Asymptotic behavior of some stochastic models in population dynamics: a Hamilton-Jacobi approach

Asymptotic beha vior of some stochastic models in population dynamics: a Hamilton-J acobi approach Anouar J edd ∗ ∗ CMAP , École pol ytechnique, Institut polytechnique de P aris, anouar .jeddi@polytechnique.ed u February 25, 2026 Abstract In this paper , we investiga te the asymptotic behavior of individual-based models describing the evol ution of a population structured by a real trait, subject to selection and m utation. W e consider two di ff erent sets of assumptions: first, the case of critical or subcritical branching population processes in a regime combining a discretization of the tr ait space, small m utations, large time and large initial population size, where we are able to characterize using a Hamilton-J acobi approach, the survival set of the population, and the asymptotic of the logarithmic scaling of subpopulation sizes. Sec- ond, we generalize by a direct method the converg ence to the classical Hamilton- J acobi equation obtained in the super -critical branching regime considered in [ 6 ] to a more general trait space and under weaker assumptions. Moreover , we establish that the stochastic and the deterministic dynamics are asymptotically equiv alent in large population. 1 Introd uction In this paper , we provide a probabilistic justification for certain Hamilton-J acobi equa- tions from stochastic individual-based models in a large population regime. Our goal is to prove that, on logarithmic time and size scales, the sizes of subpopulations converg e to a viscosity solution of a Hamilton-J acobi equation, able to account for extinction of subpopulations. 1 In recent y ears, a Hamil ton-J acobi approach has been emerged as an importan t tool in the mathematical analysis of evolutionary biology (see [ 9 ]). This approach consists in studying the evolution of populations structured by a quantitativ e trait subject to se- lection and mutation, in a regime of small muta tions and large time and considering exponential orders of magnitude of population sizes. The original approach is based on PDEs models in population dynamics, structured by phenotypes, or traits (see for in- stance [ 9 , 15 , 3 , 12 ]). More precisely , these references prove that the Hopf -Cole transfor - mation of the population density converg es to a viscosity solution of a Hamilton-J acobi equation. This approach captures the concentration of the population into Dirac masses at the dominant traits, characterized through the associated viscosity solution. This de- terministic approach takes into account the evolution of negligible populations, but not the possibility of local extinction of subpopulations, since population densities in the PDE are always positive everywhere in space. In this context, some attempts to account for extinct populations ha ve been introd uced in [ 14 ] (see also [ 11 ]), by imposing a singu- lar mortality r ate below a surviv al threshold, and in this case the limit dynamics is not an ymore described by a classical Hamilton-J acobi equation. A natural alternativ e approach to account for extinctions would be to consider stochas- tic individual based models instead PDEs models. This approach w as follow ed in sev - eral references [ 4 , 5 , 7 , 8 ]. Classically the link between individual based models and Hamilton-J acobi equations was carried out in tw o steps: first , prov e the con verg ence of the individual based model to a PDE in a limit of larg e population (see [ 4 , 5 ]); second obtain the Hamilton-J acobi equation from the PDE as explained above. Howev er , these two successive scalings do not all ow to account f or possible extinction of subpopulations, as discussed above. In [ 7 , 8 ] individ ual based models with rare m utations w ere considered in a discrete trait space and with a scaling parameter K . Defining the subpopulation size of individu- als of type i at time t as N K i ( t ) , these references establish an asymptotic behavior of the exponents β K i ( t ) defined by N K i ( t l og K ) = K β K i ( t ) . These exponents can be seen as a Hopf - Cole transf orm in a discrete trait space with logarithmic time scaling. In these ref erences a regime of rare muta tions was considered, but not small mutations. In order to model small mutations, the trait space should be discretized in steps de- pending on the muta tion sizes. Recently in [ 6 ] such a discretization of the trait space was introduced by considering a parameter δ K → 0 as K → + ∞ . This wor k introduces a birth- 2 death individual based model with mutation in the discrete trait space with small muta- tions of size of the order 1 / log K . In the limit K → + ∞ , the exponents β K i defined as abov e by the Hopf -Cole transform are proved to con verg e to a viscosity solution of a Hamilton- J acobi equation. Up to our knowledg e, this is the first and only work which establishes a direct derivation of a Hamilton-J