aa r X i v : . [ m a t h . P R ] D ec Intertwining of birth-and-death processes
Jan M. SwartNovember 20, 2018
Abstract
It has been known for a long time that for birth-and-death processes started in zero thefirst passage time of a given level is distributed as a sum of independent exponentiallydistributed random variables, the parameters of which are the negatives of the eigenvaluesof the stopped process. Recently, Diaconis and Miclo have given a probabilistic proofof this fact by constructing a coupling between a general birth-and-death process and aprocess whose birth rates are the negatives of the eigenvalues, ordered from high to low,and whose death rates are zero, in such a way that the latter process is always ahead ofthe former, and both arrive at the same time at the given level. In this note, we extendtheir methods by constructing a third process, whose birth rates are the negatives of theeigenvalues ordered from low to high and whose death rates are zero, which always lagsbehind the original process and also arrives at the same time.
MSC 2010.
Primary: 60J27; Secondary: 15A18, 37A30, 60G40, 60J35, 60J80
Keywords.
Intertwining of Markov processes; birth and death process; averaged Markov pro-cess; first passage time; coupling; eigenvalues.
Acknowledgement.
Work sponsored by GA ˇCR grant 201/09/1931.
Contents
Let X = ( X t ) t ≥ be the continuous-time Markov process in N = { , , . . . } , started from X = 0, that jumps from x − x with birth rate b x > x to x − death rate d x > x ≥ τ N := inf { t ≥ X t = N } ( N ≥
1) (1.1)denote the first passage time of N . The following result has been known at least since [KM59,Prop. 1]. Proposition 1.1 (Law of first passage times)
The first passage time τ N is distributed asa sum of independent exponentially distributed random variables whose parameters λ < · · · <λ N are the negatives of the nonzero eigenvalues of the generator of the process stopped in N . τ N bypurely algebraic methods, see [DM09] for a historical overview. In the latter paper, Diaconisand Miclo gave for the first time a probabilistic proof of Proposition 1.1, by coupling theprocess X to another birth-and-death process X + with birth rates b +1 = λ N , . . . , b + N = λ andzero death rates, in such a way that X t ∧ τ N ≤ X + t for all t ≥ X and X + arrive in N atthe same time. In the present paper, we will extend their methods by showing that X and X + can in addition be coupled to a process X − with birth rates b +1 = λ , . . . , b + N = λ N andzero death rates, in such a way that X − t ≤ X t ∧ τ N ≤ X + t for all t ≥ N at the same time. The coupling technique used by Diaconis and Miclo in [DM09] is of a special kind, which issometimes called intertwining of Markov processes . Let X and X ′ be continuous-time Markovprocesses with finite state spaces S and S ′ and generators G and G ′ , respectively, and let K be a probability kernel from S to S ′ . Then K defines a linear operator from R S ′ to R S , alsodenoted by K , by the formula Kf ( x ) := X y ∈ S ′ K ( x, y ) f ( y ) . (1.2)The following result, which is based on an observation by Rogers and Pitman [RP81], wasproved by Fill in [Fil92, Thm. 2]. (An independent proof can be found in [AS10, Prop. 4]). Proposition 1.2 (Intertwining of Markov processes)
