A non-conservative Harris ergodic theorem
Vincent Bansaye, Bertrand Cloez, Pierre Gabriel, Aline Marguet
aa r X i v : . [ m a t h . A P ] F e b A NON-CONSERVATIVE HARRIS ERGODIC THEOREM
VINCENT
BANSAYE , BERTRAND
CLOEZ , PIERRE
GABRIEL , AND ALINE
MARGUETAbstract.
We consider non-conservative positive semigroups and obtain necessary and suf-ficient conditions for uniform exponential contraction in weighted total variation norm. Thisensures the existence of Perron eigenelements and provides quantitative estimates of the spec-tral gap, complementing Krein-Rutman theorems and generalizing probabilistic approaches. Theproof is based on a non-homogenous h -transform of the semigroup and the construction of Lya-punov functions for this latter. It exploits then the classical necessary and sufficient conditionsof Harris’s theorem for conservative semigroups and recent techniques developed for the study ofabsorbed Markov processes. We apply these results to population dynamics. We obtain exponen-tial convergence of birth and death processes conditioned on survival to their quasi-stationarydistribution, as well as estimates on exponential relaxation to stationary profiles in growth-fragmentation PDEs. Contents
1. Introduction 12. Definitions and main result 52.1. Assumptions and criterion for exponential convergence 62.2. Sufficient conditions: drift and irreducibility 73. Quantitative estimates 73.1. The embedded conservative propagator 83.2. Estimation of the eigenelements 93.3. Uniform exponential convergence 94. Proofs 104.1. Preliminary inequalities 104.2. Contraction property: proofs of Section 3.1 114.3. Eigenelements: proofs of Section 3.2 134.4. Proofs of Section 3.3 184.5. Proofs of Section 2 205. Applications 235.1. Convergence to quasi-stationary distribution 235.2. The growth-fragmentation equation 245.3. Comments and a few perspectives 296. Appendix 316.1. Conservative operators 316.2. Localization argument 326.3. The growth-fragmentation semigroup 33Acknowledgments 36References 36 Introduction
Iteration of a positive linear operator is a fundamental issue in operator analysis, linear PartialDifferential Equations (PDEs), probabilities, optimization and control. In continuous time, theassociated structure is a positive semigroup. A positive semigroup is a family of linear operators( M t ) t ∈ R + acting on a space of measurable functions f on X . It satisfies M f = f , M t + s f = M t ( M s ( f )) for any t, s ≥
0, and M t f ≥ f ≥ t ≥
0. In finite dimension, i.e.
Date : February 23, 2021.2020
Mathematics Subject Classification.
Primary 47A35; Secondary 35B40, 47D06, 60J80, 92D25.
Key words and phrases.
Positive semigroups; measure solutions; ergodicity; Krein-Rutman theorem; branchingprocesses; growth-fragmentation; quasi-stationary distribution.
BANSAYE , BERTRAND
CLOEZ , PIERRE
GABRIEL , AND ALINE
MARGUET when X is a finite set, under a strong positivity assumption, the Perron-Frobenius theorem [47, 78]ensures the existence of unique right and left positive eigenvectors h = ( h ( x ) : x ∈ X ) and γ = ( γ ( x ) : x ∈ X ) associated to the maximal eigenvalue λ ∈ R : M t h = e λt h and γM t = e λt γ .Moreover, the asymptotic profile when t → + ∞ is given by M t f = e λt h γ, f i h + O (e ( λ − ω ) t ) , for any vector f , any ω > h γ, f i = P x ∈X γ ( x ) f ( x ).The generalization in infinite dimension has attracted lots of attention. It is motivated in par-ticular by the asymptotic analysis of linear PDEs counting the density of particules (or individuals) u xt ( y ) in location y at time t when initially the particles are located in x , i.e. u x = δ x . The semigroupis then given by M t f ( x ) = R X u xt ( y ) f ( y ) dy . Semigroups play also a key role in the study of theconvergence in law of Markov processes via M t f ( x ) = E x [ f ( X t )] or the study of quasi-stationaryregimes via M t f ( x ) = E x [ f ( X t ) X t ], where D is an absorbing domain. In spectral theory, theKrein-Rutman theorem [63] yields an extension to positive compact operators for the existence ofeigenelements. Generalizations have been then obtained [29, 56, 75]. The most recents providingasymptotic profiles with exponential convergence associated to a spectral gap, see in particular[74] and the references therein. When positive eigenelements exist, an alternative tool to proveconvergence is the dissipation of entropy, especially the general relative entropy introduced in [73].Our aim is to relax and simplify some assumptions involved in the spectral approach and pro-vide more quantitative results. We apply them to two classical models: growth fragmentation PDEsand birth and death processes.For that purpose, we are inspired and use techniques developed in probabilities. When M t = for any t ≥
0, the semigroup is said to be conservative . Indeed, the total number of particlesis preserved along time for the corresponding linear PDE or for the first moment semigroup ofthe branching Markov process. It holds when particles move without reproduction or death. Con-servative semigroups arise classically in the study of the law of Markov processes ( X t ) t ≥ via M t f ( x ) = E x [ f ( X t )]. In that case, the ergodic theory and stability of Markov processes can beinvoked to analyze the long time behavior of the semigroup, in the spirit of the pioneering worksof Doeblin [36] and Harris [55] and coupling techniques. We refer to [71] for an overview. Theseresults allow proving the existence of a stationary probability measure and the uniform exponen-tial convergence for the weighted total variation distance towards this latter. They rely on the twofollowing conditions, corresponding respectively to Lyapunov function and small set property, forsome time τ > K of X . First, there exists V : X → [1 , + ∞ ) such that M τ V ≤ a V + b K (1.1)for some constants a ∈ (0 ,
1) and b ≥
0. Furthermore, there exists a probability measure ν on X such that for any non-negative f and any x ∈ K , M τ f ( x ) ≥ c ν ( f ) (1.2)with c >
0. Under aperiodicity condition, these assumptions ensure that M τ admits a uniqueinvariant probability measure γ and satisfies a contraction principle for the weighted total variationdistance, see [62, 71, 72]. In our study, we will assume that the probability measure ν is supportedby the small set K , thus guaranteeing aperiodicity. Under mild conditions of local boundedness, thecontraction property captures the continuous time by iteration. Then there exist explicit constants C, ω > x ∈ X , f such that | f | ≤ V , and all t ≥ (cid:12)(cid:12) M t f ( x ) − γ ( f ) (cid:12)(cid:12) ≤ CV ( x )e − ωt , where γ ( f ) = R X f dγ . It provides a necessary and sufficient condition and quantitative estimates[39, Chapter 15] and is referred to as V -uniform ergodicity [71]. Main results and ideas of the proof.
In the present work we provide a counterpart to V -uniform ergodicity in the non-conservative framework. We obtain necessary and sufficient con-ditions for exponential convergence. We also estimate the constants involved, in particular theamplitude of the spectral gap, using only the parameters in the assumptions. The approach relies NON-CONSERVATIVE HARRIS ERGODIC THEOREM 3 on the reduction of the problem to a non-homogeneous but conservative one.In this vein, the Doob h -transform is a historical and inspiring technique and we mention [30, 37,38, 59, 75, 81, 83]. In the case when we already know some positive eigenfunction h associated withthe eigenvalue λ , a conservative semigroup is indeed defined for t ≥ P t f = M t ( hf )e λt h . The ergodic analysis of the conservative semigroup P using the theory mentioned above yields theexpected estimates on the original semigroup M t f = e λt h P t ( f /h ).Our work focuses on the case when we do not know a priori the existence of positive eigenele-ments or need more information on it to apply Harris’s theory. We simultaneously prove their ex-istence and estimate them. The novelty of the approach mainly lies in the form of the h -transformand the general and explicit construction of Lyapunov functions for this latter. It also stronglyexploits recent progresses on the convergence to quasistationnary distribution. More precisely, weintroduce the transform for s ≤ u ≤ t : P ( t ) s,u f = M u − s ( f M t − u ψ ) M t − s ψ . For a fixed time t and positive function ψ , this family is a conservative propagator (or semiflow).To mimic the stabilizing property of the eigenfunction h in the h -transform, we assume that ψ satisfies M τ ψ ≥ βψ, (1.3)for some β > τ >
0, similarly as in [24, 27, 74]. Section 5 provides examples where ψ canbe obtained, sometimes explicitly, while we do not know a priori eigenelements. We construct afamily of Lyapunov functions ( V k ) k ≥ as follows V k = ν (cid:18) M kτ ψψ (cid:19) VM kτ ψ , where ν is a probability measure. Assuming that V satisfies M τ V ≤ αV + θ K ψ, (1.4)these functions allow us to prove that P ( t ) satisfies Lyapunov type condition (1.1). To prove that P ( t ) also satisfies the small set condition (1.2) and invoke the contraction of the conservativeframework, we follow techniques due to Champagnat and Villemonais [23] for the study for theconvergence of processes conditioned on non-absorption. This yields the following generalized smallset condition. We assume that there exist c, d > x ∈ K , M τ ( f ψ )( x ) ≥ c ν ( f ) M τ ψ ( x ) (1.5)for any f positive and M nτ ψ ( x ) ≤ d ν (cid:18) M nτ ψψ (cid:19) ψ ( x ) (1.6)for any integer n . The two latter conditions involve uniform but local bounds on the space X . Thefirst inequality is restricted to a fixed time. Convenient sufficient condition can be given to restrictalso the second one to a finite time estimate, in particular via coupling, see the next paragraphand applications.Assuming that a = α/β < V ≥ ψ , we obtain a contraction result for P ( t ) in forthcomingProposition 3.3. By estimating the mass M t ψ , we then prove that a triplet ( λ, h, γ ) of eigenelementsexists and provide estimates on these elements. We get that for all x ∈ X and f such that | f | ≤ V , (cid:12)(cid:12) e − λt M t f ( x ) − h ( x ) γ ( f ) (cid:12)(cid:12) ≤ CV ( x )e − ωt , with constants ω > C >
0, that are explicit. We also prove that the conditions given aboveare actually necessary for this uniform convergence in weighted total variation distance.
VINCENT
BANSAYE , BERTRAND
CLOEZ , PIERRE
GABRIEL , AND ALINE
MARGUET
Tractable conditions and applications.
We apply our results to describe the profile ofpopulations in growth-fragmentation PDEs and birth and death processes conditioned on non-absorption. We also refer to the follow-up paper [26] for an application to mutation-selectionPDEs.Sufficient conditions are helpful for such issues. In particular, one can check Lyapunov condi-tions (1.3) and (1.4) using the generator. Loosely speaking, writing L the generator of ( M t ) t ≥ ,we check that it is enough to find V, ψ and a < b such that L V ≤ aV + ζψ and L ψ ≥ bψ. (1.7)The second part of the assumptions deals with local bounds on the semigroup. It is worth noticingthat in the case 0 < inf K ψ ≤ sup K ψ < ∞ , Assumption (1.5) is equivalent to the small setcondition (1.2). For a discrete state space X , the conditions (1.5) and (1.6) can be reduced toirreducibility properties. In the continuous setting, a coupling condition implying (1.6) is proposedand applied in [26]: M τ f ( x ) ψ ( x ) ≥ c Z τ M τ − s f ( y ) ψ ( y ) σ x,y ( ds )for all x, y ∈ K , f ≥ c , τ > σ x,y ) x,y ∈ K on [0 , τ ].This condition is stronger than Assumption (1.6), but it is easier to check for several populationmodels.Let us briefly illustrate the interest of the conditions above and the novelty of the resultfor applications on the so-called growth-fragmentation equation. This equation is central in themodeling of various physical or biological phenomena [3, 4, 40, 70, 79, 82]. In its classical form itcan be written as ∂ t u t ( x ) + ∂ x u t ( x ) + B ( x ) u t ( x ) = Z B (cid:16) xz (cid:17) u t (cid:16) xz (cid:17) ℘ ( dz ) z , where t, x >
0, and is complemented with a zero flux boundary condition u t (0) = 0. It drives thetime evolution of the population density u t of particles characterized by a structural size x ≥ x grows with speed one and splits with rate B ( x ) to produce smallerparticles of sizes zx , with 0 < z < ℘ . Thecorresponding dual equation reads ∂ t ϕ t ( x ) = L ϕ t ( x ) , where L f ( x ) = f ′ ( x ) + B ( x ) (cid:18)Z f ( zx ) ℘ ( dz ) − f ( x ) (cid:19) . It generates a semigroup ( M t ) t ≥ on the state space X = [0 , ∞ ) as follows: M t f = ϕ t is thesolution to the dual equation with initial condition ϕ = f . The duality property ensures therelation R X u t ( x ) f ( x ) dx = R X u ( x ) M t f ( x ) dx for all time t ≥
0. A direct computation ensures thatthe infinitesimal generator L verifies (1.7) with V : x x k , k > ψ : x x. Consequently the semigroup ( M t ) t ≥ satisfies (1.3)-(1.4). Using the reachability of any sufficientlylarge size and a monotonicity argument on B , we can show (1.5)-(1.6). It gives the existence ofeigenelements and the exponential convergence. It improves the existing results we know, wherea polynomial bound is assumed for B . It additionally provides estimates on the spectral gap. Werefer the reader to Section 5.2 for details and more references. State of the art and related works . Our results and method are linked to the study ofgeometric ergodicity of Feynman-Kac type semigroups M t f ( x ) = E x h f ( X t )e R t F ( X s ) ds i . The study of these non-conservative semigroups has been developed in a general setting [60, 61].We refer also to [33, 34] in discrete time. These semigroups appear in particular for the study ofbranching processes [15, 25, 68] and large deviations [44, 60, 61]. In [60] the function F is supposedto be bounded, with a norm smaller than an explicit constant. The unbounded case is addressed NON-CONSERVATIVE HARRIS ERGODIC THEOREM 5 in [61] where some weighted norm of F is required to be small, together with either a “densityassumption” or some irreducibility and aperiodicity conditions. The growth-fragmentation modelwould correspond to F = B and we relax here the assumptions on B . Nevertheless, we observethat the criteria in [60, 61] may be simpler to apply, and under the density assumption the authorsprove a stronger result of discrete spectrum.In the same vein, let us mention [45], which uses the Krein-Rutman theorem for the existenceof eigenelements and an h -transform. It requires additional regularity assumptions to derive thegeometric ergodicity of general Feynman-Kac semigroups, as well as a stronger small set condition.The regularity requirement is a local strong Feller assumption. It is well-suited for diffusive equa-tions, see also [58] for a result on non-conservative hypoelliptic diffusions. It does not seem to beadapted to our motivations with less regularizing effect.In [14, 15], a Feynman-Kac approach is developed for growth-fragmentation equation with exponen-tial individual growth. It relies on the application of the Krein-Rutman theorem on an well-suitedoperator. It provides sharp conditions on the growth rate but it requires the division rate B tobe bounded. We expect that our results provide quantitative estimates and relax the boundednessassumption on B in [14, 15] for some relevant classes of growth rates.Finally, Hilbert metric and Birkhoff contraction yield another powerful method for analysis ofsemigroup, which has been well developed [16, 76, 84]. As far as we see, it requires uniform boundson the whole space for the semigroup at fixed time, which are relaxed by the backward approachexploited here.More generally, our work is motivated by the study of structured populations and varyingenvironment. The conditions are well adapted to complex trait spaces for populations, includingdiscrete and continuous components: space, age, phenotype, genotype, size, etc., see Section 5 and e.g. [8, 9, 26, 67]. The contraction with explicit bounds allows relevant compositions appearingwhen parameters of the population dynamic vary along time. Such a method has already beenexploited in [8] for the analysis of PDEs in varying environment in the case of uniform exponentialconvergence. Several models need the more general framework considered in this paper. Indeed,the typical trait does not come back in compact sets in a bounded time for models like growth-fragmentation and mutation-selection. Outline of the paper.
We stress that the paper is self-contained. It is structured as follows.In Section 2, we gather the assumptions, the necessary and sufficient condition in weighted totalvariation distance and useful sufficient conditions. In Section 3.1, we consider the conservativeembedded propagator and we establish its contraction property. We present in Section 3.2 ourresults on the existence and the estimations of the eigenelements and in Section 3.3 complementaryconvergence estimates. Section 4 contains the proofs of these statements. Section 5 is devoted toapplications: the random walk on integers absorbed at 0 and the growth-fragmentation equation.We finally mention briefly some additional results and perspectives, including the discrete timeframework and the reducible case.2.
