On time scales and quasi-stationary distributions for multitype birth-and-death processes

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On time scales and quasi-stationary distributions for multitype birth-and-death processes

J.-R Chazottes, P. Collet, S Méléard

To cite this version:

J.-R Chazottes, P. Collet, S Méléard. On time scales and quasi-stationary distributions for multi-

type birth-and-death processes. Annales de l’Institut Henri Poincaré (B) Probabilités et Statistiques,

Institut Henri Poincaré (IHP), 2019. �hal-02357308�

On time scales and quasi-stationary distributions for multitype birth-and-death processes

J.-R. Chazottes

, P. Collet

, and S. M´el´eard

1

Centre de Physique Th´eorique, CNRS UMR 7644, F-91128 Palaiseau Cedex (France)

2

Centre de Math´ematiques Appliqu´ees, CNRS UMR 7641, F-91128 Palaiseau Cedex (France)

Dated: February 5, 2019

Abstract R´esum´e en fran¸cais :

Nous consid´erons une classe de processus de naissance-et-mort d´ecrivant une population constitu´ee de d sous-populations de types diff´erents qui int´eragissent entre elles. L’espace d’´etat estZ^d+ (il est donc non born´e).

Nous supposons que la population s’´eteint presque sˆurement, de sorte que l’unique distribution de probabilit´e stationnaire est la masse de Dirac `a l’origine. Nous faisons d´ependre ces processus d’un param`etre d’´echelle K qu’on peut interpr´eter comme l’ordre de grandeur de la taille totale de la population au temps 0. Etant donn´e un intervalle de temps, il est bien connu que de tels processus, normalis´es par K, sont proches, dans la limite K → +∞, des solutions d’une certaine ´equation diff´erentielle dansR^d+ dont le champ de vecteurs est d´etermin´e par les taux de nais- sance et de mort du processus. Nous consid´erons le cas o`u le champ de vecteurs poss`ede un unique point fixe attractif `a l’int´erieur de l’orthant positif, tandis que l’origine est un point fixe r´epulsif. On s’attend `a ce que, pourK grand, le processus reste dans le voisinage du point fixe attractif pendant tr`es longtemps avant d’ˆetre absorb´e `a l’origine. Afin de d´ecrire pr´ecis´ement ce comportement, nous d´emontrons l’existence dune distribu- tion quasi-stationnaire (dqs, en abr´eg´e). Nous ´etablissons une borne pour la distance en variation totale entre le processus conditionn´e `a ne pas s’´eteindre avant le tempstet la dqs. Cette borne est exponentiellement petite entpourtlogK. En particulier, nous obtenons une estimation du temps moyen d’extinction dans la dqs. Nous quantifions ´egalement la distance entre le processus (non conditionn´e `a la non-extinction) et une certaine combinaison convexe de la masse de Dirac `a l’origine et de la dqs, ceci pour des temps beaucoup plus grands que logKet beaucoup plus pe- tits que le temps moyen d’extinction, qui est exponentiellement grand en K. Nous attirons l’attention sur le fait que nous sommes int´eress´es par ce

∗Email: [email protected]

†Email: [email protected]

‡Email: [email protected]

qui se passe pourKfini. Nous obtenons ainsi des r´esultats hors de port´ee des techniques de grandes d´eviations.

English version:

We consider a class of birth-and-death processes describing a popula- tion made of d sub-populations of different types which interact with one another. The state space isZ^d+ (unbounded). We assume that the population goes almost surely to extinction, so that the unique station- ary distribution is the Dirac measure at the origin. These processes are parametrized by a scaling parameterKwhich can be thought as the order of magnitude of the total size of the population at time 0. For any fixed finite time span, it is well-known that such processes, when renormalized byK, are close, in the limitK→+∞, to the solutions of a certain differ- ential equation inR^d+ whose vector field is determined by the birth and death rates. We consider the case where there is a unique attractive fixed point (off the boundary of the positive orthant) for the vector field (while the origin is repulsive). What is expected is that, forKlarge, the process will 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 prove the existence of a quasi-stationary distribution (qsd, for short). In fact, we establish a bound for the total variation distance between the process conditioned to non-extinction before timetand the qsd. This bound is exponentially small int, fort logK. As a by-product, we obtain an estimate for the mean time to extinction in the qsd. We also quantify how close is the law of the process (not conditioned to non-extinction) either to the Dirac measure at the origin or to the qsd, for times much larger than logKand much smaller than the mean time to extinction, which is exponentially large as a function ofK. Let us stress that we are interested in what happens for finiteK. We obtain results much beyond what large deviation techniques could provide.

Keywords: Markov jump process, differential equations, competition models, population ecology, mean time to extinction, Lyapunov functions.

Contents

1 Introduction 4

2 Setting and standing assumptions 6

2.1 A class of vector fields . . . 6

2.2 An example . . . 7

2.3 The stochastic process and its basic properties . . . 9

3 Statements of the main results 10 4 Some preparatory results 12 4.1 A Lyapunov function . . . 12

4.2 Lemma of the four domains . . . 14

5 Proof of Theorem 3.1 16 5.1 Plan for the proof: checking conditions (A1) and (A2) . . . 16

5.2 Proof of Condition (A1) . . . 17

5.3 Proof of Lemma 5.1 . . . 18

5.4 Proof of Lemma 5.2 . . . 24

5.5 Proof of Lemma 5.3 . . . 26

5.6 Proof of Condition (A2) . . . 30

5.7 Proof of Theorem 3.1 . . . 33

6 Proof of Theorem 3.2 34 6.1 Proof of the upper bound . . . 34

6.2 Proof of the lower bound . . . 36

7 Proofs of Theorems 3.3 and 3.4 38

8 Proof of Lemma 5.10 42

A A difference between monotype and multitype birth-and-death

processes 50

1 Introduction

A fundamental question in population ecology concerns the risk of extinction of populations [16]. Stochastic models are well suited to account for the inherently discrete nature of individuals, especially when populations are “small”. Such models are often referred to as “individual-based models”. In contrast, “large populations” are traditionally modelled by ordinary differential equations, when the spatial structure, the age-structure, the fluctuations of the environment, etc, are ignored. These “population-level” models are supposed to account for the deterministic trends of large populations (the macroscale), and are inherently incapable of describing extinction phenomena.