Assume that GK = KG ′ . (1.3) Then there exists a generator ˆ G of an S × S ′ -valued Markov process with the property that if ( X, X ′ ) evolves according to ˆ G and satisfies P [ X ′ = y | X ] = K ( X , y ) ( y ∈ S ′ ) , (1.4) then P [ X ′ t = y | ( X s ) ≤ s ≤ t ] = K ( X t , y ) ( t ≥ , y ∈ S ′ ) , (1.5) and the processes X and X ′ , on their own, are Markov processes evolving according to thegenerators G and G ′ , respectively. Algebraic relations of the type (1.3) are called intertwining relations, hence the name inter-twining of Markov processes . We note that the operator K needs in general not have aninverse, and even if it does, this inverse will in general not be associated to a probabilitykernel from S ′ to S . In view of this, an intertwining of Markov processes is not a symmetricrelation. To express this, following terminology introduced in [AS10], we will also say that inthe set-up of Proposition 1.2, X ′ is an averaged Markov process on X . We are now ready to formulate our main result. Deviating slightly from our notation in Sec-tion 1.1, we let X = ( X t ) t ≥ be a continuous-time Markov process with state space { , . . . , N } ,2tarted from X = 0, that jumps from x − x with birth rate b x and from x to x − d x , where b , . . . , b N > d , . . . , d N − >
0, but d N = 0, i.e., X is the stoppedprocess from Section 1.1. We let G denote the generator of X , i.e., Gf ( x ) := b x +1 (cid:0) f ( x + 1) − f ( x ) (cid:1) + d x (cid:0) f ( x − − f ( x ) (cid:1) (0 ≤ x ≤ N ) , (1.6)where f : { , . . . , N } → R is a real function and we adopt the convention that d = 0 and b N +1 = 0 so that the corresponding terms in (1.6) are zero, regardless of the (fictive) valuesof f in − N + 1. The following theorem is our main result. Theorem 1.3 (Intertwining of birth-and-death processes)
The operator G has N + 1 distinct eigenvalues − λ > − λ > · · · > − λ N . Let X − and X + be the pure birth processesin { , . . . , N } , started from X − = X +0 = 0 , with birth rates b − := λ , . . . , b − N := λ N and b +1 := λ N , . . . , b + N := λ , respectively, and let G − and G + be their generators. Then there existprobability kernels K − and K + on { , . . . , N } satisfying K − ( x, { , . . . , x } ) = 1 , K + ( x, { , . . . , x } ) = 1 , (0 ≤ x ≤ N ) K − ( N, N ) = 1 , K + ( N, N ) = 1 , (1.7) and (i) K + G = G + K + and (ii) GK − = K − G − . (1.8) Moreover, the processes X − , X , and X + can be coupled in such a way that (i) P [ X t = y | ( X + s ) ≤ s ≤ t ] = K + ( X + t , y ) ( t ≥ , ≤ y ≤ N ) , (ii) P [ X − t = y | ( X + s , X s ) ≤ s ≤ t ] = K − ( X t , y ) ( t ≥ , ≤ y ≤ N ) . (1.9)The existence of a kernel K + such that (1.8) (i) and (1.9) (i) hold has been proved beforein [DM09, Prop. 10]. Our new contribution is the construction of the kernel K − such thatmoreover (1.8) (ii) and (1.9) (ii) hold. It is easy to see that formulas (1.7) and (1.9) implythat (i) X − t ≤ X t ≤ X + t ( t ≥ , (ii) τ − N = τ N = τ + N , (1.10)where τ N := inf { t ≥ X t = N } and τ − N and τ + N are defined similarly for X − and X + ,respectively. We note that X − and X + move, in a sense, in the slowest resp. fastest possibleway from 0 to N , given that they have to arrive at exactly the same time as X . Note that,using terminology introduced at the end of Section 1.2, X is an averaged Markov process on X + and X − is an averaged Markov process on X . In comparison to the paper by Diaconis and Miclo [DM09], the present paper does not add toomuch