acobi equa tion from an individual-based model, but under restrictiv e assum ptions. In particular their br anching model does not allow for ex- tinction, beca use the reprod uction is assumed uniformly super -critical in space (the birth rate is strictly greater than the death rate), and the initial subpopulation sizes are low er bounded. This implies the surviv al of all subpopula tions a t all time with probability con- verging to 1 . In addition this reference considers only the case of the one dimensional torus as trait space. The method used relies on compactness and Lipschitz estimates re- quiring these assumptions. Our objective in this work is to extend the derivation of Hamil ton-J acobi equations from individual-based models in two more gener al settings. The first one allows local extinctions and the second one generalizes the super -critical case. First part: In this part w e study the case of critical or subcritical spatial branching pro- cesses in the real trait space. W e assume no lower bound on the initial subpopulation sizes so that extinction of subpopulations and ultimately extinction of the whole popu- lation occur almost surely . Under the classical Hopf -Cole transform, the extinction of a subpopulation means that the exponent takes the v alue −∞ . Theref ore, we cannot expect an ymore the tightness of the exponents to hold at all times and across the entire trait space, and the limit Hamilton-J acobi equation needs to be modified to account for ex- tinction. Then the question consists in characterizing the limit Hamilton-J acobi equation by introducing appropriate cut -o ff and in proving the converg ence of exponents of sub- populations. Foll owing the same intuition as in [ 14 ], we expect to obtain the Hamilton- J acobi equa tion in a region of the spatio-tem poral space corresponding to the surviv al set and −∞ for extinct subpopulations. In the critical or subcritical case, we will prove that this equation actually takes a very simple form (much simpler than in [ 14 ]), obtained by directly applying a cut -o ff to the original Hamilton-J acobi equation when it is negativ e. Note that the conv ergence of the exponents to −∞ does not make use of the sub-criticality assumption, and woul d apply in more general cases. T o study the behavior of the population in the set of survival we introduce a new method based on comparing the stochastic dynamics with the deterministic one. In gen- 3 eral we cannot expect that the stochastic dynamics is close to its mean. However , in the subcritical case w e will prov e that this holds true which explains why w e can describe simply the limit dynamics. W e use a method relying on control of variances based on maximum principles. A sec- ond important step is to study the converg ence of the exponents of the mean dynamics obtained by the classical Hopf -Cole transform. These exponents satisfy a discrete ver - sion of the classical PDE in [ 3 ] for which converg ence to the Hamilton-J acobi equation is known. An additional di ffi culty to show the con verg ence in our case comes from the discrete nature of the PDE which w as studied in a more general setting in [ 12 ]. Com- bining these two steps we prove the con verg ence in the limit of larg e population of the exponents to the viscosity solution of the corresponding Hamilton-J acobi equation with cut -o ff . The method used in this part is direct and do not rel y on the classical tightness cri- teria, in contrast to [ 6 ] which makes use of Lipschitz estimates and martingale control. Howev er , the conv ergence here is weaker: pointwise in time and locally uniforml y in trait space. Whether a stronger converg ence holds true remains unclear because, first the ex- ponents can takes the v alue −∞ , and second our method does not all ow to determine the behavior at the boundary betw een the extinction and survival sets. Second part: In this part, we study the super -critical branching regime considered in [ 6 ] in a more g eneral trait space and under more realistic assum ptions. In this setting all the initial subpopulation sizes are large, and we work in a uniformly super -critical regime, meaning that the subpopulation sizes remain large and the total population size is alwa ys infinite. This last fact makes the construction of the model not obvious, beca use muta tions can occur from an infinite number of subpopulations. Then, to justify the ex- istence of the model, we in troduce a new localization argument. Our proof of the converg ence toward the Hamilton-J acobi equation follows a similar method as above. W e first establish a uniform control on the normalized variances us- ing the maximum principle that holds in super -critical regime. This allows us to deduce that the stochastic and deterministic dynamics are asymptotically equivalent , which im- plies that the stochastic and deterministic exponents are close. W e then conclude us- ing the converg ence of the Hopf -Cole transform of the mean dynamics to the associated Hamilton-J acobi equation. Here we obtain a stronger conv ergence, locally uniform in time and space, and we ded uce the tightness of the exponents ( β K ) K . 