# Conditional Processes Induced by Birth and  Death Processes

Authors: Masaru Iizuka; Matsuyo Tomisaki
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

For birth and death processes with finite state space, we consider stochastic
processes induced by conditioning on hitting the right boundary point before
hitting the left boundary point. We call the induced stochastic processes the
conditional processes. We show that the conditional processes are again birth
and death processes when the right boundary point is absorbing. On the other
hand, it is shown that the conditional processes do not have Markov property
and they are not birth and death processes when the right boundary point is
reflecting.


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## Body

## 1. Introduction

For one-dimensional diffusion processes on[0,1] related to diffusion models in population genetics, Ewens [1] considered stochastic processes induced by conditioning on hitting the boundary point 1 before hitting the other boundary point 0. The boundary points 0 and 1 are accessible and absorbing boundaries for the diffusion processes that he considered and the induced stochastic processes are again diffusion processes. Then the induced stochastic processes are referred to as the conditional diffusion processes by Ewens [1] (see also [2]). Motivated by this work, Iizuka et al. [3] were concerned with one-dimensional generalized diffusion processes (ODGDPs for brief) on (l1,l2) whose speed measures are right-continuous and strictly increasing functions. They considered stochastic processes induced by conditioning on hitting the right boundary point l2 before hitting the left boundary point l1. The induced stochastic processes are called the conditional processes. They showed as Theorem 2.1 that the conditional processes are again ODGDPs when the boundary point l2 is accessible with the absorbing boundary condition (Assertion 1). If the original process x(t) is a one-dimensional diffusion process with the generator(1.1)L=a(x)2d2dx2+b(x)ddx,
then the conditional process x*(t) induced by conditioning on hitting l2 before hitting l1 is again a one-dimensional diffusion process and its generator can be expressed as(1.2)L*=a(x)2d2dx2+{b(x)+a(x)s0(x)s(x)-s(l1)}ddx.
Here we put(1.3)s0(x)=exp{-2∫cxb(y)a(y)dy},s(x)=∫cxs0(y)dy,
where c is a point with l1<c<l2 (see [4, 5]). On the other hand, Iizuka et al. [3] showed as Theorem 2.2 that the probability distributions of the conditional processes do not satisfy the Chapman-Kolmogorov equation when the boundary point l2 is accessible with the reflecting boundary condition. Hence the conditional processes cannot be Markov processes when the boundary point l2 is accessible with the reflecting boundary condition (Assertion 2).An important class of ODGDPs which is used as stochastic models in various fields is that of birth and death processes. For example, Moran [6] introduced a birth and death process as one of fundamental stochastic models in population genetics called Moran model (we will consider this model in Section 5). However, the speed measure of any birth and death process is not a strictly increasing function (see [7]) and we cannot apply the results of [3] to birth and death processes.In this paper we prove that Assertions1 and 2 hold for the case that the speed measure is a nondecreasing step function. The motivation of this paper is to investigate the properties of the conditional processes induced by conditioning on hitting the right boundary point before hitting the left boundary point when the original processes are birth and death processes. The proof of Theorem 2.2 in [3] is analytical (nonprobabilistic) and it is not easy to see that the conditional processes do not satisfy Markov property when the right boundary point is accessible with the reflecting boundary condition. The proof presented in this paper for Assertion 2 is based on the fact that the state space is discrete. The proof is probabilistic and we can see intuitively that the conditional processes do not satisfy Markov property when the right boundary point is reflecting. It is our extra purpose to see this by considering birth and death processes.In Section2 we state our results more precisely. Section 3 is devoted to their proofs. In Section 4 we introduce a very simple birth and death process and present concrete expressions of its conditional processes considering all the boundary conditions. Finally we discuss some stochastic models in population genetics and their conditional processes in Section 5.