that is new. In particular the construction of the kernel K − in Theorem 1.3 is very similarto the construction of the kernel K + , which was already carried out in [DM09]. However, webelieve that the observation that both constructions are possible, with an interesting symmetrybetween them, is of some interest.The (new) construction with the process X − has in fact one advantage over the construc-tion with X + , since Proposition 1.2 and formula (1.8) imply that the process X started in any X − with the same dynamics as in Theorem 1.3, insuch a way that P [ X − t = y | ( X s ) ≤ s ≤ t ] = K − ( X t , y ) for all 0 ≤ y ≤ N and t ≥
0. This impliesthat for a general initial state X = x ∈ { , . . . , N } , the stopping time τ N is distributed as P Ny = Z σ y where σ , . . . , σ N are independent exponentially distributed random variables withparameters λ , . . . , λ N and Z is an independent { , . . . , N } -valued random variable with law K − ( x, · ). Note that the (old) coupling with the process X + forces one to start the process X in an initial law that is a convex combination of the laws K + ( x, · ) with 0 ≤ x ≤ N , henceno conclusions can be drawn for arbitrary initial states.On the other hand, the methods of [DM09] can also be used to study birth-and-deathprocesses on { , . . . , N } whose death rate d N is not zero and which, therefore, converge inlaw to a unique equilibrium. In particular, Diaconis and Miclo use a generalization of theirintertwining relation (1.8) (i) to construct a fastest strong stationary time for such processes(we refer to [DM09] for the definition). In contrast, it seems that the interwining relation(1.8) (ii) does not generalize to such a setting.On a more general level, one may ask what the advantage is of a ‘probabilistic’ proof ofProposition 1.1 as opposed to older, more algebraic proofs. Since most of the work behindTheorem 1.3 goes into proving the intertwining relations (1.8), one might even argue that thepresent proof is still rather algebraic in nature, although with a strong probabilistic flavour.In this context, it is interesting to note that the fact that G is diagonalizable with real,distinct eigenvalues follows as a result of our proofs (in particular, this follows from a repeatedapplication of the Perron-Frobenius theorem) and does not have to be provided by some extraargument (based on, for example, reversibility).In general, diagonalizing a generator of a Markov process gives very strong informationabout the process, but in practice, if the state space is large, it is hard to get good informationabout the position of eigenvalues etc. The idea of interwining generators with transitionkernels may in some cases be a good way to transform generators of complicated processesinto generators of more simple processes and thus provide a more probabilistic alternative todiagonalization.The methods of this paper can certainly be extended to one-dimensional processes withtwo traps, to dicrete-time processes, and to one-dimensional diffusions. Miclo [Mic10] hasproved a generalization of Proposition 1.1 for reversible Markov chains. In [AS10], intertwiningrelations were used to estimate the time to extinction for large hierarchical contact processes.The present work was partly motivated by an open problem from that paper. (To be precise,Question 1 ◦ from Section 3.3.) Let X be the birth-and-death