In the present work we consider birth-and-death processes (N^K(t), t ≥ 0) describing a population made of a finite number of sub-populations ofddifferent types which interact with one another. At each timet, the state of the process is thus given by a vector n = (n1, . . . , nd) ∈ Z^d+, where ni is the number of individuals of theith sub-population. We assume that these processes depend on a scaling parameterK >0 which can be thought as the order of magnitude of the total size of the population at time 0. More precisely, if at some timet,N^K(t) = n, the rate at which the population is increased (respectively decreased) by one individual of typej∈ {1, . . . , d} isKBj(n/K) (respectivelyKDj(n/K)).

On the one hand, keepingKfixed and lettingtgo 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 to model 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 it describes only the state where the population is extinct.

On the other hand, one can prove that the probability thatN^K(t)/K devi- ates, over any fixed finite time span, from the solution of the differential equation

dx

dt =B(x)−D(x) (1.1)

by more than some prescribed quantity, goes to zero, as K goes to +∞. In the previous equation x= (x1, . . . , xd) ∈R^d+, B(x) = (B1(x), . . . , Bd(x)) and D(x) = (D1(x), . . . , Dd(x)). Basically, our aim is to describe what happens “in between” 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 its basin 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 attracting fixed pointx^∗in the interior ofR^d+and the origin is a repelling fixed point. The big 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 fixed point x^∗ and will stay in the vicinity of the state bKx^∗c 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 the existence of a unique quasi-stationary distribution (qsd, for short). In fact, we prove a stronger result since we establish a bound for the total variation distance between the process conditioned to non-extinction before time t and the qsd.

This bound is exponentially small int, fortmuch larger than logK(see Theorem

3.1). Our second result is an upper bound and a lower bound for the mean time to extinction in the qsd. This mean time is exponential inK(ee Theorem 3.2).

Our third result quantifies how close, in total variation distance, the law of the process not conditioned to non-extinction, is to a convex combination of the Dirac measure at the origin and the qsd (see Theorem 3.3). Fort much larger than logK and much smaller than the mean time to extinction, this distance is very small. Then, fortmuch larger than exp(O(1)K), the law of the process not conditioned to non-extinction is very close to the Dirac measure at the origin.

Our fourth main result shows that the spectral gap of this semigroup is larger thanO(1)/logK, 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 process to “come down from infinity” and to arrive in a ball of radius of orderK and center bKx^∗c. This is contained in Sublemma 5.4. Afterwards, it takes a time of order logK to arrive in a ball of radius of order√

K and centerbKx^∗c(see Lemma 5.1). Then the process fluctuates around bKx^∗cfor 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 a certain self-adjoint operator acting on a suitable “weighted” Hilbert space. We obtained precise estimates, notably for the mean time to extinction, as well as the approximate behavior of the process in terms of a Gaussian distribution.

These spectral techniques in Hilbert spaces are lost whend≥2 since in general the generator cannot be made self-adjoint, as explained in Appendix A. Hence we are forced to follow a different route: we will exploit a theorem proved in [5]. This abstract theorem gives a necessary and sufficient condition for the exponential convergence, in total variation distance, of the process conditioned on non-extinction toward the quasi-stationary distribution. These conditions are of Doeblin type for submarkovian semigroups. In our setting, we have to verify these conditions and a substancial work we have to do is to obtain the precise dependence onK of the involved constants.

Let us mention the survey article [2] which describes how the so-called WKB method can be used to evaluate the mean time and/or probability of population extinction, fixation and switches resulting from either intrinsic (demographic) noise, etc. That article deals with much more general situations than the one we consider here, but the approach is “semi-rigorous” from the mathematical viewpoint. Let us also mention that there are other papers dealing with quanti- tative estimates of quasi-stationary distributions in contexts which are different from ours, namely [3] and [9, 10]. In particular, the state space is finite in those papers, and different methods are developed. We emphasize that, in the con- text of stochastic models in population ecology, taking a finite state space is not natural. Indeed, large fluctuations can arise in such a way that we “go out” of the state space.

The paper is structured as follows. In Section 2 we state the hypotheses we 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 Lyapunov function 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 takes to come down from one level set of the Lyapunov function to a lower one. We expect this quantitative result to be useful in more general situations. Section 5

is 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 enters a ball centered atn^∗with a radius of order√

K. Then we compare the process in this ball with an auxiliary symmetric random walk. In Section 3.1 we bound from above and below the parameter of the exponential law of the extinction time under the qsd. Section 7 is devoted to the proof of a lower bound of the spectal gap of the semigroup associated to the process.

2 Setting and standing assumptions

Throughout the paper, we will use the following notations. Elements of R^d+

will be denoted byx= (x1, . . . , xd), and those ofZ^d+ byn= (n1, . . . , nd). For x∈R^d+, we will denote by kxk its Euclidean norm, by|x| its`¹-norm, and by d(x, y) =kx−yk the Euclidean distance betweenxandy. The scalar product inR^dis denoted byh·,·i. Givenx∈R^d+ andr >0, the Euclidean ball of radius rand centerxis denoted by B(x, r).