Definitions and main result
We start by stating precisely our framework. Let X be a measurable space. For any measurablefunction ϕ : X → (0 , ∞ ) we denote by B ( ϕ ) the space of measurable functions f : X → R whichare dominated by ϕ, i.e. such that the quantity k f k B ( ϕ ) = sup x ∈X | f ( x ) | ϕ ( x )is finite. Endowed with this weighted supremum norm, B ( ϕ ) is a Banach space. Let B + ( ϕ ) ⊂ B ( ϕ )be its positive cone, namely the subset of nonnegative functions.Let M + ( ϕ ) be the cone of positive measures on X which integrate ϕ, i.e. the set of positivemeasures µ on X such that the quantity µ ( ϕ ) = R X ϕ dµ is finite. We denote by M ( ϕ ) = M + ( ϕ ) −M + ( ϕ ) the set of the differences of measures that belong to M + ( ϕ ). When inf X ϕ >
0, this set
VINCENT
BANSAYE , BERTRAND
CLOEZ , PIERRE
GABRIEL , AND ALINE
MARGUET is the subset of signed measures which integrate ϕ . When inf X ϕ = 0 it is not a subset of signedmeasures, but in all cases it can be rigorously built as the quotient space M ( ϕ ) = M + ( ϕ ) × M + ( ϕ ) (cid:30) ∼ where ( µ , µ ) ∼ (˜ µ , ˜ µ ) if µ + ˜ µ = µ + ˜ µ , and we denote µ = µ − µ to mean that µ isthe equivalence class of ( µ , µ ). The Hahn-Jordan Decomposition Theorem ensures that for any µ ∈ M ( ϕ ) there exists a unique couple ( µ + , µ − ) ∈ M + ( ϕ ) × M + ( ϕ ) of mutually singular measuressuch that µ = µ + − µ − . An element µ of M ( ϕ ) acts on B ( ϕ ) through µ ( f ) = µ + ( f ) − µ − ( f ) and M ( ϕ ) is endowed with the weighted total variation norm k µ k M ( ϕ ) = µ + ( ϕ ) + µ − ( ϕ ) = sup k f k B ( ϕ ) ≤ | µ ( f ) | , which makes it a Banach space.We are interested in semigroups ( M t ) t ≥ of kernel operators, i.e. semigroups of linear operators M t which act both on B ( ϕ ) (on the right f M t f ) and on M ( ϕ ) (on the left µ µM t ), arepositive in the sense that they leave invariant the positive cones of both spaces, and enjoy theduality relation ( µM t )( f ) = µ ( M t f ). We work in the continuous time setting, t ∈ R + , and werefer to Section 5.3 for comments on the discrete time case. Finally, we use the shorthand notation f . g when, for two functions f, g : Ω → R , there exists a constant C > f ≤ Cg on Ω . Assumptions and criterion for exponential convergence.
We consider two measurablefunctions V : X → (0 , ∞ ) and ψ : X → (0 , ∞ ), with ψ ≤ V , and a positive semigroup ( M t ) t ≥ ofkernel operators acting on B ( V ) and M ( V ). Let us state Assumption A , which gathers conditionsgiven in Introduction. Assumption A.
There exist τ > β > α > θ >
0, ( c, d ) ∈ (0 , , K ⊂ X and ν a probabilitymeasure on X supported by K such that sup K V /ψ < ∞ and(A1) M τ V ≤ αV + θ K ψ ,(A2) M τ ψ ≥ βψ ,(A3) inf x ∈ K M τ ( f ψ )( x ) M τ ψ ( x ) ≥ c ν ( f ) for all f ∈ B + ( V /ψ ),(A4) ν (cid:18) M nτ ψψ (cid:19) ≥ d sup x ∈ K M nτ ψ ( x ) ψ ( x ) for all positive integers n .Assumption A is linked and relax classical assumptions for the ergodicity of non-conservativesemigroups [8, 24, 45, 58, 60, 61, 74, 87, 88].More precisely, one can compare A to the assumptions (1-4) of [74, Theorem 5.3], where (2) corre-sponds to (A2), (1) has strong connections with (A1) (see [91]), while (4) is relaxed. Besides, thesmall set condition (A3) is more general than strong positivity or irreducibility assumptions.Assumption A provides an extension of the conditions of [8] in the homogeneous case. In that lat-ter, small set condition (A3) was required on the whole space X , which imposes uniform ergodicity.Uniformity does not hold in the two applications we consider in the present paper.Our assumptions are similar and inspired from [24], which is dedicated to convergence to quasi-stationary distribution. Our technique differs, especially for the form of the embedded propagatorand the construction of Lyapunov functions, see Section 3.1. We relax the boundedness of ψ re-quired in [24] and obtain necessary conditions for weighted exponential convergence. As far as wesee, this approach also provides more quantitative estimates, see forthcoming Section 3 and theproofs.The main result of the paper can be stated as follows. It is proved in Section 4.5, where constantsinvolved are explicit. To state the result in the continuous time setting, we need that M t V ( x ) /V ( x )is bounded on compact time intervals, uniformly on X . We refer to [71, Section 20.3] for details onthis classical assumption. NON-CONSERVATIVE HARRIS ERGODIC THEOREM 7
Theorem 2.1.
Assume that t
7→ k M t V k B ( V ) is locally bounded on [0 , ∞ ) .i) Let ( V, ψ ) be a couple of measurable functions from X to (0 , ∞ ) which satisfies Assumption A .Then, there exists a unique triplet ( γ, h, λ ) ∈ M + ( V ) × B + ( V ) × R of eigenelements of M with γ ( h ) = k h k B ( V ) = 1 satisfying for all t ≥ , γM t = e λt γ and M t h = e λt h. (2.1) Moreover, there exist
C, ω > such that for all t ≥ and µ ∈ M ( V ) , (cid:13)(cid:13) e − λt µM t − µ ( h ) γ (cid:13)(cid:13) M ( V ) ≤ C k µ k M ( V ) e − ωt . (2.2) ii) Assume that there exist a positive measurable function V, a triplet ( γ, h, λ ) ∈ M + ( V ) × B + ( V ) × R , and constants C, ω > such that (2.1) and (2.2) hold. Then, the couple ( V, h ) satisfies Assump-tion A . Sufficient conditions: drift and irreducibility.
Assumptions (A1)-(A2) can be checkedmore easily using the generator L of the semigroup ( M t ) t ≥ . We give convenient sufficient conditionsby adopting a mild formulation of L = ∂ t M t | t =0 , similar to [52]. For F, G ∈ B ( V ) we say that L F = G if for all x ∈ X the function s M s G ( x ) is locally integrable, and for all t ≥ M t F = F + Z t M s G ds.
In general for F ∈ B ( V ), there may not exist G ∈ B ( V ) such that L F = G, meaning that F is notin the domain of L . Therefore we relax the definition by saying that L F ≤ G, resp. L F ≥ G, if for all t ≥ M t F − F ≤ Z t M s G ds, resp. M t F − F ≥ Z t M s G ds.
We can now state the drift conditions on L guaranteeing the validity of Assumptions (A1)-(A2).It will be useful for the applications in Section 5. For convenience, we use the shorthand ϕ ≃ ψ tomean that ψ . ϕ . ψ , i.e. that the ratios of the two functions are bounded. Proposition 2.2.
Let
V, ψ, ϕ : X → (0 , ∞ ) such that ψ ≤ V and ϕ ≃ ψ. Assume that there existconstants a < b and ζ ≥ , ξ ∈ R such that L V ≤ aV + ζψ, L ψ ≥ bψ, L ϕ ≤ ξϕ. Then, for any τ > , there exists R > such that ( V, ψ ) satisfies (A1)-(A2) with K = { V ≤ R ψ } . We provide now a sufficient condition for (A3)-(A4).
Proposition 2.3.
Let K be a finite subset of X and assume that there exists τ > such that forany x, y ∈ K , δ x M τ ( { y } ) > . Then (A3)-(A4) are satisfied for any positive function ψ ∈ B ( V ) . This sufficient condition is relevant for the study of irreducible processes on discrete spaces.We refer to Section 5.1 for an application to the convergence to quasi-stationary distribution ofbirth and death processes. As a motivation, let us also mention the study of the first momentsemigroup of discrete branching processes in continuous time and the exponential of denumerablenon-negative matrices. For continuous state space, the irreducibility condition above is not relevantand we refer to [26] for more general conditions to check (A4) via a coupling argument.3.
Quantitative estimates
To exploit the conservative theory, we consider a relevant conservative propagator associatedto M . We then derive the expected estimates for the eigenelements and obtain the quantitativeestimates for the original semigroup M . VINCENT
BANSAYE , BERTRAND
CLOEZ , PIERRE
GABRIEL , AND ALINE
MARGUET
The embedded conservative propagator.
Let us fix a positive function ψ ∈ B ( V ) anda time t >
0. For any 0 ≤ s ≤ u ≤ t , we define the operator P ( t ) s,u acting on bounded measurablefunctions f through P ( t ) s,u f = M u − s ( f M t − u ψ ) M t − s ψ . (3.1)We observe that the family P ( t ) = ( P ( t ) s,u ) ≤ s ≤ u ≤ t is a conservative propagator (or semiflow), mean-ing that for any 0 ≤ s ≤ u ≤ v ≤ t , P ( t ) s,u P ( t ) u,v = P ( t ) s,v . It has a probabilistic interpretation in terms of particles systems, see e.g. [68] and references below.Roughly speaking, it provides the position of the backward lineage of a particle at time t sampledwith a bias ψ .The particular case ψ = corresponds to uniform sampling and has been successfully used inthe study of positive semigroups, see [6, 8, 23, 24, 32, 68]. Whenever possible, the right eigenfunctionof the semigroup provides a relevant choice for ψ . Indeed, if h is a positive eigenfunction, theconservative semigroup P ( t ) associated to ψ = h is homogeneous and does not depend on t : P s = P ( t ) u,u + s = M s ( h · ) / e λs h . This corresponds to the h -transform given in Introduction. Thistransformation provides a powerful tool for the analysis of branching processes and absorbedMarkov process, see e.g. respectively [25, 42, 64] and [28, 37]. To shed some light on the sequel, let usexplain how to apply Harris’s ergodic theorem ([53] or Theorem 6.1 in Appendix 6.1 with W = V )to ( P s ) s ≥ and get the asymptotic behavior of M . Inequality (6.1) for P τ reads P τ V ≤ a V + c andyields M τ ( V h ) ≤ α V h + θh , with α = a e λτ , θ = c e λτ . This inequality involving M τ is guaranteedby (A1) by setting V = V h and ψ = h. Additionally, Equation (6.2) corresponds exactly to (A3).In this paper, we deal with the general case and consider a positive function ψ satisfying (A2).The analogy with the h -transform above suggests to look for Lyapunov functions of the form V = V /ψ . The family of functions (
V /M kτ ψ ) k ≥ satisfies, under Assumption (A1), an extendedversion of the Lyapunov condition for P ( t ) . But their level sets may degenerate as k goes to infinity,which raises a problem to check the small set condition. We compensate the magnitude of M kτ ψ and consider for k ≥ V k = ν (cid:18) M kτ ψψ (cid:19) VM kτ ψ . (3.2)The two following lemmas, which are proved in Section 4.2, ensure that ( V k ) k ≥ provides Lyapunovfunctions whose sublevel sets are small for P ( t ) . Lemma 3.1.
For all integers k ≥ and n ≥ m ≥ k + 1 , we have P ( nτ ) kτ,mτ V n − m ≤ a V n − k + c , where a = αβ ∈ (0 , , c = θc ( β − α ) ≥ . (3.3) Lemma 3.2.
Let R > . There exists a family of probability measures { ν k,n , k ≤ n } such that forall ≤ k ≤ n − p and x ∈ { V n − k ≤ R } ,δ x P ( nτ ) kτ, ( k + p ) τ ≥ b ν k,n , where p ∈ N and b ∈ (0 , are given by p = (cid:22) log (cid:16) R ( α + θ )( β − α ) d (cid:17) log( β/α ) (cid:23) + 1 and b = d β c ( α/θ + 1)( αR + θ ) 1 P ℓj =1 ( a /cr ) j (3.4) with R = sup K V /ψ and r = ( β/ ( α ( R + θ/ ( β − α )) + θ )) . We can now state the key contraction result. Its proof lies in the two previous lemmas and aslight adaptation of Harris theorem provided in Appendix.
NON-CONSERVATIVE HARRIS ERGODIC THEOREM 9
Proposition 3.3.
Let ( V, ψ ) be a couple of measurable functions from X to (0 , ∞ ) satisfyingAssumption A . Let R > c / (1 − a ) , b ′ ∈ (0 , b ) , a ′ ∈ ( a + 2 c / R , and set κ = b ′ c , y = max (cid:26) − ( b − b ′ ) , κ Ra ′ κ R (cid:27) . Then, for any µ , µ ∈ M + ( V /ψ ) and any integers k and n such that ≤ k ≤ n − p , (cid:13)(cid:13) µ P ( nτ ) kτ, ( k + p ) τ − µ P ( nτ ) kτ, ( k + p ) τ (cid:13)(cid:13) M (1+ κV n − k − p ) ≤ y k µ − µ k M (1+ κV n − k ) . Estimation of the eigenelements.
Using the notations introduced in the previous section,we set now ρ = max { y , a p } = max (cid:26) − ( b − b ′ ) , κ Ra ′ κ R , a p (cid:27) ∈ (0 , , (3.5)with b ′ ∈ (0 , b ) and a ′ ∈ ( a + 2 c / R , . We first deal with the right eigenelement.
Lemma 3.4.
There exists h ∈ B + ( V ) and λ ∈ R such that for all t ∈ [0 , ∞ ) , M t h = e λt h. Moreover, (cid:18) ψV (cid:19) q ψ . h . V, with q = log( cr )log( a ) > and there exists C > such that for all integer k ≥ and µ ∈ M + ( V ) , (cid:12)(cid:12)(cid:12)(cid:12) µ ( h ) − µM kτ ψν ( M kτ ψ/ψ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C µ ( V ) µ ( ψ ) ρ ⌊ k/ p ⌋ . (3.6)Let us turn to the left eigenmeasure and provide a similar result. Lemma 3.5.
There exists γ ∈ M + ( V ) such that γ ( h ) = 1 and for all t ∈ [0 , ∞ ) , γM t = e λt γ. Moreover, there exists
C > such that for all integer k ≥ and µ ∈ M + ( V ) , (cid:13)(cid:13)(cid:13)(cid:13) γ ( · ψ ) γ ( ψ ) − µM kτ ( · ψ ) µM kτ ( ψ ) (cid:13)(cid:13)(cid:13)(cid:13) M (1+ κV ) ≤ C (cid:18) µ ( V ) µ ( ψ ) + θβ − α (cid:19) ρ ⌊ k/ p ⌋ . (3.7)Using Lemma 3.5, ψ ≤ V , (A1) and (A2) yields e λτ γ ( V ) = γM τ V ≤ ( α + θ ) γ ( V ) and e λτ γ ( ψ ) = γM τ ψ ≥ βγ ( ψ ). It gives the following estimate of the eigenvaluelog( β ) τ ≤ λ ≤ log( α + θ ) τ . (3.8)Theorem 2.1 follows from Lemmas 3.4 and 3.5. Indeed, existence and estimates on the eigenelementsallow checking that ( V, h ) satisfies Assumption A . Then (3.7) with ψ replaced by h yields (2.2).Details are given in Section 4.5.3.3. Uniform exponential convergence.
We give here complementary results about the con-vergence to the profile given by the eigenelements. The first result provides a more explicit rate ofconvergence than in Theorem 2.1, to the price of loosing the V -uniformity. It requires the followingadditional assumption to capture the continuous time: t sup x ∈X M t V ( x ) V ( x ) and t sup x ∈X ψ ( x ) M t ψ ( x ) are locally bounded on [0 , ∞ ) . (3.9) Theorem 3.6.
Under Assumptions A and (3.9) , there exists C > such that for all µ ∈ M + ( V ) and t ≥ , (cid:13)(cid:13) µM t − e λt µ ( h ) γ (cid:13)(cid:13) M ( V ) ≤ C µ ( V ) µ ( ψ ) e − σt min (cid:8) µM t ψ, µ ( V )e λt (cid:9) , where σ = − log ρ p τ > . BANSAYE , BERTRAND
CLOEZ , PIERRE
GABRIEL , AND ALINE
MARGUET
Finally, the renormalisation of the semigroup by its mass M t may also be relevant for applica-tions. We mention the study of the convergence of the conditional probability to a quasi-stationarydistribution and the study of the typical trait in a structured branching process, see respectivelySection 5.1 and e.g. [8, 67, 68]. We write P ( V ) for the set of probability measures which belongto M ( V ) and we define the total variation norm for finite signed measures by k µ k TV = k µ k M ( ) .The following result is direct a corollary of Theorem 2.1. Corollary 3.7.
Assume that conditions of Theorem 2.1-(i) hold and inf X V > . Then there exist C, ω > and π ∈ P ( V ) such that for every µ ∈ M + ( V ) and t ≥ , (cid:13)(cid:13)(cid:13)(cid:13) µM t µM t − π (cid:13)(cid:13)(cid:13)(cid:13) TV ≤ C µ ( V ) µ ( h ) e − ωt . (3.10)4. Proofs
Preliminary inequalities.
For all t ≥
0, let us define the following operator c M t : f M t ( K c f ) , and for convenience, we introduce the following constantsΘ = θβ − α , R = sup K Vψ ,
Ξ = α ( R + Θ) + θ, (4.1)which are well-defined and finite under Assumption A . We first give some estimates which aredirectly deduced from Assumption A . Lemma 4.1.
For all k ≥ , we havei) c M kτ M τ V ≤ α k M τ V, ii) for all µ ∈ M + ( V ) , µM kτ VµM kτ ψ ≤ a k µ ( V ) µ ( ψ ) + Θ , iii) for all x ∈ K and n ≥ k , M nτ ψ ( x ) ≤ Ξ k M ( n − k ) τ ψ ( x ) , iv) for all x ∈ K , and f ∈ B + ( V /ψ ) , M ( k +1) τ ( f ψ ) ( x ) ≥ c k +1 ν ( f ) M ( k +1) τ ψ ( x ) , with c k +1 = c k +1 ( β/ Ξ) k . (4.2) Remark 4.2.