## 2. Main Results

Let𝔢 be an exponentially distributed random variable with the mean 1 and let {𝔢1,𝔢2,𝔢3,…} be a sequence of independent copies of 𝔢. We put τ0=0 and τk=∑i=1k𝔢i  (k=1,2,…). For {𝔢1,𝔢2,𝔢3,…}, an integer N(N≥2), and points ai(i=0,1,…,N) such that a0<a1<a2<⋯<aN, we consider a birth and death process 𝔻=[X(t),Px] with the state space Σ={a0,a1,a2,…,aN} satisfying the following conditions. For ai∈Σ and τk<t<τk+1, conditional probabilities conditional on X(τk)=ai satisfy(2.1)Px(X(t)=ai∣X(τk)=ai)=1,Px(X(τk+1)=ai+1∣X(τk)=ai)=pi,Px(X(τk+1)=ai-1∣X(τk)=ai)=qi,Px(X(τk+1)=ai∣X(τk)=ai)=1-pi-qi=ri,
where 0≤p0≤1,pN=0,q0=0,0≤qN≤1, and pi>0,qi>0,ri≥0 for i=1,2,…,N-1. Here Px denotes the probability measure concentrated at the event {X(0)=x}, that is, Px(X(0)=x)=1. The end (boundary) point a0 [resp., aN] is called to be absorbing or reflecting according to p0=0 [resp., qN=0] or p0>0 [resp., qN>0].The generatorℒ of 𝔻 is given by(2.2)ℒu(ai)=pi{u(ai+1)-u(ai)}-qi{u(ai)-u(ai-1)},i=1,2,…,N-1,
for u∈D(ℒ), where D(ℒ) is the set of all functions u on Σ such that (2.3)u(a0)=0ifa0isabsorbing,ℒu(a0)=p0{u(a1)-u(a0)}ifa0isreflecting,u(aN)=0ifaNisabsorbing,ℒu(aN)=-qN{u(aN)-u(aN-1)}ifaNisreflecting.
Here is a proof of  (2.2). By means of  (2.1), we find that (2.4)Eai[u(X(t))]=Eai[u(X(t));t<τ1]+Eai[u(X(t));τ1≤t<τ2]+Eai[u(X(t));τ2≤t],Eai[u(X(t));t<τ1]=u(ai)Pai(t<τ1)=u(ai)e-t,Eai[u(X(t));τ1≤t<τ2]={u(ai+1)pi+u(ai-1)qi+u(ai)ri}(1-e-t)e-t,Eai[u(X(t));τ2≤t]=o(t)ast↓0.
Therefore we obtain the following: (2.5)ℒu(ai)=limt↓0Eai[u(X(t))]-u(ai)t=-u(ai)+u(ai+1)pi+u(ai-1)qi+u(ai)ri=pi{u(ai+1)-u(ai)}-qi{u(ai)-u(ai-1)}
(see also [7]).We show that the birth and death process𝔻 can be described as an ODGDP. We set(2.6)l1={a0ifp0=0,-∞ifp0>0,l2={aNifqN=0,∞ifqN>0,(2.7)s(x)={ρ0(x),l1<x≤a0,x-a0a1-a0,a0≤x≤a1,s(ai)+q1q2⋯qip1p2⋯pi·x-aiai+1-ai,ai≤x≤ai+1,i=1,…,N-1,s(aN)+ρN(x),aN≤x<l2,
where ρ0 [resp., ρN] is an increasing continuous function on (l1,a0] [resp., [aN,l2)] such that ρ0(a0)=0 and ρ0(l1)=-∞ [resp., ρN(aN)=0 and ρN(l2)=∞]. Further we set(2.8)m(x)={-∞,x<l1,-1/p0,l1≤x<a0,0,a0≤x<a1,1/q1,a1≤x<a2,m(ai-)+p1p2⋯pi-1q1q2⋯qi,ai≤x<ai+1,i=2,…,N-1,m(aN-)+p1p2⋯pN-1q1q2⋯qN,aN≤x<l2,∞,l2≤x.
Note that {l1≤x<a0}=∅ [resp., {aN≤x<l2}=∅] if p0=0 [resp., qN=0]. Here s is a real-valued continuous increasing function on S=(l1,l2), and m is a right-continuous nondecreasing function on ℝ. They are called the scale function and the speed measure, respectively. We set m({x})=m(x)-m(x-), mi=m({ai}), and si=s(ai),  i=0,1,…,N. We note that m0=∞ [resp., mN=∞] if a0 [resp. aN] is absorbing.For a functionf on S, we simply write f(l1) [resp., f(l2)] in place of f(l1+) [resp., f(l2-)] provided f(l1+) [resp., f(l2-)] exists. Further, f+ [resp., f-] stands for the right [resp., left] derivative of f with respect to s if it exists, that is, f+(x)=limɛ↓0{f(x+ɛ)-f(x)}/{s(x+ɛ)-s(x)} [resp., f-(x)=limɛ↓0{f(x-ɛ)-f(x)}/{s(x-ɛ)-s(x)}].We setΣ*=Σ∩S. Let D(𝒢) be the space of all bounded continuous functions u on S satisfying the following conditions.(𝒢.1)
There exist a functionf on Σ* and two constants A1,A2 such that
(2.9)u(x)=A1+A2{s(x)-s(c)}+∫(c,x]{s(x)-s(y)}f(y)dm(y),x∈S.(𝒢.2)
For eachi=1,2, u(li)=0 if |li|<∞.Throughout this paper we denote byc an arbitrarily fixed point of Σ*. The operator 𝒢 is defined by the mapping from u∈D(𝒢) to f that appeared in (2.9). The operator 𝒢 is called the one-dimensional generalized diffusion operator (ODGDO for brief) with (s,m). It is known that there exists a strong Markov process 𝔻* with the generator 𝒢, which is called an ODGDP on S (see [8, 9]). It is also known that 𝔻 can be identified with 𝔻* (see [7–9]). Indeed, it is easy to see that u∈D(𝒢) satisfies the following: (2.10)u+(ai)=u-(ai+1)=u(ai+1)-u(ai)si+1-si,i=0,1,…,N-1,𝒢u(ai)=u+(ai)-u-(ai)mi=pi{u(ai+1)-u(ai)}-qi{u(ai)-u(ai-1)},i=1,2,…,N-1,u(a0)=0ifp0=0,𝒢u(a0)=u+(a0)m0=p0{u(a1)-u(a0)}ifp0>0,u(aN)=0ifqN=0,𝒢u(aN)=-u-(aN)mN=-qN{u(aN)-u(aN-1)}ifqN>0.In order to make the boundary conditions ata0 and aN clear, we use 𝔻IJ and PxIJ in place of 𝔻 and Px, respectively. Here I,J∈{A,R}, and I=A [resp., J=A] means that a0 [resp., aN] is absorbing (i.e., p0=0 [resp., qN=0]) and I=R [resp., J=R] means that a0 [resp., aN] is reflecting (i.e., p0>0 [resp., qN>0]). It is known that there is the transition probability density pIJ(t,x,y) of 𝔻IJ with respect to m, that is,(2.11)PxIJ(X(t)=y)=pIJ(t,x,y)m({y}),t>0,x,y∈Σ*
(see [8, 10]).LetΣo={a1,…,aN} and let σa be the first hitting time at a, that is, σa=inf{t>0:X(t)=a}. In this paper we consider stochastic processes induced by the following conditional probability:(2.12)QxIJ(·)=PxIJ(·∣σaN<σa0),x∈Σo.
We set(2.13)h(x)=PxIJ(σaN<σa0),x∈Σ.
It is known that(2.14)h(x)=s(x)-s(a0)s(aN)-s(a0)=s(x)sN,x∈Σ
(see [8]). We note that h is independent of boundary conditions I,J∈{A,R}.First we show thatQIA induces a birth and death process for I∈{A,R}.Theorem 2.1.
Assume thataN is absorbing. Then  QxIA(X(t)=y) is independent of  I∈{A,R}, and it is represented as
(2.15)QxIA(X(t)=y)=h(y)h(x)PxAA(X(t)=y)=1s(x)pAA(t,x,y)s(y)m({y}),
for t>0 and x,y∈Σo∖{aN}. Further QxIA induces a birth and death process 𝔻o on Σo for which the end point a1 is reflecting, the end point aN is absorbing, and the generator ℒo is given by
(2.16)ℒou(ai)=pi{u(ai+1)-u(ai)}-qi{u(ai)-u(ai-1)}+q1⋯qip1⋯pi·pisi{u(ai+1)-u(ai-1)},i=2,…,N-1,
for u∈D(ℒo), where D(ℒo) is the set of all functions u on Σo such that
(2.17)ℒou(a1)=(p1+q1){u(a2)-u(a1)},(2.18)u(aN)=0.Theorem2.1 shows that the relation between  (1.1) and  (1.2) for diffusion processes corresponds to the relation between  (2.2) and  (2.16) for birth and death processes. We note that the generator is given by  (2.17) and the boundary condition (2.18) when N=2.Remark 2.2.
By means of  (2.7),
(2.19)pisi+1si=pi+q1…qip1…pi·pisi,qisi-1si=qi-q1…qi-1p1…pi-1·qisi,pisi+1si+qisi-1si=pi+qi≤1,1-pisi+1si-qisi-1si=ri.
Combining these with  (2.16), we find that QxIA satisfies the following. For x∈Σo∖{aN}, τk(k=1,2,…), and τk<t<τk+1,
(2.20)QxIA(X(t)=ai∣X(τk)=ai)=1,QxIA(X(τk+1)=ai+1∣X(τk)=ai)=pisi+1si,QxIA(X(τk+1)=ai-1∣X(τk)=ai)=qisi-1si,QxIA(X(τk+1)=ai∣X(τk)=ai)=ri.We turn to the case thataN is reflecting. When m(x) is strictly increasing, a representation of QxIR is given by (2.11) of [3]. We note that this representation is available even if m(x) is not strictly increasing. Therefore we obtain the following representation for birth and death processes:(2.21)QxIR(X(t)=y)=s(y)s(x)PxAA(X(t)=y)+sNs(x)ℳ(t,x,y)+sNs(x)∫0tμx(u)𝒩I(t-u,aN,y)du,
for I∈{A,R}, t>0, and x,y∈Σo∖{aN}. Here μx(u),ℳ(t,x,y) and 𝒩I(t,x,y) are given as follows. For t>0 and x∈Σo∖{aN}, let(2.22)μx(t)=-limy↑aNpAA(t,x,aN)-pAA(t,x,y)s(aN)-s(y).
For t>0 and x∈Σo, let(2.23)νx(t)=limy↓a0pAR(t,x,y)-pAR(t,x,a0)s(y)-s(a0).
It is known that μx and νx are nonnegative density functions such as(2.24)PxIJ(σaN<t,σaN<σa0)=∫0tμx(u)du,(2.25)PxIR(σa0<t)=∫0tνx(u)du
(see [8]). Note that μx is independent of I,J∈{A,R} and νx is independent of I∈{A,R}. By virtue of [11], we see that (2.26)μx(t)=pAA(t,x,aN-1)sN-sN-1,νx(t)=pAR(t,x,a1)s1.
Then we set (2.27)ℳ(t,x,y)=∫0tμx(u)PaNAR(X(t-u)=y)du,𝒩I(t,x,y)=∫0tνx(u)Pa0IR(X(t-u)=y)du.
We note that 𝒩A(·,·,·)=0.The second and the third terms of the right-hand side of  (2.21) come from sample path's behavior after hitting the boundary aN. This representation suggests that QIR does not satisfy Markov property. Indeed we obtain the following theorem.Theorem 2.3.
LetI∈{A,R} and  0<t1<t2. Then
(2.28)QxIR(X(t2)=y∣X(t1)=z)=QzIR(X(t2-t1)=y)
does not hold for some x,y,z∈Σo. This implies that QxIR does not satisfy Markov property.This theorem is proved by applying the following simple proposition for sample path's behavior after hitting the boundaryaN.Proposition 2.4.
LetI,J∈{A,R} and  t>0. Then
(2.29)PxIJ(X(t)=y,σaN<σa0)=PxIJ(X(t)=y)PxIJ(σaN<σa0)
does not hold for some x∈Σo∖{aN} and y∈Σo.We prove this proposition in the following section.


## 3. Proofs of Theorems

We use the same notations as those in Section2.