2.1 A class of vector fields

Since we want the process to stay in the positive orthant, we naturally assume the normal component of D of R^d+ is zero on the boundary. We make the following hypotheses on the vector fieldsB,D andB−D.

• The vector fieldsB and Dare locally Lipschitz functions on R^d+, and Bj(x)≥0, Dj(x)≥0, ∀j∈ {1, . . . , d},∀x∈R^d+. (H0)

• The vector fieldsB and Dvanish only at the origin:

B(x) = 0 ⇐⇒ D(x) = 0 ⇐⇒ x= 0. (H1) The fixed point 0 of the vector fieldB−D is linearly unstable.

• There existsx^∗∈int(R^d+) such that

B(x^∗)−D(x^∗) = 0. (H2)

• There existβ >0 andR > L >0 such that

(i) kx^∗k< Rand for allx∈R^d+ such thatkxk< R

hB(x)−D(x), x−x^∗i ≤ −βkxkkx−x^∗k². (H3) (ii) Pd

j=1x^∗_j < Land B

x^∗,1

2 min

1≤j≤dx^∗_j

⊂

y∈R^d+:|y| ≤L ⊂ B(0, R). (H4) We will denote byPL the hyperplane defined by

d

X

j=1

xj=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.

• Moreover we assume thatLis such that sup

s>L

Bmax(s) Dmin(s) <1

2 (H5)

where

Dmin(s) = inf

|x|=s d

X

j=1

Dj(x) and Bmax(s) = sup

|x|=s d

X

j=1

Bj(x). (2.2)

• We assume thatDminis an eventually monotone function such that Z ∞

1

ds

Dmin(s) <+∞. (H6)

• There existsξ >0 such that

x∈infR^d+

1≤j≤dinf

Dj(x) sup<sub>1≤`≤d</sub>x`

> ξ >0. (H7)

• Finally, we assume that

1≤j≤dinf ∂xjBj(0)>0. (H8) (By∂xj we mean _∂^∂

xj.)

We now comment on the different hypotheses. Notice that, because of the Lipschitz property of the vector field, the polynomial on the right-hand side in (H3) is natural locally around 0 and x^∗. Hypothesis (H3) implies that the fixed point ofB−D is unique inR^d+∩ B(0, R)\{0}. Any trajectory starting in R^d+∩ B(0, R)\{0}converges tox^∗. The fixed point 0 is unstable. In particular, this implies that the faces ofR^d+ are not globally invariant by the flow. Notice also 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 the birth rates, this will be used together with Hypothesis (H6) to prove that the process “comes down from infinity”.

We will see that Hypothesis (H7) implies that the jump rate of the process is bounded below away from zero.

Hypothesis (H8) guarantees that the birth rate of the stochastic process is not identically 0 near the origin.

Finally, notice that Hypotheses (H2), (H3) (i), (H8) are open conditions in the C²-topology of vector fields. Colloquially, this means that if we slightly perturb the vector field, these hypotheses remain valid with slightly modified constants.

2.2 An example

We defineS(x) =Pd

j=1xj and for everyj∈ {1, . . . , d} Bj=λS , Dj =xj(µ+κS)

PL

x^∗

0 x2

x₁ R

infjx^∗_j/2

Figure 1: Illustration of Hypotheses (H3) and (H4)

where λ > µ/d > 0 and κ > 0. The non trivial fixed point x^∗ is given by x^∗_j =S^∗/d whereS^∗= (λd−µ)/κ. We have

hx−x^∗, B−Di=λS(S−S^∗)−(µ+κS)

kx−x^∗k²+ (S−S^∗)S^∗ d

=−κ

dS(S−S^∗)²−(µ+κS)

kx−x^∗k²−(S−S^∗)² d

. It is now convenient to use the decomposition

x=S d1 +y

where 1 is the vector with all components equal to 1, andy is orthogonal to 1.

We obtain (sincex^∗=S^∗1/d) hx−x^∗, B−Di=−κ

dS(S−S^∗)²−(µ+κS)kyk². Forxin the positive quadrant we havekxk ≤S, hence

kyk ≤S.

It is easy to verify that there exists a constant Γ >0 such that for all S ≥0 and allkyk ≤S

kxkkx−x^∗k²= r

kyk²+S² d

(S−S^∗)² d +kyk²

≤Γκ

dS(S−S^∗)²+ (µ+κS)kyk²

which implies Hypothesis (H3) (i) withβ = 1/Γ. Checking the other hypotheses is left to the reader.

Notice that one can construct many more examples by perturbating (in the C² sense) this example.

2.3 The stochastic process and its basic properties

We consider a birth-and-death process (N^K(t), t ≥ 0) on the d-dimensional integer lattice Z^d+. So, for each t ≥ 0, N^K(t) is a vector with dcomponents, that is, N^K(t) = (N^K)1(t), . . . ,(N^K)d(t)

. The birth and death rates of this process are given by KBj

n K

andKDj n K

,j = 1, . . . , d. Givenf :Z^d+ →R with finite support, the generator of the process is given by

(LKf) (n) = (2.3)

K

d

X

j=1

hBj

n K

f(n+e^(j))−f(n) +Dj

n K

f(n−e^(j))−f(n)i , wheree^(j)= (0, . . . ,0,1,0, . . . ,0), the 1 being at thej-th position.

Proposition 2.1. For each K >0, the process(N^K(t), t≥0) goes to 0 with probability one.