process in S N := { , . . . , N } from Section 1.3 and let G : R S N → R S N be its generator, defined in (1.6). We equip R S N with the usual inner product h π | f i := P Nx =0 π ( x ) f ( x ) and let G † denote the adjoint of G with respect to this inner product.Then G † π ( x ) = b x π ( x − − b x +1 π ( x ) + d x +1 π ( x + 1) − d x π ( x ) , (2.1)where as in (1.6) we use the convention that d = 0 and b N +1 = 0 so that the correspondingterms in (2.1) are zero, regardless of the (fictive) values of π in − N + 1.4ince δ N (the delta mass in N ) is the unique invariant law of X , the eigenvalue 0 ofthe generator G has multiplicity one and its unique left and right eigenvectors are δ N andthe constant function 1, respectively. We will need the following result on the next largesteigenvalue and its left and right eigenvectors. Lemma 2.1 (Leading eigenvectors)
There exists a λ > and f, π ∈ R S N such that (i) f is strictly decreasing on { , . . . , N } and satisfies f (0) = 1 , f ( N ) = 0 , (ii) π is strictly positive on { , . . . , N − } and satisfies P N − x =0 π ( x ) = 1 = − π ( N ) , (iii) Gf = − λf and G † π = − λπ. Proof
Set e ( x ) := δ x (0 ≤ x ≤ N −
1) and e ( N ) := 1 ,ξ ( x ) := δ x − δ N (0 ≤ x ≤ N −
1) and ξ ( N ) := δ N . (2.2)Then { e (0) , . . . , e ( N ) } is a basis for R S N and { ξ (0) , . . . , ξ ( N ) } is its associated dual basis, i.e., h e ( x ) | ξ ( y ) i = 1 { x = y } . Set E := span { e (0) , . . . , e ( N − } = { f ∈ R S N : f ( N ) = 0 } , F := span { ξ (0) , . . . , ξ ( N − } = { π ∈ R S N : P Nx =0 π ( x ) = 0 } . (2.3)Since N is a trap for the process X , it is easy to see that the operator G maps the space E into itself. Since the coordinates of a vector in E with respect to the basis { e (0) , . . . , e ( N ) } are the same as its coordinates with respect to the standard basis { δ , . . . , δ N } , it follows thatwith respect to the basis { e (0) , . . . , e ( N ) } , the matrix [ G ] of G has the form[ G ] = (cid:18) A
00 0 (cid:19) , (2.4)where A ( x, y ) = G ( x, y ) for 0 ≤ x, y ≤ N −
1. The restriction of the process X to the space { , . . . , N − } is irreducible in the sense that there is a positive probability of going fromany state to any other state. Therefore, by applying the Perron-Frobenius theorem (see, e.g.Chapter XIII, §
2, Theorem 2 in [Gan00]) to A + cI and its adjoint for some sufficiently large c , one finds that A has a real eigenvalue − λ of multiplicity one, which is larger than all otherreal eigenvalues, and associated left and right eigenvectors π ∈ F and f ∈ E that are strictlypositive with respect to the bases { ξ (0) , . . . , ξ ( N − } and { e (0) , . . . , e ( N − } , respectively.Since Markov semigroups are contractive we have − λ ≤ G has multiplicity one and belongs to different left and right eigenvectors, we conclude that − λ <
0. Since we can always normalize our eigenvectors such that P N − x =0 π ( x ) = 1 andmax N − x =0 f ( x ) = 1, this proves all statements of the lemma except for the fact that f is strictlydecreasing.To prove this latter fact, we observe that by the facts that Gf = − λf and f > { , . . . , N − } , b (cid:0) f (1) − f (0) (cid:1) = − λf (0) < , (2.5)which show that f (0) > f (1). By the same argument, b x +1 (cid:0) f ( x + 1) − f ( x ) (cid:1) = − λf ( x ) − d x (cid:0) f ( x − − f ( x ) (cid:1) < ≤ x ≤ N − , (2.6)from