### 3.1. Proof of Theorem2.1

First we prepare the following lemma. The proof of this lemma is easy and we omit it.Lemma 3.1.
LetI,J∈{A,R},  t>0, and  x,y∈Σo∖{aN}. Then it holds true that
(3.1)PxIJ(X(t)=y,t<σa0∧σaN)=PxAA(X(t)=y),(3.2)PxIJ(X(t)=y)=PxAA(X(t)=y)+PxIJ(X(t)=y,t>σa0∧σaN),(3.3)PxIJ(X(t)=y,t<σa0∧σaN,σaN<σa0)=PxIJ(X(t)=y,t<σa0∧σaN)PyIJ(σaN<σa0)=PxAA(X(t)=y)h(y).Proof of Theorem2.1.
We assume thataN is absorbing. Let I∈{A,R} and t>0. Then
(3.4)QaNIA(X(t)=y)=PaNIA(X(t)=y)={1ify=aN,0otherwise.
Let x∈Σo∖{aN}. Then by means of  (2.13), and  (2.24),
(3.5)QxIA(X(t)=aN)=PxIA(X(t)=aN,σaN≤t<σa0)PxIA(σaN<σa0)=1h(x)PxIA(σaN≤t<σa0)=1h(x)∫0tμx(u)du.
Let x,y∈Σo∖{aN}. Then by using Markov property of 𝔻IA,  (2.13) and  (3.1), we see that
(3.6)QxIA(X(t)=y)=1h(x)PxIA(X(t)=y,t<σaN∧σa0,σaN<σa0)=1h(x)PxIA(X(t)=y,t<σaN∧σa0)PyIA(σaN<σa0)=h(y)h(x)PAA(X(t)=y).
The formulas  (3.4),  (3.5), and  (3.6) show that QxIA(X(t)=y) is independent of I∈{A,R}. The formula  (2.15) follows from  (2.11),  (2.14), and  (3.6).
It follows from Theorem2.2 and Propositions 3.1 and 3.4 of [12] that QxIA induces an ODGDP 𝔻o on (a0,aN), the boundary a0 is entrance in the sense of Feller (see [8, 13]), the boundary aN is absorbing, and the generator is the ODGDO 𝒢o with (so,mo), where(3.7)so(x)=∫cxs(y)-2ds(y)=1s(c)-1s(x),(3.8)mo(x)=∫(c,x]s(y)2dm(y).
Therefore,
(3.9)𝒢ou(ai)={u(ai+1)-u(ai)so(ai+1)-so(ai)-u(ai)-u(ai-1)so(ai)-so(ai-1)}{mo(ai)-mo(ai-)}-1,
for a function u on Σo∖{aN} and i=2,3,…,N-1. By means of  (2.7),  (2.8),  and (3.7), we see that
(3.10){so(ai+1)-so(ai)}{mo(ai)-mo(ai-)}=(1si-1si+1)si2mi=(si+1-si)simisi+1=sisi+1pi.
In the same way, we have
(3.11){so(ai)-so(ai-1)}{mo(ai)-mo(ai-)}=sisi-1qi.
Therefore we get
(3.12)𝒢ou(ai)=si+1sipi{u(ai+1)-u(ai)}-si-1siqi{u(ai)-u(ai-1)}=pi{u(ai+1)-u(ai)}-qi{u(ai)-u(ai-1)}+q1…qip1…pi·pisi{u(ai+1)-u(ai-1)},
for i=2,3,…,N-1. Since a0 is entrance, we see that
(3.13)𝒢ou(a1)=u(a2)-u(a1)so(a2)-so(a1){mo(ai)-mo(ai-)}-1=s2s1p1{u(a2)-u(a1)}=(p1+q1){u(a2)-u(a1)},
by virtue of general theory on ODGDOs. Thus we find that 𝔻o is a birth and death process on Σo, the generator is given by  (2.16), the end point a1 is reflecting with  (2.17), and the end point aN is absorbing. The proof is completed.


### 3.2. Proof of Theorem2.3

We introduce the Green function corresponding to𝔻IJ. For I,J∈{A,R}, k=1,2, and α>0, let gkIJ(·,α) be a continuous function on S satisfying the following properties:(3.14)gkIJ(·,α)>0onS,(3.15)g1IJ(·,α)isnondecreasingandg2IJ(·,α)isnonincreasingonS,(3.16)g1AJ(a0,α)=0,(3.17)g1RJ,+(a0,α)=g1RJ(a1,α)-g1RJ(a0,α)=αg1RJ(a0,α)m0,(3.18)g2IA(aN,α)=0,(3.19)g2IR,-(aN,α)=g2IR(aN,α)-g2IR(aN-1,α)sN-sN-1=-αg2IR(aN,α)mN,(3.20)gkIJ(x,α)=gkIJ(c,α)+gkIJ,+(c,α){s(x)-s(c)}+α∫(c,x]{s(x)-s(y)}gkIJ(y,α)dm(y),x∈S.
Here gkIJ,±(x,α)=limɛ↓0{gkIJ(x±ɛ,α)-gkIJ(x,α)}/{s(x±ɛ)-s(x)}. It is known that there exist such functions gkIJ(·,α),k=1,2 (see [8]). We set WIJ(α)=g1IJ,+(x,α)g2IJ(x,α)-g1IJ(x,α)g2IJ,+(x,α). Note that WIJ(α) is a positive number independent of x∈S. We put(3.21)GIJ(α,x,y)=GIJ(α,y,x)=WIJ(α)-1g1IJ(x,α)g2IJ(y,α),
for α>0 and l1<x≤y<l2, which is the Green function corresponding to 𝔻IJ. It is also known that(3.22)GIJ(α,x,y)=∫0∞e-αtpIJ(t,x,y)dt,
for α>0 and x,y∈Σ (see [8, 14]).First we prove Proposition2.4.Proof of Proposition2.4.
We divide the proof into four cases.Case 1.
I=J=A. Since aN is absorbing, we find that
(3.23)PxAA(X(t)=y,σaN<σa0)=PxAA(X(t)=y,t<σa0∧σaN,σaN<σa0)=PxAA(X(t)=y)h(y),
by means of  (3.3). Since there are x,y∈Σo∖{aN} such that x≠y for N≥3,  (3.23) shows that  (2.29) does not hold true for x≠y for N≥3.
LetN=2. Then(3.24)Pa1AA(X(t)=a2,σa2<σa0)=Pa1AA(σa2<σa0,σa2<t),Pa1AA(X(t)=a2)Pa1AA(σa2<σa0)=Pa1AA(σa2<σa0,σa2<t)Pa1AA(σa2<σa0),
which imply that  (2.29) is not valid for x=a1 and y=a2. Thus  (2.29) does not hold for some x,y∈Σo.Case 2.
I=R and J=A. By means of  (3.2), we also see that
(3.25)PxRA(X(t)=y,σaN<σa0)-PxRA(X(t)=y)h(x)=PxAA(X(t)=y){h(y)-h(x)}-PxRA(X(t)=y,t>σa0∧σaN)h(x).
The right-hand side of this formula is negative if x≥y. This implies that  (2.29) does not hold true for x≥y.Case 3.
I=A and J=R. By means of  (3.3),
(3.26)PxAR(X(t)=y,σaN<σa0)=PxAA(X(t)=y)h(y)+PxAR(X(t)=y,t>σa0∧σaN,σaN<σa0).
Since a0 is absorbing, we get
(3.27)PxAR(X(t)=y,t>σa0∧σaN,σaN<σa0)=PxAR(X(t)=y,σaN<t<σa0)=PxAR(X(t)=y,t>σa0∧σaN).
Combining these equalities with  (3.2) and  (3.3), we see that
(3.28)PxAR(X(t)=y,σaN<σa0)-PxAR(X(t)=y)h(x)=PxAA(X(t)=y){h(y)-h(x)}+PxAR(X(t)=y,σaN<t<σa0){1-h(x)}.
The right-hand side of this formula is positive if x≤y. This implies that  (2.29) does not hold true for x≤y.Case 4.
I=J=R. Suppose that  (2.29) holds true for t>0,x∈Σo∖{aN}, and y∈Σo. Then
(3.29)∫0∞e-αtH(t,x,y)dt=0,
for α>0,x∈Σo∖{aN}, and y∈Σo, where
(3.30)H(t,x,y)=1m({y}){PxRR(X(t)=y,σaN<σa0)-PxRR(X(t)=y)h(x)}.
By means of  (3.3),
(3.31)PxRR(X(t)=y,σaN<σa0)=PxAA(X(t)=y)h(y)+PxRR(X(t)=y,t>σa0∧σaN,σaN<σa0)=PxAA(X(t)=y)h(y)+∫0tPxRR(σaN∈du,u<σa0)PaNRR(X(t-u)=y),
for α>0 and x,y∈Σo∖{aN}.
Combining this with  (3.22), we see that(3.32)0=∫0∞e-αtH(t,x,y)dt=GAA(α,x,y)h(y)+ExRR[e-ασaN,σaN<σa0]GRR(α,aN,y)-GRR(α,x,y)h(x),
for α>0 and x,y∈Σo∖{aN}, where ExIJ stands for the expectation with respect to PxIJ. Here we note that  (3.32) is valid for y=aN. Indeed,
(3.33)PxRR(X(t)=aN,σaN<σa0)=PxRR(X(t)=aN,σaN≤t,σaN<σa0)=∫0tPxRR(σaN∈du,u<σa0)PaNRR(X(t-u)=aN).
Combining this with  (3.22), we see that
(3.34)0=∫0∞e-αtH(t,x,aN)dt=ExRR[e-ασaN,σaN<σa0]GRR(α,aN,aN)-GRR(α,x,aN)h(x),
which implies  (3.32) with y=aN.
Since  (3.32) holds true for x∈Σo∖{aN} and y∈Σo, we have(3.35)0=-GAA(α,x,aN-1)h(aN-1)+ExRR[e-ασaN,σaN<σa0]{GRR(α,aN,aN)-GRR(α,aN,aN-1)}-{GRR(α,x,aN)-GRR(α,x,aN-1)}h(x),
for x∈Σo∖{aN}. By virtue of  (3.17),  (3.20), and  (3.21), we see that
(3.36)GRR(α,aN,aN)-GRR(α,aN,aN-1)=WRR(α)-1{g1RR(aN,α)-g1RR(aN-1,α)}g2RR(aN,α)=αWRR(α)-1(sN-sN-1)g2RR(aN,α)∫[a0,aN-1]g1RR(z,α)dm(z).
We take a point x∈Σo∖{aN} such that x≤aN-1. Then by virtue of  (3.19),  (3.20), and  (3.21),
(3.37)GRR(α,x,aN)-GRR(α,x,aN-1)=WRR(α)-1g1RR(x,α){g2RR(aN,α)-g2RR(aN-1,α)}=-αWRR(α)-1(sN-sN-1)g1RR(x,α)∫(aN-1,aN]g2RR(z,α)dm(z).
Thus we obtain that
(3.38)0=-GAA(α,x,aN-1)h(aN-1)+ExRR[e-ασaN,σaN<σa0]×αWRR(α)-1(sN-sN-1)∑l=0N-1g1RR(al,α)g2RR(aN,α)ml+h(x)αWRR(α)-1(sN-sN-1)g1RR(x,α)g2RR(aN,α)mN.
It is known that
(3.39)limα↓0GAA(α,ξ,η)={s(ξ)-s(a0)}{s(aN)-s(η)}s(aN)-s(a0),a0≤ξ≤η≤aN,limα↓0αGRR(α,ξ,η)=1m(l2)-m(l1),ξ,η∈S
(see [8, 9, 14, 15]). Therefore letting α↓0 in  (3.38) leads us to
(3.40)0=s(x)sN{-(sN-sN-1)h(aN-1)}+h(x)(sN-sN-1)m(l2)-m(l1)∑l=0Nml=s(x)sN2(sN-sN-1)2.
This contradicts the fact that the last term is positive. Thus  (2.29) does not hold true for t>0, x∈Σo∖{aN}, and y∈Σo.Remark 3.2.
LetN=2, I=J∈{A,R},q0=0,p2=0, and p1=q1=1/2. Then we see that
(3.41)Pa1II(σa2<σa0)=12,Pa1II(X(t)=a1,σa2<σa0)=Pa1II(X(t)=a1,σa0<σa2),
where I∈{A,R}. Therefore
(3.42)Pa1II(X(t)=a1,σa2<σa0)=12Pa1II(X(t)=a1)=Pa1II(X(t)=a1)Pa1II(σa2<σa0),
that is,  (2.29) is valid for x=y=a1. Proposition 2.4 implies, however, that  (2.29) does not hold for x=a1 and y=a2.Proof of Theorem2.3.
LetI∈{A,R},0<t1<t2, x,z∈Σo∖{aN}, and y∈Σo. Then, by using Markov property of 𝔻IR, we obtain that
(3.43)QxIR(X(t2)=y∣X(t1)=z)=QxIR(X(t1)=z,X(t2)=y)QxIR(X(t1)=z)=PxIR(X(t1)=z,X(t2)=y,σaN<σa0)PxIR(X(t1)=z,σaN<σa0)={PxAA(X(t1)=z)PzIR(X(t2-t1)=y∣σaN<σa0)h(z)+PxIR(X(t1)=z,σaN<t1,σaN<σa0)PzIR(X(t2-t1)=y)}×{PxAA(X(t1)=z)h(z)+PxIR(X(t1)=z,σaN<t1,σaN<σa0)}-1.
Therefore  (2.28) is equivalent to the following:
(3.44)PxAA(X(t1)=z)PzIR(X(t2-t1)=y∣σaN<σa0)h(z)+PxIR(X(t1)=z,σaN<t1,σaN<σa0)PzIR(X(t2-t1)=y)={PxAA(X(t1)=z)h(z)+PxIR(X(t1)=z,σaN<t1,σaN<σa0)}×PzIR(X(t2-t1)=y∣σaN<σa0).
Again (3.44) is equivalent to the following:
(3.45)PxIR(X(t1)=z,σaN<t1,σaN<σa0)PzIR(X(t2-t1)=y)=PxIR(X(t1)=z,σaN<t1,σaN<σa0)PzIR(X(t2-t1)=y∣σaN<σa0).
Since PxIR(X(t1)=z,σaN<t1,σaN<σa0)>0,  (3.45) is equivalent to
(3.46)PzIR(X(t2-t1)=y)=PzIR(X(t2-t1)=y∣σaN<σa0).
However  (3.46) does not hold true for some y∈Σo and z∈Σo∖{aN} by virtue of Proposition 2.4. Thus  (2.28) does not hold true for some x,z∈Σo and y∈Σo.


