On time scales and quasi-stationary distributions for multitype birth-and-death processes
OOn time scales and quasi-stationary distributionsfor multitype birth-and-death processes
J.-R. Chazottes ∗ , P. Collet † , and S. M´el´eard ‡ Centre de Physique Th´eorique, CNRS UMR 7644, F-91128Palaiseau Cedex (France) Centre de Math´ematiques Appliqu´ees, CNRS UMR 7641, F-91128Palaiseau Cedex (France)Dated: November 20, 2018
Abstract
We consider a class of birth-and-death processes describing a popu-lation made of d sub-populations of different types which interact withone another. The state space is Z d + (unbounded). We assume that thepopulation goes almost surely to extinction, so that the unique station-ary distribution is the Dirac measure at the origin. These processes areparametrized by a scaling parameter K which can be thought as the orderof magnitude of the total size of the population at time 0. For any fixedfinite time span, it is well-known that such processes, when renormalizedby K , are close, in the limit K → + ∞ , to the solutions of a certain differ-ential equation in R d + whose vector field is determined by the birth anddeath rates. We consider the case where there is a unique attractive fixedpoint (off the boundary of the positive orthant) for the vector field (whilethe origin is repulsive). What is expected is that, for K large, the processwill stay in the vicinity of the fixed point for a very long time before be-ing absorbed at the origin. To precisely describe this behavior, we provethe existence of a quasi-stationary distribution (qsd, for short). In fact,we establish a bound for the total variation distance between the processconditioned to non-extinction before time t and the qsd. This bound isexponentially small in t , for t (cid:29) log K . As a by-product, we obtain anestimate for the mean time to extinction in the qsd. We also quantify howclose is the law of the process (not conditioned to non-extinction) eitherto the Dirac measure at the origin or to the qsd, for times much largerthan log K and much smaller than the mean time to extinction, which isexponentially large as a function of K . Let us stress that we are interestedin what happens for finite K . We obtain results much beyond what largedeviation techniques could provide. Keywords : Markov jump process, differential equations, competitionmodels, population ecology, mean time to extinction, Lyapunov functions. ∗ Email: [email protected] † Email: [email protected] ‡ Email: [email protected] a r X i v : . [ m a t h . P R ] N ov ontents Introduction
A fundamental question in population ecology concerns the risk of extinction ofpopulations [16]. Stochastic models are well suited to account for the inherentlydiscrete nature of individuals, especially when populations are “small”. Suchmodels are often referred to as “individual-based models”. In contrast, “largepopulations” are traditionally modelled by ordinary differential equations, whenthe spatial structure, the age-structure, the fluctuations of the environment, etc,are ignored. These “population-level” models are supposed to account for thedeterministic trends of large populations (the macroscale), and are inherentlyincapable of describing extinction phenomena.In the present work we consider birth-and-death processes ( N K ( t ) , t ≥ d differenttypes which interact with one another. At each time t , the state of the processis thus given by a vector n = ( n , . . . , n d ) ∈ Z d + , where n i is the number ofindividuals of the i th sub-population. We assume that these processes depend ona scaling parameter K > t , N K ( t ) = n , the rate at which the population is increased (respectively decreased) by oneindividual of type j ∈ { , . . . , d } is KB j ( n/K ) (respectively KD j ( n/K )).On the one hand, keeping K fixed and letting t go to + ∞ , we will show that,under appropriate assumptions, the total population goes extinct with proba-bility one. In the context of population ecology, this is a natural assumption tomodel the truism that “nothing last forever”, due to the finiteness of ressources.In the terminology of Markov chains, there is an absorbing state, so the sta-tionary distribution (the Dirac measure sitting at this state) is irrelevant as itdescribes only the state where the population is extinct.On the other hand, one can prove that the probability that N K ( t ) /K devi-ates, over any fixed finite time span, from the solution of the differential equationd x d t = B ( x ) − D ( x ) (1.1)by more than some prescribed quantity, goes to zero, as K goes to + ∞ . Inthe previous equation x = ( x , . . . , x d ) ∈ R d + , B ( x ) = ( B ( x ) , . . . , B d ( x )) and D ( x ) = ( D ( x ) , . . . , D d ( x )). Basically, our aim is to describe what happens “inbetween” these two limiting regimes.Given a differential equation as above, e.g. , a Lotka-Volterra type equation,one can have repelling fixed points, attracting fixed points (each one with itsbasin of attraction), limit cycles, “strange attractors”, etc, see for instance [17].In this work we restrict to a simple situation where there is a unique attractingfixed point x ∗ in the interior of R d + and the origin is a repelling fixed point. Thebig picture is then intuitively clear: for large (but finite) values of the parameter K , one expects that the process will “feel” the presence of the deterministic fixedpoint x ∗ and will stay in the vicinity of the state (cid:98) Kx ∗ (cid:99) for a very long time(“quasi-stationary” regime), until it is finally absorbed.Let us informally describe the main results that we obtain. We firt prove theexistence of a unique quasi-stationary distribution (qsd, for short). In fact, weprove a stronger result since we establish a bound for the total variation distancebetween the process conditioned to non-extinction before time t and the qsd.This bound is exponentially small in t , for t much larger than log K (see Theorem3.1). Our second result is an upper bound and a lower bound for the mean timeto extinction in the qsd. This mean time is exponential in K (ee Theorem 3.2).Our third result quantifies how close, in total variation distance, the law of theprocess not conditioned to non-extinction, is to a convex combination of theDirac measure at the origin and the qsd (see Theorem 3.3). For t much largerthan log K and much smaller than the mean time to extinction, this distance isvery small. Then, for t much larger than exp( O (1) K ), the law of the process notconditioned to non-extinction is very close to the Dirac measure at the origin.Our fourth main result show that the spectral gap of this semigroup is largerthan O (1) / log K , see Theorem 3.4.We emphasize that we perform a rather fine pathwise analysis of the process.Roughly speaking, we also prove that it takes a time of order one for the processto “come down from infinity” and to arrive in a ball of radius of order K andcenter (cid:98) Kx ∗ (cid:99) . This is contained in Sublemma 5.4. Afterwards, it takes a timeof order log K to arrive in a ball of radius of order √ K and center (cid:98) Kx ∗ (cid:99) (seeLemma 5.1). Then the process fluctuates around (cid:98) Kx ∗ (cid:99) for a very long time,and is almost distributed according to the qsd.This work is the natural extension of our work [6] on monotype ( i.e. , d = 1)birth-and-death processes. Therein, we used a precise spectral analysis of acertain self-adjoint operator acting on a suitable “weighted” Hilbert space. Weobtained precise estimates, notably for the mean time to extinction, as well asthe approximate behavior of the process in terms of a Gaussian distribution.These spectral techniques in Hilbert spaces are lost when d ≥ K of the involved constants.Let us mention the survey article [2] which describes how the so-called WKBmethod can be used to evaluate the mean time and/or probability of populationextinction, fixation and switches resulting from either intrinsic (demographic)noise, etc. That article deals with much more general situations than the onewe consider here, but the approach is “semi-rigorous” from the mathematicalviewpoint. Let us also mention that there are other papers dealing with quanti-tative estimates of quasi-stationary distributions in contexts which are differentfrom ours, namely [3] and [9, 10]. In particular, the state space is finite in thosepapers, and different methods are developed. We emphasize that, in the con-text of stochastic models in population ecology, taking a finite state space is notnatural. Indeed, large fluctuations can arise in such a way that we “go out” ofthe state space.The paper is structured as follows. In Section 2 we state the hypotheseswe make on the vector field B ( x ) − D ( x ) and on the birth and death rates.Section 3 contains our four main results. In Section 4, we construct a Lyapunovfunction for the generator of the process. We also prove a result (Lemma 4.3) giving quantitative bounds on the probability of the time the process takesto come down from one level set of the Lyapunov function to a lower one. Weexpect this quantitative result to be useful in more general situations. Section 54s devoted to the proof of the necessary and sufficient conditions required in [5].More precisely, we prove that the process comes down from infinity and entersa ball centered at n ∗ with a radius of order √ K . Then we compare the processin this ball with an auxiliary symmetric random walk. In Section 3.1 we boundfrom above and below the parameter of the exponential law of the extinctiontime under the qsd. Section 7 is devoted to the proof of a lower bound of thespectal gap of the semigroup associated to the process. Throughout the paper, we will use the following notations. Elements of R d + will be denoted by x = ( x , . . . , x d ), and those of Z d + by n = ( n , . . . , n d ). For x ∈ R d + , we will denote by (cid:107) x (cid:107) its Euclidean norm, by | x | its (cid:96) -norm, and by d ( x, y ) = (cid:107) x − y (cid:107) the Euclidean distance between x and y . The scalar productin R d is denoted by (cid:104)· , ·(cid:105) . Given x ∈ R d + and r >
0, the Euclidean ball of radius r and center x is denoted by B ( x, r ). Since we want the process to stay in the positive orthant, we naturally assumethe normal component of D of R d + is zero on the boundary. We make thefollowing hypotheses on the vector fields B , D and B − D . • The vector fields B and D are locally Lipschitz functions on R d + , and B j ( x ) ≥ , D j ( x ) ≥ , ∀ j ∈ { , . . . , d } , ∀ x ∈ R d + . (H0) • The vector fields B and D vanish only at the origin: B ( x ) = 0 ⇐⇒ D ( x ) = 0 ⇐⇒ x = 0 . (H1)The fixed point 0 of the vector field B − D is linearly unstable. • There exists x ∗ ∈ int( R d + ) such that B ( x ∗ ) − D ( x ∗ ) = 0 . (H2) • There exist β >
R > L > (cid:107) x ∗ (cid:107) < R and for all x ∈ R d + such that (cid:107) x (cid:107) < R (cid:104) B ( x ) − D ( x ) , x − x ∗ (cid:105) ≤ − β (cid:107) x (cid:107)(cid:107) x − x ∗ (cid:107) . (H3)(ii) (cid:80) dj =1 x ∗ j < L and B (cid:18) x ∗ ,
12 min ≤ j ≤ d x ∗ j (cid:19) ⊂ (cid:8) y ∈ R d + : | y | ≤ L (cid:9) ⊂ B (0 , R ) . (H4)We will denote by P L the hyperplane defined by d (cid:88) j =1 x j = L. (2.1)We refer to Figure 1 to help the reader visualizing how the different do-mains defined in Hypotheses (H3) and (H4) are organized.5 Moreover we assume that L is such thatsup s>L B max ( s ) D min ( s ) <
12 (H5)where D min ( s ) = inf | x | = s d (cid:88) j =1 D j ( x ) and B max ( s ) = sup | x | = s d (cid:88) j =1 B j ( x ) . (2.2) • We assume that D min is an eventually monotone function such that (cid:90) ∞ d sD min ( s ) < + ∞ . (H6) • There exists ξ > x ∈ R d + inf ≤ j ≤ d D j ( x )sup ≤ (cid:96) ≤ d x (cid:96) > ξ > . (H7) • Finally, we assume that inf ≤ j ≤ d ∂ x j B j (0) > . (H8)(By ∂ x j we mean ∂∂ xj .)We now comment on the different hypotheses. Notice that, because of theLipschitz property of the vector field, the polynomial on the right-hand sidein (H3) is natural locally around 0 and x ∗ . Hypothesis (H3) implies that thefixed point of B − D is unique in R d + ∩ B (0 , R ) \{ } . Any trajectory starting in R d + ∩ B (0 , R ) \{ } converges to x ∗ . The fixed point 0 is unstable. In particular,this implies that the faces of R d + are not globally invariant by the flow. Noticealso that Hypothesis (H5) implies that there is no fixed point in R d + \B (0 , R ).This hypothesis means that for large populations the death rates dominate thebirth rates, this will be used together with Hypothesis (H6) to prove that theprocess “comes down from infinity”.We will see that Hypothesis (H7) implies that the jump rate of the process isbounded below away from zero.Hypothesis (H8) guarantees that the birth rate of the stochastic process is notidentically 0 near the origin.Finally, notice that Hypotheses (H2), (H3) (i), (H8) are open conditions in the C -topology of vector fields. Colloquially, this means that if we slightly perturbthe vector field, these hypotheses remain valid with slightly modified constants. We define S ( x ) = (cid:80) dj =1 x j and for every j ∈ { , . . . , d } B j = λS , D j = x j ( µ + κS )6 L x ∗ x x R inf j x ∗ j / Figure 1: Illustration of Hypotheses (H3) and (H4)where λ > µ/d > κ >
0. The non trivial fixed point x ∗ is given by x ∗ j = S ∗ /d where S ∗ = ( λd − µ ) /κ . We have (cid:104) x − x ∗ , B − D (cid:105) = λS ( S − S ∗ ) − ( µ + κS ) (cid:16) (cid:107) x − x ∗ (cid:107) + ( S − S ∗ ) S ∗ d (cid:17) = − κd S ( S − S ∗ ) − ( µ + κS ) (cid:18) (cid:107) x − x ∗ (cid:107) − ( S − S ∗ ) d (cid:19) . It is now convenient to use the decomposition x = Sd y where 1 is the vector with all components equal to 1, and y is orthogonal to 1.We obtain (since x ∗ = S ∗ /d ) (cid:104) x − x ∗ , B − D (cid:105) = − κd S ( S − S ∗ ) − ( µ + κS ) (cid:107) y (cid:107) . For x in the positive quadrant we have (cid:107) x (cid:107) ≤ S , hence (cid:107) y (cid:107) ≤ S.
7t is easy to verify that there exists a constant Γ > S ≥ (cid:107) y (cid:107) ≤ S (cid:107) x (cid:107)(cid:107) x − x ∗ (cid:107) = (cid:114) (cid:107) y (cid:107) + S d (cid:18) ( S − S ∗ ) d + (cid:107) y (cid:107) (cid:19) ≤ Γ (cid:16) κd S ( S − S ∗ ) + ( µ + κS ) (cid:107) y (cid:107) (cid:17) which implies Hypothesis (H3) (i) with β = 1 / Γ. Checking the other hypothesesis left to the reader.Notice that one can construct many more examples by perturbating (in the C sense) this example. We consider a birth-and-death process ( N K ( t ) , t ≥
0) on the d -dimensionalinteger lattice Z d + . So, for each t ≥ N K ( t ) is a vector with d components,that is, N K ( t ) = (cid:0) ( N K ) ( t ) , . . . , ( N K ) d ( t ) (cid:1) . The birth and death rates of thisprocess are given by KB j (cid:0) nK (cid:1) and KD j (cid:0) nK (cid:1) , j = 1 , . . . , d . Given f : Z d + → R with finite support, the generator of the process is given by( L K f ) ( n ) = (2.3) K d (cid:88) j =1 (cid:104) B j (cid:16) nK (cid:17) (cid:0) f ( n + e ( j ) ) − f ( n ) (cid:1) + D j (cid:16) nK (cid:17) (cid:0) f ( n − e ( j ) ) − f ( n ) (cid:1)(cid:105) , where e ( j ) = (0 , . . . , , , , . . . , j -th position. Proposition 2.1.
For each
K > , the process ( N K ( t ) , t ≥ goes to withprobability one.Proof. For a fixed K , the process (cid:0) (cid:80) dj =1 (cid:104) N K ( t ) , e ( j ) (cid:105) , t ≥ (cid:1) can be stochasti-cally dominated by a monotype birth-and-death process with birth rate KB max ( m )and death rate KD min ( m ) with m ∈ Z + (see (2.2)). Hypotheses (H5) and (H7)imply that the process ( N K ( t ) , t ≥
0) goes almost surely to 0 (see [15, Theorem5.5.5]).Under mild assumptions, one-parameter families of pure jump Markov pro-cesses can be approximated, in every finite time interval, by the solutions of adifferential equation whose vector field is determined by the infinitesimal transi-tion rates. This is referred to as Kurtz’s theorem. In our framework, this resulttakes the following form.