We observe that ( c k ) k ≥ is a decreasing geometric sequence. Indeed since ψ ≤ V ,(A1) and (A2) ensure that on K , βψ ≤ M τ ψ ≤ M τ V ≤ ( αR + θ ) ψ, so that β < Ξ. Adding that c < c k ) k ≥ decreases geometrically.Points i ) and ii ) of Lemma 4.1 are sharp inequalities , while iv ) extends Assumption (A3) for anytime. Proof.
Using (A1) we readily have K c M τ V ≤ αV and i ) follows by induction.Composing respectively (A1) and (A2) with M kτ yields M ( k +1) τ V ≤ αM kτ V + θM kτ ψ ; M ( k +1) τ ψ ≥ βM kτ ψ. Combining these inequalities gives M ( k +1) τ VM ( k +1) τ ψ ≤ a M kτ VM kτ ψ + θβ and ii ) follows by induction recalling that a < R , we immediately deduce from ii ) that for any x ∈ K , M kτ V ( x ) M kτ ψ ( x ) ≤ R + Θ . Combining this inequality with M nτ ψ ≤ M ( n − τ M τ V ≤ M ( n − τ ( αV + θψ ) , NON-CONSERVATIVE HARRIS ERGODIC THEOREM 11 coming from (A1) and ψ ≤ V , yields for x ∈ K , M nτ ψ ( x ) ≤ Ξ M ( n − τ ψ ( x ) . The proof of iii ) iscompleted by induction.Finally, let x ∈ K . We have M ( n +1) τ ( f ψ ) ( x ) M ( n +1) τ ψ ( x ) = M τ ( M nτ ( f ψ ))( x ) M ( n +1) τ ψ ( x ) ≥ cν (cid:18) M nτ ( f ψ ) ψ (cid:19) M τ ψ ( x ) M ( n +1) τ ψ ( x ) , using (A3) with the function M nτ ( f ψ ) /ψ . Besides, since ν is supported by K , (A3) and (A2) yield ν (cid:18) M nτ ( f ψ ) ψ (cid:19) = ν (cid:18) M τ ( M ( n − τ ( f ψ )) ψ (cid:19) ≥ cν (cid:18) ν (cid:18) M ( n − τ ( f ψ ) ψ (cid:19) M τ ψψ (cid:19) ≥ cβν (cid:18) M ( n − τ ( f ψ ) ψ (cid:19) . Iterating the last inequality and plugging it in the previous one, we obtain M ( n +1) τ ( f ψ ) ( x ) M ( n +1) τ ψ ( x ) ≥ c n +1 β n M τ ψ ( x ) M ( n +1) τ ψ ( x ) ν ( f ) . Moreover, for x ∈ K , we have M ( n +1) τ ψ ( x ) ≤ Ξ n M τ ψ ( x ) using iii ). We obtain M ( n +1) τ ( f ψ ) ( x ) M ( n +1) τ ψ ( x ) ≥ c n +1 β n Ξ n ν ( f ) , for all x ∈ K , which completes the proof. (cid:3) Contraction property: proofs of Section 3.1.
First, we prove that ( V k ) k ≥ is a family ofLyapunov functions for the sequence of operators ( P ( nτ ) kτ, ( k +1) τ ) ≤ k ≤ n − . Lemma 4.3.
For all k ≥ and n ≥ m ≥ k , we have P ( nτ ) kτ,mτ V n − m ≤ a m − k V n − k + θcβ m − X j = k a m − j P ( nτ ) kτ,jτ ( K ) . Proof.
By definition of V k in (3.2), we have, for 0 ≤ k ≤ n , P ( nτ )( k − τ,kτ V n − k = M τ (cid:0) V n − k M ( n − k ) τ ψ (cid:1) M ( n − k +1) τ ψ = ν (cid:18) M ( n − k ) τ ψψ (cid:19) M τ VM ( n − k +1) τ ψ Using (A1) and (A2), we have M τ V ≤ αV + θψ K and M ( n − k ) τ ψ ≤ M ( n − k +1) τ ψ/β . We obtainfrom the definitions of a and V n − k +1 that P ( nτ )( k − τ,kτ V n − k ≤ a V n − k +1 + ν (cid:18) M ( n − k ) τ ψψ (cid:19) θψ K M ( n − k +1) τ ψ . Besides, combining (A2) and (A3) with f = M ( n − k ) τ ψ/ψ , we get ν (cid:18) M ( n − k ) τ ψψ (cid:19) ψ K M ( n − k +1) τ ψ ≤ K cβ . The last two inequalities yield P ( nτ )( k − τ,kτ V n − k ≤ a V n − k +1 + θ K cβ . The conclusion follows from P ( nτ ) kτ,mτ V n − m = P ( nτ ) kτ, ( k +1) τ · · · P ( nτ )( m − τ,mτ V n − m . (cid:3) Proof of Lemma 3.1.
Using that P ( nτ ) kτ, ( j − τ ( K ) ≤ a <
1, it is a direct consequence ofLemma 4.3. (cid:3)
BANSAYE , BERTRAND
CLOEZ , PIERRE
GABRIEL , AND ALINE
MARGUET
Using (A3) and (A4) and following [8, 23], we prove a small set condition (6.2) on the set K forthe embedded propagator P ( t ) . However, Theorem 6.1 requires that (6.2) is satisfied on a sublevelset of V k . Although there exists R > K ⊂ { V k ≤ R } , nothing guarantees the otherinclusion. This situation is reminiscent of [53, Assumption 3] and we adapt here their arguments.For that purpose, we need a lower bound for the Lyapunov functions ( V k ) k ≥ , which is stated inthe next lemma. Lemma 4.4.
For every n ≥ , we have d M ( n +1) τ ψ ≤ ν (cid:18) M nτ ψψ (cid:19) M τ V and V n ≥ d , with d = (cid:0) − a (cid:1) d, d = ( β − α ) d/ ( α + θ ) . Proof.
First, using (A4), d M ( n +1) τ ψ = d M τ ( K M nτ ψ + K c M nτ ψ ) ≤ ν (cid:18) M nτ ψψ (cid:19) M τ ψ + d c M τ M nτ ψ. Then, by iteration, using (A2) and ψ ≤ V,d M ( n +1) τ ψ ≤ ν (cid:18) M nτ ψψ (cid:19) n X j =0 β − j c M jτ M τ ψ ≤ ν (cid:18) M nτ ψψ (cid:19) n X j =0 β − j c M jτ M τ V. Hence by Lemma 4.1 i ) d M ( n +1) τ ψ ≤ − a ν (cid:18) M nτ ψψ (cid:19) M τ V and the first identity is proved. From the definition of V n we deduce V n ≥ d M ( n +1) τ ψM nτ ψ VM τ V ≥ d βα + θ = d by using successively (A2), (A1), and ψ ≤ V . (cid:3) We now prove the small set condition (6.2) for the embedded propagator.
Proof of Lemma 3.2.
First, we introduce the measure ν i defined by ν i ( f ) = ν (cid:18) f M iτ ψψ (cid:19) for all integer i ≥
0. For any x ∈ K , j ≤ k ≤ n , we have using Lemma 4.1 iv ) with the function f M ( n − k ) τ ψ/ψ , P ( nτ )( j − τ,kτ f ( x ) = M ( k − j +1) τ (cid:0) f M ( n − k ) τ ψ (cid:1) ( x ) M ( n − j +1) τ ψ ( x ) ≥ c k − j +1 ν n − k ( f ) M ( k − j +1) τ ψ ( x ) M ( n − j +1) τ ψ ( x ) . for any nonnegative measurable function f . Then, Lemma 4.4 and (A2) yield P ( nτ )( j − τ,kτ f ( x ) ≥ d c k − j +1 ν n − k ( f ) ν n − j ( ) M ( k − j +1) τ ψ ( x ) M τ V ( x ) ≥ d c k − j +1 β k − j ν n − k ( f ) ν n − j ( ) M τ ψ ( x ) M τ V ( x ) . Recalling from (A1) and (A2) that for x ∈ K , M τ ψ ( x ) /M τ V ( x ) ≥ β/ ( αR + θ ) , and from Lemma 4.1 iii ) and ν ( K ) = 1 that ν n − k ( ) /ν n − j ( ) ≥ Ξ − ( k − j ) , we get P ( nτ )( j − τ,kτ f ( x ) ≥ α k − j ν n − k ( f ) ν n − k ( ) , (4.3)for x ∈ K , where α i = d c i +1 βr i / ( αR + θ ) and r = ( β/ Ξ) . The previous bound holds only on K . We prove now that the propagator charges K at an intermediate time and derive the expectedlower bound. More precisely, setting ω i = a i α i and S ℓ = ℓ X j =1 ω ℓ − j = αR + θdc ( β − α ) . ℓ X j =1 (cid:16) a cr (cid:17) j , NON-CONSERVATIVE HARRIS ERGODIC THEOREM 13 we obtain for k ≤ n − ≤ ℓ ≤ n − k , P ( nτ ) kτ, ( k + ℓ ) τ f = 1 S ℓ k + ℓ X j = k +1 ω k + ℓ − j P ( nτ ) kτ, ( j − τ P ( nτ )( j − τ, ( k + ℓ ) τ ( f ) ≥ S ℓ k + ℓ X j = k +1 ω k + ℓ − j P ( nτ ) kτ, ( j − τ ( K P ( nτ )( j − τ, ( k + ℓ ) τ f ) ≥ B ( ℓ ) k,n ν n − k − ℓ ( f ) ν n − k − ℓ ( ) , where the last inequality comes from (4.3) and B ( ℓ ) k,n = 1 S ℓ k + ℓ X j = k +1 a k + ℓ − j P ( nτ ) kτ, ( j − τ K . To conclude, we need to find a positive lower bound for B ( ℓ ) k,n which does not depend on k or n . For that purpose, we first observe that the second bound of Lemma 4.4 ensures that P ( nτ ) kτ, ( k + ℓ ) τ V n − k − ℓ ≥ d . Using now Lemma 4.3 yields k + ℓ X j = k +1 a k + ℓ − j P ( nτ ) kτ, ( j − τ K ≥ cβ d − a ℓ V n − k θ , for n ≥ k + ℓ . For x ∈ { V n − k ≤ R } and ℓ = p defined in (3.4), we get B ( p ) k,n ( x ) ≥ cβ d θS p = c β d θ ( α + θ )( αR + θ ) 1 P p j =1 ( a /cr ) j , which ends the proof. (cid:3) Proof of Proposition 3.3.
Let n and k be two integers such that 0 ≤ k ≤ n − p and consider R > c / (1 − a ) . According to Lemmas 3.1 and 3.2, the conservative operator P ( nτ ) kτ, ( k + p ) τ satisfiescondition (6.1) with the functions V n − k − p and V n − k and condition (6.2) with the probabilitymeasure ν k,n . Applying Theorem 6.1 then yields the contraction result. (cid:3) Eigenelements: proofs of Section 3.2.
Let us consider, for every µ ∈ M + ( V ), the familyof operators ( Q µt ) t ≥ defined for f ∈ B ( V ) by Q µt f = µM t ( ψf ) µM t ( ψ ) . Fixing the measure µ , the operator f Q µt f is linear. Observe that Q δ x t = δ x P ( t )0 ,t so that Propo-sition 3.3 implies contraction inequalities for δ x Q δ x n p τ . Notice that µ Q µn p τ is non-linear andforthcoming Lemma 4.6 extends the contraction to a more general space of measures. Besides, forany positive measure µ , we set [ µ ] = µ ( V ) µ ( ψ ) . We prove now the existence of the eigenvector and eigenmeasure, respectively stated in Lemma 3.4and Lemma 3.5. Let us first provide a useful upper bound for V k . For that purpose, we also set p = $ log (cid:0) θ/α )(Θ + R ) (cid:1) log (1 / a ) % + 1 , C = 2 Ξ p +1 cc p − β p +1 . (4.4)where ( c k ) k ≥ is defined in (4.2). Lemma 4.5.
For all positive measure µ such that [ µ ] ≤ Θ + R, (4.5) we have for all k ≥ p , ν (cid:18) M kτ ψψ (cid:19) ≤ C µM kτ ψµ ( ψ ) . (4.6) BANSAYE , BERTRAND
CLOEZ , PIERRE
GABRIEL , AND ALINE
MARGUET
The idea is the following: condition (4.5) ensures the existence of a time p at which the prop-agator charges K . Then, (A3) yields (4.6). It will be needed in this form in the sequel, but couldbe extended to more general right-hand side in (4.5). Proof.
Recalling that c M τ = M τ ( K c · ) and using that for all g ∈ B ( V ), M ( k +1) τ g = M τ ( K M kτ g )+ c M τ ( M kτ g ), we obtain by induction M kτ g = c M k g + k X j =1 c M k − jτ M τ (cid:0) K M ( j − τ g (cid:1) . Let g = ψ K . Using Lemma 4.1 iv ) with f = K and that ν ( K ) = 1, we have M kτ ( K ψ ) ≥ k X j =1 c M k − jτ M τ (cid:0) K M ( j − τ ( K ψ ) (cid:1) ≥ k X j =1 c j − c M k − jτ M τ (cid:0) K M ( j − τ ψ (cid:1) ≥ c k − k X j =1 (cid:16) c M k − jτ M jτ ψ − c M k − jτ M τ (cid:0) K c M ( j − τ ψ (cid:1)(cid:17) = c k − (cid:16) M kτ ψ − c M kτ ψ (cid:17) , with the convention that c = 1. Then, using (A2) and the fact that c M kτ ψ ≤ c M k − τ M τ V togetherwith Lemma 4.1 i ) yields M kτ ( K ψ ) ≥ c k − (cid:0) β k ψ − α k − M τ V (cid:1) . Next, (A1) and the fact that V ≥ ψ yield M kτ ( K ψ ) ≥ c k − β k (cid:0) ψ − a k (1 + θ/α ) V (cid:1) . (4.7)Using the definition (4.4) of p and the fact that a ≤ a p θ/α ) (Θ + R ) ≤ , and(4.5) yields a p (1 + θ/α ) µ ( V ) ≤ µ ( ψ ) / . Then, (4.7) becomes µM pτ ( K ψ ) ≥ c p − β p µ ( ψ ) / . (4.8)Using µM nτ ψ ≥ µM pτ ( K M ( n − p ) τ ψ ) = µM pτ ( K M τ (( M ( n − p − τ ψ/ψ ) ψ )) for n ≥ p and succes-sively (A3) with f = M ( n − p − τ ψ/ψ , (A2) and (4.8), we get µM nτ ψ ≥ cc p − β p +1 µ ( ψ )2 ν (cid:18) M ( n − p − τ ψψ (cid:19) . Finally, combining this estimate with Lemma 4.1 iii ) ensures that ν (cid:18) M nτ ψψ (cid:19) ≤ Ξ p +1 ν (cid:18) M ( n − p − τ ψψ (cid:19) ≤ C µM nτ ψµ ( ψ ) , which ends the proof. (cid:3) We generalize now Proposition 3.3 to ( Q µn p τ ) n ≥ . Recall that p is defined in Lemma 3.2, κ and y are defined in Proposition 3.3 and ρ is defined in (3.5). Lemma 4.6.