4 This paper is organized as follows: in Section 2, we define the model. Section 3 is de- voted to the critical or subcritical regime. Finally , in Section 4, w e study the supercritical regime. W e ref er to the beginning of Sections 3 and 4 for more details about each section. 2 Model and motiv ation W e recall the branching stochastic model introduced in [ 6 ] in a more general trait space R , instead of the one-dimensional torus in [ 6 ]. W e consider a population parameterized by a scaling parameter K ∈ N , composed of individuals characterized by a tr ait which belongs to R , subject to mutation. W e consider a discretized trait space with discretization step δ K such that δ K → 0 as K → + ∞ . The trait space associated to the scaling parameter K is given by X K := { i δ K , i ∈ Z } . W e define in a probabilistic space ( Ω , F , P ) a Markov pure jump process ( N K i ( t ) , i ∈ Z , t ≥ 0) that models the size of the subpopulations, where for each i ∈ Z , the size of the subpopula tion of trait i δ K at time t is N K i ( t ) . Our model is a branching process in continuous time, so we only need to specify the individual birth and death rates. Any individual with tr ait x ∈ X K : • gives birth to a new individ ual with the same trait x at rate b ( x ); • dies at rate d ( x ); • or gives birth to a new individ ual with trait y ∈ X K at rate δ K p log K G (( y − x ) log K ) . (1) Here, b : R → R + is the birth rate, d : R → R + is the death rate and p > 0 is the mutation rate. W e assume that G is a probability kernel describing m utation e ff ects. It is scaled in ( 1 ) at logarithmic scale to make small mutations of sizes of the order 1 log K , and δ K to make the Riemann sum corresponding to the global population mutation rate conv erge. In the remaining, we define h K = δ K log K . The process ( N K i ( t ) , i ∈ Z , t ≥ 0) can be constructed using P oisson random measures in a classical wa y (cf . e.g. [ 10 , 1 ]). Let ( Q b i ) i ∈ Z , ( Q d i ) i ∈ Z , and ( Q p i ) i ∈ Z , be sequences of inde- pendents P oisson random measures on R + × R + with Lebesgue intensity , defined on the probabilistic space ( Ω , F , P ) . Informally speaking let us give some classical trajectorial representations about our process which will be justified in each setting in sections 3 and 5 4: for all i ∈ Z N K i ( t ) = N K i (0) + Z t 0 Z R + 1 y ≤ b ( i δ K ) N K i ( s − ) Q b i ( d s , d y ) − Z t 0 Z R + 1 y ≤ d ( i δ K ) N K i ( s − ) Q d i ( d s , d y ) + X j ∈ Z Z t 0 Z R + 1 y ≤ ph K G (( j − i ) h K ) N K j ( s − ) Q p j ( d s , d y ) . (2) Furthermore, by compensating the P oisson random measures, we obtain the following semi-martingale decomposition: N K i ( t ) = N K i (0) + Z t 0 ( b ( i δ K ) − d ( i δ K )) N K i ( s ) d s + X l ∈ Z ph K G ( l h K ) Z t 0 N K l + i ( s ) d s + M K i ( t ) , (3) where M K i is a local martingale with quadr atic variation (see [ 1 ]) ⟨ M K i ⟩ t = Z t 0 ( b ( i δ K ) + d ( i δ K )) N K i ( s ) d s + X l ∈ Z ph K G ( l h K ) Z t 0 N K l + i ( s ) d s . (4) Note that this representation needs a justification, since N K i appears both on the left and right hand side, and the stochastic integrals need to be well-defined. This is the case if for all T > 0 , and i ∈ Z , we hav e E  sup t ∈ [0 ,T ]  ( b ( i δ K ) + d ( i δ K )) N K i ( t ) + X j ∈ Z ph K G (( j − i ) h K ) N K j ( t )   < ∞ . (5) Under this assumption we denote for all i ∈ Z and t ≥ 0 , n K i ( t ) = E ( N K i ( t )) , and we take the expectation in ( 3 ). W e obtain the system of ordinary di ff erential equations:          d n K i ( t ) d t = ( b ( i δ K ) − d ( i δ K )) n K i ( t ) + P l ∈ Z ph K G ( l h K ) n K l + i ( t ) , ( t , i ) ∈ R + × Z , n K i (0) = n K , 0 ( i δ K ) , (6) where n K , 0 is the initial condition. Our goal is to study the exponential growth of the population when K → + ∞ , and more precisely to describe the exponent of each subpopulation in the form K β , where β ∈ {−∞} ∪ [0 , + ∞ ). T o this end we introduce following [ 6 ] the Hopf -Cole transform at a logarithmic scale β K i ( t ) = log N K i ( t l og K ) log K , (7) 6 and the corresponding transform of the mean dynamics u K i ( t ) = log n K i ( t l og K ) log K , (8) which satisfies the equation          d d t u K i ( t ) = b ( i δ K ) − d ( i δ K ) + P l ∈ Z ph K G ( l h K ) e log K ( u K l + i ( t ) − u K i ( t )) , ∀ ( t , i ) ∈ (0 , + ∞ ) × Z , u K i (0) = u K , 0 ( i δ K ) . (9) The above system is a discretized version of the integro-di ff erential equation studied in [ 3 ], where the solution converg es to a viscosity sol ution of a Hamilton-J acobi equation. W e refer to [ 12 ] for the study of such discrete models in a more g eneral setting. Following [ 6 , 12 ], we define the linear interpolation of β K i , which allows us to define a dynamics in the continuous trait space R : for all t > 0 and x ∈ R let i ∈ Z such that x ∈ [ i δ K , ( i + 1) δ K ) , e β K ( t , x ) = β K i ( t )(1 − x δ K + i ) + β K i +1 ( t )( x δ K − i ) . (10) The linear interpolation of u K i is defined in the same wa y as above and denoted by e u K . The remaining of the paper is dedicated to the study of the conv ergence of ( e β K ) K , as K → + ∞ . Assumptions A Here, we assume the general classical assumptions from [ 3 ] in a discrete setting close to [ 12 ]. 