Proposition 2.2 ([13, 14]) . Let E ⊆ R d + be an open bounded subset of R d + . Fixa bounded time interval (cid:2) , t (cid:3) with t > . Let x ∈ E be such that the trajectoryof the solution x ( t ) of the differential equation d x d t = B ( x ) − D ( x ) (2.4) with initial condition x belongs to E for all t ∈ [0 , t ] . If lim K → + ∞ N K (0) K = x hen, for every ε > , lim K → + ∞ P (cid:32) sup t ≤ t (cid:12)(cid:12)(cid:12)(cid:12) N K ( t ) K − x ( t ) (cid:12)(cid:12)(cid:12)(cid:12) > ε (cid:33) = 0 . According to Propositions 2.1 and 2.2, we thus have the following picture.On the one hand, for K fixed, the (total) population dies out with probabilityone in the limit t → + ∞ . On the other hand, for a fixed finite time span, thenumber of individuals in the population, when rescaled by K , is very close to thesolution of the differential equation (2.4) in the limit K → + ∞ . The purposeof the present work is to describe the process for finite times and for finite K . The hypotheses of Section 2 are in force in the following four theorems.We will use the following notations throughout the article.
Notation.
The first entrance time of the process ( N K ( t ) , t ≥ in a subset A of Z d + is defined by T A = inf { t > N K ( t ) ∈ A } . When A is a singleton, say { n } , we shall simply write T n . As usual, P n will denote the law of the process given that N K (0) = n , and,for a probability measure µ on Z d + and a subset A of Z d + , P µ ( A ) = (cid:88) m ∈ Z d + µ ( m ) P m ( A ) . Our first main result is about quantifying the closeness, in total variationdistance, of the process condioned to not being extinct before time t , and thequasi-stationary distribution. Recall that the total variation distance betweentwo probability measures µ and ν on Z d + is (cid:107) µ − ν (cid:107) TV = sup A ∈ P ( Z d + ) | µ ( A ) − ν ( A ) | where P ( Z d + ) is the powerset of Z d + . Theorem 3.1.
There exist K > , < c < and < a < b < + ∞ such thatthe following result holds. For all K ≥ K , there exist t ( K ) ∈ ( a log K, b log K ) and a unique probability measure m K on Z d + \{ } such that for every probabilitymeasure µ on Z d + \{ } , and for all t ≥ , we have (cid:107) P µ (cid:0) N K ( t ) ∈ · | t < T (cid:1) − m K ( · ) (cid:107) TV ≤ − c ) (cid:98) t/t ( K ) (cid:99) . This theorem tells us that for t (cid:29) log K , the process condioned to not beingextinct before time t is very close to the quasi-stationary distribution m K . As t tends to + ∞ , we get a convergence of the process conditioned to non-extinctiontowards the quasi-stationary distribution.By a general result on quasi-stationary distributions (see for instance [7]),one has P m K (cid:0) T > t (cid:1) = e − λ ( K ) t , t ≥ , (3.1)9here λ ( K ) is a positive real number called the exponential rate of extinction.In particular, the mean time to extinction, starting from the quasi-stationarydistribution is E m K [ T ] = 1 λ ( K ) . (3.2)The following theorem shows that the exponential rate of extinction is ex-ponentially small in K . Theorem 3.2.
There exists K > and two numbers d > d > such thatfor all K > K e − d K ≤ λ ( K ) ≤ e − d K . (3.3)Hence we get an estimate of the mean time to extinction (3.2):e d K ≤ E m K [ T ] ≤ e d K for all K > K . When d = 1, a more precise estimate was proved in our previouswork [6, Theorem 3.2]. Remark 3.1.
The upper bound in (3.3) could be obtained by a large deviationasymptics for jump processes (see [4, Section 4.2]). Theorem 3.2 also provides alower bound. In the present paper we are interested, among other things, in thedifferent time scales for large K and not so much in their precise asymptotics. The following theorem provides a quantitative bound for the distance (intotal variation) between the law of the process and a convex combination of thequasi-stationary distribution and the Dirac measure at the origin.
Theorem 3.3.
Let c and t ( K ) be as in Theorem 3.1. There exist positiveconstants C (3 . , c (3 . , η (3 . , K , such that for all t ≥ and all K > K , foreach n ∈ Z d + \{ } , there exists a number p K ( n ) ∈ ( c, such that sup n ∈ Z d + \{ } (cid:13)(cid:13)(cid:13) P n ( N K ( t ) ∈ · ) − e − λ ( K ) t p K ( n ) m K ( · ) − (cid:0) − e − λ ( K ) t p K ( n ) (cid:1) δ ( · ) (cid:13)(cid:13)(cid:13) TV ≤ − η (3 . K e − λ ( K ) t + C (3 . e − ω ( K ) t (3.4) where ω ( K ) = − log(1 − c ) t ( K ) ≥ c (3 . log K .
Remark 3.2.
Let us give the meaning of inequality (3.4) in two differentregimes corresponding to two different time-scales. We assume that K is largeenough to have e − η (3 . K (cid:28) . First notice that the right-hand side of (3.4) is (cid:28) provided that t (cid:29) log K . Then, for log K (cid:28) t (cid:28) /λ ( K ) , (3.4) impliesthat sup n ∈ Z d + \{ } (cid:13)(cid:13) P n ( N K ( t ) ∈ · ) − p K ( n ) m K ( · ) − (cid:0) − p K ( n ) (cid:1) δ ( · ) (cid:13)(cid:13) TV ≤ − η (3 . K e − λ ( K ) t + C (3 . e − ω ( K ) t +2(1 − e − λ ( K ) t ) (cid:28) . This means that, in that time span, the law of the process is close to a mixtureof the Dirac measure at the origin and the quasi-stationary distribution withrespective weights − p K ( n ) and p K ( n ) . For t (cid:29) /λ ( K ) , (3.4) implies thatthe law of the process is close to the Dirac measure at the origin. P K t , t ≥
0) be the semigroup of the birth and death process killed at 0.More precisely P K t f ( n ) = E n (cid:104) f ( N K ( t )) { t The resolvent of ( P K t , t ≥ in the Banach space (cid:96) ∞ ( Z d + \{ } ) is meromorphic in the set (cid:60) z > − ω ( K ) with a unique simple pole at − λ ( K ) with residue the one dimensional projection π K given by π K ( f ) = u K m K ( f ) . The sequence (cid:0) u K ( n ) (cid:1) n ∈ Z d + \{ } is such that m K ( u K ) = 1 , and, for all t ≥ , P K t u K = e − λ ( K ) t u K . Moreover, for all n ∈ Z d + \{ } , c ≤ u K ( n ) ≤ −O (1) K , where c is defined in Theorem 3.1. In particular, the spectral gap ω ( K ) − λ ( K ) is bounded below by c (3 . log K − e − d K . Remark 3.3. We will see in the proofs that the weights p K ( n ) of Theorem 3.3are equal to u K ( n ) ∧ . We first introduce the natural quantity n ∗ = (cid:98) Kx ∗ (cid:99) wich will appear throughout the article. Let ϕ : Z d + → R + defined by ϕ ( n ) = e αK (cid:107) n − n ∗ (cid:107) (4.1)where α > α small enough, the function ϕ is a Lyapunovfunction. Theorem 4.1. There exist < α < / , K > and C (4 . > such that forall K ≥ K and for all n ∈ B (0 , RK ) , we have L K ϕ ( n ) ≤ (cid:18) − αβ (cid:107) n (cid:107) K (cid:107) n − n ∗ (cid:107) K + C (4 . (cid:19) ϕ ( n ) where β and R are defined in (H3) . roof. We use the elementary fact that for all x ∈ R such that | x | ≤ R thereexists c ( R ) > ≤ e x − − x ≤ C ( R ) x . Then, for all n ∈ B (0 , RK ) we get L K ϕ ( n ) ϕ ( n )= K d (cid:88) j =1 (cid:20) B j (cid:16) nK (cid:17) (cid:18) ϕ ( n + e ( j ) ) ϕ ( n ) − (cid:19) + D j (cid:16) nK (cid:17) (cid:18) ϕ ( n − e ( j ) ) ϕ ( n ) − (cid:19)(cid:21) = K d (cid:88) j =1 (cid:104) B j (cid:16) nK (cid:17) (cid:16) exp (cid:16) αK (2( n j − n ∗ j ) + 1) (cid:17) − (cid:17) + D j (cid:16) nK (cid:17) (cid:16) exp (cid:16) αK ( − n j − n ∗ j ) + 1) (cid:17) − (cid:17)(cid:105) = K d (cid:88) j =1 (cid:20) α (cid:16) B j (cid:16) nK (cid:17) − D j (cid:16) nK (cid:17)(cid:17) (cid:18) n j − n ∗ j K (cid:19) + (cid:16) B j (cid:16) nK (cid:17) + D j (cid:16) nK (cid:17)(cid:17) C ( R ) α (cid:107) n − n ∗ (cid:107) K + (cid:16) B j (cid:16) nK (cid:17) + D j (cid:16) nK (cid:17)(cid:17) (cid:18) αK + 2 C ( R )( n j − n ∗ j ) K + C ( R ) α K (cid:19) (cid:21) . Using (H0) and (H1), there exists C ( R ) > ≤ B j (cid:16) nK (cid:17) + D j (cid:16) nK (cid:17) ≤ C ( R ) (cid:107) n (cid:107) K for all n ∈ B (0 , RK ). It is easy to verify that the third term in the squarebracket is bounded in absolute value by a constant independent of K provided K is larger than some K > 0. The second term in the square bracket is boundedby 4 dC ( R ) α (cid:107) n (cid:107) K (cid:107) n − n ∗ (cid:107) K . (4.2)We finally deal with the first term in the square bracket. Writing F = B − D for brevity, we obtain by (H3) that2 αK d (cid:88) j =1 (cid:16) B j (cid:16) nK (cid:17) − D j (cid:16) nK (cid:17) (cid:17) (cid:18) n j − n ∗ j K (cid:19) = 2 αK (cid:28) F (cid:16) nK (cid:17) , n − n ∗ K (cid:29) = 2 αK (cid:68) F (cid:16) nK (cid:17) , (cid:16) nK − x ∗ (cid:17)(cid:69) + 2 αK (cid:28) F (cid:16) nK (cid:17) , (cid:18) x ∗ − n ∗ K (cid:19)(cid:29) ≤ − α βK (cid:107) n (cid:107) K (cid:13)(cid:13)(cid:13)(cid:13) nK − n ∗ K (cid:13)(cid:13)(cid:13)(cid:13) + O (1) ≤ − αβ (cid:107) n (cid:107) K (cid:107) n − n ∗ (cid:107) K + O (1) , (cid:13)(cid:13)(cid:13) x ∗ − n ∗ K (cid:13)(cid:13)(cid:13) ≤ K and F (cid:0) nK (cid:1) is bounded on B (0 , R ), andwhere O (1) is a quantity uniformly bounded in K . To finish the proof, wechoose α small enough in such a way that the prefactor 4 dC ( R ) α in (4.2) isless than half of 2 αβ . Corollary 4.2. There exist K > and two constants ρ (4 . > and c (4 . > such that, for all K ≥ K and for all c (4 . ≤ (cid:107) n (cid:107) ≤ RK satisfying (cid:107) n − n ∗ (cid:107) ≥ ρ (4 . √ K we have L K ϕ ( n ) ≤ − αβ (cid:107) n (cid:107) K (cid:107) n − n ∗ (cid:107) K ϕ ( n ) . Proof. We choose c (4 . and ρ (4 . large enough such that for n as in the state-ment, αβ (cid:107) n (cid:107) K (cid:107) n − n ∗ (cid:107) K > C (4 . . Remark 4.1. The intuitive rate of decrease αβ (cid:107) n (cid:107) K (cid:107) n − n ∗ (cid:107) K of the Lyapunov function, given by Corollary 4.2, is uniformly bounded below bythe constant C (4 . , if c (4 . ≤ (cid:107) n (cid:107) ≤ RK and (cid:107) n − n ∗ (cid:107) ≥ ρ (4 . √ K . However,if (cid:107) n (cid:107) and (cid:107) n − n ∗ (cid:107) are of order K , this rate is also of order K . We will latertake advantage of this non uniformity of the rate by a suitable decomposition ofthe set Z d + \{ } ∩ {(cid:107) n (cid:107) ≥ c (4 . } . In this section, we formulate a lemma and a corollary of it which will help usto take advantage of the decomposition of the space Z d + . We could formulateit in a much more abstract setting. Since K plays no role here, we drop the K dependence, hence N ( t ) stands for N K ( t ), L for L K , etc. Lemma 4.3. Let D − , D − , D , D be subsets of Z d + \{ } such that D (cid:32) D (cid:32) D − (cid:32) D − (cid:32) Z d + \{ } , with D − a compact subset. Next, let H − = D − \D − , H − = D − \D , H = D \D . (See Figure 2.) Assume that for all n ∈ H we have P n (cid:0) T H − < ∞ (cid:1) = 1 , and H − ∩ D = ∅ and { n : d ( n, H ∪ H − ) = 1 } ⊂ D ∪ H − . Assume that there exists a positive function ψ defined in Z d + \{ } such that Λ := − sup H − ∪ H − ∪ H L ψ ( n ) ψ ( n ) > . H H − H − Figure 2: The four domains Let a = sup n ∈H ψ ( n ) , a (cid:48)(cid:48)− = inf n ∈H − ψ ( n ) and a (cid:48)− = inf n ∈H − ∪ H