For all measures µ , µ ∈ M + ( V /ψ ) and all n ≥ we have (cid:13)(cid:13) Q µ n p τ − Q µ n p τ (cid:13)(cid:13) M (1+ κV ) ≤ C ρ n ([ µ ] + [ µ ]) , (4.9) where C = max n a − p + κC (cid:0) a − p (cid:1) , κ Θ) a − ( p + p ) + κ o with C and p defined in (4.4) .Proof. Fix µ , µ ∈ M + ( V /ψ ), f ∈ B ( V /ψ ) with k f k B (1+ κV ) ≤ n ≥
0. Set forconvenience m = (cid:22) log ([ µ ] + [ µ ]) p log (1 / a ) (cid:23) + 1 , n = p τ n, m = p τ m. (4.10) NON-CONSERVATIVE HARRIS ERGODIC THEOREM 15
By definition of the embedded propagator in (3.1), we have µ M n ( f ψ ) µ M n ψ − µ M n ( f ψ ) µ M n ψ = Z X M n ψ ( x ) M n ψ ( y ) (cid:18) M n ( f ψ )( x ) M n ψ ( x ) − M n ( f ψ )( y ) M n ψ ( y ) (cid:19) µ ( dx ) µ ( dy ) ≤ Z X M n ψ ( x ) M n ψ ( y ) (cid:13)(cid:13)(cid:13) δ x P ( n )0 , n − δ y P ( n )0 , n (cid:13)(cid:13)(cid:13) M (1+ κV ) µ ( dx ) µ ( dy ) . Using Proposition 3.3, we get for n ≥ m , µ M n ( f ψ ) µ M n ψ − µ M n ( f ψ ) µ M n ψ (4.11) ≤ y n − m Z X M n ψ ( x ) M n ψ ( y ) (cid:13)(cid:13)(cid:13) δ x P ( n )0 , m − δ y P ( n )0 , m (cid:13)(cid:13)(cid:13) M (1+ κV ( n − m ) p ) µ ( dx ) µ ( dy ) . Using the definition of the norm on M (1 + κV ( n − m ) p ) and the definition of V ( n − m ) p in (3.2), weobtain (cid:13)(cid:13)(cid:13) δ x P ( n )0 , m − δ y P ( n )0 , m (cid:13)(cid:13)(cid:13) M (1+ κV ( n − m ) p ) ≤ Z X (1 + κV ( n − m ) p ( z )) (cid:12)(cid:12)(cid:12) δ x P ( n )0 , m − δ y P ( n )0 , m (cid:12)(cid:12)(cid:12) ( dz ) ≤ κν (cid:18) M n − m ψψ (cid:19) (cid:18) M m V ( x ) M n ψ ( x ) + M m V ( y ) M n ψ ( y ) (cid:19) . Combining this inequality with (4.11) and ρ ≥ y , we get Q µ n f − Q µ n f = µ M n ( f ψ ) µ M n ψ − µ M n ( f ψ ) µ M n ψµ M n ψ.µ M n ψ ≤ ρ n (cid:18) ρ − m + κρ − m ν (cid:18) M n − m ψψ (cid:19) (cid:18) µ M m Vµ M n ψ + µ M m Vµ M n ψ (cid:19) (cid:19) . (4.12)We now bound each term of the right-hand side. First, using that a p ≤ ρ and (4.10), a p ≤ ρ m ([ µ ] + [ µ ]) . (4.13)Second, Lemma 4.1 ii ) ensures that for µ ∈ { µ , µ } ,[ µM m ] ≤ a m p [ µ ] + Θ . (4.14)Besides (4.10) also guarantees that for µ ∈ { µ , µ } , a m p [ µ ] ≤
1. It means that the positive measure µM m satisfies inequality (4.5), since R ≥
1. Then, Lemma 4.5 applied to µM m with k = ( n − m ) p yields for all n ≥ m + p/ p , ν (cid:18) M n − m ψψ (cid:19) µM m VµM n ψ ≤ C µM m M n − m ψµM m ψ µM m VµM n ψ ≤ C [ µM m ] . Finally, using again (4.14) and (4.13), we get ν (cid:18) M n − m ψψ (cid:19) (cid:18) µ M m Vµ M n ψ + µ M m Vµ M n ψ (cid:19) ≤ C (1 + 2Θ a − p ) ρ m ([ µ ] + [ µ ]) . Plugging the last inequality in (4.12) ensures that for all n ≥ m + p/ p , Q µ n f − Q µ n f ≤ (2 a − p + κC (1 + 2Θ a − p )) ([ µ ] + [ µ ]) ρ n . To conclude, it remains to show that (4.9) also holds for n ≤ m + p/ p . We have k Q µ n − Q µ n k M (1+ κV ) ≤ k Q µ n k M (1+ κV ) + k Q µ n k M (1+ κV ) ≤ κ [ µ M n ] + κ [ µ M n ] . Using again (4.14), [ µM n ] ≤ a n p [ µ ] + Θ ≤ ρ n [ µ ] + Θ for µ ∈ { µ , µ } , so that k Q µ n − Q µ n k M (1+ κV ) ≤ κ Θ) + κρ n ([ µ ] + [ µ ]) . Finally, ρ ≥ a p and n ≤ m + p/ p and (4.10) yield 1 ≤ ρ n a − ( p + m p ) = ρ n a − ( p + p ) a − ( m − p ≤ ρ n a − ( p + p ) ([ µ ] + [ µ ]), and we get k Q µ n − Q µ n k M (1+ κV ) ≤ ρ n (cid:16) κ Θ) a − ( p + p ) + κ (cid:17) ([ µ ] + [ µ ]) , for all n ≤ m + p/ p , which ends the proof. (cid:3) BANSAYE , BERTRAND
CLOEZ , PIERRE
GABRIEL , AND ALINE
MARGUET
We have now all the ingredients to prove the existence of the eigenelements and the associatedestimates. We start with the right eigenfunction.
Proof of Lemma 3.4.
We define η ( · ) = ν ( · /ψ )and, for k ≥ h k = M kτ ψν ( M kτ ψ/ψ ) = M kτ ψηM kτ ψ . The proof is divided into four steps. We begin by giving preliminary estimates on M kτ ψ . Next,we show that h k converges in B ( V /ψ ) as k → ∞ and that its limit h satisfies h ≤ d − V, (4.15)where d is defined in Lemma 4.4. Then, we show that h is an eigenvector and we give a lowerbound for h .Using the first part of Lemma 4.4 and (A1), we obtain for any k ≥ M ( k +1) τ ψ ( x ) ≤ d − ( α + θ )( ηM kτ ψ ) V ( x ) . (4.16)Since k M kτ ψ/ψ k B (1+ κV ) = sup X M kτ ψ/ ( ψ + κV ), we get for any k ≥ (cid:13)(cid:13)(cid:13)(cid:13) M ( k +1) τ ψψ (cid:13)(cid:13)(cid:13)(cid:13) B (1+ κV ) ≤ d − ( α + θ )( ηM kτ ψ ) sup X Vψ + κV ≤ ( α + θ ) d κ ηM kτ ψ. (4.17)Besides, from Lemma 4.1 ii) and the fact that V ≤ Rψ on K , [ ηM kτ ] ≤ a k η ( V ) + Θ ≤ R + Θ , so µ = ηM kτ verifies (4.5). Lemma 4.5 applied to µ gives for all n ≥ p ,( ηM kτ )( M nτ ψ ) ≥ C − ( ηM nτ ψ )( ηM kτ ψ ) . (4.18)We can now proceed to the second step: the convergence of ( h k ) k ≥ . Let µ ∈ M + ( V ). We usethat µ ( h k + n ) = µM ( k + n ) τ ψ/µM kτ ψηM ( k + n ) τ ψ/ηM kτ ψ µ ( h k )to obtain that | µ ( h k + n ) − µ ( h k ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12) µM ( k + n ) τ ψµM kτ ψ − ηM ( k + n ) τ ψηM kτ ψ (cid:12)(cid:12)(cid:12)(cid:12) ηM ( k + n ) τ ψ/ηM kτ ψ µ ( h kτ )= (cid:12)(cid:12)(cid:12)(cid:12) Q µkτ (cid:18) M nτ ψψ (cid:19) − Q ηkτ (cid:18) M nτ ψψ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) µM kτ ψηM ( k + n ) τ ψ . Then, Lemma 4.6 yields | µ ( h k + n ) − µ ( h k ) | ≤ C ρ ⌊ k/ p ⌋ ([ µM mτ ] + [ ηM mτ ]) (cid:13)(cid:13)(cid:13)(cid:13) M nτ ψψ (cid:13)(cid:13)(cid:13)(cid:13) B (1+ κV ) µM kτ ψηM ( k + n ) τ ψ , (4.19)where m = k − ⌊ k/ p ⌋ p . First, using (A1) and (A2) we have since m ≤ p [ µM mτ ] ≤ max (cid:18) , ( α + θ ) p β p (cid:19) [ µ ] . (4.20)Next, combining (4.19) with (4.16), (4.17), (4.18), (4.20), and using (A2) yields | µ ( h k + n ) − µ ( h k ) | ≤ C C ′ ( α + θ ) d β κ ρ ⌊ k/ p ⌋ (cid:18) [ µ ] + ν (cid:18) Vψ (cid:19)(cid:19) µ ( V ) , for all n ≥ p , where C ′ = C max (cid:18) , ( α + θ ) p β p (cid:19) . (4.21)Recalling that ψ ≤ V , V ≤ Rψ on K and ν ( K c ) = 0, we get for all n ≥ p | µ ( h k + n ) − µ ( h k ) | ≤ C ρ ⌊ k/ p ⌋ µ ( V )[ µ ] , (4.22) NON-CONSERVATIVE HARRIS ERGODIC THEOREM 17 with C = C C ′ α + θ ) d β κ (1 + R ). Taking µ = δ x we deduce for all n ≥ p k h k + n − h k k B ( V /ψ ) ≤ C ρ ⌊ k/ p ⌋ . Cauchy criterion ensures that h k converges as k → ∞ in B ( V /ψ ) to a limit denoted by h . Moreoverfrom Lemma 4.4, d ≤ V m = V /h mτ which yields (4.15) by letting m → ∞ . Letting n → ∞ in(4.22) gives (3.6).We move on to the third step and check that h is an eigenfunction. For all x ∈ X , ηM τ (cid:18) M nτ ψηM nτ ψ (cid:19) M ( n +1) τ ψ ( x ) ηM ( n +1) τ ψ = δ x M τ (cid:18) M nτ ψηM nτ ψ (cid:19) . Letting n → + ∞ and using boundedness condition from (4.16), we get( ηM τ h ) . h ( x ) = M τ h ( x ) . Hence h is an eigenvector of M τ associated to the eigenvalue ηM τ h . Moreover, by the semigroupproperty, ηM kτ h = ( ηM τ h ) k . Then, for all k ≥ ηM kτ h = e λkτ , with λ = τ − log( ηM τ h ). Next,for t ≥ M τ M t h = e λτ M t h so that M t h is an eigenfunction of M τ with associated eigenvalue e λτ . By uniqueness, there exists( c t ) t ≥ such that for all t ≥ M t h = c t h . Moreover, by the semigroup property, c t + s = c t c s forall t, s ≥ t c t is locally bounded using (4.15) and t
7→ k M t V k B ( V ) is locally bounded.Therefore, there exists ˜ λ ∈ R such that c t = e ˜ λt and as c τ = e λτ , we finally get ˜ λ = λ . Then, forall t ≥ M t h = e λt h. Let us proceed to the last step and provide a lower bound for h . Combining M nτ ψ ≥ M kτ ( K M ( n − k ) τ ψ ) = M kτ ( K M τ (( M ( n − k − τ ψ/ψ ) ψ ))for all k ≤ n with (A3) and (A2), we get M nτ ψ ≥ cβ ( ηM ( n − k − τ ψ ) M kτ ( K ψ ) . Recalling (4.7) and dividing by ηM nτ ψ , we obtain M nτ ψηM nτ ψ ≥ cβc k − (cid:0) β k ψ − α k − ( α + θ ) V (cid:1) ηM ( n − k − τ ψηM nτ ψ . Let n → ∞ , the left-hand side converges to h and using from Lemma 4.1 iii) that ηM ( n − k − τ ψ/ηM nτ ψ ≥ / Ξ k +1 , and recalling the expression of c k in (4.2), we obtain h ≥ c ( cr ) k (cid:0) ψ − a k − ( α + θ ) V /β (cid:1) . Considering now k = k ( x ) = j log (cid:16) ψ ( x ) V ( x ) β α + θ ) (cid:17) / log( a ) k + 2 and recalling that r = β / Ξ yields ψ − a k − α + θβ V ≥ ψ/ . We get h ≥ c (cid:18) ψV (cid:19) q ψ, with c = c ( cr ) ( β α + θ ) ) / log( a ) / > q = log( cr ) / log( a ) >
0, which ends the proof. (cid:3)
Remark 4.7.
Notice that the eigenfunction h built in this proof satisfies ν ( h/ψ ) = 1 and theconstants in ( V /ψ ) q ψ . h . V depend on this normalization. If we normalize h such that k h k B ( V ) =1 as in Theorem 2.1 we get c d ( ψ/V ) q ψ ≤ h ≤ V. We consider now the left eigenelement.
Proof of Lemma 3.5.
Let us use again η = ν ( · /ψ ). Applying Lemma 4.6 to µ = η and µ = ηM nτ and using again (4.20), we get for k, n ≥ (cid:13)(cid:13) Q η ( k + n ) τ − Q ηkτ (cid:13)(cid:13) M (1+ κV ) ≤ C ′ ρ ⌊ k/ p ⌋ (cid:18) ν (cid:18) Vψ (cid:19) + [ ηM nτ ] (cid:19) , BANSAYE , BERTRAND
CLOEZ , PIERRE
GABRIEL , AND ALINE
MARGUET where C ′ is defined in (4.21). Then, using Lemma 4.1 ii) , V ≤ Rψ on K and that ν ( K ) = 1, wehave (cid:13)(cid:13) Q η ( k + n ) τ − Q ηkτ (cid:13)(cid:13) M (1+ κV ) ≤ C ′ ρ ⌊ k/ p ⌋ ( R + a n R + Θ) . Therefore, the sequence of probabilities ( Q ηkτ ) k ≥ satisfies the Cauchy criterion in M (1 + κV ) andit then converges to a probability measure π ∈ M (1 + κV ). Similarly, applying Lemma 4.6 to µ = µ and µ = ηM nτ , we also have (cid:13)(cid:13) Q η ( k + n ) τ − Q µkτ (cid:13)(cid:13) M (1+ κV ) ≤ C ′ ρ ⌊ k/ p ⌋ ([ µ ] + a n R + Θ)for any µ ∈ M (1 + κV ). Letting n tend to infinity yields (cid:13)(cid:13) π − Q µkτ (cid:13)(cid:13) M (1+ κV ) ≤ C ′ ([ µ ] + Θ) ρ ⌊ k/ p ⌋ . (4.23)Besides, π ( h/ψ ) ≤ π ( V /ψ ) = π ( V ) < + ∞ and we can define γ ∈ M ( ψ + κV ) by γ ( f ) = π ( f /ψ ) π ( h/ψ )for f ∈ B ( ψ + κV ) = B ( V ). Observe that γ ( h ) = 1. Next, Q η ( k +1) τ ( f /ψ ) = Q ηkτ ( M τ f /ψ ) ηM kτ ψηM ( k +1) τ ψ . (4.24)Applying (3.6) to µ = ηM τ , ηM kτ ψηM ( k +1) τ ψ −−−−−→ k → + ∞ e − λτ . Then, letting k → ∞ in (4.24), we obtain π ( f /ψ ) = π ( M τ f /ψ )e − λτ , which ensures that γ isan eigenvector for M τ . As in the proof of Lemma 3.4, using the semigroup property and that t
7→ k M t V k B ( V ) is locally bounded, we get that for all t ∈ R + that γM t = e λt γ. Adding that π ( f ) = γ ( f ψ ) /γ ( ψ ) since π is a probability measure, (3.7) follows from (4.23). (cid:3) Proofs of Section 3.3.
We start by proving a discrete version of Theorem 3.6.
Proposition 4.8.
Under Assumption A , there exists C > such that for all µ ∈ M + ( V ) and allintegers k ≥ we have (cid:13)(cid:13) µM kτ − e λkτ µ ( h ) γ (cid:13)(cid:13) M ( V ) ≤ C [ µ ] e − σkτ min (cid:8) µM kτ ψ, µ ( V )e λkτ (cid:9) , (4.25) where σ = − log ρ p τ > . Proof.
Using that k π − Q µkτ k M (1+ κV ) = sup f ∈B (1+ κV ) (cid:12)(cid:12)(cid:12)(cid:12) γ ( f ψ ) γ ( ψ ) − µM kτ ( f ψ ) µM kτ ψ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:13)(cid:13)(cid:13)(cid:13) γγ ( ψ ) − µM kτ µM kτ ψ (cid:13)(cid:13)(cid:13)(cid:13) M ( ψ + κV ) and multiplying (4.23) by µM kτ ψ , we get (cid:13)(cid:13)(cid:13)(cid:13) µM kτ ψ γγ ( ψ ) − µM kτ (cid:13)(cid:13)(cid:13)(cid:13) M ( ψ + κV ) ≤ C ′ ρ ⌊ k/ p ⌋ ([ µ ] + Θ) µM kτ ψ. (4.26)Moreover, h ∈ M ( ψ + κV ) since h . V . As γ ( h ) = 1, the previous inequality applied to theeigenfunction h yields (cid:12)(cid:12)(cid:12)(cid:12) µM kτ ψγ ( ψ ) − µ ( h )e λkτ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ′ ρ ⌊ k/ p ⌋ ([ µ ] + Θ) µM kτ ψ. Then, recalling that ψ ≤ V , we have (cid:13)(cid:13)(cid:13)(cid:13) µM kτ ψ γγ ( ψ ) − e λkτ µ ( h ) γ (cid:13)(cid:13)(cid:13)(cid:13) M ( ψ + κV ) = (cid:12)(cid:12)(cid:12)(cid:12) µM kτ ψγ ( ψ ) − e λkτ µ ( h ) (cid:12)(cid:12)(cid:12)(cid:12) k γ k M ( ψ + κV ) (4.27) ≤ C ′ ρ ⌊ k/ p ⌋ ([ µ ] + Θ) µM kτ ψ × (1 + κ ) γ ( V ) . NON-CONSERVATIVE HARRIS ERGODIC THEOREM 19
Combining (4.26) and (4.27), by triangular inequality, we get (cid:13)(cid:13) µM kτ − γ e λkτ µ ( h ) (cid:13)(cid:13) M ( ψ + κV ) ≤ C ′ ρ ⌊ k/ p ⌋ ([ µ ] + Θ) µM kτ ψ (1 + (1 + κ ) γ ( V )) . (4.28)This gives the first part of (4.25). Finally, by integration of (4.16), µM kτ ψ ≤ d − ( α + θ ) ν ( M kτ ψ/ψ ) µ ( V ) . (4.29)Adding that [ γ ] ≤ Θ according to Lemma 4.1 ii) and γM kτ = e λkτ γ from Lemma 3.5, Lemma 4.5applied to µ = γ yields ν ( M kτ ψ/ψ ) ≤ C e λkτ for k ≥ p and we obtain µM kτ ψ ≤ C d − ( α + θ ) µ ( V )e λkτ . It proves (4.25) for k ≥ p with C = C ′ (1 + Θ) (1 + (1 + κ ) γ ( V )) max(1 , C d − ( α + θ )) . The fact that (4.25) holds for some constant C also for k ≤ p comes directly from (4.28), (4.29),Lemma 4.1 iii) and ν ( K ) = 1. (cid:3) Next, we extend this result to continuous time.