1. W e assume that b and d are Lipschitz-continuous functions, the mutation rate p is constant , and there exist positive constants b and d , such that for an y x ∈ R , 0 ≤ b ( x ) ≤ b, 0 ≤ d ( x ) ≤ d . (11) 2. The kernel G has super -exponential deca y , i,e Z R G ( x ) e ax d x < + ∞ , ∀ a ∈ R . 7 3. There exists a positive constant L such that, f or any i ∈ Z and K ≥ 1       u K , 0 (( i + 1) δ K ) − u K , 0 ( i δ K ) δ K       ≤ L, (12) i.e for an y K ≥ 1 , u K , 0 is L -Lipschitz continuous in the discrete tr ait space X K . 4. The linear interpola tion e u K , 0 of u K , 0 , con verges locally unif ormly to a con tinuous function u 0 , as K → + ∞ . 5. Finally , w e assume that for all ε > 0 , we hav e 1 K ε ≪ δ K ≪ 1 log K . (13) Assumption 2 is standard in the Hamil ton–J acobi approach (see [ 3 ]) and ensures tha t the Hamilton–J acobi equation is well-defined. Kernels with slow decay were studied in [ 12 ], leading to a non-standard Hamilton–J acobi equations. Assumption 4 im plies tha t u 0 is L -Lipschitz. Finally , Assumption 5 follows from [ 6 ] and implies in particular that h K → 0 , as K → + ∞ . . In addition, f or technical reason, the muta tion ra te is assumed constant in this paper , this makes the spatial lipschitz bound of u K global. 3 Critical or sub-critical case: In this section, w e investiga te the asym ptotic behavior assuming a unif ormly critical or subcritical dynamics. In Subsection 3.1, we state our assumptions. In Subsection 3.2, we establish preliminary resul ts concerning well-posedness and the existence of second mo- ments. In Subsection 3.3, we state our main result. Subsection 3.4 is devoted to the Hamilton–J acobi approach applied to the mean dynamics, which constitutes the first step in the proof of the main result. In Subsection 3.5, we quantify the gap between the stochastic and deterministic dynamics, which is the key idea of the proof. Finally , Subsection 3.6 is devoted to the proof of the main resul t. 3.1 Assumptions B W e state our assumptions in the case of a uniformly critical or subcritical dynamics. 8 1. There exist positive constants A 1 and A 2 such that for all i ∈ Z , and K ≥ 1 u K , 0 ( i δ K ) ≤ − A 1 | i δ K | + A 2 . (14) 2. W e assume that E   X i ∈ Z N K i (0)  2  < + ∞ . (15) 3. W e assume that there exists a positiv e constant C such that sup i ∈ Z n K i (0) E        N K i (0) − n K i (0) n K i (0)  2       ≤ C . (16) 4. W e assume a subcritical branching regime for all traits: α := sup x ∈ R b ( x ) − d ( x ) + p ≤ 0 . (17) Assumption B- 1 is classical in the Hamilton-J acobi approach (see for instance [ 3 , 12 ]), it gives the a ffi ne deca y of u K . Moreover , it implies that P i ∈ Z n K i (0) < + ∞ . Therefore, by Borel-Cantelli lemma ’ s we hav e for large enough i , that N K i (0) = 0 almost surely . As- sumption B- 2 guarantees the well-posedness of the dynamics ( 2 ) and the integrability of the martingales involv ed in the semi martingale decomposition ( 3 ), as prov ed in Propo- sition 3.1. Assumption B- 3 means that, initially the variance is bounded by the mean for each subpopulation. This is for example the case if N K i (0) is a P oisson random variable with parameter n K i (0) . Note that N K i (0) ma y be correlated random variables for di ff er - ent i . Furthermore, combined with Assumption A - 4 , this implies that β K (0) converg es in probability to u 0 when u 0 > 0, and to −∞ when u 0 < 0 . This will be a consequence of the arguments of section 3.6. 3.2 Preliminary result Proposition 3.1. Under Assumptions A - 1 - 2 and Assumption B- 2 . The process ( N K i ( t ) , i ∈ Z , t ≥ 0) is well-defined, and we have for all T > 0 E sup t ∈ [0 ,T ]  X i ∈ Z N K i ( t )  2 ! ≤ C ( T , K ) , (18) where C ( T , K ) is a positive constant depending on T , K but not on i . 9 Proof. By standard coupling argument, we construct in ( Ω , F , P ) a Y ule process Z K with parameter b + p and initial condition P i ∈ Z N K i (0) such that X i ∈ Z N K i ( t ) ≤ Z K t , almost surely ∀ t ≥ 0 . (19) Under assumption B- 2 , the Y ule process Z K is well-defined, and admits a finite second moment. Hence, ( 5 ) is verified, and thus ( N K i ( t ) , i ∈ Z , t ≥ 0) is well-defined and satisfies ( 18 ). 