Proof. For a fixedK, the process Pd

j=1hN^K(t), e^(j)i, t≥0

can be stochasti- cally dominated by a monotype birth-and-death process with birth rateKBmax(m) and death rateKDmin(m) withm∈Z⁺ (see (2.2)). Hypotheses (H5) and (H7) imply that the process (N^K(t), t≥0) goes almost surely to 0 (see [15, Theorem 5.5.5]).

Under mild assumptions, one-parameter families of pure jump Markov pro- cesses can be approximated, in every finite time interval, by the solutions of a differential equation whose vector field is determined by the infinitesimal transi- tion rates. This is referred to as Kurtz’s theorem. In our framework, this result takes the following form.

Proposition 2.2([13, 14]). LetE⊆R^d+be an open bounded subset ofR^d+. Fix a bounded time interval

0, t

witht >0. Letx0∈E be such that the trajectory of the solutionx(t)of the differential equation

dx

dt =B(x)−D(x) (2.4)

with initial condition x0 belongs to E for allt∈[0, t]. If

K→+∞lim N^K(0)

K =x0

then, for everyε >0,

K→+∞lim P sup

t≤t

N^K(t) K −x(t)

> ε

!

= 0.

According to Propositions 2.1 and 2.2, we thus have the following picture.

On the one hand, forK fixed, the (total) population dies out with probability one in the limitt→+∞. On the other hand, for a fixed finite time span, the number of individuals in the population, when rescaled byK, is very close to the solution of the differential equation (2.4) in the limit K →+∞. The purpose of the present work is to describe the process for finite times and for finite K.

3 Statements of the main results

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≥0) in a subset A of Z^d+ is defined by

TA= inf{t >0 :N^K(t)∈A}. WhenA is a singleton, say{n}, we shall simply writeTn.

As usual,Pⁿ will denote the law of the process given thatN^K(0) =n, and, for a probability measureµonZ^d+ and a subsetA ofZ^d+,

Pµ(A) = X

m∈Z^d+

µ(m)Pm(A).

Our first main result is about quantifying the closeness, in total variation distance, of the process condioned to not being extinct before time t, and the quasi-stationary distribution. Recall that the total variation distance between two probability measuresµandν onZ^d+ is

kµ−νk^TV= sup

A∈P(Z^d+)

|µ(A)−ν(A)| whereP(Z^d+) is the powerset ofZ^d+.

Theorem 3.1. There existK0>1,0< c <1 and0< a < b <+∞such that the following result holds. For allK≥K0, there existt0(K)∈(alogK, blogK) and a unique probability measurem_K onZ^d+\{0}such that for every probability measureµon Z^d+\{0}, and for allt≥0, we have

kP^µ N^K(t)∈ · |t < T0

−mK(·)k^TV≤2(1−c)^bt/t0(K)c.

This theorem tells us that fortlogK, the process condioned to not being extinct before timetis very close to the quasi-stationary distributionm_K. Ast tends to +∞, we get a convergence of the process conditioned to non-extinction towards the quasi-stationary distribution.

By a general result on quasi-stationary distributions (see for instance [7]), one has

P^mK T0> t

= e^−λ0(K)t, t≥0, (3.1)

whereλ0(K) is a positive real number called the exponential rate of extinction.

In particular, the mean time to extinction, starting from the quasi-stationary distribution is

EmK[T0] = 1

λ0(K). (3.2)

The following theorem shows that the exponential rate of extinction is ex- ponentially small inK.

Theorem 3.2. There exists K0 >0 and two numbers d1 > d2 >0 such that for allK > K0

e^−d1K ≤λ0(K)≤e^−d2K. (3.3) Hence we get an estimate of the mean time to extinction (3.2):

e^d2K ≤E^mK[T0]≤e^d1K

for allK > K0. Whend= 1, a more precise estimate was proved in our previous work [6, Theorem 3.2].

Remark 3.1. The upper bound in (3.3)could be obtained by a large deviation asymptics for jump processes (see [4, Section 4.2]). Theorem 3.2 also provides a lower bound. In the present paper we are interested, among other things, in the different time scales for large K and not so much in their precise asymptotics.

The following theorem provides a quantitative bound for the distance (in total variation) between the law of the process and a convex combination of the quasi-stationary distribution and the Dirac measure at the origin.

Theorem 3.3. Let c and t0(K) be as in Theorem 3.1. There exist positive constants C(3.3),c(3.3), η(3.3),K0, such that for all t≥0 and allK > K0, for each n∈Z^d+\{0}, there exists a numberpK(n)∈(c,1]such that

sup

n∈Z^d+\{0}

Pn(N^K(t)∈ ·)−e^−λ0(K)tpK(n)m_K(·)− 1−e^−λ0(K)tpK(n) δ0(·)

TV

≤2 e^−η(3.3)Ke^−λ0(K)t+C(3.3)e^−ω(K)t (3.4)

where

ω(K) =−log(1−c)

t0(K) ≥ c(3.3)

logK.

Remark 3.2. Let us give the meaning of inequality (3.4) in two different regimes corresponding to two different time-scales. We assume that K is large enough to have e^−η(3.3)K 1. First notice that the right-hand side of (3.4)is 1 provided that t logK. Then, for logK t 1/λ0(K), (3.4) implies that

sup

n∈Z^d₊\{0}

Pⁿ(N^K(t)∈ ·)−pK(n)mK(·)− 1−pK(n) δ0(·)

TV

≤2 e^−η(3.3)Ke^−λ0(K)t+C(3.3)e^−ω(K)t+2(1−e^−λ0(K)t)1.