## 3.1. Proof of Theorem2.1

First we prepare the following lemma. The proof of this lemma is easy and we omit it.Lemma 3.1.
LetI,J∈{A,R},  t>0, and  x,y∈Σo∖{aN}. Then it holds true that
(3.1)PxIJ(X(t)=y,t<σa0∧σaN)=PxAA(X(t)=y),(3.2)PxIJ(X(t)=y)=PxAA(X(t)=y)+PxIJ(X(t)=y,t>σa0∧σaN),(3.3)PxIJ(X(t)=y,t<σa0∧σaN,σaN<σa0)=PxIJ(X(t)=y,t<σa0∧σaN)PyIJ(σaN<σa0)=PxAA(X(t)=y)h(y).Proof of Theorem2.1.
We assume thataN is absorbing. Let I∈{A,R} and t>0. Then
(3.4)QaNIA(X(t)=y)=PaNIA(X(t)=y)={1ify=aN,0otherwise.
Let x∈Σo∖{aN}. Then by means of  (2.13), and  (2.24),
(3.5)QxIA(X(t)=aN)=PxIA(X(t)=aN,σaN≤t<σa0)PxIA(σaN<σa0)=1h(x)PxIA(σaN≤t<σa0)=1h(x)∫0tμx(u)du.
Let x,y∈Σo∖{aN}. Then by using Markov property of 𝔻IA,  (2.13) and  (3.1), we see that
(3.6)QxIA(X(t)=y)=1h(x)PxIA(X(t)=y,t<σaN∧σa0,σaN<σa0)=1h(x)PxIA(X(t)=y,t<σaN∧σa0)PyIA(σaN<σa0)=h(y)h(x)PAA(X(t)=y).
The formulas  (3.4),  (3.5), and  (3.6) show that QxIA(X(t)=y) is independent of I∈{A,R}. The formula  (2.15) follows from  (2.11),  (2.14), and  (3.6).
It follows from Theorem2.2 and Propositions 3.1 and 3.4 of [12] that QxIA induces an ODGDP 𝔻o on (a0,aN), the boundary a0 is entrance in the sense of Feller (see [8, 13]), the boundary aN is absorbing, and the generator is the ODGDO 𝒢o with (so,mo), where(3.7)so(x)=∫cxs(y)-2ds(y)=1s(c)-1s(x),(3.8)mo(x)=∫(c,x]s(y)2dm(y).
Therefore,
(3.9)𝒢ou(ai)={u(ai+1)-u(ai)so(ai+1)-so(ai)-u(ai)-u(ai-1)so(ai)-so(ai-1)}{mo(ai)-mo(ai-)}-1,
for a function u on Σo∖{aN} and i=2,3,…,N-1. By means of  (2.7),  (2.8),  and (3.7), we see that
(3.10){so(ai+1)-so(ai)}{mo(ai)-mo(ai-)}=(1si-1si+1)si2mi=(si+1-si)simisi+1=sisi+1pi.
In the same way, we have
(3.11){so(ai)-so(ai-1)}{mo(ai)-mo(ai-)}=sisi-1qi.
Therefore we get
(3.12)𝒢ou(ai)=si+1sipi{u(ai+1)-u(ai)}-si-1siqi{u(ai)-u(ai-1)}=pi{u(ai+1)-u(ai)}-qi{u(ai)-u(ai-1)}+q1…qip1…pi·pisi{u(ai+1)-u(ai-1)},
for i=2,3,…,N-1. Since a0 is entrance, we see that
(3.13)𝒢ou(a1)=u(a2)-u(a1)so(a2)-so(a1){mo(ai)-mo(ai-)}-1=s2s1p1{u(a2)-u(a1)}=(p1+q1){u(a2)-u(a1)},
by virtue of general theory on ODGDOs. Thus we find that 𝔻o is a birth and death process on Σo, the generator is given by  (2.16), the end point a1 is reflecting with  (2.17), and the end point aN is absorbing. The proof is completed.