3.3 Conv ergence result in the critical or subcritical dynamics Let us state our main theorem. Theorem 3.2. Under Assumptions A -B, in the limit of K → + ∞ , the stochastic process ( e β K ) K ⊂ D ( R + , C ( R , R )) converges in the sense of finite-dimensional distributions in probability locally in space to the deterministic function β ( t , x ) =          u ( t , x ) for ( t , x ) ∈ { ( t , x ) ∈ [0 , + ∞ ) × R , u ( t , x ) > 0 } , −∞ for ( t , x ) ∈ { ( t , x ) ∈ [0 , + ∞ ) × R , u ( t , x ) < 0 } , (20) where u is the unique Lipschitz continuous viscosity solution of the Hamilton-J acobi equation:          ∂ t u ( t , x ) = b ( x ) − d ( x ) + p R R G ( y ) e ∇ u ( t ,x ) .y d y , ∀ ( t , x ) ∈ (0 , + ∞ ) × R , u (0 , . ) = u 0 ( . ) . (21) More pr ecisely , we prove that for all η > 0 , t > 0 and real numbers a < b , if { t } × [ a, b ] ⊂ { u > 0 } , we have lim K → + ∞ P ( sup x ∈ [ a,b ] | e β K ( t , x ) − u ( t , x ) | > η ) = 0 , (22) if { t } × [ a, b ] ⊂ { u < 0 } , we have lim K → + ∞ P ( sup x ∈ [ a,b ] e β K ( t , x ) = −∞ ) = 1 . (23) This theorem provides a probabilistic derivation of a Hamilton-J acobi equation with cut -o ff . It characterizes the space- time domain of surviv al and describes within this set the limit behavior of exponents of the population using a Hamilton-J acobi approach. In 10 addition this result shows that the exponents go to −∞ in the set where the viscosity sol u- tion is negativ e. In other words ( 23 ) means that for large K the subpopulations are extinct. The proof of this resul t is divided into two main steps. The first step consists in prov - ing that the linear interpolation of the Hopf -Cole transform of the mean dynamics e u K , conv erges to the viscosity solution u of the associated Hamilton-J acobi equation. This is based on the Arzela- Ascoli theorem, using local uniform bounds and local uniform Lip- schitz estimates in both time and space as in [ 3 ] but in discretized setting. The second step, the key step of the proof. It consists in comparing the stochastic dynamics with the deterministic one. This is achieved by establishing a uniform control of variances ov er all subpopulations, which allows us to prove that , in probability β K and u K remain close on the set { u > 0 } , when the parameter K is large. As a consequence of the first step, we conclude tha t the stochastic exponent β K conv erges to the viscosity sol ution of the corre- sponding Hamilton-J acobi equation on the set where this solution is positive. On the set where the viscosity sol ution is negativ e, the first step implies that the mean dynamics n K conv erges to zero. The in teger -v alued nature of the subpopulation sizes then ensures that the stochastic dynamics also converg es to zero. Finally , we note that the converg ence of the exponen t to −∞ in { u < 0 } hol ds true in g eneral and in particular does not require any sub-criticality or super -criticality assumption. 3.4 Hamilton-J acobi approach for the mean dynamics In this subsection, we investig ate the Hamilton-J acobi approach for the mean dynamics as a first step to analyze the stochastic dynamics. Theorem 3.3. Under Assumptions A and B- 1 , in the limit K → + ∞ , the sequence of functions ( e u K ) K converges locally uniformly in time and space to the unique viscosity solution of ( 21 ) . Proof. T o prove this result, we use the Arzelà–Ascoli theorem. Let T > 0 . From Assump- tions A - 1 - 3 , and B- 1 , Lemma 4.1 and Proposition 4.2 in [ 12 ], we have for all ( t , i ) ∈ [0 , T ] × Z that − L | i δ K | + A 4 + A 5 t ≤ u K i ( t ) ≤ − A 1 | i δ K | + A 2 + A 3 t , (24) and | u K i +1 ( t ) − u K i ( t ) | ≤ ∥ p ∥ Li p p | u K i +1 ( t ) | δ K + C Li p ( T ) δ K = C Li p ( T ) δ K , (25) where A 3 , A 4 , A 5 , A 6 are constants independent of K , and C Li p ( T ) is a positive constant independent of K , but depends on T . W e deduce a local bound, and spatial Lipschitz 11 estimates. Now , we prov e the Lipschitz estimates in time. For K large enough, w e hav e | ∂ t u K i ( t ) | ≤ b + d + p X l ∈ Z h K G ( l h K ) e C Li p ( T ) | l h K | ≤ b + d + 2 p Z R G ( x ) e C Li p ( T ) | x | d x . Thus, we obtain uniform l ocal Lipschitz estimates in time. Then, by the Arzelà- Ascoli theorem, we deduce the converg ence along a subsequence of K to a continuous function u , and by classical arguments of viscosity solutions and Assum ption A - 4 , we conclude that u is a viscosity solution of ( 21 ). Moreover , by uniqueness of viscosity solutions of ( 21 ), we ded uce the converg ence of the full sequence ( e u K ) K to u . 