This means that, in that time span, the law of the process is close to a mixture of the Dirac measure at the origin and the quasi-stationary distribution with respective weights 1−pK(n) and pK(n). For t 1/λ0(K), (3.4) implies that the law of the process is close to the Dirac measure at the origin.

Let (P^Kt, t≥0) be the semigroup of the birth and death process killed at 0.

More precisely

P^Ktf(n) =Eⁿh

f(N^K(t))1{t<T0}

i

where f : Z^d+\{0} → R is any bounded measurable function. We now prove that the spectral gap of this semigroup is larger thanO(1)/logK, which is what we obtained in dimension one [6, Theorem 3.3].

Theorem 3.4. The resolvent of(P^Kt, t≥0) in the Banach space`^∞(Z^d+\{0}) is meromorphic in the set <z >−ω(K)with a unique simple pole at −λ0(K) with residue the one dimensional projectionπK given by

πK(f) =u_Km_K(f). The sequence u_K(n)

n∈Z^d₊\{0} is such thatm_K(uK) = 1, and, for all t≥0, P^Ktu_K= e^−λ0(K)tu_K.

Moreover, for all n∈Z^d+\{0},

c≤u_K(n)≤1 + e^−O(1)K,

wherecis defined in Theorem 3.1. In particular, the spectral gapω(K)−λ0(K) is bounded below by

c(3.3)

logK −e^−d2K.

Remark 3.3. We will see in the proofs that the weightspK(n)of Theorem 3.3 are equal touK(n)∧1.

4 Some preparatory results

4.1 A Lyapunov function

We first introduce the natural quantity n^∗=bKx^∗c

wich will appear throughout the article. Letϕ:Z^d+→R⁺ defined by

ϕ(n) = e^{Kαkn−n∗k2} (4.1)

where α > 0 is a parameter to be chosen later on. We now prove that under the previous assumptions and forαsmall enough, the functionϕis a Lyapunov function.

Theorem 4.1. There exist0 < α <1/2, K0 >0 and C(4.1) >0 such that for all K≥K0 and for alln∈ B(0, RK), we have

LKϕ(n)≤

−αβknk K

kn−n^∗k²

K +C(4.1)

ϕ(n) whereβ andR are defined in (H3).

Proof. We use the elementary fact that for all x∈R such that|x| ≤R there existsc1(R)>0 such that

0≤e^x−1−x≤C1(R)x². Then, for alln∈ B(0, RK) we get

LKϕ(n) ϕ(n)

=K

d

X

j=1

Bj

n K

ϕ(n+e^(j)) ϕ(n) −1

+Dj

n K

ϕ(n−e^(j)) ϕ(n) −1

=K

d

X

j=1

hBj

n

K expα

K(2(nj−n^∗_j) + 1)

−1 +Dj

n

K expα

K(−2(nj−n^∗_j) + 1)

−1i

=K

d

X

j=1

2α

Bj

n K

−Dj

n K

nj−n^∗_j K

+ Bj

n K

+Dj

n K

4C1(R)α²kn−n^∗k² K²

+ Bj

n K

+Dj

n K

α

K +2C1(R)(nj−n^∗_j)

K² +C1(R)α²

K² .

Using (H0) and (H1), there exists C2(R)>0 such that 0≤Bj

n K

+Dj

n K

≤C2(R)knk K

for all n ∈ B(0, RK). It is easy to verify that the third term in the square bracket is bounded in absolute value by a constant independent ofK provided Kis larger than someK0>0. The second term in the square bracket is bounded by

4dC2(R)α²knk K

kn−n^∗k²

K . (4.2)

We finally deal with the first term in the square bracket. Writing F =B−D for brevity, we obtain by (H3) that

2αK

d

X

j=1

Bj

n K

−Dj

n K

nj−n^∗_j K

= 2αK

Fn K

,n−n^∗ K

= 2αKD Fn

K ,n

K −x^∗E + 2αK

Fn

K ,

x^∗−n^∗

K

≤ −2α βKknk K

n K −n^∗

K

2

+O(1)

≤ −2αβknk K

kn−n^∗k²

K +O(1),

where we used that x^∗−ⁿK^∗

≤ K¹ and F _Kⁿ

is bounded on B(0, R), and where O(1) is a quantity uniformly bounded in K. To finish the proof, we choose αsmall enough in such a way that the prefactor 4dC2(R)α² in (4.2) is less than half of 2αβ.

Corollary 4.2. There existK0>0 and two constantsρ(4.2)>0 andc(4.2)>4 such that, for all K≥K0 and for allc(4.2)≤ knk ≤RK satisfying

kn−n^∗k ≥ρ(4.2)

√K we have

LKϕ(n)≤ −αβ 2

knk K

kn−n^∗k²

K ϕ(n).

Proof. We choosec(4.2) andρ(4.2)large enough such that fornas in the state- ment, ^αβ₂ ^knk_K ^kn−n_K^∗k2 > C(4.1).

Remark 4.1. The intuitive rate of decrease αβ

2 knk

K

kn−n^∗k² K

of the Lyapunov function, given by Corollary 4.2, is uniformly bounded below by the constant C(4.1), if c(4.2)≤ knk ≤RK andkn−n^∗k ≥ρ(4.2)

√K. However, if knk andkn−n^∗k are of orderK, this rate is also of orderK. We will later take advantage of this non uniformity of the rate by a suitable decomposition of the setZ^d+\{0} ∩ {knk ≥c(4.2)}.

4.2 Lemma of the four domains

In this section, we formulate a lemma and a corollary of it which will help us to take advantage of the decomposition of the space Z^d+. We could formulate it in a much more abstract setting. SinceK plays no role here, we drop theK dependence, henceN(t) stands forN^K(t),LforLK, etc.

