papers construct a strong Markov chain corresponding to its transition function using Ray-Knight compact method [3] [4] , which is named regular chain. The papers give an orbit construction of birth and death process [5] [6] . The papers solve the construction problem of two-sided birth and death process [3] -[11] . The papers prove that the appended points in the compacting and the points on the Martin entrance boundary are monogamy, under the condition of finite entrance boundary [12] -[14] . This paper makes a strong Markov process by Ray-Knight compacting, discusses its orbit nature and explains the significance probability of Kolmogorov forward and backward equations.

2. The Orbit Natures of Regular Chain

Assume is a honest transition function on, is its density func-

tion, is its resolvent, is the Ray-Knight compacting of, and is the Ray re-

solvent and the semi-group correspondence, denote as non-ramification point set,

, , then E is Borel algebras on, is the

regular chain of correspondence to. Denote and

respectively as escape time and return time, by Blumenthal 0 - 1 law, for arbitrary, or 1,

or 1, if, x is called absorption state, if, is called sojourn state, if, is called regular state, if, is called temporary state.

Theorem 1 Let, then

(1) is a regular state,

(2) on, the distribution of escape time is the exponential distribution of,

(3) on, and is mutual independent.

(4) if, for arbitrary,.

Proof (1) Assume is not a regular state, then. for arbitrary, it is easy to check, and when, , thus


this is a contradictory proposition.

(2) The proof is same as Theorem 5 in [15] .

(3) If or, then or, the conclusion is true, if, for arbitrary Borel subset and,

Let, we have.

then, on, , and is mutual independent.

(4) If, for arbitrary, According to the strong Markov properties of and (3), we can obtain that

Give arbitrary, and continuous function on with,

but, in addition,


Remark 1 (3), (4) in the Theorem 1 are equivalence with the Theorem 6 in [15] , but it require, do not incloude.

Remark 2 According to (2) in Theorem 1, is a temporary state, if and only if is a sojourn state of

the regular chain.

Definition 1 Let is the constant set of, the interval in is called i-interval of

Theorem 2 If, then for arbitrary, we can get a stopping time squence, with, when, we have, when, we have. And for arbitrary,

For arbitrary, denote as the number of belong to, we have


Proof Let


, ,

where are the stoping time of. for arbitrary, if, since is right continuity, , and

then we have almost sure on.

Since is strong Markov chain, and for arbitrary,

then we have almost sure on.

For arbitrary, obviously, by Theorem 3.1 in [15]

According to Fatou lemma, for arbitrary, , then almost sure there are only finite in a finite interval, such that, this means.

Theorem 3 If, then

(1) Almost sure, do not contain any interval,

(2) Almost sure, is a dense set in itself.

Proof (1) Obviously, is a optional set, denote (where we assume), then is a monotone increasing left continuous process, and adapt in, denote, thus is a optional right continuous process. Let


It is easy to check that, thus is a optional set adapt in.

Assume is debut time, If, by Section Theorem, exists a stopping time in, such that, and on, by (2) in the theorem 1,

this is a contradictory proposition, thus and almost sure do not contain any interval.

(2) The proof is similar to (1).

3. The Significance Probability of Kolmogorov Equations

Theorem 4 For arbitrary, and

, (1)

if and only if.

Proof For arbitrary,

then (1) and the following equation is equivalence.


According to Theorem 1, we have

and the necessary and sufficient condition of equality is.

For arbitrary let is the first k i-interval of,

Corollary 1 The following conditions are equivalence [16] [17] .

(1) The backward equation of Kolmogorov is true,

(2) For arbitrary,

(3) Density matrix is conservative,

(4) Almost sure, for all and, we have.

Theorem 5 For arbitrary


if and only if for all -interval almost sure.

Proof (1) Asumme,

Obviously are not intersection. It is easy to check if there are infinite to make,

then, and if, then existing, when, we have, thus that, and

(2) For arbitrary, by (1),


and the necessary and sufficient condition of equality is.

Thus we get the equation

, (5)

let go to in Equation (5), we can obtain Equation (3).

Corollary 2 The Kolmogorov forward equations are true if and only if for all and i-interval, almost sure.

Remark 3 Equation(3) is equivalent to

. (6)

Remark 4 If contains some transient state, then Equation (1) is true if and only if

Remark 5 Under the condition of, Equation (1) is not probably true. for the example

in Remark 1, the Ray-Knight compaction of under the resolvent is, thus, the corresponding

regular chain meets the equation, but according to Corollary 2, Doob process does not

satisfy Kolmogorov forward equation, then also does not satisfy forward equation.

If is an non-honest transition function with total stability, then we can construct a

honest transition function on such that

, (7)

where the density matrix of is such that


the resolvent of is, then


Assume is a regular chain corresponding to. For, by Theorem 1, is a absorption state, this is

Set, obviously is a killing Markov process, for arbitrary

, and, we known the transition function

of is.

For arbitrary, since then for arbitrary the following equations are Equivalence.

It is easy to get:

Proposition 1 Assume is an non-honest transition function with total stability, is corresponding Markov process with killing, then satisfy Kolmogorov backward equation if and only if almost sure for all and,.

Proposition 2 Assume is an non-honest transition function with total stability, is corresponding Markov process with killing, then satisfy Kolmogorov forward equation if and only if almost sure for all and,.

Conflicts of Interest

The authors declare no conflicts of interest.


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