Proof of Theorem 3.6.
We introduce the following constant C = sup s ≤ τ max (cid:26)(cid:13)(cid:13)(cid:13)(cid:13) M s VV (cid:13)(cid:13)(cid:13)(cid:13) ∞ , (cid:13)(cid:13)(cid:13)(cid:13) ψM s ψ (cid:13)(cid:13)(cid:13)(cid:13) ∞ (cid:27) , which is finite under Assumption (3.9). Let t ≥ k, ε ) ∈ N × [0 , τ ) be such that t = kτ + ε .Using that µM t h = e λt µ ( h ), we have (cid:13)(cid:13) µM t − e λt µ ( h ) γ (cid:13)(cid:13) M ( V ) = k µM kτ + ε − µ ( M kτ + ε h ) γ k M ( V ) . Then, from (4.25) with µ := µM ε , there exists C > k ≥ (cid:13)(cid:13) µM t − e λt µ ( h ) γ (cid:13)(cid:13) M ( V ) ≤ C [ µM ε ]e − σkτ min (cid:8) µM kτ + ε ψ, µM ε V e λkτ (cid:9) ≤ CC e στ max n , C e | λ | τ o [ µ ]e − σt min (cid:8) µM t ψ, µ ( V )e λt (cid:9) , which ends the proof. (cid:3) Now we turn to the proof of Corollary 3.7.
Proof of Corollary 3.7.
For convenience and without loss of generality, we assume that V ≥ .Then, γ ( ) < ∞ . Next, if γ ( ) = 0, then γ ( X ) = 0. In this case, γ = 0, which is absurd because γ ( ψ ) > ψ >
0. Therefore, γ ( ) > π ( · ) = γ ( · ) /γ ( ) and we have by triangular inequality (cid:13)(cid:13)(cid:13)(cid:13) µM t µM t − π (cid:13)(cid:13)(cid:13)(cid:13) TV ≤ e λt µM t (cid:13)(cid:13) e − λt µM t − π e − λt µM t (cid:13)(cid:13) M ( V ) ≤ e λt µM t (cid:16)(cid:13)(cid:13) e − λt µM t − γµ ( h ) (cid:13)(cid:13) M ( V ) + (cid:12)(cid:12) γ ( ) µ ( h ) − e − λt µM t (cid:12)(cid:12) π ( V ) (cid:17) . From (2.2), we have (cid:13)(cid:13) e − λt µM t − γµ ( h ) (cid:13)(cid:13) M ( V ) ≤ Cµ ( V )e − ωt . Using this estimate with V ≥ , wealso have (cid:12)(cid:12) γ ( ) µ ( h ) − e − λt µM t (cid:12)(cid:12) ≤ Cµ ( V )e − ωt . (4.30)Combining the three last estimates yields (cid:13)(cid:13)(cid:13)(cid:13) µM t µM t − π (cid:13)(cid:13)(cid:13)(cid:13) TV ≤ C e λt µM t µ ( V )e − ωt (1 + π ( V )) . (4.31)Now on the first hand, Equation (4.30) also gives e − λt µM t ≥ γ ( ) µ ( h ) − Cµ ( V )e − ωt , and for any t ≥ t ( µ ) = ω log (cid:16) Cγ ( ) µ ( V ) µ ( h ) (cid:17) , we havee − λt µM t ≥ µ ( h ) γ ( ) / . (4.32)Plugging (4.32) in (4.31) yields (3.10) when t ≥ t ( µ ). Otherwise, (cid:13)(cid:13)(cid:13)(cid:13) µM t µM t − π (cid:13)(cid:13)(cid:13)(cid:13) TV ≤ ≤ e − ωt e ωt ( µ ) ≤ C µ ( V ) µ ( h ) e − ωt , BANSAYE , BERTRAND
CLOEZ , PIERRE
GABRIEL , AND ALINE
MARGUET which ends the proof. (cid:3)
Proofs of Section 2.
Proof of Theorem 2.1 i).
We assume that Assumption A is satisfied by ( V, ψ ) for a set K , constants α, β, θ, c, d and a probability ν . Then, from Lemmas 3.4 and 3.5, there exist eigenelements ( γ, h, λ )such that β ≤ e λτ ≤ α + θ, c d ( ψ/V ) q ψ ≤ h ≤ V. We check now that (
V, h ) satisfies also Assumption A with the same set K and constant α as( V, ψ ) but other constants β ′ , θ ′ , c ′ , d ′ and an other probability measure ν ′ .First the inequality V /h ≤ c d ( V /ψ ) q +1 ensures that sup K V /h < ∞ since sup K V /ψ < ∞ .Moreover h satisfies (A2) with β ′ = exp( λτ ) ≥ β > α . Recalling that R = sup K V /ψ < ∞ , we alsohave sup K ψh ≤ R q c d . Adding that (
V, ψ ) satisfies (A1) with constants α, θ yields M τ V ≤ αV + θ ′ K h with θ ′ = θ R q c d , which gives (A1) for ( V, h ). We use now (A3) and (A2) for (
V, ψ ) and get for x ∈ Kδ x M τ ( f h ) ≥ cν (cid:18) f hψ (cid:19) δ x M τ ψ ≥ cβν (cid:18) f hψ (cid:19) ψ ( x ) . Using again that M τ h = e λτ h , we obtain for x ∈ K , δ x M τ ( f h ) ≥ c ′ ν ′ ( f ) δ x M τ h, with ν ′ = ν (cid:0) · hψ (cid:1) ν (cid:0) hψ (cid:1) , c ′ = cβ e − λτ ν (cid:18) hψ (cid:19) inf K ψh ≥ βα + θ c d R q +1 > . Finally, (A4) is satisfied since sup x ∈ K M nτ h ( x ) h ( x ) = e λnτ = ν ′ (cid:18) M nτ hh (cid:19) . Therefore, every result stated above holds replacing ψ by h and the constants β, θ, c, d ofAssumption A by β ′ , θ ′ , c ′ , d ′ defined above. In particular, for all µ ∈ M + ( V ), (3.7) becomes (cid:13)(cid:13)(cid:13)(cid:13) γ ( · h ) − µM kτ ( · h )e λkτ µ ( h ) (cid:13)(cid:13)(cid:13)(cid:13) M (1+ κV/h ) ≤ C (cid:18) µ ( V ) µ ( h ) + Θ ′ (cid:19) ρ ⌊ k/ p ⌋ , where Θ ′ = θ ′ / ( β ′ − α ) and C = max n a − p + κC (cid:0) a − p (cid:1) , κ Θ) a − ( p + p ) + κ o max (cid:18) , ( α + θ ′ ) p β ′ p (cid:19) and ρ is defined in (3.5), p in (3.4), a in (3.3), κ in Proposition 3.3, Θ in (4.1), and p, C in (4.4)(replacing β, θ, c, d by β ′ , θ ′ , c ′ , d ′ in each definition). Using that (cid:13)(cid:13)(cid:13)(cid:13) γ ( · h ) − µM kτ ( · h )e λkτ µ ( h ) (cid:13)(cid:13)(cid:13)(cid:13) M (1+ κV/h ) = (cid:13)(cid:13)(cid:13)(cid:13) γ − µM kτ e λkτ µ ( h ) (cid:13)(cid:13)(cid:13)(cid:13) M ( h + κV ) ≥ κ (cid:13)(cid:13)(cid:13)(cid:13) γ − µM kτ e λkτ µ ( h ) (cid:13)(cid:13)(cid:13)(cid:13) M ( V ) , multiplying by e λkτ µ ( h ) and using that h ≤ V , we get (cid:13)(cid:13) e λkτ µ ( h ) γ − µM kτ (cid:13)(cid:13) M ( V ) ≤ Cκρ µ ( V ) (1 + Θ ′ ) e − ωkτ e λkτ , where ω = − log ρ/ p τ . Let t ≥ k, ε ) ∈ N × [0 , τ ) be such that t = kτ + ε . Applying the aboveinequality to the measure µM ε in place of µ yields (cid:13)(cid:13) e λt µ ( h ) γ − µM t (cid:13)(cid:13) M ( V ) ≤ Cκρ µ ( M ε V ) (1 + Θ ′ ) e − ωkτ e λkτ ≤ C ′ µ ( V )e − ωt , where C ′ = Cκρ (cid:0) ′ (cid:1) e ωτ e | λ | τ sup s ≤ τ (cid:13)(cid:13) M s VV (cid:13)(cid:13) ∞ . Adding that uniqueness is a direct consequenceof ω > i) . (cid:3) NON-CONSERVATIVE HARRIS ERGODIC THEOREM 21
Proof of Theorem 2.1 ii).
The proof follows the usual equivalence in Harris Theorem [39, Chapter15]. It also used the necessary condition of small set obtained in [23, Theorem 2.1]. Assume thatthere exist a positive measurable function V, a triplet ( γ, h, λ ) ∈ M + ( V ) × B + ( V ) × R , andconstants C, ω > k h k B ( V ) = γ ( h ) = 1. It remains to check that ( V, h ) satisfies Assumption A .Fix R > γ ( V ) and τ > − ωτ/ C ( R + γ ( V )) < − γ ( V ) R . (4.33)It ensures that α := e λτ (cid:18) C e − ωτ + γ ( V ) R (cid:19) < β := e λτ . Using (2.1), we obtain that (A2) and (A4) are satisfied by h with d = 1 and any probabilitymeasure ν , which ends the proof.By (2.2), we have for all x ∈ X , e − λt M t V ( x ) − h ( x ) γ ( V ) ≤ CV ( x )e − ωt . We define K = { x ∈X , V ( x ) ≤ Rh ( x ) } , which is not empty since k h k B ( V ) = 1 and R > γ ( V ) ≥ γ ( h ) = 1 . Writing θ = γ ( V )e λτ and using h ( x ) = K c ( h ( x ) /V ( x )) V ( x ) + K h ( x ), we get M τ V ( x ) ≤ αV ( x ) + K θh ( x )for all x ∈ X . Therefore, (A1) holds for ( V, h ) and it remains to prove (A3).We define the probability measure π := γ ( · h ) and we use the Hahn-Jordan decomposition ofthe following family of signed measure indexed by x ∈ X , ν x = δ x M τ/ ( · h )e λτ/ h ( x ) − π = ν x + − ν x − . As h ≤ V , Equation (2.2) with t = τ / µ = δ x yields ν x ± ( ) ≤ ν x ± ( V /h ) ≤ k ν x k M ( V/h ) = (cid:13)(cid:13)(cid:13)(cid:13) δ x M τ/ e λτ/ h ( x ) − γ (cid:13)(cid:13)(cid:13)(cid:13) M ( V ) ≤ C V ( x ) h ( x ) e − ωτ/ . (4.34)For every f ∈ B + ( V /h ) and x ∈ X we have δ x M τ ( hf )e λτ h ( x ) = δ x M τ/ e λτ/ h ( x ) (cid:18) M τ/ ( hf )e λτ/ h h (cid:19) ≥ ( π − ν x − ) (cid:18) M τ/ ( hf )e λτ/ h (cid:19) . (4.35)Next, π (cid:18) M τ/ ( hf )e λτ/ h (cid:19) = γM τ/ ( hf )e λτ/ = e λτ/ γ ( hf )e λτ/ = π ( f ) (4.36)and writing ( ν y ( f )) + the positive part of the real number ν y ( f ), ν x − (cid:18) M τ/ ( hf )e λτ/ h (cid:19) = Z X δ y M τ/ ( hf )e λτ/ h ( y ) ν x − ( dy ) ≤ Z X ( π ( f ) + ( ν y ( f )) + ) ν x − ( dy ) . (4.37)Combining (4.35) with (4.36) and (4.37), we get δ x M τ ( hf )e λτ h ( x ) ≥ π ( f )(1 − ν x − ( )) − Z X ( ν y ( f )) + ν x − ( dy ) . The minimality property of the Hahn-Jordan decomposition entails that ν x − ≤ π and (4.34) yieldst ν x − ( ) ≤ CR e − ωτ/ for x ∈ K. We deduce that for all x ∈ K , δ x M τ ( hf )e λτ h ( x ) ≥ π ( f )(1 − CR e − ωτ/ ) − Z ( ν y ( f )) + π ( dy ) =: η ( f ) . BANSAYE , BERTRAND
CLOEZ , PIERRE
GABRIEL , AND ALINE
MARGUET
We consider now the infimum measure ν of the left hand side. More precisely, using [23, Lemma5.2] we can define the mesure ν by ν ( A ) = inf ( n X i =1 δ x i M τ ( h B i )e λτ h ( x i ) : n ≥
1; ( x , . . . , x n ) ∈ X n ; ( B , . . . , B n ) ∈ P ( A ) ) , where A is a measurable subset of X and P ( A ) is the set of finite partitions of A formed by measur-able sets of X . Besides for any measurable partition B , . . . , B n of X , we have P ni =1 ( ν y ( B i )) + ≤ ν y + ( X ) ≤ C e − ωτ/ V ( y ) /h ( y ) from (4.34). Then n X i =1 η ( B i ) ≥ − CR e − ωτ/ − Z X n X i =1 ( ν y ( B i )) + π ( dy ) ≥ c, where c = 1 − C e − ωτ/ ( R + γ ( V )) > η ( K c ) ≤ η (cid:18) VRψ (cid:19) ≤ π (cid:18) VRψ (cid:19) = γ ( V ) R , we obtain that ν ( X ) ≥ c and ν ( K c ) < c . As a conclusion, ν is a positive measure such that ν ( K ) > x ∈ X and f ∈ B + ( V /h ), δ x M τ ( hf )e λτ h ( x ) ≥ ν ( f )which yields (A3) and ends the proof. (cid:3) We end this section by proving the sufficient conditions of Section 2.2.
Proof of Proposition 2.2.
Let
C > C − ψ ≤ ϕ ≤ Cψ.
By assumptions L ψ ≥ bψ and L ϕ ≤ ξϕ so that we have for all t ≥ M t ψ ≥ ψ + b Z t M s ψ ds and M t ϕ ≤ ϕ + ξ Z t M s ϕ ds, which yields by Gr¨onwall’s lemma M t ψ ≥ e bt ψ and M t ψ ≤ CM t ϕ ≤ C e ξt ϕ ≤ C e ξt ψ. Similarly,setting φ = V − ζb − a ψ , we have L φ ≤ aV + ζψ − b ζb − a ψ = aφ, so by Gr¨onwall’s lemma M t φ ≤ e at φ and M t V ≤ e at (cid:16) V − ζb − a ψ (cid:17) + ζb − a M t ψ ≤ e at V + C ζb − a e ξt ψ. Consider now τ, R > , and α = e aτ + C ζ ( b − a ) R e ξτ , β = e bτ , θ = C ζb − a e ξτ , R = θ e bτ − e aτ positive constants. Defining K = { x ∈ X , V ( x ) ≤ R ψ ( x ) } , we get ψ ≤ V /R on K c and M τ V ≤ αV + θ K ψ . Besides for R > R , α < β . So Assumptions (A1)-(A2) are verified for K = { V ≤ R ψ } ,R > R , and constants α, β, θ defined just above. (cid:3) Proof of Proposition 2.3.
Let ψ : X → (0 , ∞ ) and define ν = ( K ) − P x ∈ K δ x the uniform mea-sure on K, where K stands for the cardinal of K. We have for all f ≥ x, y ∈ K , δ x M τ ( f ψ ) ≥ δ x M τ ( { y } ) f ( y ) ψ ( y ) ≥ cf ( y ) M τ ψ ( x ) . where c = min x,y ∈ K ψ ( y ) δ x M τ ( { y } ) M τ ψ ( x ) > δ x M τ ( { y } ) > . Integrating with respect to ν shows that (A3)holds and Assumption (A4) is trivially satisfied with d = 1 / K . (cid:3) NON-CONSERVATIVE HARRIS ERGODIC THEOREM 23 Applications
Convergence to quasi-stationary distribution.
Let ( X t ) t ≥ be a c`ad-l`ag Markov processon the state space X ∪{ ∂ } , where X is measurable space and ∂ is an absorbing state. In this section,we apply the results to the (non-conservative) semigroup defined for any measurable boundedfunction f on X and any x ∈ X by M t f ( x ) = E x [ f ( X t ) X t = ∂ ] . We assume that there exists a positive function V such that for any t >
0, there exists C t > E x [ V ( X t )] ≤ C t V ( x ) for any x ∈ X . This ensures that the semigroup M acts on B ( V ) andthat we can use the framework of Section 2.A quasi-stationary distribution (QSD) is a probability law π on X such that ∀ t ≥ , P π ( X t ∈ · | X t = ∂ ) = π ( · ) . Corollary 3.7 directly gives existence and uniqueness of a QSD and quantitative estimates for theconvergence. Recall that P ( V ) stands for the set of probability measures which integrate V. Theorem 5.1.