## 3.2. Proof of Theorem2.3

We introduce the Green function corresponding to𝔻IJ. For I,J∈{A,R}, k=1,2, and α>0, let gkIJ(·,α) be a continuous function on S satisfying the following properties:(3.14)gkIJ(·,α)>0onS,(3.15)g1IJ(·,α)isnondecreasingandg2IJ(·,α)isnonincreasingonS,(3.16)g1AJ(a0,α)=0,(3.17)g1RJ,+(a0,α)=g1RJ(a1,α)-g1RJ(a0,α)=αg1RJ(a0,α)m0,(3.18)g2IA(aN,α)=0,(3.19)g2IR,-(aN,α)=g2IR(aN,α)-g2IR(aN-1,α)sN-sN-1=-αg2IR(aN,α)mN,(3.20)gkIJ(x,α)=gkIJ(c,α)+gkIJ,+(c,α){s(x)-s(c)}+α∫(c,x]{s(x)-s(y)}gkIJ(y,α)dm(y),x∈S.
Here gkIJ,±(x,α)=limɛ↓0{gkIJ(x±ɛ,α)-gkIJ(x,α)}/{s(x±ɛ)-s(x)}. It is known that there exist such functions gkIJ(·,α),k=1,2 (see [8]). We set WIJ(α)=g1IJ,+(x,α)g2IJ(x,α)-g1IJ(x,α)g2IJ,+(x,α). Note that WIJ(α) is a positive number independent of x∈S. We put(3.21)GIJ(α,x,y)=GIJ(α,y,x)=WIJ(α)-1g1IJ(x,α)g2IJ(y,α),
for α>0 and l1<x≤y<l2, which is the Green function corresponding to 𝔻IJ. It is also known that(3.22)GIJ(α,x,y)=∫0∞e-αtpIJ(t,x,y)dt,
for α>0 and x,y∈Σ (see [8, 14]).First we prove Proposition2.4.Proof of Proposition2.4.
We divide the proof into four cases.Case 1.
I=J=A. Since aN is absorbing, we find that
(3.23)PxAA(X(t)=y,σaN<σa0)=PxAA(X(t)=y,t<σa0∧σaN,σaN<σa0)=PxAA(X(t)=y)h(y),
by means of  (3.3). Since there are x,y∈Σo∖{aN} such that x≠y for N≥3,  (3.23) shows that  (2.29) does not hold true for x≠y for N≥3.
LetN=2. Then(3.24)Pa1AA(X(t)=a2,σa2<σa0)=Pa1AA(σa2<σa0,σa2<t),Pa1AA(X(t)=a2)Pa1AA(σa2<σa0)=Pa1AA(σa2<σa0,σa2<t)Pa1AA(σa2<σa0),
which imply that  (2.29) is not valid for x=a1 and y=a2. Thus  (2.29) does not hold for some x,y∈Σo.Case 2.
I=R and J=A. By means of  (3.2), we also see that
(3.25)PxRA(X(t)=y,σaN<σa0)-PxRA(X(t)=y)h(x)=PxAA(X(t)=y){h(y)-h(x)}-PxRA(X(t)=y,t>σa0∧σaN)h(x).
The right-hand side of this formula is negative if x≥y. This implies that  (2.29) does not hold true for x≥y.Case 3.
I=A and J=R. By means of  (3.3),
(3.26)PxAR(X(t)=y,σaN<σa0)=PxAA(X(t)=y)h(y)+PxAR(X(t)=y,t>σa0∧σaN,σaN<σa0).
Since a0 is absorbing, we get
(3.27)PxAR(X(t)=y,t>σa0∧σaN,σaN<σa0)=PxAR(X(t)=y,σaN<t<σa0)=PxAR(X(t)=y,t>σa0∧σaN).
Combining these equalities with  (3.2) and  (3.3), we see that
(3.28)PxAR(X(t)=y,σaN<σa0)-PxAR(X(t)=y)h(x)=PxAA(X(t)=y){h(y)-h(x)}+PxAR(X(t)=y,σaN<t<σa0){1-h(x)}.
The right-hand side of this formula is positive if x≤y. This implies that  (2.29) does not hold true for x≤y.Case 4.
I=J=R. Suppose that  (2.29) holds true for t>0,x∈Σo∖{aN}, and y∈Σo. Then
(3.29)∫0∞e-αtH(t,x,y)dt=0,
for α>0,x∈Σo∖{aN}, and y∈Σo, where
(3.30)H(t,x,y)=1m({y}){PxRR(X(t)=y,σaN<σa0)-PxRR(X(t)=y)h(x)}.
By means of  (3.3),
(3.31)PxRR(X(t)=y,σaN<σa0)=PxAA(X(t)=y)h(y)+PxRR(X(t)=y,t>σa0∧σaN,σaN<σa0)=PxAA(X(t)=y)h(y)+∫0tPxRR(σaN∈du,u<σa0)PaNRR(X(t-u)=y),
for α>0 and x,y∈Σo∖{aN}.
Combining this with  (3.22), we see that(3.32)0=∫0∞e-αtH(t,x,y)dt=GAA(α,x,y)h(y)+ExRR[e-ασaN,σaN<σa0]GRR(α,aN,y)-GRR(α,x,y)h(x),
for α>0 and x,y∈Σo∖{aN}, where ExIJ stands for the expectation with respect to PxIJ. Here we note that  (3.32) is valid for y=aN. Indeed,
(3.33)PxRR(X(t)=aN,σaN<σa0)=PxRR(X(t)=aN,σaN≤t,σaN<σa0)=∫0tPxRR(σaN∈du,u<σa0)PaNRR(X(t-u)=aN).
Combining this with  (3.22), we see that
(3.34)0=∫0∞e-αtH(t,x,aN)dt=ExRR[e-ασaN,σaN<σa0]GRR(α,aN,aN)-GRR(α,x,aN)h(x),
which implies  (3.32) with y=aN.
Since  (3.32) holds true for x∈Σo∖{aN} and y∈Σo, we have(3.35)0=-GAA(α,x,aN-1)h(aN-1)+ExRR[e-ασaN,σaN<σa0]{GRR(α,aN,aN)-GRR(α,aN,aN-1)}-{GRR(α,x,aN)-GRR(α,x,aN-1)}h(x),
for x∈Σo∖{aN}. By virtue of  (3.17),  (3.20), and  (3.21), we see that
(3.36)GRR(α,aN,aN)-GRR(α,aN,aN-1)=WRR(α)-1{g1RR(aN,α)-g1RR(aN-1,α)}g2RR(aN,α)=αWRR(α)-1(sN-sN-1)g2RR(aN,α)∫[a0,aN-1]g1RR(z,α)dm(z).
We take a point x∈Σo∖{aN} such that x≤aN-1. Then by virtue of  (3.19),  (3.20), and  (3.21),
(3.37)GRR(α,x,aN)-GRR(α,x,aN-1)=WRR(α)-1g1RR(x,α){g2RR(aN,α)-g2RR(aN-1,α)}=-αWRR(α)-1(sN-sN-1)g1RR(x,α)∫(aN-1,aN]g2RR(z,α)dm(z).
Thus we obtain that
(3.38)0=-GAA(α,x,aN-1)h(aN-1)+ExRR[e-ασaN,σaN<σa0]×αWRR(α)-1(sN-sN-1)∑l=0N-1g1RR(al,α)g2RR(aN,α)ml+h(x)αWRR(α)-1(sN-sN-1)g1RR(x,α)g2RR(aN,α)mN.
It is known that
(3.39)limα↓0GAA(α,ξ,η)={s(ξ)-s(a0)}{s(aN)-s(η)}s(aN)-s(a0),a0≤ξ≤η≤aN,limα↓0αGRR(α,ξ,η)=1m(l2)-m(l1),ξ,η∈S
(see [8, 9, 14, 15]). Therefore letting α↓0 in  (3.38) leads us to
(3.40)0=s(x)sN{-(sN-sN-1)h(aN-1)}+h(x)(sN-sN-1)m(l2)-m(l1)∑l=0Nml=s(x)sN2(sN-sN-1)2.
This contradicts the fact that the last term is positive. Thus  (2.29) does not hold true for t>0, x∈Σo∖{aN}, and y∈Σo.Remark 3.2.
LetN=2, I=J∈{A,R},q0=0,p2=0, and p1=q1=1/2. Then we see that
(3.41)Pa1II(σa2<σa0)=12,Pa1II(X(t)=a1,σa2<σa0)=Pa1II(X(t)=a1,σa0<σa2),
where I∈{A,R}. Therefore
(3.42)Pa1II(X(t)=a1,σa2<σa0)=12Pa1II(X(t)=a1)=Pa1II(X(t)=a1)Pa1II(σa2<σa0),
that is,  (2.29) is valid for x=y=a1. Proposition 2.4 implies, however, that  (2.29) does not hold for x=a1 and y=a2.Proof of Theorem2.3.
LetI∈{A,R},0<t1<t2, x,z∈Σo∖{aN}, and y∈Σo. Then, by using Markov property of 𝔻IR, we obtain that
(3.43)QxIR(X(t2)=y∣X(t1)=z)=QxIR(X(t1)=z,X(t2)=y)QxIR(X(t1)=z)=PxIR(X(t1)=z,X(t2)=y,σaN<σa0)PxIR(X(t1)=z,σaN<σa0)={PxAA(X(t1)=z)PzIR(X(t2-t1)=y∣σaN<σa0)h(z)+PxIR(X(t1)=z,σaN<t1,σaN<σa0)PzIR(X(t2-t1)=y)}×{PxAA(X(t1)=z)h(z)+PxIR(X(t1)=z,σaN<t1,σaN<σa0)}-1.
Therefore  (2.28) is equivalent to the following:
(3.44)PxAA(X(t1)=z)PzIR(X(t2-t1)=y∣σaN<σa0)h(z)+PxIR(X(t1)=z,σaN<t1,σaN<σa0)PzIR(X(t2-t1)=y)={PxAA(X(t1)=z)h(z)+PxIR(X(t1)=z,σaN<t1,σaN<σa0)}×PzIR(X(t2-t1)=y∣σaN<σa0).
Again (3.44) is equivalent to the following:
(3.45)PxIR(X(t1)=z,σaN<t1,σaN<σa0)PzIR(X(t2-t1)=y)=PxIR(X(t1)=z,σaN<t1,σaN<σa0)PzIR(X(t2-t1)=y∣σaN<σa0).
Since PxIR(X(t1)=z,σaN<t1,σaN<σa0)>0,  (3.45) is equivalent to
(3.46)PzIR(X(t2-t1)=y)=PzIR(X(t2-t1)=y∣σaN<σa0).
However  (3.46) does not hold true for some y∈Σo and z∈Σo∖{aN} by virtue of Proposition 2.4. Thus  (2.28) does not hold true for some x,z∈Σo and y∈Σo.