Lemma 4.3. Let D⁻²,D⁻¹,D⁰,D¹ be subsets ofZ^d+\{0} such that D¹ D⁰ D⁻¹ D⁻² Z^d+\{0},

with D⁻² a compact subset. Next, let

H⁻²=D⁻²\D⁻¹, H⁻¹=D⁻¹\D⁰, H⁰=D⁰\D¹. (See Figure 2.) Assume that for all n∈ H⁰ we have

Pⁿ TH−2<∞

= 1, and

H⁻²∩ D¹=∅ and {n:d(n,H⁰∪ H⁻¹) = 1} ⊂ D¹∪ H⁻². Assume that there exists a positive functionψ defined in Z^d+\{0} such that

Λ :=− sup

H−2∪ H−1∪ H0

Lψ(n) ψ(n) >0.

D1

H0

H−1

H−2

Figure 2: The four domains

Let

a0= sup

n∈H₀ψ(n), a⁰⁰₋₂= inf

n∈H−2

ψ(n) and a⁰₋₁= inf

n∈H−1∪ H0

ψ(n).

Assume that a0/a⁰⁰₋₂<1. Then

n∈Hinf0Pⁿ TD1 ≤t , TH−2 > TD1)≥1− a0

a⁰⁰₋₂− a0

a⁰₋₁ e^−Λt.

Note thata0/a⁰₋₁≥1. In practice we will use for H⁻² some kind of outer boundary of D⁻¹.

Proof. Using Dynkin’s formula, we have for a path issued fromn∈ H⁰ e^Λ(^t∧TD1∧T_H−2) ψ N(t∧TD1∧TH−2)

=

Z t∧TD1∧T_H−2 0

e^Λs Λψ(N(s)) +Lψ(N(s))

ds+M(t∧TD1∧TH−2) where M(t∧TD1∧TH−2)

t≥0is a martingale. Using the assumptions and the fact thatψis bounded bya0 onH⁰ we obtain

En

he^Λ(^t∧TD1∧T_H−2) ψ N(t∧TD1∧TH−2)i

≤ψ n

≤a0. (4.3) Sinceψis positive we deduce that

a0≥Eⁿ

ψ N(t∧TD1∧TH−2)

≥En

hψ N(t∧TD1∧TH−2)

1{TD1≥T_H−2}1{T_H−2≤t}

i

=En

hψ N(TH−2)

1{TD1≥T_H−2}1{T_H−2≤t}

i

≥a⁰⁰₋₂Pⁿ TD₁ ≥TH−2, TH−2 ≤t .

Lettingttend to infinity and using our hypothesis (and Lebesgue’s dominated convergence theorem) we get that for alln∈ H⁰

Pⁿ TH−2 ≤TD1

≤ a0

a⁰⁰₋₂. Using again (4.3) we also have that for alln∈ H⁰

Eⁿh

e^Λtψ N(t)

1{^TH−2>TD1>t} i≤a0, which implies that for allt≥0

Pⁿ TH−2 > TD1> t

≤ a0

a⁰₋₁e^−Λt. We have for alln∈ H⁰

Pn TD1 ≤t , TH−2 > TD1) =Pn TH−2 > TD1)−Pn TD1 > t , TH−2> TD1)

= 1−Pn TH−2≤TD1)−Pn TH−2 > TD1 > t).

The lemma follows from the above estimates.

Corollary 4.4. Under the assumptions of Lemma 4.3 we have

n∈Hinf0Pⁿ TD1 ≤tD1, TH−2 > TD1)≥1−ηD1

with

tD1 = 1 Λ log

a⁰⁰₋₂ a⁰₋₁

and ηD1 = 2a0

a⁰⁰₋₂. The estimate also holds with

ηD₁= 1 2 + a0

2a⁰⁰₋₂ and

tD1 =−1 Λ log

a⁰₋₁ 2a0

1− a0

a⁰⁰₋₂

.

5 Proof of Theorem 3.1

5.1 Plan for the proof: checking conditions (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. Suppose there exists a probability measureν onE such that

• There existt0, c1>0 such that Pn N^K(t0)∈ · |t0< T0

≥c1ν(·), ∀n∈Z^d+\{0}. (A1)

• There existsc2>0 such that

P^ν(t < T0)≥c2Pⁿ(t < T0), ∀n∈Z^d+\{0}, ∀t≥0. (A2)

Then there exists a unique quasi-stationary distributionm_K such that for every initial distributionµ,

kP^µ(N^K(t)∈ · |t < T0)−mK(·)k^TV ≤2(1−c1c₂)^t/t0.

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 c1 and c2 are independent ofK, and thatt0is of order logK.

5.2 Proof of Condition (A1)

Let

∆ =B n^∗,2ρ(4.2)

√K

, (5.1)

whereB(n, r) denotes the ball centered innwith radiusrandρ(4.2)the constant introduced in Corollary 4.2. Sincen^∗is of orderK, the set ∆ is included in the interior ofZ^d+ forK large enough.

Notation. We shall denote byν the uniform probability measure supported on

∆.

This discrete measure thus gives each point of ∆ a mass proportional to K^−d/2.

The proof of Condition (A1) relies on the following three lemmas whose proofs are given later on.

The first lemma shows that the descent (from infinity) into the set ∆ happens with a time scale of at most logK.

Lemma 5.1. There exist C(5.1) > 0 and η(5.1) <1 such that for all K large enough

n∈∆inf^cPⁿ T∆< C(5.1)logK

≥1−η(5.1).

The second lemma shows that on a time span of order logK, the process starting in ∆ stays near ∆, more precisely in a ball with a radius of order √

K centered atn^∗.