which we see by induction that f ( x ) > f ( x + 1) for all 0 ≤ x ≤ N − .2 Intertwining the fast process In this section, we prove the existence of a kernel K + satisfying (1.7) and (1.8). Our proofis basically the same as the proof given in [DM09], but as a preparation for the next sectionit will be convenient to review their proof and shorten it somewhat. The proof in [DM09] iswritten in such a way as to make clear how the authors arrived at their argument and usesdiscrete derivatives that are presumably also useful if one wants to generalize the theory toone-dimensional diffusions. If our only aim is Theorem 1.3, however, we can summarize theirarguments quite a bit.The kernel K + will be constructed as the concatenation of an inductively defined sequenceof kernels K ( N −
1) + , . . . , K (1) + . Associated with these kernels is a sequence of generators G ( N − , + , . . . , G (0) + of birth-and-death processes in { , . . . , N } satisfying the intertwiningrelations K ( M ) + G ( M ) + = G ( M −
1) + K ( M ) + (1 ≤ M ≤ N − , (2.7)where the process with generator G ( M ) has birth rates b ( M )1 , . . . , b ( M ) N > d ( M )1 , . . . , d ( M ) M > d ( M ) M +1 = · · · = d ( M ) N = 0; see Figure 1 for a picture. In particular, we willchoose G ( N −
1) + := G and setting G + := G (0) + will yield the desired pure birth process withbirth rates b +1 = λ N , . . . , b + N = λ .The core the proof is the following proposition, which corresponds to the inductive step inthe argument. Proposition 2.2 (Inductive step)
Let ≤ M ≤ N − and let G be the generator of a birth-and-death process in { , . . . , N } with birth rates b , . . . , b N > and death rates d , . . . , d M > , d M +1 = · · · = d N = 0 . Then there exists a probability kernel K on { , . . . , N } satisfying K ( x, { , . . . , x } ) = 1 (0 ≤ x ≤ N ) and K ( x, x ) = 1 ( M + 1 ≤ x ≤ N ) , (2.8) and a generator G ′ of a birth-and-death process in { , . . . , N } with birth rates b ′ , . . . , b ′ N > and death rates d ′ , . . . , d ′ M − > , d ′ M = · · · = d ′ N = 0 , such that KG = G ′ K . Proof
It follows from Lemma 2.1 applied to the process stopped at M + 1 that there existsa function ρ : { , . . . , N } → R such that ρ > { , . . . , M } , ρ = 0 on { M + 1 , . . . , N } , P Nx =0 ρ ( x ) = 1, and G † ρ ( x ) = − λρ ( x ) + λδ M +1 ( x ) (0 ≤ x ≤ N ) , (2.9)where λ = b M +1 ρ ( M ) > . (2.10)The law ρ is sometimes called a quasi-stationary law . Using ρ , we define the kernel K on { , . . . , N } by K ( x, y ) := { y ≤ x } ρ ( y ) H ( x ) if x ≤ M, { y = x } if M + 1 ≤ x, (2.11)where H ( x ) := x X y =0 ρ ( y ) (0 ≤ x ≤ M ) . (2.12)6 b b b λ λ λ λ λ λ λ λ λ λ λ λ λ λ ˙ b ¨ b ˙ b ¨ b ˙ b K (3) + d d b ′ d ′ b ′ d ′ b ′′ d ′′ d b ′′ b ′ G (2) + K (1) − K (2) − K (2) + K (1) + G (0) + G (1) + G (1) − G (3) − G (2) − G K (3) − ˙ d ˙ d ¨ d Figure 1: Intertwining of birth and death processes. In this picture N = 4. All nonzerotransition rates and probabilities have been indicated with arrows.7ince K is a lower triangular matrix, it is invertible, so there exists a unique linear operator G ′ satisfying KG = G ′ K and G ′ is in fact given by G ′ = KGK − . Since G ′ G ′ K KG G ′ ( x, x ) = − X y = x G ′ ( x, y ) . (2.13)In view of this, to prove our claim, it suffices to check that the off-diagonal entries of G ′ coincide with those of a birth-and-death process in { , . . . , N } with birth rates b ′ , . . . , b ′ N > d ′ , . . . , d ′ M − > d ′ M = · · · = d ′ N = 0.To determine the off-diagonal entries of G ′ , we calculate, using (2.1) and (2.9),( KG )( x, y ) = G † K ( x, · )( y )= − λ { y ≤ x } ρ ( y ) H ( x ) − d x +1 ρ ( x + 1) H ( x ) δ x ( y ) + b x +1 ρ ( x ) H ( x ) δ x +1 ( y ) if x < M, − λρ ( y ) + λδ M +1 ( y ) if x = M, − b x +1 δ x ( y ) + b x +1 δ x +1 ( y ) if x > M. (2.14)In order to find G ′ , we need to express these formulas, as functions of y , as linear combinationsof the basis vectors ( K ( x, · )) ≤ x ≤ N . To that aim, we observe that δ x = K ( x, · ) ( M + 1 ≤ x ≤ N ) , (2.15)while for 1 ≤ x ≤ M , we have δ x ( y ) = (cid:0) { y ≤ x } − { y ≤ x − } (cid:1) ρ ( y ) ρ ( x )= H ( x ) ρ ( x ) 1 { y ≤ x } ρ ( y ) H ( x ) − H ( x − ρ ( x ) 1 { y ≤ x − } ρ ( y ) H ( x − H ( x ) ρ ( x ) K ( x, y ) − H ( x − ρ ( x ) K ( x − , y ) . (2.16)Inserting this into (2.14), we find that N X x ′ =0 G ′ ( x, x ′ ) K ( x ′ , y ) := ( KG )( x, y )= − λK (0 , y ) − d ρ (1) H (0) K (0 , y )+ b ρ (0) H (0) (cid:16) H (1) ρ (1) K (1 , y ) − H (0) ρ (1) K (0 , y ) (cid:17) if x = 0 , − λK ( x, y ) − d x +1 ρ ( x + 1) H ( x ) (cid:16) H ( x ) ρ ( x ) K ( x, y ) − H ( x − ρ ( x ) K ( x − , y ) (cid:17) + b x +1 ρ ( x ) H ( x ) (cid:16) H ( x + 1) ρ ( x + 1) K ( x + 1 , y ) − H ( x ) ρ ( x + 1) K ( x, y ) (cid:17) if 0 < x < M, − λK ( M, y ) + λK ( M + 1 , y ) if x = M, − b x +1 K ( x, y ) + b x +1 K ( x + 1 , y ) if x > M. (2.17)8rom this, we can read off the off-diagonal entries of G ′ . Indeed, b ′ x +1 = G ′ ( x, x + 1) = b x +1 ρ ( x ) H ( x + 1) H ( x ) ρ ( x + 1) if x < M,λ if x = M,b x +1 if x > M,d ′ x = G ′ ( x, x −
1) = d x +1 ρ ( x + 1) H ( x − H ( x ) ρ ( x ) if 0 < x < M, x ≥ M, (2.18)and all other off-diagonal entries are zero. Remark
The proof of Proposition 2.2 is straightforward except for the clever choice of K in(2.11)–(2.12). For some motivation of this choice and the way the authors arrived at it werefer to [DM09].Using Proposition 2.2 we can construct a sequence of kernels K ( N −
1) + , . . . , K (1) + and gener-ators G ( N − , + , . . . , G (0) + satisfying the intertwining relations (2.7), such that G + := G (0) + is a pure birth process with birth rates b +1 , . . . , b + N >
0, say. It is now easy to see that thecomposed kernel K + := K (1) + · · · K ( N −
1) + (2.19)satisfies K + ( x, { , . . . , x } ) = 1 (0 ≤ x ≤ N ), K + ( N, N ) = 1 and K + G = G + K + . It is straight-forward to check that the eigenvalues of G ′ are − b +1 , . . . , − b + N ,
0. Since G = ( K + ) − G + K + ,the operators G and G + have the same spectrum.We claim that b +1 > · · · > b + N >
0. To see this, recall from the proofs of Lemma 2.1 andProposition 2.2 that − b + M is the Perron-Frobenius eigenvalue of the process with generator G ( M ) + stopped at M + 1. It follows from the intertwining relation (2.7) that − b + M − is alsoan eigenvalue of this process, corresponding to a different eigenvector, hence by the Perron-Frobenius theorem, b M − > b M . In the previous section, we have constructed a kernel K + and generator of a pure birth process G + such that (1.8) (i) holds. In this section, we construct a kernel K − and generator of apure birth process G − satisfying (1.8) (ii). The proof will be very similar to the previouscase, except that some things will ‘go he other way around’. In