## 4. Examples

In this section, we consider a simple birth and death process. LetN=3,ai=i(i=0,1,2,3), q0=0,p3=0, and pi=qi=1/K(i=1,2), where K≥2. For τk(k=1,2,…) given in Section 2, the transition law of this birth and death process 𝔻 satisfies that (4.1)Px(X(τk+1)=0∣X(τk)=1)=Px(X(τk+1)=2∣X(τk)=1)=1K,Px(X(τk+1)=1∣X(τk)=2)=Px(X(τk+1)=3∣X(τk)=2)=1K.


### 4.1. Case That the End Point 1 Is Absorbing

We first consider𝔻AA, that is, (4.2)Px(X(τk+1)=0∣X(τk)=0)=1,Px(X(τk+1)=3∣X(τk)=3)=1.
Then l1=0 and l2=3. Further  (2.7) and  (2.8) are reduced to(4.3)s(x)=x,m(x)={-∞,x<0,0,0≤x<1,K,1≤x<2,2K,2≤x<3,∞,3≤x.
By virtue of [11], we obtain that(4.4)PxAA(X(t)=y)=pAA(t,x,y)m({y}),pAA(t,x,y)=12Kχ{1,2}(x)χ{1,2}(y){e-t/K+(-1)x+ye-3t/K},
where χΛ(ξ)=1 [resp., χΛ(ξ)=0] if ξ∈Λ [resp., ξ∉Λ]. Further by virtue of Theorem 2.1,(4.5)QxIA(X(t)=y)=KyxpAA(t,x,y),
for x,y∈{1,2}, where I∈{A,R}. Note that QxRA(X(t)=y) is expressed by using pAA(t,x,y). As in Remark 2.2, this induces a birth and death process 𝔻o on {1,2,3} with the transition law (4.6)QxIA(X(τk+1)=2∣X(τk)=1)=1,QxIA(X(τk+1)=3∣X(τk)=2)=32K,QxIA(X(τk+1)=1∣X(τk)=2)=12K,QxIA(X(τk+1)=3∣X(τk)=3)=1.


### 4.2. Case That the End Point 0 Is Absorbing and the End Point 3 Is Reflecting

We next consider𝔻AR with q3=1, that is, (4.7)Px(X(τk+1)=0∣X(τk)=0)=1,Px(X(τk+1)=2∣X(τk)=3)=1.
For simplicity, we put K=2. Then l1=0 and l2=∞. Further  (2.7) and  (2.8) are reduced to(4.8)s(x)=x,m(x)={-∞,x<0,0,0≤x<1,2,1≤x<2,4,2≤x<3,5,3≤x.
By virtue of [11], we obtain that (4.9)PxAR(X(t)=y)=pAR(t,x,y)m({y}),x,y∈{1,2,3},pAR(t,x,y)=112e-(2-3)t/2ψ1AR(x)ψ1AR(y)+13e-tψ2AR(x)ψ2AR(y)+112e-(2+3)t/2ψ3AR(x)ψ3AR(y),ψ1AR(x)={x,x=0,1,3,x=2,2,x=3,ψ2AR(x)={x,x=0,1,0,x=2,-1,x=3,ψ3AR(x)={x,x=0,1,-3,x=2,2,x=3,m({y})={2,y=1,2,1,y=3.
By means of  (2.21), we obtain that (4.10)QxAR(X(t)=y)=yxPxAA(X(t)=y)+3xℳ(t,x,y)=yxpAA(t,x,y)m({y})+3xM(t,x,y)m({y}),
for x,y∈{1,2}, where pAA(t,x,y) is given by  (4.4) with K=2, and (4.11)M(t,x,y)=∫0tμx(u)pAR(t-u,3,y)du,μx(t)=pAA(t,x,2)=14{e-t/2+(-1)xe-3t/2},pAR(t,3,y)=16{e-(2-3)t/2ψ1AR(y)-2e-tψ2AR(y)+e-(2+3)t/2ψ3AR(y)}.


### 4.3. Case That the End Points 0 and 3 Are Reflecting

We finally consider𝔻RR with p0=1 and q3=1, that is, (4.12)Px(X(τk+1)=1∣X(τk)=0)=1,Px(X(τk+1)=2∣X(τk)=3)=1.
For simplicity, we put K=2. Then l1=-∞ and l2=∞. Further  (2.7) and  (2.8) are reduced to(4.13)s(x)=x,m(x)={-1,x<0,0,0≤x<1,2,1≤x<2,4,2≤x<3,5,3≤x.
By virtue of [11], we obtain that (4.14)PxRR(X(t)=y)=pRR(t,x,y)m({y}),x,y∈{0,1,2,3},pRR(t,x,y)=16ψ1RR(x)ψ1RR(y)+13e-t/2ψ2RR(x)ψ2RR(y)+13e-3t/2ψ3RR(x)ψ3RR(y)+16e-2tψ4RR(x)ψ4RR(y),ψ1RR(x)=1,x=0,1,2,3,ψ2RR(x)={(-1)x,x=0,3,(-1)x-12,x=1,2,ψ3RR(x)={1,x=0,3,-12,x=1,2,ψ4RR(x)=(-1)x,x=0,1,2,3,m({y})={1,y=0,3,2,y=1,2.
By means of  (2.21), we obtain that (4.15)QxRR(X(t)=y)=yxPxAA(X(t)=y)+3xℳ(t,x,y)+3x∫0tμx(u)𝒩R(t-u,3,y)du=yxpAA(t,x,y)m({y})+3xM(t,x,y)m({y})+3x∫0tμx(u)NR(t-u,3,y)m({y})du,forx,y∈{1,2}, where pAA(t,x,y) is given by (4.4) with K=2, and M(t,x,y) and μx(t) are given by (4.11). Further NR(t,3,y) is given as follows: (4.16)NR(t,x,y)=∫0tν3(u)pRR(t-u,0,y)du,ν3(t)=pAR(t,3,1)=16{e-(2-3)t/2-2e-t+e-(2+3)t/2},pRR(t,0,y)=16{1-(-1)ye-t/2-e-3t/2+(-1)ye-2t}.