Lemma 5.2. There exists C(5.2) >2ρ(4.2) and η(5.2) <1 such that for all K large enough

n∈∆inf inf

0≤t≤C_(5.1)logK+1Pⁿ N^K(t)∈∆⁰

≥1−η(5.2)

where

∆⁰ =B n^∗, C(5.2)

√K

⊃∆.

The third lemma says that the probability measureνis a significant compo- nent of the distribution of the process at time 1 starting near ∆. This lemma does not seem to be available in the literature. The main difference with existing results (see for instance [8]) is that our generator is not symmetric.

Lemma 5.3. There exists η(5.3) <1 such that for all K large enough and all A⊂∆

n∈∆inf⁰ Pn N^K(1)∈A

≥(1−η(5.3))ν(A), where∆⁰ is defined in Lemma 5.2.

Proof of Condition(A1). Applying the three preceding lemmas, we can prove that condition (A1) holds forK large enough with

c1= (1−η(5.1)) (1−η(5.2)) (1−η(5.3))<1, (5.2) t0=t0(K) = 1 +C(5.1)logK.t0(K) (5.3) Indeed, for all n∈Z^d+\{0} and for allA⊂∆ we can write

Pⁿ N^K(t0)∈A

=Eⁿ

1^A N^K(t0)

≥Eⁿh

1{T∆<C_(5.1)logK}1^A N^K(t0)i . Now by the Markov property we have

Pⁿ N^K(t0)∈A

≥Eⁿh

1{T∆<C_(5.1)logK}EN^K(T∆)

1^A N^K(t0−T∆)i

≥En

h1{T∆<C_(5.1)logK}EN^K(T_∆)

1∆⁰ N^K(t0−T∆−1)

YA,∆(t0)i , where

YA,∆(t0) =EN^K(t0−T∆−1)

1^A N^K(1) . Using successively Lemma 5.3, Lemma 5.2 and Lemma 5.1 we get

Pⁿ N^K(t0)∈A

≥(1−η(5.1)) (1−η(5.2)) (1−η(5.3))ν(A). (5.4) Since 0 is an absorbing point we have Pn N^K(t0)∈A, T0≤t0

= 0, and using the trivial estimatePⁿ T0> t0

≤1 we get Pⁿ N^K(t0)∈A

T0> t0

≥(1−η(5.1)) (1−η(5.2)) (1−η(5.3)) Pⁿ T0> t0

ν(A)

≥(1−η(5.1)) (1−η(5.2)) (1−η(5.3))ν(A).

Thus we have proved that Condition (A1) holds.

5.3 Proof of Lemma 5.1

The proof of Lemma 5.1 is based on the fine description of the trajectories of the process. For this purpose, we need to introduce a decomposition ofZ^d+\{0} according to the different time scales at which the process goes down from infinity to ∆.

Let

R∗= 1 2

R+ sup

y∈P_L∩R^d+

ky−x^∗k , wherePL is the hyperplane defined in (2.1).

Note thatR∗< R by hypothesis (H4). We define the sets E¹=

n∈Z^d+\{0}:

d

X

j=1

nj> LK

H−5=

n∈Z^d+\{0}:R∗K≤ kn−n^∗k< RK H−4=

n∈Z^d+\{0}:

d

X

j=1

nj> LK , kn−n^∗k< R∗K

H−3=

n∈Z^d+\{0}:

d

X

j=1

nj≤LK ,kn−n^∗k ≥ kn^∗k −c(4.2)

H−2=

n∈Z^d+\{0}:kn^∗k −(c(4.2)+ 4)≤ kn−n^∗k<kn^∗k −c(4.2)

H−1=

n∈Z^d+\{0}:kn^∗k −(c(4.2)+ 8)≤ kn−n^∗k<kn^∗k −(c(4.2)+ 4) H0=

n∈Z^d+\{0}:kn^∗k −(c(4.2)+ 12)≤ kn−n^∗k<kn^∗k −(c(4.2)+ 8) E²=

n∈Z^d+\{0}:knk< c(4.2)+ 17 .

These sets are well-defined provided thatK is large enough.

The proof of Lemma 5.1 will result from a series of sublemmas which quantify the probability of coming down from infinity and crossing the various level sets of the Lyapunov function.

Sublemma 5.4. There exist two constants t(5.4)>0 andη(5.4)<1 (indepen- dent of K) such that forK large enough

n∈Einf₁Pn TE₁^c≤t(5.4)

≥1−η(5.4). Proof. The process Pd

j=1hN^K(t), e^(j)i, t ≥ 0

can be coupled with a one- dimensional birth-and-death process (Z(t), t ≥ 0) with birth rate Λ(m) = KBmax m

K

and death rateM(m) =KDmin m K

. The coupling is such that Z(t)≥

d

X

j=1

hN^K(t), e^(j)i if Z(0)≥

d

X

j=1

hN^K(0), e^(j)i.

Let us introduce pK=bLKcand denote byTbpK its hitting time. We are going to prove thatAK:= sup_p>pKE^p(Tbp_K) is bounded uniformly inK. As shown in [18, p.384] or in [1, Chap.3], one has

AK=

∞

X

m=pK+1

1 M(m)+

∞

X

i=m+1

Λ(m)· · ·Λ(i−1) M(m)· · ·M(i)

! . By assumption (H5), forq≥pK, Λ(q)/M(q)≤1/2. Then

AK≤

∞

X

m=pK+1

1 M(m)+

∞

X

i=m+1

2^m−i M(i)

!

≤2

∞

X

m=pK+1

1 M(m),

where we have interchanged the order of the sums to get the second inequality.