ψ ( n ) . Assume that a /a (cid:48)(cid:48)− < . Then inf n ∈H P n (cid:0) T D ≤ t , T H − > T D ) ≥ − a a (cid:48)(cid:48)− − a a (cid:48)− e − Λ t . Note that a /a (cid:48)− ≥ 1. In practice we will use for H − some kind of outerboundary of D − . Proof. Using Dynkin’s formula, we have for a path issued from n ∈ H e Λ ( t ∧ T D ∧ T H− ) ψ (cid:0) N ( t ∧ T D ∧ T H − ) (cid:1) = (cid:90) t ∧ T D ∧ T H− e Λ s (cid:0) Λ ψ ( N ( s )) + L ψ ( N ( s )) (cid:1) d s + M ( t ∧ T D ∧ T H − )where (cid:0) M ( t ∧ T D ∧ T H − ) (cid:1) t ≥ is a martingale. Using the assumptions and thefact that ψ is bounded by a on H we obtain E n (cid:104) e Λ ( t ∧ T D ∧ T H− ) ψ (cid:0) N ( t ∧ T D ∧ T H − ) (cid:1)(cid:105) ≤ ψ (cid:0) n (cid:1) ≤ a . (4.3)Since ψ is positive we deduce that a ≥ E n (cid:2) ψ (cid:0) N ( t ∧ T D ∧ T H − ) (cid:1)(cid:3) ≥ E n (cid:104) ψ (cid:0) N ( t ∧ T D ∧ T H − ) (cid:1) { T D ≥ T H− } { T H− ≤ t } (cid:105) = E n (cid:104) ψ (cid:0) N ( T H − ) (cid:1) { T D ≥ T H− } { T H− ≤ t } (cid:105) ≥ a (cid:48)(cid:48)− P n (cid:0) T D ≥ T H − , T H − ≤ t (cid:1) . t tend to infinity and using our hypothesis (and Lebesgue’s dominatedconvergence theorem) we get that for all n ∈ H P n (cid:0) T H − ≤ T D (cid:1) ≤ a a (cid:48)(cid:48)− . Using again (4.3) we also have that for all n ∈ H E n (cid:104) e Λ t ψ (cid:0) N ( t ) (cid:1) { T H− >T D >t } (cid:105) ≤ a , which implies that for all t ≥ P n (cid:0) T H − > T D > t (cid:1) ≤ a a (cid:48)− e − Λ t . We have for all n ∈ H P n (cid:0) T D ≤ t , T H − > T D ) = P n (cid:0) T H − > T D ) − P n (cid:0) T D > t , T H − > T D )= 1 − P n (cid:0) T H − ≤ T D ) − P n (cid:0) T H − > T D > t ) . The lemma follows from the above estimates. Corollary 4.4. Under the assumptions of Lemma 4.3 we have inf n ∈H P n (cid:0) T D ≤ t D , T H − > T D ) ≥ − η D with t D = 1Λ log (cid:18) a (cid:48)(cid:48)− a (cid:48)− (cid:19) and η D = 2 a a (cid:48)(cid:48)− . The estimate also holds with η D = 12 + a a (cid:48)(cid:48)− and t D = − 1Λ log (cid:18) a (cid:48)− a (cid:18) − a a (cid:48)(cid:48)− (cid:19)(cid:19) . (A1) and (A2) Our proof relies on a general theorem proved in [5]. We formulate it in our set-ting. Let ( N K ( t ) , t ≥ 0) be the birth-and-death process defined above. Supposethere exists a probability measure ν on E such that • There exist t , c > P n (cid:0) N K ( t ) ∈ · | t < T (cid:1) ≥ c ν ( · ) , ∀ n ∈ Z d + \{ } . (A1) • There exists c > P ν ( t < T ) ≥ c P n ( t < T ) , ∀ n ∈ Z d + \{ } , ∀ t ≥ . (A2)15hen there exists a unique quasi-stationary distribution m K such that for everyinitial distribution µ , (cid:107) P µ ( N K ( t ) ∈ · | t < T ) − m K ( · ) (cid:107) TV ≤ − c c ) t/t . We shall take ν as the uniform probability measure supported on a ball cen-tered at n ∗ with radius of order √ K . We shall also prove that c and c areindependent of K , and that t is of order log K . (A1) Let ∆ = B (cid:0) n ∗ , ρ (4 . √ K (cid:1) , (5.1)where B ( n, r ) denotes the ball centered in n with radius r and ρ (4 . the constantintroduced in Corollary 4.2. Since n ∗ is of order K , the set ∆ is included in theinterior of Z d + for K large enough. Notation. We shall denote by ν the uniform probability measure supported on ∆ . This discrete measure thus gives each point of ∆ a mass proportional to K − d/ .The proof of Condition (A1) relies on the following three lemmas whoseproofs are given later on.The first lemma shows that the descent (from infinity) into the set ∆ happenswith a time scale of at most log K . Lemma 5.1. There exist C (5 . > and η (5 . < such that for all K largeenough inf n ∈ ∆ c P n (cid:0) T ∆ < C (5 . log K (cid:1) ≥ − η (5 . . The second lemma shows that on a time span of order log K , the processstarting in ∆ stays near ∆, more precisely in a ball with a radius of order √ K centered at n ∗ . Lemma 5.2. There exists C (5 . > ρ (4 . and η (5 . < such that for all K large enough inf n ∈ ∆ inf ≤ t ≤ C (5 . log K +1 P n (cid:0) N K ( t ) ∈ ∆ (cid:48) (cid:1) ≥ − η (5 . where ∆ (cid:48) = B (cid:0) n ∗ , C (5 . √ K (cid:1) ⊃ ∆ . The third lemma says that the probability measure ν is a significant compo-nent of the distribution of the process at time 1 starting near ∆. This lemmadoes not seem to be available in the literature. The main difference with existingresults (see for instance [8]) is that our generator is not symmetric. Lemma 5.3. There exists η (5 . < such that for all K large enough and all A ⊂ ∆ inf n ∈ ∆ (cid:48) P n (cid:0) N K (1) ∈ A (cid:1) ≥ (1 − η (5 . ) ν ( A ) , where ∆ (cid:48) is defined in Lemma 5.2. roof of Condition (A1). Applying the three preceding lemmas, we can provethat condition (A1) holds for K large enough with c = (1 − η (5 . ) (1 − η (5 . ) (1 − η (5 . ) < , (5.2) t = t ( K ) = 1 + C (5 . log K.t ( K ) (5.3)Indeed, for all n ∈ Z d + \{ } and for all A ⊂ ∆ we can write P n (cid:0) N K ( t ) ∈ A (cid:1) = E n (cid:2) A (cid:0) N K ( t ) (cid:1)(cid:3) ≥ E n (cid:104) { T ∆ The proof of Lemma 5.1 is based on the fine description of the trajectories ofthe process. For this purpose, we need to introduce a decomposition of Z d + \{ } according to the different time scales at which the process goes down frominfinity to ∆.Let R ∗ = 12 (cid:16) R + sup y ∈ P L ∩ R d + (cid:107) y − x ∗ (cid:107) (cid:17) , where P L is the hyperplane defined in (2.1).17ote that R ∗ < R by hypothesis (H4). We define the sets E = (cid:26) n ∈ Z d + \{ } : d (cid:88) j =1 n j > LK (cid:27) H − = (cid:8) n ∈ Z d + \{ } : R ∗ K ≤ (cid:107) n − n ∗ (cid:107) < RK (cid:9) H − = (cid:26) n ∈ Z d + \{ } : d (cid:88) j =1 n j > LK , (cid:107) n − n ∗ (cid:107) < R ∗ K (cid:27) H − = (cid:26) n ∈ Z d + \{ } : d (cid:88) j =1 n j ≤ LK , (cid:107) n − n ∗ (cid:107) ≥ (cid:107) n ∗ (cid:107) − c (4 . (cid:27) H − = (cid:8) n ∈ Z d + \{ } : (cid:107) n ∗ (cid:107) − ( c (4 . + 4) ≤ (cid:107) n − n ∗ (cid:107) < (cid:107) n ∗ (cid:107) − c (4 . (cid:9) H − = (cid:8) n ∈ Z d + \{ } : (cid:107) n ∗ (cid:107) − ( c (4 . + 8) ≤ (cid:107) n − n ∗ (cid:107) < (cid:107) n ∗ (cid:107) − ( c (4 . + 4) (cid:9) H = (cid:8) n ∈ Z d + \{ } : (cid:107) n ∗ (cid:107) − ( c (4 . + 12) ≤ (cid:107) n − n ∗ (cid:107) < (cid:107) n ∗ (cid:107) − ( c (4 . + 8) (cid:9) E = (cid:8) n ∈ Z d + \{ } : (cid:107) n (cid:107) < c (4 . + 17 (cid:9) . These sets are well-defined provided that K is large enough.The proof of Lemma 5.1 will result from a series of sublemmas which quantifythe probability of coming down from infinity and crossing the various level setsof the Lyapunov function. Sublemma 5.4. There exist two constants t (5 . > and η (5 . < (indepen-dent of K ) such that for K large enough inf n ∈E P n (cid:0) T E c ≤ t (5 . (cid:1) ≥ − η (5 . . Proof. The process (cid:0) (cid:80) dj =1 (cid:104) N K ( t ) , e ( j ) (cid:105) , t ≥ (cid:1) can be coupled with a one-dimensional birth-and-death process ( Z ( t ) , t ≥ 0) with birth rate Λ( m ) = KB max (cid:0) mK (cid:1) and death rate M ( m ) = KD min (cid:0) mK (cid:1) . The coupling is such that Z ( t ) ≥ d (cid:88) j =1 (cid:104) N K ( t ) , e ( j ) (cid:105) if Z (0) ≥ d (cid:88) j =1 (cid:104) N K (0) , e ( j ) (cid:105) . Let us introduce p K = (cid:98) LK (cid:99) and denote by (cid:98) T p K its hitting time. We are goingto prove that A K := sup p>p K E p ( (cid:98) T p K ) is bounded uniformly in K . As shown in[18, p.384] or in [1, Chap.3], one has A K = ∞ (cid:88) m = p K +1 (cid:32) M ( m ) + ∞ (cid:88) i = m +1 Λ( m ) · · · Λ( i − M ( m ) · · · M ( i ) (cid:33) . By assumption (H5), for q ≥ p K , Λ( q ) /M ( q ) ≤ / 2. Then A K ≤ ∞ (cid:88) m = p K +1 (cid:32) M ( m ) + ∞ (cid:88) i = m +1 m − i M ( i ) (cid:33) ≤ ∞ (cid:88) m = p K +1 M ( m ) , where we have interchanged the order of the sums to get the second inequality.By Hypothesis (H6), we know that1 K ∞ (cid:88) m = p K +1 D min ( mK ) −−−−→ K →∞ (cid:90) ∞ L d sD min ( s ) < + ∞ . n ∗ n n E H H − H − H − H − H − P L Figure 3: The various subsets when d = 2 when K is large enough.Then there exists K such that for all K ≥ K , for all p ≥ p K , we have E p ( (cid:98) T p K ) ≤ (cid:90) ∞ L d sD min ( s ) . The result follows by Markov inequality with t (5 . = 6 (cid:90) ∞ L d sD min ( s ) and η (5 . = 12 . Sublemma 5.5. There exist two constants t (5 . > and η (5 . < (indepen-dent of K ) such that for K large enough inf n ∈ H − ∪ H − ∪ H − P n (cid:0) T H ≤ t (5 . (cid:1) ≥ − η (5 . . roof. We define D − = (cid:8) n ∈ Z d + \{ } : (cid:107) n − n ∗ (cid:107) < RK (cid:9) D − = (cid:8) n ∈ Z d + \{ } : (cid:107) n − n ∗ (cid:107) < R ∗ K (cid:9) D = n ∈ Z d + \{ } : d (cid:88) j =1 n j ≤ LK D = (cid:8) n ∈ Z d + \{ } : (cid:107) n − n ∗ (cid:107) < (cid:107) n ∗ (cid:107) − ( c (4 . + 8) (cid:9) . (5.5)We now apply Corollary 4.4 with D i = D i , i = − , − , , 1. For K largeenough and using (H4), the Lyapunov function ϕ defined in Theorem 4.1 andthe geometry of the sets, we have a (cid:48)(cid:48)− a (cid:48)− ≤ e O (1) K , a a (cid:48)(cid:48)− < . Moreover we have Λ = O (1) K by Theorem 4.1. The result follows since H − ∪ H − ∪ H − = D \ D andsince for K large enough, D can be reached from D \ D only through H .We need a specific estimate near 0. Sublemma 5.6. There exists η (5 . < (independent of K ) such that for K large enough inf n ∈E \ D P n (cid:0) T H ≤ (cid:1) ≥ − η (5 . . Proof. For all n ∈ E \ D , for all j ∈ { , . . . , d } , there exists s ≤ 17, such that n + se ( j ) ∈ H . Since n (cid:54) = 0, there exists j with n j > V = (cid:8) m ( t ) , t = 0 , ∃ t < s , . . . , t s < s such that m ( t ) = n + q e ( j ) , ∀ t q ≤ t < t q +1 , ≤ q ≤ s − (cid:9) . Let us compute the probability for the birth and death process to belong to V .Note that by assumption KB j (cid:16) nK (cid:17) = d (cid:88) (cid:96) =1 n (cid:96) ∂ x (cid:96) B j (0) + O (cid:18) K (cid:19) and KD j (cid:16) nK (cid:17) = d (cid:88) (cid:96) =1 n (cid:96) ∂ x (cid:96) D j (0) + O (cid:18) K (cid:19) . Therefore, for K large enough, the birth probability of an individual with type j is bounded below byinf n ∈E KB j ( nK ) K (cid:80) d(cid:96) =1 B (cid:96) ( nK ) + K (cid:80) d(cid:96) =1 D (cid:96) ( nK ) > 12 inf n ∈E ∂ x j B j (0) (cid:80) d(cid:96) =1 n (cid:96) ∂ x (cid:96) B j (0) + (cid:80) d(cid:96) =1 n (cid:96) ∂ x (cid:96) D j (0) = ζ, ζ > ≤ (cid:96) ≤ d n (cid:96) ≤ 17 for n ∈ E . Note also that thedenominator (which is the jump rate) is bounded below by ζ (cid:48) = inf j ∂ x j B j (0) > P n ( N K ∈ V ) ≥ ζ s (cid:16) − e − ζ (cid:48) /s (cid:17) s ≥ ζ (cid:16) − e − ζ (cid:48) / (cid:17) . The results follows.In the following lemma we will partition more finely the disk D to fit aswell as possible the speed of decrease of the distance between the process and n ∗ . Sublemma 5.7. There exists two constants t (5 . > and η (5 . < such thatfor K large enough inf n ∈ D \ ∆ P (cid:0) T ∆ ≤ t (5 . log K (cid:1) ≥ − η (5 . , where ∆ is defined in (5.1) and D in (5.5) .Proof. We start by defining a decreasing (finite) sequence of numbers ( R j ) asfollows: R − = (cid:107) n ∗ (cid:107) − c (4 . , R − = (cid:107) n ∗ (cid:107) − ( c (4 . + 4) ,R = (cid:107) n ∗ (cid:107) − ( c (4 . + 8) , R = (cid:107) n ∗ (cid:107) − ( c (4 . + 12) . Define j ∗ = inf (cid:26) j : R − j − + 1 ≤ 12 inf (cid:96) n ∗ (cid:96) (cid:27) . Note that j ∗ = O (1) log K . For 2 ≤ j ≤ j ∗ we define R j = R − j − + 1 . Note that for 1 ≤ j ≤ j ∗ , R j ≥ R j ∗ = O (1) K . Define j ∗∗ = sup { j > j ∗ : R j ∗ − ( j − j ∗ ) > ρ (4 . √ K } − . Note that j ∗∗ = O (1) log K . For j ∗ ≤ j ≤ j ∗∗ + 1, let R j = R j ∗ − ( j − j ∗ ) . Note that ρ (4 . √ K ≤ R j ∗∗ − ≤ ρ (4 . √ K and that for j ≤ j ∗ , B ( n ∗ , R j ) ⊂B (0 , (cid:107) n ∗ (cid:107) / c . We now define a (finite) decreasing sequence of domains ( D j ),where − ≤ j ≤ j ∗∗ + 1, by D j = B ( n ∗ , R j ) ∩ Z d + \{ } . We also define a finite sequence of annuli ( H j ), where − ≤ j ≤ j ∗∗ , by H j = D j \ D j +1 . Recall that the Lyapunov function ϕ has been defined in Theorem 4.1. Wedefine the following sequences of positive numbers:( A j ) − ≤ j ≤ j ∗∗ by A j = sup n ∈ H j ϕ ( n )21 A (cid:48) j ) − ≤ j ≤ j ∗∗ by A (cid:48) j = inf n ∈ H j ∪ H j − ϕ ( n )( A (cid:48)(cid:48) j ) ≤ j ≤ j ∗∗ by A (cid:48)(cid:48) j = inf n ∈ H j − ϕ ( n )(Λ j ) ≤ j ≤ j ∗∗ by Λ j = − sup n ∈ H j − ∪ H j − ∪ H j L K ϕ ( n ) ϕ ( n ) . ( η j ) ≤ j ≤ j ∗∗ by η j = 2 A j A (cid:48)(cid:48) j (5.6)( t j ) ≤ j ≤ j ∗∗ by t j = 1Λ j log A (cid:48)(cid:48) j A (cid:48) j . (5.7)It is left to the reader to check that there exists a constant c > 1, independentof j and K , such that c − exp (cid:32) αR j K (cid:33) ≤ A j ≤ c exp (cid:32) αR j K (cid:33) (5.8) c − exp (cid:32) αR j +1 K (cid:33) ≤ A (cid:48) j ≤ c exp (cid:32) αR j +1 K (cid:33) (5.9) c − exp (cid:32) αR j − K (cid:33) ≤ A (cid:48)(cid:48) j ≤ c exp (cid:32) αR j − K (cid:33) . (5.10)If j ≤ j ∗ , we have by Corollary 4.2Λ j ≥ c − ( (cid:107) n ∗ (cid:107) − R j − ) (5.11)and if j ∗ < j ≤ j ∗∗ we have by Theorem 4.1Λ j ≥ R j +1 cK . (5.12)Let us introduce t s = j ∗∗ (cid:88) j =0 t j , with t j > . Using the Markov property and the monotonicity of P n (cid:0) T ∆ ≤ t (cid:1) as a functionof t , we have, for all 0 ≤ (cid:96) ≤ j ∗∗ and for n ∈ D (cid:96) \ D (cid:96) +1 , P n T ∆ ≤ j ∗∗ (cid:88) j = (cid:96) t j ≥ P n T ∆ ≤ j ∗∗ (cid:88) j = (cid:96) t j , T D (cid:96) +1 ≤ t (cid:96) = E n { T D(cid:96) +1 ≤ t (cid:96) } P N K ( T D(cid:96) +1 ) T ∆ ≤ j ∗∗ (cid:88) j = (cid:96) +1 t j + t (cid:96) − T D (cid:96) +1 P n (cid:0) T D (cid:96) +1 ≤ t (cid:96) (cid:1) inf n ∈ D (cid:96) +1 \ D (cid:96) +2 P n T ∆ ≤ j ∗∗ (cid:88) j = (cid:96) +1 t j . Using this estimate recursively together with D j ∗∗ ⊂ ∆ we obtain for all n ∈ D (cid:96) \ D (cid:96) +1 P n T ∆ ≤ j ∗∗ (cid:88) j = (cid:96) t j ≥ j ∗∗ (cid:89) j = (cid:96) inf n ∈ D j \ D j +1 P n (cid:0) T D j +1 ≤ t j (cid:1) . Therefore, from the monotonicity of t (cid:55)→ P n (cid:0) T ∆ ≤ t (cid:1) we have for all n ∈ D P n (cid:0) T ∆ ≤ t s (cid:1) ≥ j ∗∗ (cid:89) j =0 inf n ∈ D j \ D j +1 P n (cid:0) T D j +1 ≤ t j (cid:1) . We now derive a lower bound for each term in the product and an upper boundfor each t j , hence for t s .By elementary computations using the explicit form for R j , (5.7),(5.8), (5.9)(5.10), (5.11) and (5.12), we obtain that for j = 0 to j ∗∗ + 1, t j is of order 1.Therefore j ∗∗ +1 (cid:88) j =0 t j = O (log K ) . One can also check by considering (5.6) that j ∗∗ +1 (cid:88) j =0 η j = O (1) . The result follows by applying Corollary 4.4.We can now prove Lemma 5.1. We give the proof for n ∈ E , the other casesare similar and left to the reader. Using Sublemmas 5.4, 5.5, 5.6, 5.7 and theMarkov property we have, for all K large enough and all n ∈ E , P n (cid:0) T ∆ ≤ ( t (5 . + t (5 . + t (5 . ) log K (cid:1) ≥ η (5 . η (5 . η (5 . . The result follows. Sublemma 5.8. Let D = B (cid:16) n ∗ , 12 inf j n ∗ j (cid:17) ∩ Z d + \{ } (5.13) and define ˜ ρ = ˜ ρ ( K ) = K min( n ∗ (cid:96) )2 .For K large enough and for all n ∈ ∆ and t ≥ , P n (cid:0) t > T D c (cid:1) ≤ ( O (1) + O (1) t ) e − α ˜ ρ K . roof. Let ϕ defined in (4.1) and t > 0. We apply Dynkin’s Theorem to ϕ ( N K ( T D c ∧ t )) (in the spirit of the proof of Lemma 4.3). Using Theorem4.1, we obtain for K large enoughe α ˜ ρ K P n ( T D c < t ) ≤ O (1) + O (1) t and the result follows. Proof of Lemma 5.2. We will in fact prove a stronger result with t ≤ K whichwill imply the result if K is large enough.Let us define the ball (cid:101) B = (cid:101) B ( n ∗ , ρ K ). Let us consider the function ψ ( n ) = (cid:107) n − n ∗ (cid:107) { n ∈ (cid:101) B} . Assuming n, n + e ( j ) , n − e ( j ) ∈ (cid:101) B and using (2.3), (H0), (H1) and (H3), weobtain L K ψ ( n )= K d (cid:88) j =1 (cid:104) (cid:16) B j (cid:16) nK (cid:17) − D j (cid:16) nK (cid:17) (cid:17) ( n j − n ∗ j ) + (cid:16) B j (cid:16) nK (cid:17) + D j (cid:16) nK (cid:17) (cid:17)(cid:105) ≤ − σψ ( n ) + O (1) K + O (1) K (cid:101) B c for all n ∈ (cid:101) B , where σ := β ˜ ρ . From Itˆo’s formula and for t > 0, we have E n (cid:104) e σ ( t ∧ T ˜ B c ) ψ (cid:0) N K ( t ∧ T (cid:101) B c ) (cid:1)(cid:105) ≤ ψ ( N K (0)) + O (1) K E n (cid:20) e σ ( t ∧ T (cid:101) B c ) − σ (cid:21) . Then E n (cid:104) e σt ψ (cid:0) N K ( t ) (cid:1) { T (cid:101) B c >t } (cid:105) ≤ ψ ( n ) + O (1) e σt σ K. On another hand, for ∀ t ≤ K , E n (cid:104) ψ (cid:0) N K ( t ) (cid:1) { T (cid:101) B c 0) defined by P (cid:0) (cid:101) N ( t + d t ) = n + e ( j ) | ˜ N ( t ) = n (cid:1) = K B j ( x ∗ ) d t , P (cid:0) (cid:101) N ( t + d t ) = n − e ( j ) | ˜ N ( t ) = n (cid:1) = K D j ( x ∗ ) d t. The embedded chain will be the symmetric random walk ( N (cid:96) ) (cid:96) ∈ Z + with statespace Z d and transition matrix p ∗ defined by r j = p ∗ (cid:0) n, n + e ( j ) (cid:1) = B j ( x ∗ ) λ ( x ∗ ) = p ∗ (cid:0) n, n − e ( j ) (cid:1) = D j ( x ∗ ) λ ( x ∗ ) . (5.18)To prove Lemma 5.3, we need to obtain a lower bound for P n (cid:0) N K (1) = m (cid:1) with n ∈ ∆ (cid:48) and m ∈ ∆. We have P n (cid:0) N K (1) = m (cid:1) = (cid:88) q (cid:88) γ : γ (0)= n, γ ( q )= m (cid:90) · · · (cid:90) t + ··· + t q − < q − (cid:89) (cid:96) =0 p (cid:0) γ ( (cid:96) ) , γ ( (cid:96) + 1) (cid:1) q − (cid:89) (cid:96) =0 Λ (cid:0) γ ( (cid:96) ) (cid:1) q − (cid:89) (cid:96) =0 e − t (cid:96) Λ( γ ( (cid:96) )) e − Λ( m ) (cid:0) − (cid:80) q − (cid:96) =0 t (cid:96) (cid:1) q − (cid:89) (cid:96) =0 d t (cid:96) . We restrict our attention to the paths whose number of jumps between 0 and1 belongs to [ Λ ∗ − √ K, Λ ∗ + √ K ] (and is then of order K ) and whose valuesbelong to B ( n ∗ , √ K log K ). Moreover, we make a change of law and write akind of Girsanov formula with respect to the law of ˜ N . We obtain P n (cid:0) N K (1) = m (cid:1) ≥ (cid:88) q ∈ [Λ ∗ −√ K, Λ ∗ + √ K ] (cid:88) γ : γ (0)= n, γ ( q )= m sup ≤ (cid:96) ≤ q − | γ ( (cid:96) ) − n ∗ |≤√ K log K (cid:90) · · · (cid:90) t + ··· + t q − < q − (cid:89) (cid:96) =0 p ( γ ( (cid:96) ) , γ ( (cid:96) + 1)) q − (cid:89) (cid:96) =0 Λ( γ ( (cid:96) )) q − (cid:89) (cid:96) =0 e − t (cid:96) Λ( γ ( (cid:96) )) e − Λ( m ) (cid:0) − (cid:80) q − (cid:96) =0 t (cid:96) (cid:1) q − (cid:89) (cid:96) =0 d t (cid:96) = (cid:88) q ∈ [Λ ∗ −√ K, Λ ∗ + √ K ] Λ q ∗ e − Λ ∗ (cid:88) γ : γ (0)= n, γ ( q )= m sup ≤ (cid:96) ≤ q − | γ ( (cid:96) ) − n ∗ |≤√ K log K (cid:90) · · · (cid:90) t + ··· + t q − < q − (cid:89) (cid:96) =0 p ∗ ( γ ( (cid:96) ) , γ ( (cid:96) + 1)) Π q (cid:0) γ, t , . . . , t q − (cid:1) q − (cid:89) (cid:96) =0 d t (cid:96) q = Π q Π q Π q where Π q (cid:0) γ, t , . . . , t q − (cid:1) = q − (cid:89) (cid:96) =0 p (cid:0) γ ( (cid:96) ) , γ ( (cid:96) + 1) (cid:1) p ∗ (cid:0) γ ( (cid:96) ) , γ ( (cid:96) + 1) (cid:1) Π q (cid:0) γ, t , . . . , t q − (cid:1) = q − (cid:89) (cid:96) =0 Λ (cid:0) γ ( (cid:96) ) (cid:1) Λ ∗ q − (cid:89) (cid:96) =0 e − t (cid:96) (cid:2) Λ (cid:0) γ ( (cid:96) ) (cid:1) − Λ ∗ (cid:3) and Π q (cid:0) γ, t , . . . , t q − (cid:1) = e − (cid:2) Λ (cid:0) m (cid:1) − Λ ∗ (cid:3) (cid:0) − (cid:80) q − (cid:96) =0 t (cid:96) (cid:1) . The structure of the previous expression is as follows: (cid:88) q ∈ [ Λ ∗ −√ K, Λ ∗ + √ K ] E ∗ n → m,q (Π q ( ˜ N ) C q ) , (5.19)where C q describes the restriction of the q states of the process to B ( n ∗ , √ K log K )and Π q = Π q Π q Π q and E ∗ n → m,q denotes the expectation related to the law ofthe process ˜ N going from n to m in q jumps.Equation (5.19) writes (cid:88) q ∈ [ Λ ∗ −√ K, Λ ∗ + √ K ] E ∗ n → m,q ( C q ) E ∗ n → m,q (Π q ( ˜ N ) C q ) E ∗ n → m,q ( C q ) . To get a lower bound of this expression, we use Jensen’s inequality and obtain (cid:88) q ∈ [ Λ ∗ −√ K, Λ ∗ + √ K ] E ∗ n → m,q (Π q ( ˜ N ) C q ) ≥ (cid:88) q ∈ [ Λ ∗ −√ K, Λ ∗ + √ K ] E ∗ n → m,q ( C q ) exp (cid:32) E ∗ n → m,q (log Π q ( ˜ N ) C q ) E ∗ n → m,q ( C q ) (cid:33) . Replacing each term with its complete expression, we obtain P n (cid:0) N K (1) = m (cid:1) ≥ (cid:88) q ∈ [Λ ∗ −√ K, Λ ∗ + √ K ] (cid:32) e − Λ ∗ Λ q ∗ q ! e S q ( n,m ) × (cid:88) γ : γ (0)= n, γ ( q )= m sup ≤ (cid:96) ≤ q − | γ ( (cid:96) ) − n ∗ |≤√ K log K q − (cid:89) (cid:96) =0 p ∗ (cid:0) γ ( (cid:96) ) , γ ( (cid:96) + 1) (cid:1)(cid:33) (5.20)with S q ( n, m ) = 1 Z q ( n, m ) e − Λ ∗ Λ ∗ q (cid:88) γ : γ (0)= n, γ ( q )= m sup ≤ (cid:96) ≤ q − | γ ( (cid:96) ) − n ∗ |≤√ K log K (cid:90) · · · (cid:90) t + ··· + t q − < − (cid:89) (cid:96) =0 p ∗ (cid:0) γ ( (cid:96) ) , γ ( (cid:96) + 1) (cid:1) log Π q (cid:0) γ, t , . . . , t q − (cid:1) q − (cid:89) (cid:96) =0 dt (cid:96) and Z q ( n, m ) = e − Λ ∗ Λ q ∗ q ! (cid:88) γ, γ (0)= n, γ ( q )= m sup ≤ (cid:96) ≤ q − | γ ( (cid:96) ) − n ∗ |≤√ K log K q − (cid:89) (cid:96) =0 p ∗ (cid:0) γ ( (cid:96) ) , γ ( (cid:96) + 1) (cid:1) . Our aim is now to give a lower bound for the right-hand side term in (5.20). Itwill be deduced from the three next lemmas which show that E ∗ n → m,q ( C q ) = Z q ( n, m ) ∼ /K ( d +1) / , as K → ∞ and E ∗ n → m,q (log Π q ( ˜ N ) C q ) E ∗ n → m,q ( C q ) = S q ( n, m )is of order one uniformly in q .Let us first estimate Z q ( n, m ). Lemma 5.9. We have the following estimates. (i) There exists a constant C (5 . > independent of K such that for K largeenough, for all q ∈ (cid:2) Λ ∗ − √ K, Λ ∗ + √ K (cid:3) and for all m ∈ ∆ , n ∈ ∆ (cid:48) C − . K − d/ − / ≤ Z q ( n, m ) ≤ C (5 . K − d/ − / . (ii) There exists a constant C (cid:48) (5 . > independent of K such that for K largeenough inf n ∈ ∆ (cid:48) m ∈ ∆ (cid:88) q ∈ [Λ ∗ −√ K, Λ ∗ + √ K ] e − Λ ∗ Λ q ∗ q ! × (cid:88) γ, γ (0)= n, γ ( q )= m sup ≤ (cid:96) ≤ q − | γ ( (cid:96) ) − n ∗ |≤√ K log K q − (cid:89) (cid:96) =0 p ∗ (cid:0) γ ( (cid:96) ) , γ ( (cid:96) + 1) (cid:1) ≥ C (cid:48) (5 . K − d/ . (5.21) Proof. (i) Note first, using Stirling’s