Assume that ( M t ) t ≥ satisfies Assumption A with inf X V > . Then, there exista unique quasi-stationary distribution π ∈ P ( V ) , and λ > , h ∈ B + ( V ) , C, w > such that forall µ ∈ P ( V ) and t ≥ (cid:13)(cid:13) e λ t P µ ( X t ∈ · ) − µ ( h ) π (cid:13)(cid:13) TV ≤ Cµ ( V )e − ωt , and k P µ ( X t ∈ · | X t = ∂ ) − π k TV ≤ C µ ( V ) µ ( h ) e − ωt . It extends and complements recent results, see e.g. [24] for various interesting examples and discus-sions below for comparisons of statements. In particular, it relaxes the conditions of boundednessfor ψ and provides a quantitative estimate for exponential convergence of the conditional distribu-tion to the QSD using the eigenfunction h .As an application, we consider the simple but interesting case of a continuous time random walkon integers, with jumps +1 and −
1, absorbed at 0. We obtain optimal results for the exponentialconvergence to the QSD. Let us consider the Markov process X whose transition rates and generatorare given by the linear operator defined for any n ∈ N and f : N → R by L f ( n ) = b n ( f ( n + 1) − f ( n )) + d n ( f ( n − − f ( n )) , where b i = b > d i = d > i ≥ b , d > b = d = 0. This process is a Birthand Death (BD) process which follows a simple random walk before reaching 1. If d ≥ b , thisprocess is almost surely absorbed at 0. The convergence in law of such processes conditionally onnon-absorption has been studied in many works [1, 51, 57, 66, 69, 85, 86, 89, 93].The necessary and sufficient condition for ξ − positive recurrence of BD processes is known fromthe work of van Doorn [86]. More precisely here, the fact that there exists λ > x > i >
0, e − λt P x ( X t = i ) converges to a positive finite limit as t → ∞ is given by thefollowing condition ( H ) ∆ := ( √ b − √ d ) + b (cid:16)p d/b − (cid:17) − d > . We notice that b = d is excluded by condition ( H ) and indeed in this case t P ( X t = 0) decreasespolynomially. Similarly, the case b = b and d = d is excluded and there is an additional linearterm in the exponential decrease of P x ( X t = i ).Moreover we know from [85] that condition ( H ) ensures that P x ( X t ∈ ·| X t = 0) converges to theunique QSD π for any x >
0. To the best of our knowledge, under Assumption ( H ), the speed ofconvergence of e − λt P x ( X t = i ) or P x ( X t = i | X t = 0) and the extension of the convergence to infinitesupport masses were not proved, see e.g. [85, page 695]. For a subset of parameters satisfying (H) ,[89] obtains the convergence to QSDs for non-compactly supported initial measures µ such that µ ( V ) < ∞ . We obtain below quantitative exponential estimates for the full range of parametersgiven by ( H ) and non-compactly supported initial measures. BANSAYE , BERTRAND
CLOEZ , PIERRE
GABRIEL , AND ALINE
MARGUET
More precisely, we set X = N \ { } = { , . . . } , V : n p d/b n , ψ : n η n , for n ∈ X , where η = p d/b − ∆ / b ∈ (0 , p d/b ). Corollary 5.2.
Under Assumption ( H ) , there exists a unique QSD π ∈ P ( V ) , and λ > , h ∈B + ( V ) and C, w > such that for all µ ∈ P ( V ) and t ≥ , (cid:13)(cid:13) e λ t P µ ( X t ∈ · ) − µ ( h ) π (cid:13)(cid:13) TV ≤ Cµ ( V )e − ωt and k P µ ( X t ∈ · | X t = 0) − π k TV ≤ C µ ( V ) µ ( h ) e − ωt . Note that the constants above can be explicitly derived from Lemma 3.4, that these esti-mates hold for non-compactly supported initial laws and that V and ψ are not eigenelements.As perspectives, we expect this statement to be generalized to BD processes where b n , d n areconstant outside some compact set of N . Finally, we hope that the proof will help to study thenon-exponential decrease of the non-absorption probability, in particular for random walks, corre-sponding to b = b , d = d . Proof of Corollary 5.2.
For u ≥
1, let ϕ u : n u n for n ≥ ϕ u (0) = 0. We have L ϕ u ( n ) = λ u ( n ) ϕ u ( n ) , for any n ∈ N , where λ u ( n ) = λ u = b ( u −
1) + d (1 /u −
1) ( n ≥ , λ u (1) = b ( u − − d . We set a = inf u> λ u = λ √ d/b = − ( √ d − √ b ) , ζ = ∆ V (1) ψ (1) . Note that from ( H ), ζ >
0. Then, setting V (0) = ψ (0) = 0, we have on N = X ∪ { } that V = ϕ √ d/b and L V = aV + ζ { n =1 } ψ ≤ aV + ζψ. (5.1)Moreover, on N , ψ = ϕ η and bψ ≤ L ψ ≤ ξψ (5.2)with b = min( λ η , λ η (1)) = min (cid:0) λ η , a + ∆2 (cid:1) > a = inf u> λ u , and ξ = max( λ η , λ η (1)) . Using now a classical localization argument (see Appendix 6.2 for details), the drift conditions(5.1)-(5.2) ensure that for any n ≥ t ≥ E n [ V ( X t )] ≤ V ( x ) + Z t E n [( aV + ζψ )( X s )] ds, (5.3) ψ ( x ) + Z t E n [ bψ ( X s )] ds ≤ E x [ ψ ( X t )] ≤ ψ ( x ) + Z t E n [ ξψ ( X s )] ds. (5.4)Considering the generator L of the semigroup M t f ( x ) = E [ f ( X t )1 X t =0 ] for x ∈ X and f ∈ B ( V )and recalling the definition of Section 2.2, these inequalities ensure L V ≤ aV + ζψ, L ψ ≥ bψ, L ψ ≤ ξψ. Finally, the fact that b i , d i > i ≥ δ i M t ( { j } ) > i, j ∈ X and t >
0. Thencombining Propositions 2.2 and 2.3 ensures that Assumption A holds for M with the functions( V, ψ ). Applying Theorem 5.1 ends the proof. (cid:3)
The growth-fragmentation equation.
In this section we apply our general result to thegrowth-fragmentation partial differential equation ∂ t u t ( x ) + ∂ x u t ( x ) + B ( x ) u t ( x ) = Z B (cid:16) xz (cid:17) u t (cid:16) xz (cid:17) ℘ ( dz ) z (5.5)for t, x >
0. This nonlocal partial differential equation is complemented with the zero flux boundarycondition u t (0) = 0 for all t > u = µ. The unknown u t ( x ) represents thepopulation density at time t of some “particles” with “size” x > , which can be for instance thesize of a cell [35, 54], the length of a polymer fibril [43], the window size in data transmission overthe Internet [10, 22], the carbon content in a forest [18], or the time elapsed since the last discharge NON-CONSERVATIVE HARRIS ERGODIC THEOREM 25 of a neuron [19, 77]. Each particle grows with speed 1 , and splits with rate B ( x ) to produce smallerparticles of sizes zx with 0 < z < ℘. We assume that B : [0 , ∞ ) → [0 , ∞ ) is a continuously differentiable increasing function and ℘ is a finite positive measure on [0 ,
1] for which there exist z ∈ (0 , , ǫ ∈ [0 , z ] and c > ℘ ( dz ) ≥ c ǫ [ z − ǫ,z ] ( z ) dz if ǫ > ℘ ≥ c δ z if ǫ = 0 . (5.6)For any r ∈ R we denote by ℘ r ∈ (0 , + ∞ ] the moment of order r of ℘℘ r = Z z r ℘ ( dz ) . Notice that Assumption (5.6) implies that r ℘ r is strictly decreasing. The conservation of massduring the fragmentation leads to impose ℘ = 1 . The zero order moment ℘ represents the mean number of fragments. The conditions above en-sure that ℘ > ℘ = 1 is replaced by ℘ = 1 also appears in somesituations [10, 18, 19, 22, 77]. In this case, the eigenelements are given by h ( x ) = 1 , λ = 0 , and theclassical theory of the conservative Harris theorem applies [17, 18]. Here we are interested in themore challenging non-conservative case.An important feature in the long time behavior of the (non-conservative) growth-fragmentationequation is the property of asynchronous exponential growth [90], which refers to a separation ofthe variables t and x when time t becomes large: the size repartition of the population stabilizesand the total mass grows exponentially in time. This question attracted a lot of attention in thelast decades, [15, 20, 35, 73, 74, 80] to mention only a few. As far as we know, the existing literatureassume either that the state space is a bounded interval instead of the whole (0 , ∞ ), as in [5, 35], orthat the fragmentation rate has at most a polynomial growth, as in [2, 13, 20, 74]. We can considerhere unbounded state space and we relax the latter condition by not assuming any upper boundon the division rate with a similar approach through h -transform). We obtain thus the existence ofthe Perron eigentriplet for super-polynomial fragmentation rates. Second, an explicit spectral gapwas known only in the case of a constant division rate [10, 22, 65, 74, 80, 92]. Our method allowsus to get it for more general fragmentation rates. Finally, it guarantees exponential convergencefor measure solutions, thus drastically improving [31].Let us now state the main result of this section. Theorem 5.3.
Let k > and V ( x ) = 1+ x k . Under the above assumptions, there exists of a uniquetriplet ( γ, h, λ ) ∈ M + ( V ) ×B + ( V ) × R of Perron-Frobenius eigenelements with γ ( h ) = k h k B ( V ) = 1 , i.e. satisfying L h = λh and γ ( L f ) = λγ ( f ) for all f ∈ C c ([0 , ∞ )) , where L : C ([0 , ∞ )) → C ([0 , ∞ )) is defined by L f ( x ) = f ′ ( x ) + B ( x ) (cid:18) Z f ( zx ) ℘ ( dz ) − f ( x ) (cid:19) . Moreover there exist constants
C, ω > such that for all µ ∈ M ( V ) the solution to Equation (5.5) with u = µ satisfies for all t ≥ , (cid:13)(cid:13) e − λt u t − u ( h ) γ (cid:13)(cid:13) M ( V ) ≤ C e − ωt k u k M ( V ) . (5.7)The constants λ, ω , and C can be estimated quantitatively, and the eigenfunction h satisfies(1 + x ) − q ( k − . h . (1 + x ) k , for some explicit q >
0. Note also that we cannot expect theconvergence (5.7) to hold true in M ( h ) in general. In particular it is known to be wrong when B is bounded [12]. BANSAYE , BERTRAND
CLOEZ , PIERRE
GABRIEL , AND ALINE
MARGUET
The end of the section is devoted to the proof of Theorem 5.3. We can associate to Equation (5.5)a semigroup ( M t ) t ≥ , to which we will apply our result in Theorem 2.1 after having checked thatit verifies Assumption A with the functions V ( x ) = 1 + x k and ψ ( x ) = 12 (1 + x )with k >
1. The factor in the definition of ψ is to satisfy the inequality ψ ≤ V . We onlygive here the definition of this semigroup as well as its main properties which are useful to verifyAssumption A , and we refer to Appendix 6.3 for the proofs. For any f : [0 , ∞ ) → R measurable andlocally bounded on [0 , ∞ ), we define the family ( M t f ) t ≥ as the unique solution to the equation M t f ( x ) = f ( x + t )e − R t B ( x + s ) ds + Z t e − R s B ( x + s ′ ) ds ′ B ( x + s ) Z M t − s f ( z ( x + s )) ℘ ( dz ) ds. This semigroup is positive and preserves C ([0 , ∞ )). More precisely if f ∈ C ([0 , ∞ )) , then thefunction ( t, x ) M t f ( x ) is continuously differentiable on [0 , ∞ ) and ∂ t M t f ( x ) = L M t f ( x ) = M t L f ( x ) , where L is defined in Theorem 5.3. The space B ( V ) is invariant under ( M t ) t ≥ and for any µ ∈ M ( V ), we can define by duality µM t ∈ M ( V ) . The family ( µM t ) t ≥ is solution to (5.5) withinitial data µ, in a weak sense made precise in Appendix 6.3.Let x ≥ B > x ≥ x , B ( x ) ≥ B. Now define t = 1 + z + (1 + ǫ ) x − z + 12 , t = 1 − z z , τ = t + t , (5.8)and for all integer n ≥ y n = (cid:16) z z (cid:17) n + x . Lemma 5.4. i) Setting ϕ ( x ) = 1 − √ x + x , we have ψ ≤ ϕ ≤ ψ and there exist ζ > and a < b < ξ such that L V ≤ aV + ζψ, L ψ ≥ bψ, L ϕ ≤ ξϕ, where L is the generator of ( M t ) t ≥ in the sense defined in Section 2.2.ii) For all n ≥ , all x ∈ [0 , y n ] , and all f : [0 , ∞ ) → [0 , ∞ ) locally bounded we have M τ f ( x ) ≥ e − τB ( y n + τ ) ( c B ) n +1 − z t n n ! ν ( f ) , where ν is the probability measure defined by ν ( f ) = Z z ( y + τ )+1 z ( y + τ ) f ( y ) dy. iii) For all η > there exists c η > such that for all t, x ≥ and y ∈ [ ηx, x ] c η ≤ M t ψ ( y ) M t ψ ( x ) ≤ . iv) For all n ≥ , there exists d > such that d M t ψ ( x ) ψ ( x ) ≤ M t ψ ( y ) ψ ( y ) for all t ≥ , x ∈ [0 , y n ] and y ∈ [ z ( y + τ ) , z ( y + τ ) + 1] = supp ν. Proof of Lemma 5.4 i).
Since by Proposition 6.3 the identity ∂ t M t = M t L is valid for all C functions and the semigroup M is positive, we only need to prove that L V ≤ aV + ζψ, L ψ ≥ bψ, L ϕ ≤ ξϕ. First, L x r = rx r − + ( ℘ r − B ( x ) x r NON-CONSERVATIVE HARRIS ERGODIC THEOREM 27 for any r ≥
0. We deduce that 2 L ψ ( x ) = 1 + ( ℘ − B ( x ) ≥ , so that b = 0 suits. For ϕ we have L ϕ ( x ) = 1 − √ x + ( ℘ − B ( x ) − ( ℘ − B ( x ) √ x. Since x ( ℘ − − ( ℘ − √ x is negative for x > (cid:0) ℘ − ℘ / − (cid:1) and B is increasing we deduce L ϕ ( x ) ≤ ℘ − B (cid:18)(cid:16) ℘ − ℘ − (cid:17) (cid:19) =: ξ ≤ ξϕ ( x ) . For V we have L V ( x ) = kx k − + ( ℘ − B ( x ) + ( ℘ k − B ( x ) x k = h(cid:0) ( ℘ k −
1) + ( ℘ − x − k (cid:1) B ( x ) + kx i| {z } → l :=( ℘ k −
1) lim x → + ∞ B ( x ) when x → + ∞ x k . Since ℘ k < B is increasing, the limit l belongs to [ −∞ ,
0) and we can find x > x ≥ x , L V ( x ) ≤ ax k = aV ( x ) − a, where a = max { l/ , − } < . For all x ∈ [0 , x ] we have L V ( x ) ≤ kx k − + ( ℘ − B ( x )and finally setting ζ = 2( kx k − + ( ℘ − B ( x ) − a ), we get that for all x ≥ L V ( x ) ≤ aV ( x ) + ζ ≤ aV ( x ) + ζψ ( x ) . It ends the proof of i) . (cid:3) Before proving ii) , let us briefly comment on the definition of t , t and y n . The time t andthe sequence y n are chosen in such a way that y > , y ≥ x , lim n →∞ y n = + ∞ , and z ( y n +1 + t ) ≤ y n . The choice of t appears in the proof of the case n = 0 and the definition of ν. Since τ is independent of n and y n → + ∞ when n → ∞ we can find R and n large enoughso that supp ν ⊂ K ⊂ [0 , y n ] , where K = { x, V ( x ) ≤ Rψ ( x ) } , and thus ii) guarantees thatAssumption (A3) is satisfied with time τ on K. More precisely it suffices to take R and n largeenough so that 1 + ( z ( y + τ ) + 1) k z ( y + τ ) + 1 ≤ R ≤ y kn y n . (5.9) Proof of Lemma 5.4 ii).
Let f ≥ . We prove by induction on n that for all x ∈ [0 , y n ] and all t ∈ [0 , t ] we have M t + t f ( x ) ≥ e − ( t + t ) B ( y n + τ ) ( c B ) n +1 − z t n n ! ν ( f ) , (5.10)which yields the desired result by taking t = t . We start with the case n = 0 . The Duhamel formula M t f ( x ) = f ( x + t )e − R t B ( x + s ) ds + Z t e − R s B ( x + s ′ ) ds ′ B ( x + s ) Z M t − s f ( z ( x + s )) ℘ ( dz ) ds ensures, using the positivity of M t and the growth of B, that for all t, x ≥ M t f ( x ) ≥ e − tB ( x + t ) Z t B ( x + s ) Z f ( z ( x + s ) + t − s ) ℘ ( dz ) ds. BANSAYE , BERTRAND
CLOEZ , PIERRE
GABRIEL , AND ALINE
MARGUET
Thus for t ≥ x we have for all x ≥ , using Assumption (5.6) for the second inequality, M t f ( x ) ≥ e − tB ( x + t ) B Z tx Z f ( z ( x + s ) + t − s ) ℘ ( dz ) ds ≥ e − tB ( x + t ) B c − z Z ( z − ǫ )( x + x )+ t − x z ( x + t ) f ( y ) dy. We deduce that for t ∈ [ t , t + t ] and x ∈ [0 , y ] M t f ( x ) ≥ e − tB ( x + τ ) B c − z Z ( z − ǫ ) x + t − x z ( x + t + t ) f ( y ) dy. The time t has been defined in such a way that ( z − ǫ ) x + t − x = z ( x + t + t ) + 1 so R ( z − ǫ ) x + t − x z ( x + t + t ) f ( y ) dy = ν ( f ) and this finishes the proof of the case n = 0 . Assume that (5.10) holds for n and let’s check it for n + 1 . By Duhamel formula, using that y n ≥ x and z ( y n +1 + t ) ≤ y n , we get for x ∈ [ y n , y n +1 ] and t ∈ [0 , t ], M t + t f ( x ) ≥ Z t e − R s B ( x + s ′ ) ds ′ B ( x + s ) Z M t + t − s f ( z ( x + s )) ℘ ( dz ) ds ≥ B Z t e − sB ( x n +1 + t ) Z z M t + t − s f ( z ( x + s )) ℘ ( dz ) ds ≥ B n +2 c n +10 − z ν ( f ) Z t e − sB ( x n +1 + t ) e − ( t + t − s ) B ( x n + τ ) ( t − s ) n n ! Z z ℘ ( dz ) ds ≥ e − ( t + t ) B ( y n +1 + τ ) B n +2 c n +20 − z t n +1 ( n + 1)! ν ( f )and the proof is complete. (cid:3) We now turn to the proof of iii) and iv) , which uses the monotonicity results proved inLemma 6.5, see Appendix 6.3.