## 4.1. Case That the End Point 1 Is Absorbing

We first consider𝔻AA, that is, (4.2)Px(X(τk+1)=0∣X(τk)=0)=1,Px(X(τk+1)=3∣X(τk)=3)=1.
Then l1=0 and l2=3. Further  (2.7) and  (2.8) are reduced to(4.3)s(x)=x,m(x)={-∞,x<0,0,0≤x<1,K,1≤x<2,2K,2≤x<3,∞,3≤x.
By virtue of [11], we obtain that(4.4)PxAA(X(t)=y)=pAA(t,x,y)m({y}),pAA(t,x,y)=12Kχ{1,2}(x)χ{1,2}(y){e-t/K+(-1)x+ye-3t/K},
where χΛ(ξ)=1 [resp., χΛ(ξ)=0] if ξ∈Λ [resp., ξ∉Λ]. Further by virtue of Theorem 2.1,(4.5)QxIA(X(t)=y)=KyxpAA(t,x,y),
for x,y∈{1,2}, where I∈{A,R}. Note that QxRA(X(t)=y) is expressed by using pAA(t,x,y). As in Remark 2.2, this induces a birth and death process 𝔻o on {1,2,3} with the transition law (4.6)QxIA(X(τk+1)=2∣X(τk)=1)=1,QxIA(X(τk+1)=3∣X(τk)=2)=32K,QxIA(X(τk+1)=1∣X(τk)=2)=12K,QxIA(X(τk+1)=3∣X(τk)=3)=1.


## 4.2. Case That the End Point 0 Is Absorbing and the End Point 3 Is Reflecting

We next consider𝔻AR with q3=1, that is, (4.7)Px(X(τk+1)=0∣X(τk)=0)=1,Px(X(τk+1)=2∣X(τk)=3)=1.
For simplicity, we put K=2. Then l1=0 and l2=∞. Further  (2.7) and  (2.8) are reduced to(4.8)s(x)=x,m(x)={-∞,x<0,0,0≤x<1,2,1≤x<2,4,2≤x<3,5,3≤x.
By virtue of [11], we obtain that (4.9)PxAR(X(t)=y)=pAR(t,x,y)m({y}),x,y∈{1,2,3},pAR(t,x,y)=112e-(2-3)t/2ψ1AR(x)ψ1AR(y)+13e-tψ2AR(x)ψ2AR(y)+112e-(2+3)t/2ψ3AR(x)ψ3AR(y),ψ1AR(x)={x,x=0,1,3,x=2,2,x=3,ψ2AR(x)={x,x=0,1,0,x=2,-1,x=3,ψ3AR(x)={x,x=0,1,-3,x=2,2,x=3,m({y})={2,y=1,2,1,y=3.
By means of  (2.21), we obtain that (4.10)QxAR(X(t)=y)=yxPxAA(X(t)=y)+3xℳ(t,x,y)=yxpAA(t,x,y)m({y})+3xM(t,x,y)m({y}),
for x,y∈{1,2}, where pAA(t,x,y) is given by  (4.4) with K=2, and (4.11)M(t,x,y)=∫0tμx(u)pAR(t-u,3,y)du,μx(t)=pAA(t,x,2)=14{e-t/2+(-1)xe-3t/2},pAR(t,3,y)=16{e-(2-3)t/2ψ1AR(y)-2e-tψ2AR(y)+e-(2+3)t/2ψ3AR(y)}.


## 4.3. Case That the End Points 0 and 3 Are Reflecting

We finally consider𝔻RR with p0=1 and q3=1, that is, (4.12)Px(X(τk+1)=1∣X(τk)=0)=1,Px(X(τk+1)=2∣X(τk)=3)=1.
For simplicity, we put K=2. Then l1=-∞ and l2=∞. Further  (2.7) and  (2.8) are reduced to(4.13)s(x)=x,m(x)={-1,x<0,0,0≤x<1,2,1≤x<2,4,2≤x<3,5,3≤x.
By virtue of [11], we obtain that (4.14)PxRR(X(t)=y)=pRR(t,x,y)m({y}),x,y∈{0,1,2,3},pRR(t,x,y)=16ψ1RR(x)ψ1RR(y)+13e-t/2ψ2RR(x)ψ2RR(y)+13e-3t/2ψ3RR(x)ψ3RR(y)+16e-2tψ4RR(x)ψ4RR(y),ψ1RR(x)=1,x=0,1,2,3,ψ2RR(x)={(-1)x,x=0,3,(-1)x-12,x=1,2,ψ3RR(x)={1,x=0,3,-12,x=1,2,ψ4RR(x)=(-1)x,x=0,1,2,3,m({y})={1,y=0,3,2,y=1,2.
By means of  (2.21), we obtain that (4.15)QxRR(X(t)=y)=yxPxAA(X(t)=y)+3xℳ(t,x,y)+3x∫0tμx(u)𝒩R(t-u,3,y)du=yxpAA(t,x,y)m({y})+3xM(t,x,y)m({y})+3x∫0tμx(u)NR(t-u,3,y)m({y})du,forx,y∈{1,2}, where pAA(t,x,y) is given by (4.4) with K=2, and M(t,x,y) and μx(t) are given by (4.11). Further NR(t,3,y) is given as follows: (4.16)NR(t,x,y)=∫0tν3(u)pRR(t-u,0,y)du,ν3(t)=pAR(t,3,1)=16{e-(2-3)t/2-2e-t+e-(2+3)t/2},pRR(t,0,y)=16{1-(-1)ye-t/2-e-3t/2+(-1)ye-2t}.


## 5. Conditional Processes in Population Genetics

Here we consider two stochastic models in population genetics and their conditional processes. In this section, we use notations different from those of the previous sections to emphasize the difference between the original models and the induced models of conditional processes. We denote the conditional process byx*(t) [resp., X*(t)] when the original process is x(t) [resp., X(t)] as we did in Section 1.


### 5.1. Diffusion Model

We consider the following diffusion model for a randomly mating population consisting ofN haploid individuals with two types (alleles) A1 and A2. Let x(t) be the relative frequency of A1 at time t. Then x(t) is a one-dimensional diffusion process on [0,1] with the generator(5.1)L=x(1-x)2Nd2dx2+{u2(1-x)-u1x}ddx,
where u1 [resp., u2] is mutation rate from A1 [resp., A2] to A2 [resp., A1] (see [4]).First we consider the case thatu1=0. The point 1 is accessible and exit boundary if u1=0 (see [16]). For this diffusion process, consider a stochastic process x*(t) induced by conditioning on hitting the boundary point 1 before hitting the other boundary point 0. The induced stochastic process is again a diffusion process with the generator(5.2)L*=x(1-x)2Nd2dx2+(u2+1N)(1-x)ddx
by  (1.2) and Theorem 2.1 of [3]. Note that the effect of conditioning is that it inflates the mutation rate u2 to u2+1/N. Ewens [1] considered the case that u1=u2=0 and the induced diffusion process is referred to as the conditional diffusion process by Ewens [1] (see also [2]).Next we consider the case that0<2Nu1<1. The point 1 is regular boundary in this case (see [16]) and we can pose various boundary conditions there. If we pose the absorbing boundary condition, then the induced process is again a diffusion process with the generator(5.3)L*=x(1-x)2Nd2dx2+{(u2+1N)(1-x)-u1x}ddx
by  (1.2) and Theorem 2.1 of [3]. On the other hand, if we pose the reflecting boundary condition as it is usually done in population genetics (see [17–19]), then the induced conditional process does not satisfy the Chapman-Kolmogorov equation and this process is not a diffusion process due to Theorem 2.2 of [3]. These results imply that we cannot use the diffusion model whose generator is given by  (5.3) as the conditional process when we pose the reflecting boundary condition at the boundary point 1.