formula, that for K large enough12 √ Λ ∗ e − γ ≤ e − Λ ∗ Λ q ∗ q ! ≤ √ Λ ∗ e − γ where γ is Euler’s constant. Then e − Λ ∗ Λ q ∗ q ! is of order 1 / √ K . Now, we notethat for all K large enough and all q ∈ [Λ ∗ − √ K, Λ ∗ + √ K ], we havesup n ∈ ∆ (cid:48) m ∈ ∆ (cid:88) γ : γ (0)= n, γ ( q )= m sup ≤ (cid:96) ≤ q − | γ ( (cid:96) ) − n ∗ | > √ K log K q − (cid:89) (cid:96) =0 p ∗ (cid:0) γ ( (cid:96) ) , γ ( (cid:96) + 1) (cid:1) ≤ sup n ∈ ∆ (cid:48) q − (cid:88) (cid:96) =1 P n (cid:16) | N (cid:96) − n ∗ | > √ K log K (cid:17) . N (cid:96) ) (cid:96) ∈ Z + we getsup n ∈ ∆ (cid:48) m ∈ ∆ (cid:88) γ : γ (0)= n, γ ( q )= m sup ≤ (cid:96) ≤ q − | γ ( (cid:96) ) − n ∗ | > √ K log K q − (cid:89) (cid:96) =0 p ∗ (cid:0) γ ( (cid:96) ) , γ ( (cid:96) +1) (cid:1) ≤ e −O (1)(log K ) . (5.22)We deduce that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z q ( n, m ) − e − Λ ∗ Λ q ∗ q ! (cid:88) γ, γ (0)= n, γ ( q )= m q − (cid:89) (cid:96) =0 p ∗ (cid:0) γ ( (cid:96) ) , γ ( (cid:96) + 1) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ e −O (1)(log K ) . To finish the proof, we apply the local limit theorem [11, Chapter 3] to therandom walk ( N (cid:96) ) (cid:96) ∈ Z + . Statement (ii) immediately follows at once from (i). Lemma 5.10. There exists a constant C (5 . > independent of K such thatfor K large enough sup q ∈ [Λ ∗ −√ K, Λ ∗ + √ K ] (cid:12)(cid:12) S q ( n, m ) (cid:12)(cid:12) ≤ C (5 . . Proof. Observe that (cid:12)(cid:12) S q ( n, m ) (cid:12)(cid:12) ≤ K d/ q ! O (1) × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) γ, γ (0)= n, γ ( q )= m sup ≤ (cid:96) ≤ q − | γ ( (cid:96) ) − n ∗ |≤√ K log K (cid:90) · · · (cid:90) t + ··· + t q − < q − (cid:89) (cid:96) =0 p ∗ (cid:0) γ ( (cid:96) ) , γ ( (cid:96) + 1) (cid:1) log Π q (cid:0) γ, t , . . . , t q − (cid:1) q − (cid:89) (cid:96) =0 dt (cid:96) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Since Π q = Π q Π q Π q ,log Π q (cid:0) γ, t , . . . , t q − (cid:1) = (cid:88) i =1 log Π iq (cid:0) γ, t , . . . , t q − (cid:1) and we have to estimate separately the three terms. The result follows fromseveral technical lemmas which are postponed to Section 8.It follows from (5.20) and (5.21) and Lemma 5.10 that there exists η (5 . < P n (cid:0) N K (1) = m (cid:1) ≥ (1 − η (5 . ) ν ( { m } ) , where ν is the measure defined in Subsection 5.2.28 .6 Proof of Condition (A2) Our aim is to show the existence of a constant c such that for all t ≥ n ∈ ∆, m ∈ ∆ c P n ( T > t ) ≥ c P m ( T > t ) (5.23)where ∆ is defined in (5.1). For all t ≥ g ( t ) = sup n ∈ ∆ P n (cid:0) T > t (cid:1) and f ( t ) = sup n ∈ ∆ c P n (cid:0) T > t (cid:1) . The proof of Condition (A2) will be the consequence of the four following lemmaswhich we prove hereafter. Lemma 5.11. There exist η > and δ > such that η δ < whence δ < , and, for all K large enough, there exists t ∗ = t ∗ ( K ) such that ∀ t ≥ t ∗ , g ( t − t ∗ ) ≤ ηg ( t ) and sup n ∈ ∆ c P n (cid:0) T ∆ ∧ T > t ∗ (cid:1) < δ. Proof. The proof consists in several steps.We first show that there exists a constant η (5 . ∈ (0 , 1) such that for all K large enough and t ∆ = C (5 . log K ,sup n ∈ Z d + \{ } P n (cid:0) T ∆ ∧ T > t ∆ (cid:1) ≤ − η (5 . . (5.24)We have P n (cid:0) T ∆ > t ∆ , T > t ∆ (cid:1) = P n (cid:0) T > t ∆ (cid:1) − P n (cid:0) T ∆ ≤ t ∆ , T > t ∆ (cid:1) ≤ − P n (cid:0) T ∆ ≤ t ∆ , T > t ∆ (cid:1) . We also have, using the Markov property, the monotonicity of P n (cid:0) T > t (cid:1) andSublemma 5.8 P n (cid:0) T ∆ ≤ t ∆ , T > t ∆ (cid:1) = E n (cid:2) { T ∆ ≤ t ∆ } P N K ( T ∆ ) (cid:0) T > t ∆ − T ∆ (cid:1)(cid:3) ≥ E n (cid:2) { T ∆ ≤ t ∆ } P N K ( T ∆ ) (cid:0) T > t ∆ (cid:1)(cid:3) ≥ P n (cid:0) T ∆ ≤ t ∆ (cid:1) inf m ∈ ∆ P m (cid:0) T D c > t ∆ (cid:1) ≥ P n (cid:0) T ∆ ≤ t ∆ (cid:1) (cid:0) − ( O (1) + O (1) t ∆ ) e − α ˜ ρ K (cid:1) , and the result follows for K large enough using Lemma 5.1.Let us now prove recursively that for K large enough and for all integer q ,sup n ∈ Z d + \{ } P n (cid:0) T ∆ ∧ T > q t ∆ (cid:1) ≤ (1 − η (5 . ) q . (5.25)29he inequality is true for q = 1. For q > 1, we can write using (5.24) P n (cid:0) T ∆ ∧ T > q t ∆ (cid:1) = E n (cid:104) { T ∆ ∧ T > ( q − t ∆ } P N K (( q − t ∆ ) (cid:0) T ∆ ∧ T > t ∆ (cid:1)(cid:105) ≤ E n (cid:104) { T ∆ ∧ T > ( q − t ∆ } (cid:105) (1 − η (5 . )= (1 − η (5 . ) P n (cid:0) T ∆ ∧ T > ( q − t ∆ (cid:1) and the result follows.By a similar proof as for (5.24), (use first Lemma 5.1 and after the firstentrance in ∆ use the Markov property and Sublemma 5.8), we also obtain thatthere exists a number C (5 . > K large enoughinf n ∈ Z d + \{ } P n (cid:0) T > K (cid:1) ≥ C (5 . . (5.26)It implies that for all s ≥ 0, for all n ∈ ∆ and for K large enough, P n (cid:0) T > s + K ) = E n (cid:104) { T >s } P N K ( s ) (cid:0) T > K (cid:1)(cid:105) ≥ P n (cid:0) T > s (cid:1) inf n ∈ Z d + \{ } P n (cid:0) T > K (cid:1) ≥ C (5 . P n (cid:0) T > s (cid:1) . Then, for all s ≥ n ∈ ∆, we have g ( s + K ) ≥ C (5 . P n (cid:0) T > s (cid:1) and g ( s ) ≤ C (5 . g ( s + K ) with C (5 . = C − . > 0. We have thus proved thatfor all K large enough and all t ≥ Kg ( t − K ) ≤ C (5 . g ( t ) . (5.27)Note that C (5 . is necessarily strictly greater than 1.Let us now take t ∗ = K and let q ∗ be the smallest positive integer such that(1 − η (5 . ) q ( C (5 . ) < . We take η = C (5 . and δ = (1 − η (5 . ) q ∗ . We ofcourse have ηδ < 1. Inequality (5.27) implies g ( t − t ∗ ) ≤ ηg ( t ) for all t ≥ t ∗ .Moreover, since for K large enough q ∗ t ∆ < K and by (5.25), we havesup n ∈ ∆ c P n (cid:0) T ∆ ∧ T > t ∗ (cid:1) ≤ sup n ∈ ∆ c P n (cid:0) T ∆ ∧ T > q ∗ t ∆ (cid:1) < (1 − η (5 . ) q ∗ = δ. The lemma is proved. Lemma 5.12. With t ∗ , η and δ > defined in Lemma 5.11, we get for allinteger n f ( n t ∗ ) ≤ (cid:18) η − δ η (cid:19) g ( n t ∗ ) . Proof. For all m ∈ ∆ c and t ≥ t ∗ we have using the Markov property P m (cid:0) T > t (cid:1) = P m (cid:0) T > t, T ∆ ≤ t ∗ (cid:1) + P m (cid:0) T > t, T ∆ > t ∗ (cid:1) = E m (cid:2) { T ∆ ≤ t ∗ } P N K ( T ∆ ) (cid:0) T > t − T ∆ (cid:1)(cid:3) + E m (cid:104) { T ∆ ∧ T >t ∗ } P N K ( t ∗ ) (cid:0) T > t − t ∗ (cid:1)(cid:105) ≤ g ( t − t ∗ ) + δf ( t − t ∗ ) ≤ η g ( t ) + δf ( t − t ∗ ) , n ≥ f ( n t ∗ ) ≤ η g ( n t ∗ ) + δ f (( n − t ∗ ) . It is easy to verify recursively that this implies f ( n t ∗ ) ≤ δ n + η − δ η g ( n t ∗ ) . The result follows by observing that from Lemma 5.11 we have g ( n t ∗ ) ≥ η − n for all integers n . Lemma 5.13. With notations of Lemma 5.11 and Lemma 5.12, for all t > f ( t ) ≤ η (cid:18) η − δ η (cid:19) g ( t ) . Proof. We first consider the case t > t ∗ . Let n = [ t/t ∗ ]. We have by Lemma5.12, the monotonicity of f , Lemma 5.11 and the monotonicity of gf ( t ) ≤ f ( n t ∗ ) ≤ (cid:18) η − δ η (cid:19) g ( n t ∗ ) ≤ η (cid:18) η − δ η (cid:19) g (( n + 1) t ∗ ) ≤ η (cid:18) η − δ η (cid:19) g ( t ) . For 0 ≤ t ≤ t ∗ we have by Lemma 5.11 and using the monotonicity of g ( t ) f ( t ) ≤ ≤ η g ( t ∗ ) ≤ η g ( t ) . Lemma 5.14. There exists a constant < C (5 . < , such that for all K large enough and all t > we have inf n ∈ ∆ P n (cid:0) T > t (cid:1) ≥ C (5 . sup n ∈ ∆ P n (cid:0) T > t (cid:1) . Proof. Let α (5 . = 1+ C (5 . . With use of Lemma 5.1, (5.14), Lemma 5.3, Sub-lemma 5.8 and twice the Markov property, we obtain that there exists C (5 . > K large enoughsup m ∈ ∆ sup n ∈ D P n (cid:0) N K ( α (5 . log K ) = m (cid:1) ≥ C (5 . K d/ . (5.28)Indeed, for n ∈ D and α (cid:48) < α (5 . , we have P n (cid:0) N K ( α (5 . log K ) = m (cid:1) ≥ P n (cid:0) T ∆ <α (cid:48) log K ; N K ( α (5 . log K ) = m (cid:1) ≥ E n (cid:2) { T ∆ <α (cid:48) log K } P N K ( s ) (cid:0) N K ( α (5 . log K − T ∆ ) = m (cid:1)(cid:3) ≥ O (1) inf m ∈ ∆ P m (cid:0) N K ( α (5 . log K − T ∆ − ∈ ∆ ; N K ( α (5 . log K − T ∆ ) = m (cid:1) and (5.28) follows. 31e have for all n ∈ ∆ and all m ∈ ∆ P n (cid:0) T m > α (5 . C (5 . K d/ log K ) ≤ P n (cid:0) T m > α (5 . C (5 . K d/ log K , T D c > α (5 . C (5 . K d/ log K (cid:1) + P n (cid:0) T D c < α (5 . C (5 . K d/ log K (cid:1) ≤ P n C (5 . K d/ (cid:92) q =1 (cid:26) N K ( qα (5 . log K ) ∈ D , N K ( qα (5 . log K ) (cid:54) = m (cid:27) + P n (cid:0) T D c < α (5 . C (5 . K d/ log K (cid:1) and using the Markov property and (5.28), we obtain ≤ (cid:18) − C (5 . K d/ (cid:19) C (5 . K d/ + P n (cid:0) T D c < α (5 . C (5 . K d/ log K (cid:1) . Using Sublemma 5.8, we deduce that there exists η (5 . > m ∈ ∆ sup n ∈ ∆ P n (cid:0) T m > α (5 . C (5 . K d/ log K (cid:1) ≤ − η (5 . . (5.29)For all t > 0, let us now define n t = argmax m ∈ ∆ P m (cid:0) T > t (cid:1) . For t > α (5 . C (5 . K d/ log K , we have for all n ∈ ∆ P n (cid:0) T > t (cid:1) ≥ E n (cid:104) { T nt <α (5 . C (5 . K d/ log K } P N K ( T nt ) (cid:0) T > t − T n t (cid:1)(cid:105) ≥ E n (cid:104) { T nt <α (5 . C (5 . K d/ log K } P N K ( T nt ) (cid:0) T > t (cid:1)(cid:105) ≥ P n (cid:104) T n t < α (5 . C (5 . K d/ log K (cid:105) sup m ∈ ∆ P m (cid:0) T > t (cid:1) and the result follows from (5.29).For t ≤ α (5 . C (5 . K d/ log K , we use thatsup m ∈ ∆ P m (cid:0) T > t (cid:1) ≤ c in (5.23) is given by c = C (5 . η (cid:16) η − δη (cid:17) < . (5.30) The proof of Theorem 3.1 follows from Conditions (A1) and (A2) using theresult in [5]. The constant c is equal to c c < 1, where c and c have beendefined in (5.2) and (5.30). The number t ( K ) defined in (5.3) is of order log K .32 Proof of Theorem 3.2 The proof will be the consequence of the following lemma. Lemma 6.1. Recall that D has been defined in Sublemma 5.8. There exist K > , b (6 . > and < C (6 . < such that for all K ≥ K inf n ∈ D P n (cid:0) T D c > e b (6 . K (cid:1) ≥ C (6 . . Proof. As in the proof of Sublemma 5.8, we use Dynkin’s Theorem applied to ϕ (cid:0) N K ( t ∧ T D c ) (cid:1) to obtain E n (cid:2) ϕ (cid:0) N K ( t ∧ T D c ) (cid:1)(cid:3) = ϕ ( n ) + E n (cid:34)(cid:90) t ∧ T Dc L K ϕ ( N K ( s )) d s (cid:35) . (6.1)We distinguish the cases n ∈ ∆ and n / ∈ ∆.Let us introduce the set (cid:101) ∆ = { n : (cid:107) n − n ∗ (cid:107) ≤ ρ (4 . √ K } , where the constant ρ (4 . has been defined in Corollary 4.2.For an initial state n ∈ ∆, we remark that T (cid:101) ∆ c < T D c . For any t > E n (cid:34)(cid:90) t ∧ T (cid:101) ∆ c L K ϕ ( N K )( s ) d s (cid:35) ≤ O (1) t. We can write (cid:90) t ∧ T Dc L K ϕ ( N K ( s )) d s = (cid:90) t ∧ T c (cid:101) ∆ L K ϕ ( N K ( s )) d s + (cid:90) t ∧ T Dc t ∧ T c (cid:101) ∆ L K ϕ ( N K ( s )) d s . Using Theorem 4.1, we remark that the first term of the rhs is bounded by O (1) t . Corollary 4.2 implies that the second term is non positive. In the otherhand, there exists b > n ∈ ∂D c ϕ ( n ) ≥ e bK By (6.1) we finally obtaine bK P n ( T D c < t ) ≤ O (1) t + ϕ ( n ) . Since sup n ∈ ∆ ϕ ( n ) = O (1)we conclude that for K large enough sup n ∈ ∆ P n (cid:0) T D c < e bK/ (cid:1) < . Thereforeinf n ∈ ∆ P n (cid:16) T D c ≥ e bK/ (cid:17) > . (6.2)33or n ∈ D \ ∆ we have for K large enough (in particular e b K/ > C (5 . log K )by Lemma 5.1 and the Markov property and monotonicity of t (cid:55)→ P n ( T D c ≥ t )and (6.2) P n (cid:16) T D c ≥ e b K/ (cid:17) ≥ E n (cid:104) { T ∆ 0. For n ∈ D and for all integer q > P n (cid:0) T D c ≥ q e b (6 . K (cid:1) = E n (cid:20) (cid:110) T Dc > e b (6 . K (cid:111) P N (cid:16) e b (6 . K (cid:17) (cid:0) T D c ≥ ( q − 1) e b (6 . K (cid:1)(cid:21) ≥ P n (cid:0) T D c > e b (6 . K (cid:1) inf m ∈ D P m (cid:0) T D c ≥ ( q − 1) e b (6 . K (cid:1) . Using Lemma 6.1 we get for all q ≥ n ∈ D P n (cid:0) T D c ≥ q e b (6 . K (cid:1) ≥ C q (6 . . Therefore inf n ∈ D P n (cid:0) T ≥ q e b (6 . K (cid:1) ≥ C q (6 . . By Property (A1) proved in Subsection 5.2, we know that P n ( N K ( t ) ∈ . ) ≥ c ν ( . ) P n ( t < T ) . Integrating by m K and using that P m K ( t < T ) = e − λ ( K ) t > 0, we obtain m K ( . ) ≥ c ν ( . ) . Then for all point n ∈ ∆, m K ( n ) ≥ c ν ( n ) > m K weget for all q ≥ P m K (cid:16) T ≥ q e b (6 . K (cid:17) ≥ m K ( n ) C q (6 . . From (3.1) and this bound we deduce thatlim t →∞ − t log P m K (cid:0) T ≥ t (cid:1) = lim q →∞ − q e b (6 . K log P m K (cid:0) T ≥ q e b (6 . K (cid:1) ≤ − log C (6 . e − b (6 . K . We have used the fact that, since the limit exists, we can compute it along alldiverging sequence. Therefore we have proved the upper bound in Theorem 3.2with d = b (6 . / K is large enough.34 .2 Proof of the lower bound The proof will result from the following two lemmas. Lemma 6.2. There exists b (6 . > and t (6 . > independent of K such thatfor K large enough sup n ∈E c P n (cid:0) T ≥ t (6 . (cid:1) ≤ − e − b (6 . K . Recall that E c = (cid:8) n ∈ Z d + \{ } : (cid:80) dj =1 n j ≤ LK (cid:9) . Proof. Starting from n ∈ E c , we consider a path from n to 0 obtained by de-creasing successively the maximum of the components. We denote this path by( m ( p )) ≤ p ≤ Q ( n ) with m (0) = n and m ( Q ( n )) = 0. We observe that from theconstruction of ( m ( p )) ≤ p ≤ Q ( n ) , the sequence of integers max( m (cid:96) ( p )) is nonin-creasing (with jumps of size 1) and can have plateaus of length at most d . Notealso that Q ( n ) ≤ dLK .Using Hypotheses (H0), (H1) and (H7) there exists ξ (cid:48) > n ∈ E c and for all j such that n j = max (cid:96) =1 ,...,d n (cid:96) , KD j (cid:16) nK (cid:17) ≥ ξ max( n (cid:96) ) and K (cid:16) B j (cid:16) nK (cid:17) + D j (cid:16) nK (cid:17)(cid:17) ≤ ξ (cid:48) max( n (cid:96) ) . Therefore, inf n ∈E c inf j ∈ argmax { n (cid:96) } D j (cid:0) nK (cid:1)(cid:80) d(cid:96) =1 (cid:0) B (cid:96) (cid:0) nK (cid:1) + D (cid:96) (cid:0) nK (cid:1)(cid:1) ≥ ξdξ (cid:48) > . This implies that the probability of the path of the embedded chain is largerthan ( ξ/dξ (cid:48) ) dLK .If Θ denotes the first jump time of the process, it follows from the aboveinequalities that for all we have for all n ∈ E c and for all n P n (cid:16) Θ < ξK (cid:17) = 1 − exp (cid:34) − ξ d (cid:88) (cid:96) =1 (cid:16) B (cid:96) (cid:16) nK (cid:17) + D (cid:96) (cid:16) nK (cid:17)(cid:17)(cid:35) ≥ − exp (cid:16) − max ≤ (cid:96) ≤ d n (cid:96) K (cid:17) . For all n ∈ E c , we define a (measurable) set of trajectories T n of the stochasticprocess by T n = (cid:26) n ( · ) (cid:12)(cid:12)(cid:12)(cid:12) ∃ s = 0 < s < s < · · · < s Q ( n ) , ∀ ≤ p ≤ Q ( n ) − ,s p +1 − s p < ξ K , and n ( t ) = m ( p ) for t ∈ [ s p , s p +1 [ , n ( s Q ( n ) ) = 0 (cid:27) . Let t (6 . = dLξ . Q ( n ) ≤ dLK , we have for all n ∈ E c P n (cid:0) T < t (6 . (cid:1) ≥ P n (cid:0) T n (cid:1) ≥ (cid:18) ξdξ (cid:48) (cid:19) dLK Q ( n ) − (cid:89) p =0 (cid:16) − e − max ≤ (cid:96) ≤ d m (cid:96) ( p ) /K (cid:17) ≥ (cid:18) ξdξ (cid:48) (cid:19) dLK [ LK ] (cid:89) q =1 (cid:16) − e − q/K (cid:17) d . Let θ be the unique solution of 1 − e − θ = θ/ 2. Note that θ > − e − x ≥ x , ∀ x ∈ [0 , θ ] , − e − x ≥ − e − θ , ∀ x > θ. For L > θ we have (cid:98) LK (cid:99) (cid:89) q =1 (cid:16) − e − q/K (cid:17) = (cid:98) θK (cid:99) (cid:89) q =1 (cid:16) − e − q/K (cid:17) (cid:98) LK (cid:99) (cid:89) q =1+ (cid:98) θK (cid:99) (cid:16) − e − q/K (cid:17) ≥ −(cid:98) θK (cid:99) (cid:98) θK (cid:99) ! K (cid:98) θK (cid:99) (cid:0) − e − θ (cid:1) (cid:98) LK (cid:99)−(cid:98) θK (cid:99) ≥ −(cid:98) θK (cid:99) e −(cid:98) θK (cid:99) (cid:0) − e − θ (cid:1) (cid:98) LK (cid:99)−(cid:98) θK (cid:99) using Stirling’s formula and K large enough to obtain the last estimate. For L ≤ θ we have by a similar argument (cid:98) LK (cid:99) (cid:89) q =1 (cid:16) − e − q/K (cid:17) ≥ −(cid:98) LK (cid:99) e −(cid:98) LK (cid:99) . We finally get P n (cid:0) T < t (6 . (cid:1) ≥ (cid:18) ξd ξ (cid:48) (cid:19) dLK (2 e ) − d K ( L ∧ θ ) (cid:0) − e − θ (cid:1) d (1+ K (( L − θ ) ∨ , and the result follows since ξd ξ (cid:48) < Lemma 6.3. There exists t (6 . > independent of K such that for K largeenough sup n ∈ Z d + \{ } P n (cid:0) T ≥ t (6 . (cid:1) ≤ − (1 − η (5 . ) e − b (6 . K . Proof. Let t (6 . = t (6 . + t (5 . . By Lemma 6.2 we havesup n ∈E c P n (cid:0) T ≥ t (6 . (cid:1) ≤ − e − b (6 . K . n ∈ E using the Markov property we obtain the following estimate P n (cid:0) T > t (6 . (cid:1) = P n (cid:0) T > t (6 . , T E c > t (5 . (cid:1) + P n (cid:0) T > t (6 . , T E c ≤ t (5 . (cid:1) ≤ P n (cid:0) T E c > t (5 . (cid:1) + P n (cid:0) T > t (6 . , T E c ≤ t (5 . (cid:1) = P n (cid:0) T E c > t (5 . (cid:1) + E n (cid:104) { T E c ≤ t (5 . } P N K ( T E c ) (cid:0) T > t (6 . − T E c (cid:1)(cid:105) ≤ P n (cid:0) T E c > t (5 . (cid:1) + E n (cid:104) { T E c ≤ t (5 . } P N K ( T E c ) (cid:0) T > t (6 . (cid:1)(cid:105) ≤ P n (cid:0) T E c > t (5 . (cid:1) + P n (cid:0) T E c ≤ t (5 . (cid:1) (cid:0) − e − b (6 . K (cid:1) where we made use of Lemma 6.2. Using Lemma 5.4 we get P n (cid:0) T > t (6 . (cid:1) ≤ − P n (cid:0) T E c ≤ t (5 . (cid:1) e − b (6 . K ≤ − (1 − η (5 . ) e − b (6 . K . We can now prove the lower bound. Proof of the lower bound in Theorem 3.2. Using Lemma 6.3 and the Markovproperty we get for all q ≥ n ∈ Z d + \{ } P n (cid:0) T ≥ q t (6 . (cid:1) ≤ (cid:0) − (1 − η (5 . ) e − b (6 . K (cid:1) q . This implies P m K (cid:0) T ≥ q t (6 . (cid:1) ≤ (cid:0) − (1 − η (5 . ) e − b (6 . K (cid:1) q . Therefore lim t →∞ t log P m K (cid:0) T ≥ t (cid:1) = lim q →∞ q log P m K (cid:0) T ≥ q t (6 . (cid:1) ≤ log (cid:0) − (1 − η (5 . ) e − b (6 . K (cid:1) . and the lower bound follows by taking for instance d = 1 + b (6 . . In the sequel we will assume that K is large enough. We first observe thatsince Z d + \{ } is discrete and countable, the Banach space of bounded complexmeasures on Z d + \{ } equipped with the norm of total variation is identical to (cid:96) ( Z d + \{ } ). Its dual is therefore (cid:96) ∞ ( Z d + \{ } ).We establish a consequence of Theorem 3.1. To simplify the notation wewrite P = P K and P † denotes the adjoint semigroup. Corollary 7.1. There exists a constant C (7 . > such that for all K > K ,and for any t > (cid:13)(cid:13) R t − P † t (cid:13)(cid:13) (cid:96) ( Z d + \{ } ) ≤ C (7 . e − ωt here R t is the rank one operator given by (cid:104) R t µ , f (cid:105) = m K ( f ) P µ (cid:0) t < T (cid:1) and ω = ω ( K ) = − log(1 − c ) t ( K ) ≥ d log K where d > is independent of K .Proof. If µ is a probability measure, we get from Theorem 3.1 multiplying theestimate by P µ (cid:0) t < T (cid:1)(cid:13)(cid:13) R t µ − P † t µ (cid:13)(cid:13) (cid:96) ( Z d + \{ } ) ≤ − c P µ (cid:0) t < T (cid:1) e − ωt ≤ − c e − ωt . By standard arguments this implies that for any sequence f ∈ (cid:96) ( Z d + \{ } ) wehave (cid:13)(cid:13) R t f − P † t f (cid:13)(cid:13) (cid:96) ( Z d + \{ } ) ≤ C (7 . e − ωt (cid:107) f (cid:107) (cid:96) ( Z d + \{ } ) , with C (7 . = 4 / (1 − c ).We now derive some consequences of this estimate. Lemma 7.2. For any s > , the operator P † s has only one eigenvalue of moduluslarger than exp( − ωs ) which is equal to exp( − λ s ) . This eigenvalue is simple,the corresponding eigenvector is m K .Proof. Assume f is an eigenvector of P † s with eigenvalue z such that | z | > exp( − ωs ). From P † sn f = z n f, we get using Corollary 7.1 and the semi-group property (cid:13)(cid:13) z n f − R ns f (cid:13)(cid:13) (cid:96) ( Z d + \{ } ) = (cid:13)(cid:13) P † sn f − R ns f (cid:13)(cid:13) (cid:96) ( Z d + \{ } ) = (cid:13)(cid:13) P † ns f − R ns f (cid:13)(cid:13) (cid:96) ( Z d + \{ } ) ≤ C (7 . e − n ωs (cid:107) f (cid:107) (cid:96) ( Z d + \{ } ) . In other words (cid:13)(cid:13) f − z − n R ns f (cid:13)(cid:13) (cid:96) ( Z d + \{ } ) ≤ C (7 . | z | − n e − nωs (cid:107) f (cid:107) (cid:96) ( Z d + \{ } ) . If | z | > exp( − ωs ) the right-hand side tends to zero when n tends to infinity and f must be proportional to m K (since R ns f is proportional to m K ) which is aneigenvector of P † s with eigenvalue exp( − λ s ).We now prove (by contradiction) that the equation P † s f − e − λ s f = m K has no solution. Assume there exists such an f ∈ (cid:96) ( Z d + \{ } ) (which is neces-sarily non zero). We get P † sn f = n e − ( n − λ s m K + e − nλ s f. (cid:13)(cid:13)(cid:13) n e − ( n − λ s m K + e − nλ s f − R ns f (cid:13)(cid:13)(cid:13) (cid:96) ( Z d + \{ } ) ≤ C (7 . e − n ωs (cid:107) f (cid:107) (cid:96) ( Z d + \{ } ) which implies (cid:13)(cid:13) f + n e λ s m K − e nλ s R ns f (cid:13)(cid:13) (cid:96) ( Z d + \{ } ) ≤ C (7 . e − n ( ω − λ ) s . Since ω > λ the right hand side tends to zero when n tends to infinity and wededuce that f must be proportional to m K , a contradiction.The following result completes the description of the spectrum of P † s outsidethe disk in the complex plane of radius exp( − ωs ). Proposition 7.3. For any s > the operator P † s ( as an operator in (cid:96) ( Z d + \{ } )) has spectral radius exp( − λ s ) and essential spectral radius at most exp( − ωs ) .Outside the disk | z | ≤ exp( − ω s ) the spectrum consists of only one simple eigen-value exp( − λ s ) with eigenvector m K .Proof. From Corollary 7.1 and the semi-group property, we have for any integer n (cid:13)(cid:13) R ns − P † sn (cid:13)(cid:13) (cid:96) ( Z d + \{ } ) ≤ C (7 . e − nωs . Therefore since all operators R ns have rank one and therefore are compact,it follows from Corollaries I.4.9 and I.4.11 in [12] (page 44) that the essentialspectral radius of P † s is at most exp( − ωs ). The rest of the proposition followsfrom Lemma 7.2 since outside of the essential spectrum, the spectrum can onlyconsists of isolated eigenvalues with finite algebraic and geometric multiplicities. Proposition 7.4. For any s > , the operator P s acting in (cid:96) ∞ ( Z d + \{ } ) hasspectral radius exp( − λ s ) and essential spectral radius at most exp( − ωs ) . Out-side the disk | z | ≤ exp( − ω s ) the spectrum consists of only one eigenvalue exp( − λ s ) with a simple strictly positive