Proof of Lemma 5.4 iii).
The second inequality readily follows from Lemma 6.5 ii ). For the firstone, we start with a technical result on ℘. Due to the assumption we made on ℘, if we set z > max( z , − c (cid:0) z − ǫ/ ̺ := Z z ℘ ( dz ) ≤ z (cid:16) − Z z z ℘ ( dz ) (cid:17) ≤ − c (cid:0) z − ǫ/ z < . Using Lemma 6.5 ii) and iii) , we deduce that for all t ≥ s ≥ x ≥ Z M t − s ψ ( z ( x + s )) ℘ ( dz ) ≤ Z M t ψ ( zx ) ℘ ( dz ) ≤ ℘ M t ψ ( z x ) + ̺M t ψ ( x ) . Now from the Duhamel formula we get, using that t t e − R t B ( s ) ds is bounded on [0 , ∞ ) ,M t ψ ( x ) = ψ ( x + t )e − R t B ( x + s ) ds + Z t e − R s B ( x + s ′ ) ds ′ B ( x + s ) Z M t − s ψ ( z ( x + s )) ℘ ( dz ) ds ≤ (1 + x + t )e − R t B ( s ) ds + (cid:16) − e − R t B ( x + s ) ds (cid:17)(cid:16) ℘ M t ψ ( z x ) + ̺M t ψ ( x ) (cid:17) ≤ C ψ ( x ) + ℘ M t ψ ( z x ) + ̺M t ψ ( x ) . Choosing an integer n such that z n ≤ η we obtain M t ψ ( x ) ≤ C − ̺ n − X k =0 (cid:16) ℘ − ̺ (cid:17) k ψ ( x ) + (cid:16) ℘ − ̺ (cid:17) n M t ψ ( ηx ) = C ψ ( x ) + C M t ψ ( ηx )and since M t ψ ( ηx ) ≥ ψ ( ηx ) ≥ ηψ ( x ) we obtain for any y ∈ [ ηx, x ], M t ψ ( y ) M t ψ ( x ) ≥ M t ψ ( ηx ) C ψ ( x ) + C M t ψ ( ηx ) ≥ ηC + C η , NON-CONSERVATIVE HARRIS ERGODIC THEOREM 29 which ends the proof. (cid:3)
Proof of Lemma 5.4 iv).
Point iii) applied with η = z ( y + τ ) /y n gives, together with Lemma 6.5 ii) ,that for all x ∈ [0 , y n ] and y ∈ [ z ( y + τ ) , z ( y + τ ) + 1], M t ψ ( y ) ≥ M t ψ ( z ( y + τ )) ≥ c η M t ψ ( y n ) ≥ c η M t ψ ( x ) . Besides, ψ ( x ) /ψ ( y ) = (1 + x ) / (1 + y ) ≥ / ( z ( y + τ ) + 2) and we get M t ψ ( y ) ψ ( y ) ≥ c η z ( y + τ ) + 2 M t ψ ( x ) ψ ( x ) , which ends the proof. (cid:3) We are now in position to prove Theorem 5.3.
Proof of Theorem 5.3.
Fix τ defined in (5.8). In Lemma 5.4 i) we have verified the assumptions ofProposition 2.2, so we can find a real R > n ≥ K = { V ≤ Rψ } . Then, points ii) and iv) in Lemma 5.4ensure that Assumptions (A3) and (A4) are also satisfied. So Assumption A is verified for ( V, ψ )and by virtue of Theorem 2.1 inequality (5.7) is proved, as well as the bounds on h . It remainsto check that h is continuously differentiable and that h and γ satisfy the eigenvalue equations L h = λh and γ L = λγ. By definition of h , the Duhamel formula gives h ( x )e λt = h ( x + t )e − R t B ( x + s ) ds + Z t e − R s B ( x + s ′ ) ds ′ B ( x + s ) Z e λ ( t − s ) h ( z ( x + s )) ℘ ( dz ) ds and we deduce that for any x > t h ( t + x ) is continuous and then continuouslydifferentiable. Moreover, we have the identity ∂ t M t h = M t L h and since M t h = e λt h we deduce L h = λh. For the equation on γ , we start from Proposition 6.4 which ensures thate λt γ ( f ) = ( γM t )( f ) = γ ( f ) + Z t ( γM s )( L f ) ds = γ ( f ) + e λt − λ γ ( L f ) . for any f ∈ C c ([0 , ∞ )). Differentiating with respect to t yields the result. (cid:3) Comments and a few perspectives.
The proof of Theorem 2.1 consists in first provingthe V -uniform ergodicity of the discrete time semigroup ( M nτ ) n ≥ = ( M nτ ) n ≥ , and then extendit to the continuous setting. We can thus state analogous results for a discrete time semigroup( M n ) n ∈ N by making the following assumption for a couple of positive functions ( V, ψ ) with ψ ≤ V . Assumption B.
There exist some integer τ >
0, real numbers β > α > θ >
0, ( c, d ) ∈ (0 , ,some set K ⊂ X such that sup K V /ψ < ∞ , and some probability measure ν on X supported by K such that(B1) M τ V ≤ αV + θ K ψ ,(B2) M τ ψ ≥ βψ ,(B3) inf x ∈ K M τ ( f ψ )( x ) M τ ψ ( x ) ≥ c ν ( f ) for all f ∈ B + ( V /ψ ),(B4) ν (cid:18) M nτ ψψ (cid:19) ≥ d sup x ∈ K M nτ ψ ( x ) ψ ( x ) for all positive integers n .The discrete time counterpart of Theorem 2.1 is stated below. Theorem 5.5. i) Let ( V, ψ ) be a couple of measurable functions from X to (0 , ∞ ) such that ψ ≤ V and which satisfies Assumption B . Then, there exists a unique triplet ( γ, h, λ ) ∈ M + ( V ) ×B + ( V ) × R of eigenelements of M with γ ( h ) = k h k B ( V ) = 1 , i.e. satisfying γM = λγ and M h = λh. (5.11) BANSAYE , BERTRAND
CLOEZ , PIERRE
GABRIEL , AND ALINE
MARGUET
Moreover, there exists
C > and ρ ∈ (0 , such that for all n ≥ and µ ∈ M ( V ) , (cid:13)(cid:13) λ − n µM n − µ ( h ) γ (cid:13)(cid:13) M ( V ) ≤ C k µ k M ( V ) ρ − n . (5.12) ii) Assume that there exist a positive measurable function V, a triplet ( γ, h, λ ) ∈ M + ( V ) × B + ( V ) × R , and constants C, ρ > such that (5.11) and (5.12) hold. Then, the couple ( V, h ) satisfiesAssumption B . When inf X V >
C > π ∈ P ( V ) suchthat for all µ ∈ P ( V ) and n ≥ (cid:13)(cid:13)(cid:13)(cid:13) µM n µM n − π (cid:13)(cid:13)(cid:13)(cid:13) TV ≤ C µ ( V ) µ ( h ) ρ n . As a consequence, Assumption B gives sufficient conditions to have the existence, uniqueness andconvergence to a QSD for a Markov chain. The convergence of the Q -process, the description ofthe domain of attraction and the bounds on the extinction times can be then obtained by usualprocedure, see e.g. [24, 88].For the sake of simplicity, we have not allowed ψ to vanish in this paper. This excludes reduciblestructures. To illustrate this, let us consider the simple case of [11, Example 3.5] where X = { , } and M = (cid:18) a b c (cid:19) with a, b, c >
0. For any µ such that µ ( { } ) = 0 we have c − n µM n = µ ( { } ) δ . If µ ( { } ) > c > a , we can apply Theorem 5.5 with the right eigenvector ψ = ( b, c − a ) = b { } + ( c − a ) { } and show that, up to normalisation, µM n converges to δ . When c ≤ a , we cannot use Theorem 5.5and there is no positive right eigenvector. However, allowing ψ to vanish enables handling the case c < a , as in [24, Section 6]. Focusing on initial measures µ such that µ ( ψ ) >
0, a large part ofour results actually holds when ψ ≥
0. Indeed (B2) gives that if µ ( ψ ) > µM n ψ > n ≥
0. The critical case a = c is a situation where there is no spectral gap. We believe thatour approach could also be extended to the study of semigroups without spectral gap by allowing n -dependent constants d = d n in (B4), similarly as in [11, Assumption ( H )].The discrete time result allows us to deduce V -uniform ergodicity, not only for continuous timesemigroups as in Theorem 2.1, but also in a time periodic setting. We say that an propagator( M s,t ) ≤ s ≤ t is T -periodic if M s + T,t + T = M s,t for all t ≥ s ≥
0, and Theorem 5.5 allows derivingsome extension of the Floquet theory of periodic matrices [46] to such periodic propagators. Wesay that ( γ s,t , h s,t , λ F ) ≤ s ≤ t is a Floquet eigenfamily for the T -periodic propagator ( M s,t ) ≤ s ≤ t ifthe families ( γ s,t ) ≤ s ≤ t and ( h s,t ) ≤ s ≤ t are T -periodic in the sense that γ s + T,t + T = γ s,t = γ s,t + T and h s + T,t + T = h s,t = h s,t + T for all t ≥ s ≥
0, and are associated to the Floquet eigenvalue λ F in the sense that γ s,s M s,t = e λ F ( t − s ) γ s,t and M s,t h t,t = e λ F ( t − s ) h s,t for all t ≥ s ≥
0. Starting from Theorem 5.5 and following the proof of [8, Theorem 3.15], weobtain the periodic result stated below.
Theorem 5.6.
Let ( M s,t ) ≤ s ≤ t be a T -periodic propagator such that ( s, t )
7→ k M s,t V k B ( V ) islocally bounded, and suppose that M s,s + T satisfies Assumption B for some functions V ≥ ψ > and some s ∈ [0 , T ) . Then there exist a unique T -periodic Floquet family ( γ s,t , h s,t , λ F ) ≤ s ≤ t ⊂M + ( V ) × B + ( V ) × R such that γ s,s ( h s,s ) = k h s,s k B ( V ) = 1 for all s ≥ , and there exist C ≥ ,ω > such that for all t ≥ s ≥ and all µ ∈ M ( V ) , (cid:13)(cid:13) e − λ F ( t − s ) µM s,t − µ ( h s,s ) γ s,t (cid:13)(cid:13) M ( V ) ≤ C e − ω ( t − s ) k µ − µ ( h s,s ) γ s,s k M ( V ) . NON-CONSERVATIVE HARRIS ERGODIC THEOREM 31
This theorem may prove useful for investigating models that arise in biology when taking intoaccount the time periodicity of the environment. An example is given in [48, Section 6] which leadsto a periodic growth-fragmentation equation, and the Floquet eigenvalue is compared to some timeaverages of Perron’s.Beyond the periodic case, several extensions to the fully non-homogeneous setting are expected,in the same vein as [8]. We can now relax the “coming down from infinity” property imposed bythe generalized Doeblin condition of [8] and thus capture a larger class of non-autonomous linearPDEs. This would allow extending some ergodic results of optimal control problems, as the onesin [21], to the infinite dimension. Similarly, let us recall that the expectation of a branching processyields the first moment semigroup, which usually drives the extinction of the process (criticality)and provides its deterministic renormalization (Kesten-Stigum theorem). The method of this papershould provide a powerful tool to analyse the first moment semigroup of a branching process withinfinite number of types, including in varying environment, see [6, 7, 67] for some motivations inpopulation dynamics and queuing systems. We also mention that time inhomogeneity provides anatural point of view to deal with non-linearity in large population approximations of systems withinteraction. These points should be partially addressed in forthcoming works.6.
Appendix
Conservative operators.
The following useful result is a direct generalization of the con-traction result of Hairer and Mattingly [53], in the same vein as [24, 67]. We consider a positiveoperator P acting both on bounded measurable functions f : X → R on the right and on boundedmeasures µ of finite mass on the left, and such that ( µP ) f = µ ( P f ) . Note that the right action of P extends trivially to any measurable function f : X → [0 , + ∞ ] . We assume that P is conservativein the sense that P = . In other words, if µ is a probability measure, then so does µP. Theorem 6.1.
Assume that there exist two measurable functions V , W : X → [0 , ∞ ) , ( a , b ) ∈ (0 , , c > , R > c / (1 − a ) and a probability measure ν on X such that: • for all x ∈ X , P V ( x ) ≤ a W + c , (6.1) • for all x ∈ { W ≤ R } , δ x P ≥ b ν. (6.2) Then, there exist y ∈ (0 , and κ > such that for all probability measures µ , µ , k µ P − µ P k M (1+ κ V ) ≤ y k µ − µ k M (1+ κ W ) . In particular, for any b ′ ∈ (0 , b ) and a ′ ∈ ( a + 2 c / R , , one can choose κ = b ′ c , y = max (cid:26) − ( b − b ′ ) , κ Ra ′ κ R (cid:27) . Usually, for conservative semigroups and Markov chains [53, 71], Theorem 6.1 is stated andused with one single function V = W . For the sake of completeness, we briefly give the proof ofTheorem 6.1, which is an adaptation of [53, Theorem 1.3]. For any function V : X → [0 , ∞ ), letus define the following distance on X :dist V ( x, y ) = ( x = y V ( x ) + V ( y ) x = y We also introduce a semi-norm on measurable functions f : X → R defined by k f k Lip( V ) = sup x = y | f ( x ) − f ( y ) | dist V ( x, y ) , and the associated (Wasserstein) metric on the set of probability measures given bydist V ( µ , µ ) = sup k f k Lip( V ) ≤ Z X f ( x )( µ − µ )( dx ) = k µ − µ k M (1+ V ) , using [53, Lemma 2.1]. We prove now Theorem 6.1, which is a simple adaptation of the proofs ofTheorems 1.3 and 3.1 in [53]. BANSAYE , BERTRAND
CLOEZ , PIERRE
GABRIEL , AND ALINE
MARGUET
Proof of Theorem 6.1.
Let f be a test function such that k f k Lip( κ V ) ≤ x = y . Fix a ′ ∈ ( a + 2 c / R ,
1) and b ′ ∈ (0 , b ) , and set κ = b ′ / c and y = max { − b + b ′ , (2 + κ Ra ′ ) / (2 + κ R ) } . Considering successively the cases W ( x ) + W ( y ) ≥ R and W ( x ) + W ( y ) ≤ R as in [53, Proof ofTheorem 3.1], and using (6.1) and (6.2), we obtain | P f ( x ) − P f ( y ) | ≤ y dist κ W ( x, y ) , so that k P f k Lip( κ W ) ≤ y . Finally, for any probability measure µ and µ ,dist κ V ( µ P, µ P ) = sup k f k Lip( κ V ) ≤ Z X P f ( x )( µ − µ )( dx ) ≤ sup k P f k Lip( κ W ) ≤ y Z X P f ( x )( µ − µ )( dx ) = y dist κ W ( µ , µ )and the proof is complete. (cid:3) Localization argument.