particular, using terminologyintroduced at the end of Section 1.2, G − will be the generator of an avaraged Markov process X − on X while in the previous section we constructed a pure birth process X + such that X is an averaged Markov process on X + .As in the previous section, the kernel K − will be constructed as the concatention of aninductively defined sequence of kernels K (1) − , . . . , K ( N − − . Associated with these kernelsis a sequence of generators G (1) , − , . . . , G ( N − − of birth-and-death processes in { , . . . , N } satisfying the intertwining relations G ( M − − K ( M ) − = K ( M ) − G ( M ) − (1 ≤ M ≤ N − , (2.20)where the process with generator G ( M ) has birth rates b ( M )1 , . . . , b ( M ) N > d ( M )1 = · · · = d ( M ) M = 0, d ( M ) M +1 , . . . , d ( M ) N − >
0, and d ( M ) N = 0. We again refer to Figure 1 for anillustration.The core of the argument is the following proposition.9 roposition 2.3 (Inductive step) Let ≤ M ≤ N − and let G be the generator of abirth-and-death process in { , . . . , N } with birth rates b , . . . , b N > and death rates d = · · · = d M = 0 , d M +1 , . . . , d N − > , and d N = 0 . Then there exists a probability kernel K on { , . . . , N } satisfying K ( x, { , . . . , x } ) = 1 (0 ≤ x ≤ N ) and K ( x, x ) = 1 ( x
6∈ {
M, . . . , N − } ) , (2.21) and a generator ˙ G of a birth-and-death process in { , . . . , N } with birth rates ˙ b , . . . , ˙ b N > and death rates ˙ d = · · · = ˙ d M +1 = 0 , ˙ d M +2 , . . . , ˙ d N − > , and ˙ d N = 0 , such that GK = K ˙ G . Proof
It follows from Lemma 2.1 applied to the process restricted to { M, . . . , N } that thereexists a function f : { , . . . , N } → R such that f = 0 on { , . . . , M − } , f is strictly decreasingon { M, . . . , N } , f ( M ) = 1, f ( N ) = 0, and Gf ( x ) = − λf ( x ) + b M δ M − ( x ) (0 ≤ x ≤ N ) , (2.22)where λ = b M +1 (cid:0) − f ( M + 1) (cid:1) > . (2.23)We set K ( x, y ) := 1 { x = y } ( y
6∈ {
M, . . . , N − } ) . (2.24)For y = M, . . . , N −
1, we claim that we can inductively define the kernel K ( x, y ) and contants C y > K ( x, y ) := C y { y ≤ x } f ( x ) , (ii) y X y ′ = M K ( y, y ′ ) = 1 , ( M ≤ y ≤ N − . (2.25)To see that this is all right, note that for y = M (2.25) (i) and (ii) are satisfied by choosing C M := 1, while for M + 1 ≤ y ≤ N − C y := 1 f ( y ) (cid:16) − y − X y ′ = M K ( y, y ′ ) (cid:17) . (2.26)Since f is strictly decreasing on { M, . . . , N } , one has, by induction, y − X y ′ = M K ( y, y ′ ) = y − X y ′ = M C y ′ f ( y ) < y − X y ′ = M C y ′ f ( y −
1) = y − X y ′ = M K ( y − , y ′ ) = 1 , (2.27)which shows that C y >
0. We now calculate( GK )( x, y ) = GK ( · , y )( x )= b y δ y − ( x ) − b y +1 δ y ( x ) if 0 ≤ y ≤ M − , − λf ( x ) + b M δ M − ( x ) if y = M, − λC y { y ≤ x } f ( x ) + b y C y δ y − ( x ) − d y C y (cid:0) f ( y − − f ( y ) (cid:1) δ y ( x ) if M + 1 ≤ y ≤ N − ,b N δ N − ( x ) if y = N. (2.28)10y the same arguments as those in the previous section, there exists a unique linear operator˙ G such that GK = K ˙ G . In order to check that ˙ G is the generator of a birth-and-deathprocess in { , . . . , N } with birth rates ˙ b , . . . , ˙ b N > d = · · · = ˙ d M +1 = 0,˙ d M +2 , . . . , ˙ d N − >