eigenvector u K satisfying m K ( u K ) = 1 and independent of s .Proof. The result follows from Theorem IX.1.1 in [12] and Proposition 7.3 exceptfor the properties of the eigenvector. We have (where is the constant functionone on Z d + \{ } ) lim n →∞ e n λ s P n s = v s , From (cid:104) m K , (cid:105) = 1, we conclude that (cid:104) m K , v s (cid:105) = 1, and hence v s is an eigen-vector of P s with eigenvalue exp( − λ s ). Since the operator P s maps positivefunctions to positive functions we conclude that v s is positive.Let t (cid:48) > s . By the semi-group propertywe have P s (cid:0) P t (cid:48) v s (cid:1) = e − λ s P t (cid:48) v s . Since P t (cid:48) v s is positive and the eigenvalue exp( − λ s ) of P s is simple, this functionmust be proportional to v s . From (cid:104) m K , P t (cid:48) v s (cid:105) = e − λ t (cid:48) 39e conclude that P t (cid:48) v s = e − λ t (cid:48) v s . The independence of v s on s follows and we denote this vector by u K . Proposition 7.5. There exists a positive constant C (7 . such that (cid:107) u K (cid:107) (cid:96) ∞ ( Z d + \{ } ) ≤ − C (7 . K . Moreover inf n ∈ Z d + \{ } u K ( n ) ≥ c where the constant c is defined in Theorem 3.1.Proof. Form Corollary 7.1 it follows that for any t > (cid:13)(cid:13) R † t − P t (cid:13)(cid:13) (cid:96) ∞ ( Z d + \{ } ) ≤ C (7 . e − ω t . (7.1)From the definition of R t and m K ( u K ) = 1 we have R † t u K = P ( · ) (cid:0) t < T (cid:1) . Hence (cid:13)(cid:13) P ( · ) (cid:0) t < T (cid:1) − e − λ t u K (cid:13)(cid:13) (cid:96) ∞ ( Z d + \{ } ) = (cid:13)(cid:13) R † t u K − P t u K (cid:13)(cid:13) (cid:96) ∞ ( Z d + \{ } ) ≤ C (7 . e − ω t (cid:107) u K (cid:107) (cid:96) ∞ ( Z d + \{ } ) . (7.2)Therefore if t is large enough so that exp( − λ t ) > C (7 . exp( − ω t ) we obtain (cid:107) u K (cid:107) (cid:96) ∞ ( Z d + \{ } ) ≤ (cid:13)(cid:13) P ( · ) (cid:0) t < T (cid:1)(cid:13)(cid:13) (cid:96) ∞ ( Z d + \{ } ) e − λ t − C (7 . e − ω t ≤ − λ t − C (7 . e − ω t . The first result follows by taking t = K log K since λ = exp( −O (1) K ) (seeTheorem 3.2) and ω = O (1) / log K .From the positivity of u K and by (5.19) we get (with t = t ( K ))e − λ t u K ≥ P t ( u K ) ≥ P t ( ∆ u K ) ≥ c ν ( u K ) . (7.3)For any t > m K we get c e − λ t = c P m K (cid:0) t < T (cid:1) ≤ P ν (cid:0) t < T (cid:1) = (cid:88) m ν ( m ) P m (cid:0) t < T (cid:1) . From the estimate (7.2) and the first result we get c e − λ t ≤ (cid:88) m ν ( m ) e − λ t u K ( m ) + O (1) e − ω t . Multiplying by exp( λ t ) and letting t tend to infinity, we get (since λ < ω ) c ≤ ν ( u K ) . The second result follows by combining this estimate with the lower bound(7.3) 40or each n ∈ Z d + \{ } we define p K ( n ) = u K ( n ) ∧ . Proof of Theorem 3.3. Using the estimate (7.2) and Proposition 7.5, we get forany n ∈ Z d + \{ } (cid:12)(cid:12) P n (cid:0) t < T (cid:1) − e − λ t p K ( n ) (cid:12)(cid:12) ≤ e − λ t e − C (7 . K + C (7 . e − ω t (cid:0) − C (7 . K (cid:1) ≤ e − C (7 . K e − λ t +2 C (7 . e − ω t . The result follows from Corollary 7.1 Proof of Theorem 3.4. Combining the estimates (7.1) and (7.2) we obtain (cid:13)(cid:13) P t − e − ω t π K (cid:13)(cid:13) (cid:96) ∞ ( Z d + \{ } ) ≤ C (7 . e − ω t (cid:0) − C (7 . K (cid:1) . Therefore if (cid:60) z > − ω and z (cid:54) = − λ ( K ) we have (cid:90) ∞ P t e − t z d t = π K z + λ ( K ) + M z where M z is analytic in (cid:60) z > − ω . The result follows. The proof of Lemma 5.10 follows from a series of sublemmas. We first estimatethe contribution of Π q to S q ( n, m ). Sublemma 8.1. For all K large enough and all q ∈ [ Λ ∗ − √ K, Λ ∗ + √ K ]sup ≤ s ≤ q sup n ∈ ∆ (cid:48) m ∈ ∆ (cid:88) γ : γ (0)= n, γ ( q )= m sup ≤ (cid:96) ≤ q − | γ ( (cid:96) ) − n ∗ | < √ K log K q − (cid:89) (cid:96) =0 p ∗ (cid:0) γ ( (cid:96) ) , γ ( (cid:96) + 1) (cid:1) | γ ( s ) − n ∗ | ≤ O (1) K − d/ . Proof. We havesup ≤ s ≤ q sup n ∈ ∆ (cid:48) m ∈ ∆ (cid:88) γ : γ (0)= n, γ ( q )= m sup ≤ (cid:96) ≤ q − | γ ( (cid:96) ) − n ∗ | < √ K log K q − (cid:89) (cid:96) =0 p ∗ (cid:0) γ ( (cid:96) ) , γ ( (cid:96) + 1) (cid:1) | γ ( s ) − n ∗ | ≤ sup ≤ s ≤ q sup n ∈ ∆ (cid:48) m ∈ ∆ (cid:88) γ : γ (0)= n : γ ( q )= m q − (cid:89) (cid:96) =0 p ∗ (cid:0) γ ( (cid:96) ) , γ ( (cid:96) + 1) (cid:1) | γ ( s ) − n ∗ | . For 0 ≤ s ≤ q we have (cid:88) γ : γ (0)= n, γ ( q )= m q − (cid:89) (cid:96) =0 p ∗ (cid:0) γ ( (cid:96) ) , γ ( (cid:96) + 1) (cid:1) | γ ( s ) − n ∗ | = (cid:88) u P n ( N s = u ) | u − n ∗ | P u ( N q − s = m )= (cid:88) u P n ( N s = u ) | u − n ∗ | P m ( N q − s = u )41here the second equality follows from the reversibility of the random walk( N (cid:96) ) (cid:96) ∈ Z + . For s ≤ q/ N ) wehave P m ( N q − s = u ) ≤ O (1) K d/ . It follows that (cid:88) γ, γ (0)= n, γ ( q )= m q − (cid:89) (cid:96) =0 p ∗ (cid:0) γ ( (cid:96) ) , γ ( (cid:96) + 1) (cid:1) | γ ( s ) − n ∗ | ≤O (1) K d/ (cid:88) u P n ( N s = u ) | u − n ∗ | ≤ O (1) K d/ E (cid:0) | N s + n − n ∗ | (cid:1) ≤ O (1) K d/ − . For s > q/ n and m ). Sublemma 8.2. There exists a constant C (8 . > independent of K such thatfor K large enough and all q ∈ [Λ ∗ − √ K, Λ ∗ + √ K ] K d/ q ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) γ, γ (0)= n, γ ( q )= m sup ≤ (cid:96) ≤ q − | γ ( (cid:96) ) − n ∗ |≤√ K log K (cid:90) · · · (cid:90) t + ··· + t q − < q − (cid:89) (cid:96) =0 p ∗ (cid:0) γ ( (cid:96) ) , γ ( (cid:96) + 1) (cid:1) log Π q (cid:0) γ, t , . . . , t q − (cid:1) q − (cid:89) (cid:96) =0 d t (cid:96) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (8 . . Proof. We first observe that for all 0 ≤ s ≤ q − (cid:90) · · · (cid:90) t + ··· t q − < t s q − (cid:89) (cid:96) =0 d t (cid:96) = (cid:90) · · · (cid:90) t + ··· + t q − < t q − (cid:89) (cid:96) =0 d t (cid:96) = (cid:90) t d t (cid:90) · · · (cid:90) t + ··· + t q − < − t q − (cid:89) (cid:96) =1 d t (cid:96) = 1( q − (cid:90) t (1 − t ) q − d t = 1( q − (cid:18)(cid:90) (1 − t ) q − d t − (cid:90) (1 − t ) q d t (cid:19) = 1( q − (cid:18) q − q + 1) (cid:19) = 1( q + 1)! . K d/ (cid:88) γ, γ (0)= n, γ ( q )= m sup ≤ (cid:96) ≤ q − | γ ( (cid:96) ) − n ∗ |≤√ K log K q − (cid:89) (cid:96) =0 p ∗ (cid:0) γ ( (cid:96) ) , γ ( (cid:96) + 1) (cid:1) × q − (cid:88) (cid:96) =0 (cid:34) log (cid:32) Λ (cid:0) γ ( (cid:96) ) (cid:1) Λ ∗ (cid:33) − q + 1 (cid:0) Λ (cid:0) γ ( (cid:96) ) (cid:1) − Λ ∗ (cid:1)(cid:35) . Since | γ ( (cid:96) ) − n ∗ | ≤ √ K log K one haslog (cid:32) Λ (cid:0) γ ( (cid:96) ) (cid:1) Λ ∗ (cid:33) = Λ (cid:0) γ ( (cid:96) ) (cid:1) − Λ ∗ Λ ∗ − (cid:0) Λ (cid:0) γ ( (cid:96) ) (cid:1) − Λ ∗ (cid:1) ∗ + O (cid:0) K − / (log K ) (cid:1) . Therefore,log (cid:32) Λ (cid:0) γ ( (cid:96) ) (cid:1) Λ ∗ (cid:33) − q + 1 (cid:0) Λ (cid:0) γ ( (cid:96) ) (cid:1) − Λ ∗ (cid:1) = (cid:0) Λ (cid:0) γ ( (cid:96) ) (cid:1) − Λ ∗ (cid:1) (cid:0) q + 1 − Λ ∗ (cid:1) Λ ∗ ( q + 1) − (cid:0) Λ (cid:0) γ ( (cid:96) ) (cid:1) − Λ ∗ (cid:1) ∗ + O (cid:0) K − / (log K ) (cid:1) . This implies using the binomial inequality that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log (cid:32) Λ (cid:0) γ ( (cid:96) ) (cid:1) Λ ∗ (cid:33) − q + 1 (cid:0) Λ (cid:0) γ ( (cid:96) ) (cid:1) − Λ ∗ (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:0) q + 1 − Λ ∗ (cid:1) q + 1) + (cid:0) Λ (cid:0) γ ( (cid:96) ) (cid:1) − Λ ∗ (cid:1) Λ ∗ ≤ (cid:0) Λ (cid:0) γ ( (cid:96) ) (cid:1) − Λ ∗ (cid:1) Λ ∗ + O (cid:0) K − (cid:1) . The result follows from Sublemma 8.1, the Lipschitz continuity of λ and the factthat λ ( x ∗ ) is independent of K . (These quantities are defined in (5.15).)Let us now estimate the contribution of Π q to S q ( n, m ). Sublemma 8.3. For all K large enough sup w ∈ ∆ (cid:48) , v ∈ ∆ , ≤ j ≤ d, q/ ≤ (cid:96) ≤ q (cid:88) u P v ( N q − (cid:96) − = u ) (cid:12)(cid:12) P w ( N (cid:96) = u ) − P w ( N (cid:96) = u ± e ( j ) ) (cid:12)(cid:12) (cid:107) u − n ∗ (cid:107)≤ O (1) K d/ . Proof. The reader can check that P ( N (cid:96) = n ) = 1(2 π ) d (cid:90) [ − π,π ] d e − i (cid:104) n,θ (cid:105) e − (cid:96)S ( θ ) d θ where S ( θ ) = − log (cid:32) d (cid:88) m =1 r m cos( θ m ) (cid:33) . n | P ( N (cid:96) = n ) − P ( N (cid:96) = n ± e ( j ) ) | ≤ π ) d (cid:90) [ − π,π ] d e − (cid:96)S ( θ ) (cid:12)(cid:12)(cid:12) sin θ j (cid:12)(cid:12)(cid:12) d θ. Next using a Taylor expansion of S around 0, we havesup q/ ≤ (cid:96) ≤ q sup n | P ( N (cid:96) = n ) − P ( N (cid:96) = n ± e ( j ) ) |≤ e −O (1) (log K ) + 1(2 π ) d e O (1)(log K )4 K (cid:90) (cid:107) θ (cid:107)≤ log K √ K e − q (cid:80) dm =1 r m θ m | θ j | d θ ≤ O (1) K d +12 . Thereforesup w ∈ ∆ (cid:48) , v ∈ ∆ , ≤ j ≤ d, q/ ≤ (cid:96) ≤ q (cid:88) u P v ( N q − (cid:96) − = u ) (cid:12)(cid:12) P w ( N (cid:96) = u ) − P w ( N (cid:96) = u ± e ( j ) ) (cid:12)(cid:12) (cid:107) u − n ∗ (cid:107)≤ O (1) K d +12 sup v ∈ ∆ sup q/ ≤ (cid:96) ≤ q (cid:88) u P v ( N q − (cid:96) − = u ) (cid:107) u − n ∗ (cid:107) = O (1) K d +12 sup v ∈ ∆ sup q/ ≤ (cid:96) ≤ q (cid:88) z P ( N q − (cid:96) − = z ) (cid:107) z + v − n ∗ (cid:107) ≤ O (1) K d where we used the triangle inequality and Cauchy-Schwarz inequality. Sublemma 8.4. For all K large enough and all q ∈ [Λ ∗ − √ K, Λ ∗ + √ K ]sup ≤ r ≤ d sup n ∈ ∆ (cid:48) m ∈ ∆ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) γ : γ (0)= n, γ ( q )= m sup ≤ (cid:96) ≤ q − | γ ( (cid:96) ) − n ∗ | < √ K log K q − (cid:89) (cid:96) (cid:48) =0 p ∗ (cid:0) γ ( (cid:96) (cid:48) ) , γ ( (cid:96) (cid:48) + 1) (cid:1) q − (cid:88) (cid:96) =0 d (cid:88) j =1 d (cid:88) r =1 K r j (cid:20) B j,r ( γ ( (cid:96) ) r − n ∗ r ) δ γ j ( (cid:96) +1) − γ j ( (cid:96) ) , + D j,r ( γ ( (cid:96) ) r − n ∗ r ) δ γ j ( (cid:96) +1) − γ j ( (cid:96) ) , − (cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ O (1) K − d/ . Proof. For all s ∈ N we define Q n ( N s = u ) = (cid:88) γ : γ (0)= n, γ ( s )= u sup ≤ (cid:96) ≤ s − | γ ( (cid:96) ) − n ∗ | < √ K log K s − (cid:89) (cid:96) (cid:48) =0 p ∗ (cid:0) γ ( (cid:96) (cid:48) ) , γ ( (cid:96) (cid:48) + 1) (cid:1) = E n (cid:16) { sup ≤ (cid:96) ≤ s − | N (cid:96) − n ∗ | < √ K log K } { N s = u } (cid:17) . (cid:88) γ : γ (0)= n, γ ( q )= m sup ≤ (cid:96) ≤ q − | γ ( (cid:96) ) − n ∗ | < √ K log K q − (cid:89) (cid:96) (cid:48) =0 p ∗ (cid:0) γ ( (cid:96) (cid:48) ) , γ ( (cid:96) (cid:48) + 1) (cid:1) q − (cid:88) (cid:96) =0 d (cid:88) j =1 d (cid:88) r =1 Kr j (cid:20) B j,r ( γ ( (cid:96) ) r − n ∗ r ) δ γ j ( (cid:96) +1) − γ j ( (cid:96) ) , + D j,r ( γ ( (cid:96) ) r − n ∗ r ) δ γ j ( (cid:96) +1) − γ j ( (cid:96) ) , − (cid:21) = q − (cid:88) (cid:96) =0 d (cid:88) j =1 d (cid:88) r =1 K (cid:88) | u − n ∗ | < √ K log K Q n ( N (cid:96) = u ) × (cid:20) B j,r ( u r − n ∗ r ) Q u + e ( j ) ( N q − (cid:96) − = m ) + D j,r ( u r − n ∗ r ) Q u − e ( j ) ( N q − (cid:96) − = m ) (cid:21) = q − (cid:88) (cid:96) =0 d (cid:88) j =1 d (cid:88) r =1 K (cid:88) | u − n ∗ | < √ K log K Q n ( N (cid:96) = u ) × (cid:104) B j,r ( u r − n ∗ r ) Q m ( N q − (cid:96) − = u + e ( j ) ) + D j,r ( u r − n ∗ r ) Q m ( N q − (cid:96) − = u − e ( j ) ) (cid:105) , where we used the reversibility property of ( N (cid:96) ) (cid:96) ∈ Z + under Q . Using Sublemmas8.7 and 8.8 to bound the corrections, it is enough to estimate q − (cid:88) (cid:96) =0 d (cid:88) j =1 d (cid:88) r =1 K (cid:88) u P n ( N (cid:96) = u ) × (cid:104) B j,r ( u r − n ∗ r ) P m ( N q − (cid:96) − = u + e ( j ) ) + D j,r ( u r − n ∗ r ) P m ( N q − (cid:96) − = u − e ( j ) ) (cid:105) . Using Sublemma 8.3 we have q/ − (cid:88) (cid:96) =0 d (cid:88) j =1 d (cid:88) r =1 K − (cid:88) u P n ( N (cid:96) = u ) × (cid:104) B j,r ( u r − n ∗ r ) P m ( N q − (cid:96) − = u + e ( j ) ) + D j,r ( u r − n ∗ r ) P m ( N q − (cid:96) − = u − e ( j ) ) (cid:105) = q/ − (cid:88) (cid:96) =0 d (cid:88) j =1 d (cid:88) r =1 K − (cid:88) u P n ( N (cid:96) = u ) × [ B j,r ( u r − n ∗ r ) + D j,r ( u r − n ∗ r ) ] P m ( N q − (cid:96) − = u ) + O (1) K d/ = O (1) K d/ . Indeed, for all 1 ≤ r ≤ d , we have (cid:80) dj =1 ( B j,r + D j,r ) = 0 since d (cid:88) j =1 B j ( x ) + D j ( x ) λ ( x ) = 1 . 