We detail here how the drift conditions (5.1)-(5.2) ensure (5.3)-(5.4), namely E n [ V ( X t )] ≤ V ( x ) + Z t E n [( aV + ζψ )( X s )] ds,ψ ( x ) + Z t E n [ bψ ( X s )] ds ≤ E x [ ψ ( X t )] ≤ ψ ( x ) + Z t E n [ ξψ ( X s )] ds, for all n ≥ t ≥
0. Following [72], for m ≥
1, we let T m = inf { t > X t ≥ m } and ( X mt ) t ≥ be the Markov process defined by X mt = X t t V, ψ on N by setting V (0) = ψ (0) = 0. Using (5.1), (5.2), V (0) ≤ V ( m ) and writing O m = { , , . . . m − } , the stronggenerator L m of X m satisfies L m V ≤ aV + ζψ and L m ψ ≤ ξψ on O m L m ψ ( m − 1) = L ψ ( m − − bψ ( m ) ≥ b (1 − η ) ψ ( m − L m ψ ≥ bψ on O m − . First, using L m V ≤ ( a + ζ ) V on O m and V ( n ) → ∞ as n → ∞ , [72, Theorem 2.1] ensures thatlim m →∞ T m = ∞ and E n [ V ( X t )] ≤ e ( a + ζ ) t V ( n )for every n ∈ N . Second L m ψ ≤ ξψ on O m and ψ is bounded on O m . Using that X m coincideswith X on [0 , T m ), Fatou’s lemma and Kolmogorov equation give E n [ ψ ( X t )] ≤ lim inf m →∞ E x [ ψ ( X mt )]= ψ ( n ) + lim inf m →∞ E n "Z t ∧ T m L m ψ ( X s ) ds ≤ ψ ( x ) + ξ Z t E n [ ψ ( X s )] ds. Moreover ψ ( X mt ) = t ≤ T m ψ ( X t ) ≤ ψ ( X t ) and X mt → X t as m → ∞ . Using L m ψ ≥ bψ − bηψ m − on O m and bounded convergence twice yields E ( R t ψ ( X s )1 X s = m − ds ) → m → ∞ and E n [ ψ ( X t )] = lim m →∞ E x [ ψ ( X mt )]= ψ ( n ) + lim m →∞ E n "Z t ∧ T m L m ψ ( X s ) ds ≥ ψ ( x ) + b Z t E n [ ψ ( X s )] ds. Using Fatou’s lemma as above for V ends the proof of (5.3)-(5.4). NON-CONSERVATIVE HARRIS ERGODIC THEOREM 33 The growth-fragmentation semigroup. We give here the details of the construction ofthe growth-fragmentation semigroup and prove its basic properties, along the lines of [41, 49, 50].For a function f ∈ B loc ([0 , ∞ )), we define the family ( M t f ) t ≥ ⊂ B loc ([0 , ∞ )) through the Duhamelformula M t f ( x ) = f ( x + t )e − R t B ( x + s ) ds + Z t e − R s B ( x + s ′ ) ds ′ B ( x + s ) Z M t − s f ( z ( x + s )) ℘ ( dz ) ds. We first prove that this indeed defines uniquely the family ( M t f ) t ≥ . Then, we verify that theassociated family ( M t ) t ≥ is a semigroup of linear operators, which provides the unique solutionto the growth-fragmentation (5.5) on the space M ( V ) with V ( x ) = 1 + x k , k > . Finally weprovide some useful monotonicity properties for this semigroup, which are consequences of themonotonicity assumption on B. Lemma 6.2. For any f ∈ B loc ([0 , ∞ )) there exists a unique ¯ f ∈ B loc ([0 , ∞ ) ) such that for all t ≥ and x ≥ f ( t, x ) = f ( x + t )e − R t B ( x + s ) ds + Z t e − R s B ( x + s ′ ) ds ′ B ( x + s ) Z ¯ f ( t − s, z ( x + s )) ℘ ( dz ) ds. Moreover if f is nonnegative/continuous/continuously differentiable, then so does ¯ f . In the lattercase ¯ f satisfies the partial differential equation ∂ t ¯ f ( t, x ) = L ¯ f ( t, x ) = ∂ x ¯ f ( t, x ) + B ( x ) (cid:20) Z ¯ f ( t, zx ) ℘ ( dz ) − ¯ f ( t, x ) (cid:21) . Proof. Let f ∈ B loc ([0 , ∞ ) and define on B loc ([0 , ∞ ) ) the mapping Γ byΓ g ( t, x ) = f ( x + t )e − R t B ( x + s ) ds + Z t e − R s B ( x + s ′ ) ds ′ B ( x + s ) Z g ( t − s, z ( x + s )) ℘ ( dz ) ds. Now for T, A > T,A = { ( t, x ) ∈ [0 , T ] × [0 , ∞ ) , x + t < A } and denote by B b (Ω T,A )the Banach space of bounded measurable functions on Ω T,A , endowed with the supremum norm k · k ∞ . Clearly Γ induces a mapping B b (Ω T,A ) → B b (Ω T,A ) , still denoted by Γ . To build a fixedpoint of Γ in B loc ([0 , ∞ ) ) we prove that it admits a unique fixed point in any B b (Ω A,A ) . Let A > T < / ( ℘ B ( A )) . For any g , g ∈ B b (Ω T,A ) we have k Γ g − Γ g k ∞ ≤ ℘ T B ( A ) k g − g k ∞ and Γ is a contraction. The Banach fixed point theorem then guarantees the existence of a uniquefixed point g T,A of Γ in B b (Ω T,A ) . The same argument on Ω T,A − T ensures that g T,A can be extendedinto a unique fixed point g T,A of Γ on Ω T,A . Iterating the procedure we finally get a unique fixedpoint g A of Γ in B b (Ω A,A ) . For A ′ > A > g A ′ | Ω A = g A by uniqueness of the fixed point in B b (Ω A ) , and we candefine ¯ f by setting ¯ f | Ω A = g A for any A > . Clearly the function ¯ f thus defined is the uniquefixed point of Γ in B loc ([0 , ∞ ) ) . Since Γ preserves the closed cone of nonnegative functions if f isnonnegative, the fixed point ¯ f necessarily belongs to this cone when f is nonnegative. Similarly,the space C ([0 , ∞ ) ) of continuous functions being a closed subspace of B loc ([0 , ∞ ) ) , the fixedpoint ¯ f is continuous when f is so.Consider now that f is continuously differentiable on [0 , ∞ ). The space C ([0 , ∞ ) ) is notclosed in B loc ([0 , ∞ ) ) for the norm k · k ∞ . For proving the continuous differentiability of ¯ f werepeat the fixed point argument in { g ∈ C (Ω T,A ) , g (0 , · ) = f } , endowed with the norm k g k C = k g k ∞ + k ∂ t g k ∞ + k ∂ x g k ∞ . Differentiating Γ g with respect to t we get ∂ t Γ g ( t, x ) = L f ( x + t )e − R t B ( x + s ) ds + Z t e − R s B ( x + s ′ ) ds ′ B ( x + s ) Z ∂ t g ( t − s, z ( x + s )) ℘ ( dz ) ds and differentiating the alternative formulationΓ g ( t, x ) = f ( x + t )e − R x + tx B ( y ) dy + Z x + tx e − R yx B ( y ′ ) dy ′ B ( y ) Z g ( t + x − y, zy ) ℘ ( dz ) dy BANSAYE , BERTRAND CLOEZ , PIERRE GABRIEL , AND ALINE MARGUET with respect to x we obtain ∂ x Γ g ( t, x ) = L f ( x + t )e − R x + tx B ( y ) dy + B ( x ) (cid:16) f ( x + t )e − R x + tx B ( y ) dy − Z g ( t, zx ) ℘ ( dz ) (cid:17) + B ( x ) Z x + tx e − R yx B ( y ′ ) dy ′ B ( y ) Z g ( t + x − y, zy ) ℘ ( dz ) dy + Z x + tx e − R yx B ( y ′ ) dy ′ B ( y ) Z ∂ t g ( t + x − y, zy ) ℘ ( dz ) dy = h L f ( x + t ) + B ( x ) f ( x + t ) − B ( x ) Z f ( zx ) ℘ ( dz ) i e − R x + tx B ( y ) dy + Z x + tx e − R yx B ( y ′ ) dy ′ (cid:0) B ( y ) − B ( x ) (cid:1) Z ∂ t g ( t + x − y, zy ) ℘ ( dz ) dy + B ( x ) Z x + tx e − R yx B ( y ′ ) dy ′ Z ∂ x g ( t + x − y, zy ) z℘ ( dz ) dy. On the one hand using the second expression of ∂ x Γ g ( t, x ) above we deduce that for g , g ∈ C (Ω T,A ) such that g (0 , · ) = g (0 , · ) = f we have k Γ g − Γ g k C ≤ ℘ T B ( A ) k g − g k C . Thus Γ is a contraction for T < / (2 ℘ T B ( A )) and this guarantees that the fixed point ¯ f necessarilybelongs to C ([0 , ∞ ) ) . On the other hand using the first expression of ∂ x Γ g ( t, x ) we have ∂ t Γ g ( t, x ) − ∂ x Γ g ( t, x ) = B ( x ) (cid:20) Z g ( t, zx ) ℘ ( dz ) − Γ g ( t, x ) (cid:21) and accordingly the fixed point satisfies ∂ t ¯ f = L ¯ f . (cid:3) With Lemma 6.2 at hand we can define for any t ≥ M t on B loc (0 , ∞ ) by setting M t f ( x ) = ¯ f ( t, x ) . Proposition 6.3. The family ( M t ) t ≥ defined above is a positive semigroup of linear operatorson B loc ([0 , ∞ )) . If f ∈ C ([0 , ∞ )) then the function ( t, x ) M t f ( x ) is continuously differentiableand satisfies ∂ t M t f ( x ) = L M t f ( x ) = M t L f ( x ) . Additionally for any k > the space B ( V ) with V ( x ) = 1 + x k is invariant under ( M t ) t ≥ , and forall t ≥ the restriction of M t to B ( V ) is a bounded operator.Proof. The linearity and the semigroup property readily follow from the uniqueness of the fixedpoint in Lemma 6.2. The positivity and the stability of C ([0 , ∞ )) are direct consequences ofLemma 6.2, as well as the relation ∂ t M t f = L M t f. For getting the second one ∂ t M t f = M t L f, it suffices to remark from the computation of ∂ t Γ g in the proof of Lemma 6.2 that ∂ t M t f is theunique fixed point of Γ with initial data L f. For the invariance of B ( V ) we compute L V ( x ) = 1 + kx k − + ( ℘ − B ( x ) + ( ℘ k − B ( x ) x k which is bounded on [0 , ∞ ) since ( ℘ − B ( x ) + ( ℘ k − B ( x ) x k ≤ x ≥ (cid:0) ℘ − − ℘ k (cid:1) k . Wededuce that there exists C > L V ≤ CV and since V ∈ C ([0 , ∞ )) we get M t V ( x ) = V ( x ) + Z t M s ( L V )( x ) ds ≤ e Ct V ( x ) . Positivity of M t then yields k M t f k B ( V ) ≤ e Ct k f k B ( V ) . (cid:3) Now we define, for t ≥ µ ∈ M + ( V ), the positive measure µM t by setting for anymeasurable set A ⊂ [0 , ∞ ) ( µM t )( A ) = µ ( M t A ) . Then, for µ ∈ M ( V ) we define µM t ∈ M ( V ) as the equivalence class of ( µ + M t , µ − M t ) . NON-CONSERVATIVE HARRIS ERGODIC THEOREM 35 Proposition 6.4. The family ( M t ) t ≥ defined above is a positive semigroup of bounded linearoperators on M ( V ) . Moreover for any µ ∈ M ( V ) the family ( µM t ) t ≥ is solution to Equation (5.5) in the sense that for all f ∈ C c ([0 , ∞ )) and all t ≥ µM t )( f ) = µ ( f ) + Z t ( µM s )( L f ) ds. Proof. Let µ ∈ M ( V ) and f ∈ C c ([0 , ∞ )). From Proposition 6.3 we know that ∂ t M t f = M t L f which gives by integration in time M t f ( x ) = f ( x ) + Z t M s L f ( x ) ds = f ( x ) + Z t M s ( f ′ − Bf )( x ) ds + Z t M s F f ( x ) ds for all x ≥ 0, where we have set F f ( x ) = B ( x ) Z f ( zx ) ℘ ( dz ) . Since f ′ − Bf ∈ B ( V ) we have | M s ( f ′ − Bf ) | ≤ k f ′ − Bf k B ( V ) e Cs V and Fubini’s theorem ensuresthat µ (cid:16) Z t M s ( f ′ − Bf ) ds (cid:17) = Z t ( µM s )( f ′ − Bf ) ds. The last term deserves a bit more attention since F f can be not bounded by V. Consider g ∈ C c ([0 , ∞ )) such that g ≥ | f | . By positivity of M s and F we have | M s F f | ≤ M s F| f | ≤ M s F g andsince g ∈ C c (0 , ∞ ) µ ± (cid:16) Z t M s F g ds (cid:17) = µ ± (cid:16) M t g − g − Z t M s ( g ′ − Bg ) ds (cid:17) < + ∞ . Then ( s, x ) M s F f ( x ) is ( ds × µ )-integrable and Fubini’s theorem yields µ (cid:16) Z t M s F f ds (cid:17) = Z t ( µM s )( F f ) ds, which ends the proof. (cid:3) We end this appendix by giving some monotonicity results on ( M t ) t ≥ , which are useful forverifying (A4) in Section 5.2. They are valid under the monotonicity assumption we made on thefragmentation rate B. Lemma 6.5. i) For any x ≥ , t M t ψ ( x ) is increasing.ii) For any t ≥ , x M t ψ ( x ) is increasing.iii) For any T > , z ∈ [0 , and x ≥ , t M t ψ (cid:0) z ( x + T − t ) (cid:1) increases on [0 , T ] . Proof. The point i) readily follows from ∂ t M t ψ = M t ( L ψ ) , since M t is positive and 2 L ψ ( x ) =1 + ( ℘ − B ( x ) ≥ . Let us prove ii) . Define f ( t, x ) = ∂ x M t ψ ( x ) which satisfies ∂ t f ( t, x ) = ∂ x f ( t, x ) − B ( x ) f ( t, x ) + B ( x ) Z f ( t, zx ) z℘ ( dz ) + C ( x ) , with C ( x ) = − B ′ ( x ) M t ψ ( x ) + B ′ ( x ) R M t ψ ( zx ) ℘ ( dz ). Since ∂ t M t ψ ( x ) = L M t ψ ( x ) , ∂ t M t ψ ( x ) ≥ , and B ′ ≥ , we have C ( x ) = B ′ ( x ) B ( x ) (cid:0) ∂ t M t ψ ( x ) − ∂ x M t ψ ( x ) (cid:1) ≥ − B ′ ( x ) B ( x ) f ( t, x )and as a consequence ∂ t f ( t, x ) ≥ A f ( t, x ) := ∂ x f ( t, x ) − (cid:18) B ( x ) + B ′ ( x ) B ( x ) (cid:19) f ( t, x ) + B ( x ) Z f ( t, zx ) z℘ ( dz ) . Similarly to L the operator A generates a positive semigroup ( U t ) t ≥ . It is a standard result thatit enjoys the following maximum principle ∂ t f ( t, x ) ≥ A f ( t, x ) = ⇒ f ( t, x ) ≥ U t f ( x ), where f = f (0 , · ) . Since f (0 , x ) = ψ ′ ( x ) = ≥ U t that f ( t, x ) ≥ t, x > , which ends the proof of ii) . BANSAYE , BERTRAND CLOEZ , PIERRE GABRIEL , AND ALINE MARGUET We turn to the proof of iii) . The case z = 0 corresponds to ii) and we consider now z ∈ (0 , . Setting f ( t, x ) = M t ψ (cid:0) z ( x + T − t ) (cid:1) we have using ii) , ∂ x f ( t, x ) = z ∂ x M t ψ (cid:0) z ( x + T − t ) (cid:1) ≥ ∂ t f ( t, x ) = ( ∂ t M t ψ ) (cid:0) z ( x + T − t ) (cid:1) − z ( ∂ x M t ψ ) (cid:0) z ( x + T − t ) (cid:1) = 1 − zz ∂ x f ( t, x ) − B (cid:0) z ( x + T − t ) (cid:1) f ( t, x )+ B (cid:0) z ( x + T − t ) (cid:1) Z f (cid:0) t, z ′ x − (1 − z ′ )( T − t ) (cid:1) ℘ ( dz ′ ) . (6.3)Now define g ( t, x ) = ∂ t f ( t, x ) and differentiate the above equation with respect to t and use again(6.3) to get ∂ t g ( t, x ) = 1 − zz ∂ x g ( t, x ) − B (cid:0) z ( x + T − t ) (cid:1) g ( t, x )+ B (cid:0) z ( x + T − t ) (cid:1) Z g (cid:0) t, z ′ x − (1 − z ′ )( T − t ) (cid:1) ℘ ( dz ′ )+ z B ′ B (cid:0) z ( x + T − t ) (cid:1)(cid:16) − zz ∂ x f ( t, x ) − g ( t, x ) (cid:17) + B (cid:0) z ( x + T − t ) (cid:1) Z (1 − z ′ ) ∂ x f (cid:0) t, z ′ x − (1 − z ′ )( T − t ) (cid:1) ℘ ( dz ′ )and using the positivity of ∂ x f, B and B ′ we finally obtain ∂ t f ( t, x ) ≥ − zz ∂ x g ( t, x ) − (cid:16) B + z B ′ B (cid:17)(cid:0) z ( x + T − t ) (cid:1) g ( t, x )+ B (cid:0) z ( x + T − t ) (cid:1) Z g (cid:0) t, z ′ x − (1 − z ′ )( T − t ) (cid:1) ℘ ( dz ′ ) . Since g (0 , x ) = − z + ℘ − B (cid:0) z ( x + T ) (cid:1) ≥ g ( t, x ) ≥ . 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Zhu. Domain of attraction of the quasistationary distribution for birth-and-death processes. J. Appl. Probab. , 50(1):114–126, 03 2013.(V. Bansaye ) CMAP, ´Ecole Polytechnique, Route de Saclay, 91128 Palaiseau cedex, France. Email address : [email protected] (B. Cloez ) MISTEA, INRA, Montpellier SupAgro, Univ. Montpellier, 2 place Pierre Viala, 34060Montpellier, France. Email address : [email protected] (P. Gabriel ) Universit´e Paris-Saclay, UVSQ, CNRS, Laboratoire de Math´ematiques de Versailles,78000, Versailles, France. Email address : [email protected] (A. Marguet ) Univ. Grenoble Alpes, INRIA, 38000 Grenoble, France Email address ::