By Hypothesis (H6), we know that 1

K

∞

X

m=p_K+1

1

Dmin(^m_K) −−−−→

K→∞

Z ∞ L

ds

Dmin(s) <+∞.

E

n

0 n

n

E

H

H

H

H

H

H

P

Figure 3: The various subsets whend= 2 whenK is large enough.

Then there existsK0 such that for allK≥K0, for allp≥pK, we have Ep(TbpK)≤3

Z ∞ L

ds Dmin(s). The result follows by Markov inequality with

t(5.4)= 6 Z ∞

L

ds

Dmin(s) and η(5.4)=1 2.

Sublemma 5.5. There exist two constants t(5.5)>0 andη(5.5)<1 (indepen- dent of K) such that forK large enough

n∈H−3∪infH−2∪H−1Pⁿ TH₀ ≤t(5.5)

≥1−η(5.5).

Proof. We define D−2=

n∈Z^d+\{0} : kn−n^∗k< RK D−1=

n∈Z^d+\{0} : kn−n^∗k< R∗K D0=







n∈Z^d+\{0} :

d

X

j=1

nj ≤LK





 D1=

n∈Z^d+\{0} : kn−n^∗k<kn^∗k −(c(4.2)+ 8) . (5.5) We now apply Corollary 4.4 with Dⁱ = Di, i = −2,−1,0,1. For K large enough and using (H4), the Lyapunov function ϕdefined in Theorem 4.1 and the geometry of the sets, we have

a⁰⁰₋₂

a⁰₋₁ ≤e^O(1)K, a0

a⁰⁰₋₂ < 1 4. Moreover we have

Λ =O(1)K

by Theorem 4.1. The result follows since H−3 ∪ H−2 ∪ H−1 = D0\D1 and since forKlarge enough,D1can be reached fromD0\D1 only throughH0.

We need a specific estimate near 0.

Sublemma 5.6. There exists η(5.6) <1 (independent of K) such that for K large enough

n∈Einf2\D1

Pⁿ TH₀ ≤1

≥1−η(5.6).

Proof. For all n∈ E²\D1, for allj ∈ {1, . . . , d}, there existss≤17, such that n+se^(j)∈H0. Sincen6= 0, there existsj0 withnj₀ >0.

Let

V =

m(t), t0= 0,∃t1<1

s, . . . , ts<1

s such that m(t) =n+q e^(j0),∀tq ≤t < tq+1,0≤q≤s−1 .

Let us compute the probability for the birth and death process to belong toV. Note that by assumption

KBj₀

n K

=

d

X

`=1

n`∂x<sub>`</sub>Bj₀(0) + O 1

K

and

KDj₀

n K

=

d

X

`=1

n`∂x<sub>`</sub>Dj₀(0) +O 1

K

.

Therefore, forK large enough, the birth probability of an individual with type j0 is bounded below by

n∈Einf2

KBj(_Kⁿ) KPd

`=1B`(_Kⁿ) +KPd

`=1D`(_Kⁿ)

> 1 2 inf

n∈E2

∂xj0Bj0(0) Pd

`=1n`∂x_`Bj0(0) +Pd

`=1n`∂x_`Dj0(0) =ζ,

and ζ >0 by (H8) and since max1≤`≤dn` ≤17 for n∈ E². Note also that the denominator (which is the jump rate) is bounded below byζ⁰= infj∂xjBj(0)>

0 by (H8). Therefore,

Pⁿ(N^K∈ V)≥ζ^s

1−e^−ζ0/ss

≥ζ¹⁷

1−e^−ζ0/1717 . The results follows.

In the following lemma we will partition more finely the disk D1 to fit as well as possible the speed of decrease of the distance between the process and n^∗.

Sublemma 5.7. There exists two constantst(5.7)>0 andη(5.7)<1 such that forK large enough

n∈Dinf₁\∆P T∆≤t(5.7) logK

≥1−η(5.7), where∆ is defined in (5.1)andD1 in (5.5).

Proof. We start by defining a decreasing (finite) sequence of numbers (Rj) as follows:

R−2=kn^∗k −c(4.2), R−1=kn^∗k −(c(4.2)+ 4), R0=kn^∗k −(c(4.2)+ 8), R1=kn^∗k −(c(4.2)+ 12).

Define

j∗= inf

j:R1−2^j−1+ 1≤1 2 inf

` n<sup>∗</sup><sub>`</sub>

. Note thatj∗=O(1) logK. For 2≤j≤j∗ we define

Rj=R1−2^j−1+ 1.

Note that for 1≤j≤j∗,Rj ≥Rj^∗=O(1)K. Define j∗∗= sup{j > j^∗:Rj^∗2^{−(j−j∗)}> ρ(4.2)

√K} −1.

Note thatj∗∗=O(1) logK. Forj^∗≤j≤j^∗∗+ 1, let Rj =Rj^∗2^{−(j−j∗)}. Note that ρ(4.2)

√K ≤Rj^∗∗−1 ≤2ρ(4.2)

√K and that for j ≤j^∗, B(n^∗, Rj)⊂ B(0,kn^∗k/2)^c. We now define a (finite) decreasing sequence of domains (Dj), where−2≤j ≤j∗∗+ 1, by

Dj =B(n^∗, Rj)∩Z^d+\{0}.

We also define a finite sequence of annuli (Hj), where −2≤j≤j∗∗, by Hj=Dj\Dj+1.

Recall that the Lyapunov function ϕ has been defined in Theorem 4.1. We define the following sequences of positive numbers:

(Aj)−2≤j≤j∗∗ by

Aj= sup

n∈H_jϕ(n)