3.5 The gap between the stochastic and the deterministic dynamics In this subsection, we state and prove the next result which compares the stochastic and the deterministic dynamics. Proposition 3.4. Let T > 0 . W e have for all ( t , i ) ∈ [0 , T ] × Z , n K i ( t l og K ) E        N K i ( t l og K ) − n K i ( t l og K ) n K i ( t l og K )  2       ≤ C ( K ( α + o (1)) t + C ( T ) log K ) , (26) where C is a positive constant independent of K , C ( T ) is a positive constant independent of K , and depending on T , and o (1) a constant depending only on K , which goes to zero when K → + ∞ . Recall that α has been defined in ( 17 ) . Mor eover , at a logarithmic time scale N K / n K converges in L 2 to 1 in { u > 0 } , i,e, for all t > 0 and real numbers a < b, such that { t } × [ a, b ] ⊂ { u > 0 } , we have E  sup { i ∈ Z ,i δ K ∈ [ a,b ] }  N K i ( t l og K ) n K i ( t l og K ) − 1  2  → 0 , as K → + ∞ . (27) This result is a consequence of the branching property , and implies that the normal- ized variances are small when the mean dynamics is large. The proof relies on Itô ’ s lemma and the maxim um principle. This is an original proof which, to the best of our knowl- edge, is non standard in the theory of branching processes. Proof. Let T > 0 . Let us define for all ( t , i ) ∈ [0 , T ] × Z , S K i ( t ) = ( N K i ( t ) − n K i ( t )) 2 n K i ( t ) . Using the semi martingale decom position in ( 3 ) which is well-defined by ( 18 ), and by Ito ’ s lemma, we 12 obtain for all t > s > 0 , S K i ( t ) = S K i ( s ) + Z t s ( b ( i δ K ) − d ( i δ K )) S K i ( τ ) d τ + 2 Z t s p X l ∈ Z h K G ( l h K ) ( N K i ( τ ) − n K i ( τ ))( N K l + i ( τ ) − n K l + i ( τ )) n K i ( τ ) d τ − Z t s p X l ∈ Z h K G ( l h K ) ( N K i ( τ ) − n K i ( τ )) 2 n K l + i ( τ ) ( n K i ( τ )) 2 d τ + Z t s d ⟨ M K i ⟩ τ n K i ( τ ) + e M K i ( t ) − e M K i ( s ) , where t 7→ e M K i ( t ) is a martingale, since all quantities in the above decomposition are bounded at most by second moments of the population sizes, and by ( 18 ) w e hav e E  sup i ∈ Z sup t ∈ [0 ,T ]  N K i ( t )  2  ≤ C ( T , K ) . W e define Y K i ( t ) = E ( S K i ( t )) , which is well-defined, and we take the expectation in the above equa tion, and by Y oung’ s inequality we deduce tha t Y K i ( t ) ≤ Y K i ( s ) + Z t s  b ( i δ K ) − d ( i δ K ) + p X l ∈ Z h K G ( l h K )  Y K i ( τ )) d τ + Z t s p X l ∈ Z h K G ( l h K ) n K l + i ( τ ) n K i ( τ ) ( Y K l + i ( τ )) − Y K i ( τ )) d τ + Z t s d E ( ⟨ M K i ⟩ τ ) n K i ( τ ) . Moreover , as mentioned in the proof of Theorem 3.3, since the mutation rate is assumed to be constant , the Lipchitz bound in space ( 25 ) is global, and then X l ∈ Z h K G ( l h K ) n K l + i ( τ ) n K i ( τ ) ≤ X l ∈ Z h K G ( l h K ) e C Li p ( T / log K ) | l h K | . By Assumptions A - 1 and A - 2 , we ded uce that 1 n K i ( τ ) d E ( ⟨ M K i ⟩ τ ) d τ = b ( i δ K ) + d ( i δ K ) + p X l ∈ Z h K G ( l h K ) n K l + i ( τ ) n K i ( τ ) ≤ C ( T / log K ) , (28) where C ( . ) is a positiv e function independent of K , t and i . Moreover , by Assumption B- 4 , we ha ve b ( i δ K ) − d ( i δ K ) + p X l ∈ Z h K G ( l h K )) ≤ α + o (1) , (29) where o (1) is a negligible constant depending only on K and going to zero when K goes 13 to infinity . Therefore d d t Y K i ( t ) ≤ ( α + o (1)) Y K i ( t ) + p X l ∈ Z h K G ( l h K ) n K l + i ( t ) n K i ( t ) ( Y K l + i ( t ) − Y K i ( t )) + C ( T / log K ) . (30) By the maximum principle (see Appendix A), w e conclude that Y K i ( t ) ≤ e ( α + o (1)) t sup i ∈ Z Y K i (0) + C ( T / log K ) t . (31) Hence ( 26 ) is proved. Let us now prove the last point of the Proposition 3.4. Let t > 0 and real numbers a < b , such tha t { t } × [ a, b ] ⊂ { u > 0 } , we have by the conv ergence of e u K to u that f or all i δ K ∈ [ a, b ] n K i ( t log K ) = K u K i ( t ) ≥ K inf x ∈ [ a,b ] u ( t ,x ) / 2 . Therefore, by ( 26 ) w e obtain that E  sup { i ∈ Z ,i δ K ∈ [ a,b ] }  N K i ( t log K ) n K i ( t log K ) − 1  2  ≤ X i ∈ Z , i δ K ∈ [ a,b ] C K − inf x ∈ [ a,b ] u ( t ,x ) / 2 ( K ( α + o (1)) t ) + C ( t ) log K ) ≤ K − inf x ∈ [ a,b ] u ( t ,x ) / 4 δ K . From ( 13 ), w e conclude the L 2 conv ergence. 3.6 Proof of Theorem 3.2 Conv ergence in { u < 0 } : Let t ≥ 0 and a ≤ b real numbers such that { t } × [ a, b ] ⊂ { u < 0 } . W e hav e by the uniform conv ergence of e u K to u and the fact tha t sup x ∈ [ a,b ] u ( t , x ) < 0 that n K i ( t log K ) = K u K i ( t ) ≤ K sup { x ∈ [ a,b ] } u ( t ,x ) / 2 → 0 , as K → + ∞ , ∀ i ∈ Z , i δ K ∈ [ a, b ] . (32) 14 Thus P ( sup { i ∈ Z / i δ K ∈ [ a,b ] } N K i ( t log K ) ≥ 1) ≤ E  sup { i ∈ Z / i δ K ∈ [ a,b ] } N K i ( t log K )  ≤ X { i ∈ Z / i δ K ∈ [ a,b ] } n K i ( t log K ) ≤ C K sup { x ∈ [ a,b ] } u ( t ,x ) / 2 δ K . By ( 13 ) this probability g oes to zero when K → + ∞ . Moreov er , the suprem um of N K i is integer -v alued, hence sup { i ∈ Z / i δ K ∈ [ a,b ] } N K i ( t log K )) = 0 , (33) with probability converging to 1 when K → + ∞ . Furthermore, { u < 0 } is an open set we can deduce using similar arguments that max( e β K ( t , a ) , e β K ( t , b )) conv erges to −∞ with probability conv erging to 1. Hence, we deduce ( 23 ). Conv ergence in { u > 0 } . Let η > 0 , t ≥ 0 and a ≤ b real numbers such that { t } × [ a, b ] ⊂ { u > 0 } . Let i ∈ Z such that i δ K ∈ [ a, b ] , w e have by Proposition 3.4 and Theorem 3.3 f or large enough K that P     β K i ( t ) − u K i ( t )    > η  = P       1 log K       log 1 + N K i ( t log K ) − n K i ( t log K ) n K i ( t log K ) !       > η       ≤ P       1 log K       N K i ( t log K ) − n K i ( t log K ) n K i ( t log K )       > η       + P             N K i ( t log K ) − n K i ( t log K ) n K i ( t log K )       > 1 / 2       ≤  1 η 2 log 2 K + 4  E       N K i ( t log K ) − n K i ( t log K ) n K i ( t log K )       2 ≤ C  1 η 2 log 2 K + 4  K − u K i ( t ) ( K ( α + o (1)) t + C ( t ) log K ) ≤ C  1 η 2 log 2 K + 4  K − inf { x ∈ [ a,b ] } u ( t ,x ) / 2 ( K ( α + o (1)) t + C ( t ) log K ) ≤ C K − inf { x ∈ [ a,b ] } u ( t ,x ) / 4 . 15 In the same wa y we can prove that max(    e β K ( t , a ) − e u K ( t , a )    ,    e β K ( t , b ) − e u K ( t , b )    ) conv erges in probability to zero. Thus,from ( 13 ), we deduce tha t P       sup { x ∈ [ a,b ] }    e β K ( t , x ) − e u K ( t , x )    > η       → 0 , as K → + ∞ . (34) Therefore, by Theorem 3.3, w e conclude the proof of ( 22 ). 4 Super -critical case This section is devoted to the asymptotic analysis of the supercritical dynamics i,e the birth rate is greater than or equal to the dea th rate. In Subsection 4.1, w e state our as- sumptions. Subsection 4.2 is devoted to preliminary results concerning the existence of the stochastic model and the Hamilton–J acobi approach for the mean dynamics. In Sub- section 4.3, we state our main resul t and provide a resul t anal ogous to Proposition 3.4, which is the key in the proof of the main theorem. Finally , Subsection 4.4 contains the proof of the conv ergence toward the viscosity sol ution. 4.1 Assumptions C 1. W e assume that there exist two positiv e constants A, B such that for all i ∈ Z u K i (0) ≤ A | i δ K | + B. (35) 2. W e assume that there exists a > 0 , such that for all i ∈ Z , we hav e n K i (0) ≥ K a . (36) 3. W e assume a uniformly supercritical dynamics, i,e, f or all x ∈ R b ( x ) ≥ d ( x ) . (37) 4. Finally , we assume a similar condition as ( 16 ), i.e, there exists a positive constant C such that sup i ∈ Z E        N K i (0) − n K i (0) n K i (0)  2       ≤ C K a . (38) 16 Assumption C- 1 is new in the framework of Hamilton-J acobi approach. The classical as- sumption ( 14 ) is stronger and allows to perform the anal ysis in the classical functional spaces. Here, we will introduce a new functional space depending on apriori bounds on the solution of the Hamilton-J acobi equation. Assumption C- 2 is a strong assump- tion meaning that initially the density of all populations is large, but this assumption is w eaker than that one in [ 6 ] where a minimal initial popula tion was assumed for the stochastic dynamics N K (0). In ( 37 ), we include the possibility that the birth rate and the death rate can be equal which includes the setting of [ 6 ]. Furthermore, combining ( 36 ) with ( 37 ) we obtain tha t for all t > 0 and i ∈ Z n K i ( t ) ≥ K a . (39) In this setting the initial total population size can be infinite, i.e, P i ∈ Z N K i (0) = + ∞ almost surely . This is clear in the case where the subpopulation sizes N K i are independent. Let us prove tha t. By P aley–Zygm und inequality and from C- 2 , we ha ve for all i ∈ Z that P ( N K i (0) > K a/ 2 ) = P  N K i (0) > K a/ 2 n K i (0) n K i (0)  ≥  1 − K a/ 2 n K i (0)  2 n K i (0) 2 E ( N K i (0) 2 ) ≥ (1 − K − a/ 2 ) 2 n K i (0) 2 E ( N K i (0) 2 ) . Moreover , from C- 4 w e have E ( N K i (0) 2 ) / ( n K i (0) 2 ) ≤ C (1 + 1 K a ) . Then for K large enough w e hav e P ( ∀ i ∈ Z , N K i (0) > K a/ 2 ) = 1 . (40) Thus, P i ∈ Z N K i (0) = + ∞ almost surely . 4.2 Preliminary results and Hamilton-J acobi approach for the mean dynamics The next proposition shows the existence of the model, and provides the integrability of the first and second moments inv olved in the mutation term. Note that due the fact that the initial population maybe infinite there is no first jum p time in ( 2 ). Hence the notion of solution should be carefully defined. W e will say that a process ( N K i ( t ) , i ∈ Z , t ≥ 0) defined on the probability space ( Ω , F , P ) is solution to our branching population dynamics if for all i ∈ Z , the process ( N K i ( t ) , t ≥ 0) is almost surely finite and satisfies 17 almost surely ( 2 ). Proposition 4.1. Under Assumptions A -C, there exist a process ( N K i ( t ) , i ∈ Z , t ≥ 0) solution to our branching dynamics, and for all T > 0 , we have max       E  sup t ∈ [0 ,T ] X i ∈ Z e − C A | i h K | N K i ( t )  , E  sup t ∈ [0 ,T ] X i ∈ Z e − C A | i h K | ( N K i ( t )) 2        < + ∞ , (41) where C A := 2 A + 1 . Proof. T o prove that the process N K is well-defined, we introduce new arguments. First , we approximate the process by a well-defined process that implies the existence of the process as a ma thematical object , then w e prove that it is finite by introd ucing a judicious stopping time and applying localiza tion arguments. This model was studied in [ 6 ] in the torus, in which there is a finite number of finite subpopulations which makes the existence standard. But here, the initial total population size is infinite. Then, first we justify the existence of the process N K and hence