45e also have q − (cid:88) (cid:96) =2 q/ d (cid:88) j =1 d (cid:88) r =1 K − (cid:88) u P n ( N (cid:96) = u ) × (cid:104) B j,r ( u r − n ∗ r ) P m ( N q − (cid:96) − = u + e ( j ) ) + D j,r ( u r − n ∗ r ) P m ( N q − (cid:96) − = u − e ( j ) ) (cid:105) = q − (cid:88) (cid:96) =2 q/ d (cid:88) j =1 d (cid:88) r =1 K − (cid:88) v P m ( N q − (cid:96) − = v ) × (cid:104) B j,r ( v r − e ( j ) r − n ∗ r ) P n ( N (cid:96) = v − e ( j ) ) + D j,r ( v r + e ( j ) r − n ∗ r ) P n ( N (cid:96) = v + e ( j ) ) (cid:105) . Using again Sublemma 8.3 and the same cancellation as before this is equal to O (1) K d/ + q − (cid:88) (cid:96) =2 q/ d (cid:88) j =1 K − (cid:88) v P m ( N q − (cid:96) − = v ) × (cid:104) − B j,j P n ( N (cid:96) = v − e ( j ) ) + D j,j P n ( N (cid:96) = v + e ( j ) ) (cid:105) = O (1) K d/ since sup u ∈ Z d sup (cid:96) ≥ q/ sup n ∈ ∆ (cid:48) P n ( N (cid:96) = u ) ≤ O (1) K d/ , using once again the local limit theorem. Sublemma 8.5. There exists a constant C (8 . > independent of K such thatfor K large enough and all q ∈ [Λ ∗ − √ K, Λ ∗ + √ K ] K d/ q ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) γ : γ (0)= n, γ ( q )= m sup ≤ (cid:96) ≤ q − | γ ( (cid:96) ) − n ∗ |≤√ K log K (cid:90) · · · (cid:90) t + ··· + t q − < q − (cid:89) (cid:96) =0 p ∗ (cid:0) γ ( (cid:96) ) , γ ( (cid:96) + 1) (cid:1) log Π q (cid:0) γ, t , . . . , t q − (cid:1) q − (cid:89) (cid:96) =0 d t (cid:96) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (8 . . Proof. By a similar computation as in the proof of Sublemma 8.2 we get q ! × (cid:88) γ : γ (0)= n, γ ( q )= m sup ≤ (cid:96) ≤ q − | γ ( (cid:96) ) − n ∗ |≤√ K log K (cid:90) · · · (cid:90) t + ··· + t q − < q − (cid:89) (cid:96) =0 p ∗ (cid:0) γ ( (cid:96) ) , γ ( (cid:96) + 1) (cid:1) × (cid:0) Λ (cid:0) m (cid:1) − Λ ∗ (cid:1) (cid:16) − q − (cid:88) (cid:96) =0 t (cid:96) (cid:17) q − (cid:89) (cid:96) =0 d t (cid:96) (cid:88) γ : γ (0)= n, γ ( q )= m sup ≤ (cid:96) ≤ q − | γ ( (cid:96) ) − n ∗ |≤√ K log K (cid:0) Λ (cid:0) m (cid:1) − Λ ∗ (cid:1) (cid:18) − q q ! 1( q + 1)! (cid:19) q − (cid:89) (cid:96) =0 p ∗ (cid:0) γ ( (cid:96) ) , γ ( (cid:96) +1) (cid:1) = (cid:88) γ : γ (0)= n, γ ( q )= m sup ≤ (cid:96) ≤ q − | γ ( (cid:96) ) − n ∗ |≤√ K log K (cid:0) Λ (cid:0) m (cid:1) − Λ ∗ (cid:1) q + 1 q − (cid:89) (cid:96) =0 p ∗ (cid:0) γ ( (cid:96) ) , γ ( (cid:96) + 1) (cid:1) . We have since m ∈ ∆ K − (cid:0) Λ (cid:0) m (cid:1) − Λ ∗ (cid:1) = (cid:2) λ (cid:0) m/K (cid:1) − λ (cid:0) n ∗ /K (cid:1)(cid:3) = O (cid:0) K − / (cid:1) . Since q is of order K we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Λ (cid:0) m (cid:1) − Λ ∗ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ O (cid:0) K − / (cid:1) . The result follows from (5.22) and from the local limit theorem applied to N .Let us finally estimate the contribution of Π q . Sublemma 8.6. There exists a constant C (8 . > independent of K such thatfor K large enough and all q ∈ [Λ ∗ − √ K, Λ ∗ + √ K ] K d/ q ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) γ : γ (0)= n, γ ( q )= m sup ≤ (cid:96) ≤ q − | γ ( (cid:96) ) − n ∗ |≤√ K log K (cid:90) · · · (cid:90) t + ··· + t q − < q − (cid:89) (cid:96) =0 p ∗ (cid:0) γ ( (cid:96) ) , γ ( (cid:96) + 1) (cid:1) log Π q (cid:0) γ, t , . . . , t q − (cid:1) q − (cid:89) (cid:96) =0 d t (cid:96) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (8 . . Proof. We have p (cid:0) γ ( (cid:96) ) , γ ( (cid:96) ) + e ( j ) (cid:1) = p ∗ (cid:0) γ ( (cid:96) ) , γ ( (cid:96) ) + e ( j ) (cid:1) + K − d (cid:88) r =1 B j,r ( γ ( (cid:96) ) r − n ∗ r ) + K − O (cid:0) | γ ( (cid:96) ) − n ∗ | (cid:1) where B j,r = ∂ x r (cid:18) B j ( x ) λ ( x ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) x = x ∗ . Similarly p (cid:0) γ ( (cid:96) ) , γ ( (cid:96) ) − e ( j ) (cid:1) = p ∗ (cid:0) γ ( (cid:96) ) , γ ( (cid:96) ) + e ( j ) (cid:1) + K − d (cid:88) r =1 D j,r ( γ ( (cid:96) ) r − n ∗ r ) + K − O (cid:0) | γ ( (cid:96) ) − n ∗ | (cid:1) D j,r = ∂ x r (cid:18) D j ( x ) λ ( x ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) x = x ∗ . We then havelog (cid:32) p (cid:0) γ ( (cid:96) ) , γ ( (cid:96) ) + e ( j ) (cid:1) p ∗ (cid:0) γ ( (cid:96) ) , γ ( (cid:96) ) + e ( j ) (cid:1) (cid:33) = 1 K r j d (cid:88) r =1 B j,r ( γ ( (cid:96) ) r − n ∗ r )+ K − O (cid:0) | γ ( (cid:96) ) − n ∗ | (cid:1) , with r j defined in (5.18), andlog (cid:32) p (cid:0) γ ( (cid:96) ) , γ ( (cid:96) ) − e ( j ) (cid:1) p ∗ (cid:0) γ ( (cid:96) ) , γ ( (cid:96) ) − e ( j ) (cid:1) (cid:33) = 1 K r j d (cid:88) r =1 D j,r ( γ ( (cid:96) ) r − n ∗ r )+ K − O (cid:0) | γ ( (cid:96) ) − n ∗ | (cid:1) . The result follows using Sublemma 8.4 and Sublemma 8.1. Sublemma 8.7. For all K large enough sup n ∈ ∆ (cid:48) sup q ≥ (cid:96)>q/ sup (cid:107) u − n ∗ (cid:107) < √ K log K | Q n ( N (cid:96) = u ) − P n ( N (cid:96) = u ) | (cid:107) u − n ∗ (cid:107) ≤ e −O (1)(log K ) . Proof. We havesup n ∈ ∆ (cid:48) sup q ≥ (cid:96)>q/ sup (cid:107) u − n ∗ (cid:107) < √ K log K | Q n ( N (cid:96) = u ) − P n ( N (cid:96) = u ) | (cid:107) u − n ∗ (cid:107)≤ √ K log K sup n ∈ ∆ (cid:48) sup q ≥ (cid:96)>q/ sup (cid:107) u − n ∗ (cid:107) < √ K log K | Q n ( N (cid:96) = u ) − P n ( N (cid:96) = u ) | . Using Bonferroni’s inequality we have (cid:12)(cid:12) Q n ( N (cid:96) = u ) − P n ( N (cid:96) = u ) (cid:12)(cid:12) ≤ (cid:96) (cid:88) s =1 E n (cid:104) {| N s − n ∗ | > √ K log K } { N (cid:96) = u } (cid:105) ≤ q (cid:88) s =1 d (cid:88) j =1 P n (cid:18) | ( N s ) j − n ∗ j | > √ K log Kd (cid:19) = q (cid:88) s =1 d (cid:88) j =1 P (cid:18) | ( N s ) j + n j − n ∗ j | > √ K log Kd (cid:19) ≤ e −O (1)(log K ) where the last inequality follows by applying Hoeffding’s inequality. Sublemma 8.8. For all K large enough sup n ∈ ∆ (cid:48) sup q ≥ (cid:96)>q/ (cid:88) (cid:107) u − n ∗ (cid:107)≥√ K log K P n ( N (cid:96) = u ) (cid:107) u − n ∗ (cid:107) ≤ e −O (1)(log K ) . roof. We havesup n ∈ ∆ (cid:48) sup q ≥ (cid:96)>q/ (cid:88) (cid:107) u − n ∗ (cid:107)≥√ K log K P n ( N (cid:96) = u ) (cid:107) u − n ∗ (cid:107)≤ sup n ∈ ∆ (cid:48) sup q ≥ (cid:96)>q/ d (cid:88) j =1 (cid:88) | u j − n ∗ j |≥√ K log K/ √ d P n ( N (cid:96) = u ) d (cid:88) k =1 | u k − n ∗ k | = sup n ∈ ∆ (cid:48) sup q ≥ (cid:96)>q/ d (cid:88) j =1 d (cid:88) k =1 (cid:88) | u j − n ∗ j |≥√ K log K/ √ d P n ( N (cid:96) = u ) | u k − n ∗ k |≤ e −O (1)(log K ) where the last estimate can be easily obtained by using again Hoeffding’s in-equality since N (cid:96) is a sum of independent identically distributed random vari-ables, distinguishing the cases j = k and j (cid:54) = k . A A difference between monotype and multi-type birth-and-death processes In the one-dimensional case [6], the method is based on the existence of a ref-erence measure on N ∗ and an associated (cid:96) space such that the generator L isself-adjoint in the space. This is not the case in dimension strictly larger thanone for a generator L defined by L f ( n ) = d (cid:88) j =1 λ j ( n ) (cid:0) f ( n + e ( j ) ) − f ( n ) (cid:1) + µ j ( n ) (cid:0) f ( n − e ( j ) ) − f ( n ) (cid:1) , where λ j , µ j : Z d + \{ } → R + , n ∈ Z d + and f : Z d + → R has finite support.For a positive measure π on Z d + \{ } and two functions f, g : Z d + \{ } → R with finite support, define (cid:104) g, f (cid:105) π = (cid:88) n ∈ Z d + \{ } f ( n ) g ( n ) π ( n ) . Proposition A.1. A positive measure π on Z d + \{ } satisfies (cid:104) g, L f (cid:105) π = (cid:104) L g, f (cid:105) π (A.1) for all f, g with finite support, if and only if π ( n ) λ j ( n ) = µ j (cid:0) n + e ( j ) (cid:1) π (cid:0) n + e ( j ) (cid:1) , ∀ j ∈ { , . . . , d } , ∀ n ∈ Z d + \{ } . Proof. Equation (A.1) will be satisfied for all functions with finite support ifand only if the equality is true for f = p and g = q for all p and all q in Z d + \{ } . The result follows immediately by direct computations.This proposition has a consequence which can be cast in terms of circuitsin Z d + \{ } . A circuit C is a path of the form (cid:0) n (1) , . . . , n ( k ) , n ( k +1) (cid:1) , for some49ositive integer k , such that n ( k +1) = n (1) and n ( (cid:96) +1) = n ( (cid:96) ) + ε ( j (cid:96) ) , for (cid:96) =1 , . . . , k , where ε j = ± e ( j ) , with the constraint that ε ( j ) + · · · + ε ( j k ) = 0. Nowdefine ρ ( ε ( j ) , n ) = λ j ( n ) µ j ( n + e ( j ) ) if ε ( j ) = e ( j ) µ j ( n ) λ j ( n − e ( j ) ) if ε ( j ) = − e ( j ) . Corollary A.2. There exists a positive measure π on Z d + \{ } such that (A.1) holds if and only if for all circuits C contained in Z d + \{ } one has k (cid:89) (cid:96) =1 ρ (cid:0) ε ( j (cid:96) ) , n ( (cid:96) ) (cid:1) = 1 . (A.2) Proof. The proof is elementary and left to the reader. It is enough to observethat for all n ∈ Z d + \{ } and all 1 ≤ j ≤ d , π (cid:0) n + ε ( j ) (cid:1) π ( n ) = ρ (cid:0) ε ( j ) , n (cid:1) . Condition (A.2) is always satisfied in dimension one, but it imposes verystringent conditions on the demographic parameters in higher dimensions. Thisis why we had to follow another route in the present work. Let us illustrate thisfact in dimension two with the following example modelling two populationswith both intra- and inter-specific competition: (cid:40) λ ( n , n ) = λ n , µ ( n , n ) = n ( µ + c n + c n ) λ ( n , n ) = λ n , µ ( n , n ) = n ( µ + c n + c n ) (A.3)where λ k , µ k , c k(cid:96) , k, (cid:96) = 1 , 2, are nonnegative parameters. In this case, condition(A.2) reads λ ( n , n ) µ ( n + 1 , n ) λ ( n + 1 , n ) µ ( n + 1 , n + 1) µ ( n + 1 , n + 1) λ ( n , n + 1) µ ( n , n + 1) λ ( n , n ) = 1for all ( n , n ) ∈ Z \{ (0 , } . 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