### 5.2. Moran Model

Moran [6] introduced the following birth and death process as one of the fundamental stochastic models in population genetics called continuous-time Moran model (see [4] for discrete time Moran model). We refer to this model as Moran model for brief. Let N be the number of individuals in a haploid population with two types A1 and A2, where N is an integer greater than 2. Let τ0=0 and τk, k=1,2,⋯ be a sequence of random times introduced in Section 2. At time τk an individual is chosen randomly and it reproduces a new individual (k≥1). The type of the newborn individual is A1 [resp., A2] with probability 1-ν1 [resp., 1-ν2] and it is A2 [resp., A1] with probability ν1 [resp., ν2] if the parent is A1 [resp., A2], where 0≤ν1,ν2≤1. Then at this time τk an individual except newborn individual is chosen randomly to die. There is no change at time t≠τk (k≥1). Denoting by X(t) the relative frequency of A1 at time t, X(t) is a birth and death process on {0,1/N,2/N,…,(N-1)/N,1} with the transition law(5.4)P(X(τk+1)=j+1N∣X(τk)=jN)=(1-ν1)j(N-j)+ν2(N-j)2N2=pj,P(X(τk+1)=j-1N∣X(τk)=jN)=(1-ν2)(N-j)j+ν1j2N2=qj,P(X(τk+1)=jN∣X(τk)=jN)=rj,
where pj+qj+rj=1 (0≤j≤N). Note that p0=ν2, q0=0, r0=1-ν2, pN=0, qN=ν1, and rN=1-ν1. Note also that pj>0 unless ν1=1 and ν2=0, qj>0 unless ν1=0 and ν2=1, and rj>0 for 0<j<N. The process X(t) does not jump at time τk if the types of newborn individual and the dead are the same even though a “birth and death” event occurs at time τk. One of the end points 0 is absorbing [resp., reflecting] when ν2=0 [resp., ν2>0] and the other end point 1 is absorbing [resp., reflecting] when ν1=0 [resp., ν1>0]. Let σi be the first hitting time to i (i=0,1).First we consider the case thatν1=ν2=0. This is the case without mutation and both boundary points are absorbing with(5.5)pj=qj=j(N-j)N2,rj=1-pj-qj=1-2j(N-j)N2.
By  (2.7) and  (2.8) we have(5.6)s(x)=x,m(x)={-∞,x<0,0,0≤x<1N,N2N-1+N22(N-2)+⋯+N2i(N-i),iN≤x<i+1N,i=1,…,N-1,∞,1≤x.
Then Theorem 2.1 implies that the conditional process X*(t) conditional on {σ1<σ0} is again a birth and death process on {1/N,2/N,…,(N-1)/N,1} with the transition law(5.7)P(X*(τk+1)=j+1N∣X*(τk)=jN)=pj*,P(X*(τk+1)=j-1N∣X*(τk)=jN)=qj*,P(X*(τk+1)=jN∣X*(τk)=jN)=rj*,
where pN*=qN*=0 and(5.8)pj*=j+1jpj=(j+1)(N-j)N2,qj*=j-1jqj=(j-1)(N-j)N2,rj*=rj=1-2j(N-j)N2,
for 1≤j<N. The end point 1/N is reflecting since p1*=2(N-1)/N2>0. Note that m(x) of the original Moran model X(t) reduces to(5.9)m(x)={-∞,x<0,0,0≤x<13,92,13≤x<23,9,23≤x<1∞,1≤x,
if N=3 and this is essentially the same as the simple birth and death process discussed in Section 4.1.Next we consider the case that0<ν1<1. The boundary point 1 is reflecting. Then the induced conditional process does not satisfy Markov property and this conditional process is not a birth and death process by Theorem 2.3.


## 5.1. Diffusion Model

We consider the following diffusion model for a randomly mating population consisting ofN haploid individuals with two types (alleles) A1 and A2. Let x(t) be the relative frequency of A1 at time t. Then x(t) is a one-dimensional diffusion process on [0,1] with the generator(5.1)L=x(1-x)2Nd2dx2+{u2(1-x)-u1x}ddx,
where u1 [resp., u2] is mutation rate from A1 [resp., A2] to A2 [resp., A1] (see [4]).First we consider the case thatu1=0. The point 1 is accessible and exit boundary if u1=0 (see [16]). For this diffusion process, consider a stochastic process x*(t) induced by conditioning on hitting the boundary point 1 before hitting the other boundary point 0. The induced stochastic process is again a diffusion process with the generator(5.2)L*=x(1-x)2Nd2dx2+(u2+1N)(1-x)ddx
by  (1.2) and Theorem 2.1 of [3]. Note that the effect of conditioning is that it inflates the mutation rate u2 to u2+1/N. Ewens [1] considered the case that u1=u2=0 and the induced diffusion process is referred to as the conditional diffusion process by Ewens [1] (see also [2]).Next we consider the case that0<2Nu1<1. The point 1 is regular boundary in this case (see [16]) and we can pose various boundary conditions there. If we pose the absorbing boundary condition, then the induced process is again a diffusion process with the generator(5.3)L*=x(1-x)2Nd2dx2+{(u2+1N)(1-x)-u1x}ddx
by  (1.2) and Theorem 2.1 of [3]. On the other hand, if we pose the reflecting boundary condition as it is usually done in population genetics (see [17–19]), then the induced conditional process does not satisfy the Chapman-Kolmogorov equation and this process is not a diffusion process due to Theorem 2.2 of [3]. These results imply that we cannot use the diffusion model whose generator is given by  (5.3) as the conditional process when we pose the reflecting boundary condition at the boundary point 1.


## 5.2. Moran Model

Moran [6] introduced the following birth and death process as one of the fundamental stochastic models in population genetics called continuous-time Moran model (see [4] for discrete time Moran model). We refer to this model as Moran model for brief. Let N be the number of individuals in a haploid population with two types A1 and A2, where N is an integer greater than 2. Let τ0=0 and τk, k=1,2,⋯ be a sequence of random times introduced in Section 2. At time τk an individual is chosen randomly and it reproduces a new individual (k≥1). The type of the newborn individual is A1 [resp., A2] with probability 1-ν1 [resp., 1-ν2] and it is A2 [resp., A1] with probability ν1 [resp., ν2] if the parent is A1 [resp., A2], where 0≤ν1,ν2≤1. Then at this time τk an individual except newborn individual is chosen randomly to die. There is no change at time t≠τk (k≥1). Denoting by X(t) the relative frequency of A1 at time t, X(t) is a birth and death process on {0,1/N,2/N,…,(N-1)/N,1} with the transition law(5.4)P(X(τk+1)=j+1N∣X(τk)=jN)=(1-ν1)j(N-j)+ν2(N-j)2N2=pj,P(X(τk+1)=j-1N∣X(τk)=jN)=(1-ν2)(N-j)j+ν1j2N2=qj,P(X(τk+1)=jN∣X(τk)=jN)=rj,
where pj+qj+rj=1 (0≤j≤N). Note that p0=ν2, q0=0, r0=1-ν2, pN=0, qN=ν1, and rN=1-ν1. Note also that pj>0 unless ν1=1 and ν2=0, qj>0 unless ν1=0 and ν2=1, and rj>0 for 0<j<N. The process X(t) does not jump at time τk if the types of newborn individual and the dead are the same even though a “birth and death” event occurs at time τk. One of the end points 0 is absorbing [resp., reflecting] when ν2=0 [resp., ν2>0] and the other end point 1 is absorbing [resp., reflecting] when ν1=0 [resp., ν1>0]. Let σi be the first hitting time to i (i=0,1).First we consider the case thatν1=ν2=0. This is the case without mutation and both boundary points are absorbing with(5.5)pj=qj=j(N-j)N2,rj=1-pj-qj=1-2j(N-j)N2.
By  (2.7) and  (2.8) we have(5.6)s(x)=x,m(x)={-∞,x<0,0,0≤x<1N,N2N-1+N22(N-2)+⋯+N2i(N-i),iN≤x<i+1N,i=1,…,N-1,∞,1≤x.
Then Theorem 2.1 implies that the conditional process X*(t) conditional on {σ1<σ0} is again a birth and death process on {1/N,2/N,…,(N-1)/N,1} with the transition law(5.7)P(X*(τk+1)=j+1N∣X*(τk)=jN)=pj*,P(X*(τk+1)=j-1N∣X*(τk)=jN)=qj*,P(X*(τk+1)=jN∣X*(τk)=jN)=rj*,
where pN*=qN*=0 and(5.8)pj*=j+1jpj=(j+1)(N-j)N2,qj*=j-1jqj=(j-1)(N-j)N2,rj*=rj=1-2j(N-j)N2,
for 1≤j<N. The end point 1/N is reflecting since p1*=2(N-1)/N2>0. Note that m(x) of the original Moran model X(t) reduces to(5.9)m(x)={-∞,x<0,0,0≤x<13,92,13≤x<23,9,23≤x<1∞,1≤x,
if N=3 and this is essentially the same as the simple birth and death process discussed in Section 4.1.Next we consider the case that0<ν1<1. The boundary point 1 is reflecting. Then the induced conditional process does not satisfy Markov property and this conditional process is not a birth and death process by Theorem 2.3.



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*Source: 784567-2010-07-07.xml*