0, and ˙ d N = 0, it suffices to check that the off-diagonal entries ˙ G ( x, y ) havethe desired form. In order to do this, we must express the formulas in (2.28), as functions of x , as linear combinations of the basis vectors ( K ( · , y )) ≤ y ≤ N . We observe that δ y ( x ) = K ( · , y ) (cid:0) y
6∈ {
M, . . . , N − } (cid:1) , (2.29)while for M ≤ y ≤ N −
2, we have δ y ( x ) = (cid:0) { y ≤ x } − { y +1 ≤ x } (cid:1) f ( x ) f ( y )= 1 f ( y ) C y C y { y ≤ x } f ( x ) − f ( y ) C y +1 C y +1 { y +1 ≤ x } f ( x )= 1 f ( y ) C y K ( x, y ) − f ( y ) C y +1 K ( x, y + 1) , (2.30)and δ N − ( x ) = 1 f ( N − C N − K ( x, N − . (2.31)Inserting this into (2.28), we obtain X y ′ K ( x, y ′ ) ˙ G ( y ′ , y ) = ( GK )( x, y )= b y K ( x, y − − b y +1 K ( x, y ) if 0 ≤ y ≤ M − , − λK ( x, M ) + b M K ( x, M −
1) if y = M, − λK ( x, y ) + b y C y (cid:16) K ( x, y − f ( y − C y − − K ( x, y ) f ( y + 1) C y (cid:17) − d y C y (cid:0) f ( y − − f ( y ) (cid:1)(cid:16) K ( x, y ) f ( y ) C y − K ( x, y + 1) f ( y ) C y +1 (cid:17) if M + 1 ≤ y ≤ N − , − λK ( x, y ) + b y C y (cid:16) K ( x, y − f ( y − C y − − K ( x, y ) f ( y + 1) C y (cid:17) − d y C y (cid:0) f ( y − − f ( y ) (cid:1) K ( x, y ) f ( y ) C y if y = N − ,b N K ( x, N − f ( N − C N − if y = N, (2.32)where we use the convention that b = 0 and hence b K ( x, −
1) = 0, regardless of the (fictive)value of K ( x, − G . Indeed,˙ b y = ˙ G ( y − , y ) = b y if 1 ≤ y ≤ M,b y C y f ( y − C y − if M + 1 ≤ y ≤ N, ˙ d y +1 = ˙ G ( y + 1 , y ) = y
6∈ { M + 1 , . . . , N − } ,d y C y ( f ( y − − f ( y )) C y +1 f ( y ) if M + 1 ≤ y ≤ N − , (2.33)11nd all other off-diagonal entries are zero. We note that in particular, by (2.23) and thedefinition of the C y ’s,˙ b M +1 = b M +1 C M +1 f ( M ) C M = b M +1 C M +1 = b M +1 (cid:0) − f ( M + 1)) = λ. (2.34) Remark
As in the case of Proposition 2.2, the proof of Proposition 2.3 is straightforwardexcept for the choice of the kernel K . We have guessed formula (2.25) by analogy withformula (2.11), which is due to [DM09].With the help of Proposition 2.3, we can inductively define kernels K (1) − , . . . , K ( N − − andoperators G (1) , − , . . . , G ( N − − . Setting G − := G ( N − − and K − = K (1) − · · · K ( N − − (2.35)now yields a generator of a pure birth process with birth rates b − , . . . , b − N and a kernel K − with the properties described in (1.7)–(1.8).In the same way as in the previous section, we see that 0 , − b − , . . . , − b − N are the eigenvaluesof G . To see that 0 < b < · · · < b N we observe from (2.34) that − b − M is the Perron-Frobeniuseigenvalue of the process with generator G ( M ) − restricted to { M, . . . , N } . It follows from theintertwining relation (2.20) that − b + M +1 is also an eigenvalue of this process, corresponding toa different eigenvector, hence by the Perron-Frobenius theorem, b M < b M +1 . Proof of Theorem 1.3
The existence of generators G − , G + and kernels K − , K + satisfying(1.7)–(1.8) has been proved in the previous sections. By Proposition 1.2, it follows that X + and X can be coupled such that (1.9) (i) holds. By applying Proposition 1.2 to the kernel L from { , . . . , N } to { , . . . , N } given by L (cid:0) ( x, y ) , z (cid:1) := K − ( y, z ) (0 ≤ x, y, z ≤ N ) , (2.36)we see that ( X + , X ) and X − can be coupled in such a way that both (1.9) (i) and (ii) hold. References [AS10] S.R. Athreya and J.M. Swart. Survival of contact processes on the hierarchical group.
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