we prove that the size of each subpopulation is finite and satisfies ( 41 ). Let us justify the existence of a process N K solution to ( 2 ). T o this end let M > 0 . W e define on the same probability space the stochastic process N K ,M defined as follows f or all t > 0 , i ∈ Z , | i δ K | < M N K ,M i ( t ) = N K i (0) + Z t 0 Z R + 1 y ≤ b ( i δ K ) N K ,M i ( s − ) Q b i ( d s , d y ) − Z t 0 Z R + 1 y ≤ d ( i δ K ) N K ,M i ( s − ) Q d i ( d s , d y ) + X j ∈ Z , | j δ K | 0 , the process N K ,M is well- defined because the initial population is finite and with finite second order moment and there is a t most a finite n umber of jumps in the stochastic integrals above. This can be proved under ( 35 ) by classical coupling arguments with a Y ule Process as in the proof of Proposition 3.1. Moreover , N K ,M i ( t ) is non-decreasing in M , so the limit when M goes to infinity exists almost surely for all i ∈ Z and t ∈ R + . Hence the existence of N K , as increasing limit of N K ,M . Moreover letting M go to infinity in ( 42 ) we deduce that N K is a solution to ( 2 ). Let us now show that N K is finite and satisfies ( 41 ). W e introduce the stopping time τ M , 18 defined for all θ > 0 , by τ θ = inf { t > 0 , X i ∈ Z e − C A | i h K | N K i ( t ) ≥ θ } . (43) Let us prove tha t for any T > 0 , we ha ve P (sup θ ≥ 1 τ θ ≥ T ) = 1 . (44) Let T > 0 . For all t ∈ [0 , τ θ ∧ T ] , w e have N K i ( t ∧ τ θ ) ≤ N K i (0)+ Z t ∧ τ θ 0 Z N 1 y ≤ b ( i δ K ) N K i ( s − ) Q b i ( d s , d y ) + X j ∈ Z Z t ∧ τ θ 0 Z R + 1 y ≤ ph K G (( j − i ) h K ) N K j ( s − ) Q p j ( d s , d y ) . Then, sup t ∈ [0 ,T ∧ τ θ ] X i ∈ Z e − C A | i h K | N K i ( t ) ≤ X i ∈ Z e − C A | i h K | N K i (0) + X i ∈ Z e − C A | i h K | Z T ∧ τ θ 0 Z R + 1 y ≤ b ( i δ K ) N K i ( s − ) Q b i ( d s , d y ) + X i ∈ Z X j ∈ Z e − C A | i h K | Z T ∧ τ θ 0 Z R + 1 y ≤ ph K G (( j − i ) h K ) N K j ( s − ) Q p j ( d s , d y ) . (45) Furthermore for an y ( t , i ) ∈ R + × Z , we hav e N K i ( t ∧ τ θ ) ≤ e C A | i h K | θ , and X j ∈ Z h K G (( j − i ) h K ) N K j ( t ∧ τ θ ) = X j ∈ Z h K e C A | j h K | G (( j − i ) h K ) e − C A | j h K | N K j ( t ∧ τ θ ) ≤ e C A | i h K | θ X j ∈ Z h K e C A | l h K | G ( l h K ) . Hence, the local martingale involv ed in the decomposition of N K i , is a martingale before 19 T ∧ τ θ . Therefore, w e take the expectation in ( 45 ), to obtain E  sup t ∈ [0 ,T ∧ τ θ ] X i ∈ Z e − C A | i h K | N K i ( t )  ≤ E  X i ∈ N e − C A | i h K | N K i (0)  + E  Z T ∧ τ θ 0 X i ∈ Z e − C A | i h K | b ( i δ K ) N K i ( s )  + E  Z T ∧ τ θ 0 X i ∈ Z X j ∈ Z ph K G (( j − i ) h K ) e − C A | i h K | N K j ( s ) d s  ≤ E  X i ∈ N e − C A | i h K | N K i (0)  + b E  Z T ∧ τ θ 0 X i ∈ Z e − C A | i h K | N K i ( s ) d s  + p E  Z T ∧ τ θ 0 X j ∈ Z X i ∈ Z h K e C A | ( j − i ) h K | G (( j − i ) h K ) e − C A | j h K | N K j ( s ) d s  ≤ E  X i ∈ Z e − C A | i h K | N K i (0)  + C Z T 0 E  X i ∈ Z e − C A | i h K | N K i ( s ∧ τ θ ) d s  . Note that T 7→ E  sup t ∈ [0 ,T ∧ τ θ ] P i ∈ Z e − C A | i h K | N K i ( t )  is measurable and bounded by θ . Thus, by Gronwall’ s inequality and ( 35 ), we obtain that f or any θ > 0 E  sup t ∈ [0 ,T ∧ τ θ ] X i ∈ Z e − C A | i h K | N K i ( t )  ≤ e C T E  X i ∈ Z e − C A | i h K | N K i (0)  ≤ C ( T , K ) . (46) By contradiction we assume that ( 44 ) f ails, then there exists T > 0 such that P (sup θ ≥ 1 τ θ < T ) := a T > 0 . Therefore E  sup t ∈ [0 ,T ∧ τ θ ] P i ∈ N e − C A | i h K | N K i ( t )  > a T θ → + ∞ , as θ → + ∞ . This is in contradiction with ( 46 ). Finall y , by F atou’ s Lemma and ( 46 ), we deduce that E  sup t ∈ [0 ,T ] X i ∈ Z e − C A | i h K | N K i ( t )  ≤ C ( T , K ) . The proof of the second moment follows similar arguments. Let us giv e the main points. W e introduce a new stopping time as τ ′ θ = inf { t > 0 , X i ∈ Z e − C A | i h K | ( N K i ( t )) 2 ≥ θ } . 20 The representation of ( N K ) 2 is given as f ollows  N K i ( t )  2 =  N K i (0)  2 + Z t 0 Z R + (2 N K i ( s − ) + 1) 1 y ≤ b ( i δ K ) N K i ( s − ) Q b i ( d s , d y ) − Z t 0 Z R + (2 N K i ( s − ) − 1) 1 y ≤ d ( i δ K ) N K i ( s − ) Q d i ( d s , d y ) + X j ∈ Z Z t 0 Z R + (2 N K i ( s − ) + 1) 1 y ≤ ph K G (( j − i ) h K ) N K j ( s − ) Q p j ( d s , d y ) . (47) The only point which di ff ers from the previous proof is the fact that the local martingale appearing in the mutation term in ( 47 ) is a martingale before τ ′ θ . For all ( t , i ) ∈ (0 , + ∞ ) × Z , we ha ve X j ∈ Z N K i ( t ∧ τ ′ θ ) h K G (( j − i ) h K ) N K j ( t ∧ τ ′ θ ) ≤ X j ∈ Z 1 2 (( N K i ( t ∧ τ ′ θ )) 2 + ( N K j ( t ∧ τ ′ θ )) 2 ) h K G (( j − i ) h K ) ≤ 1 2 θ e C A | i h K | X j ∈ Z h K (1 + e −| j h K | ) G ( j h K ) . Hence, w e can take the expectation in ( 47 ), the remainder of the proof is similar as above, using the fact that E  X i ∈ N e − C A | i h K |  N K i (0)  2  ≤ C ( T , K ) , which results from ( 35 ) and ( 38 ). That completes the proof of ( 41 ). Proposition 4.2. Under Assumptions A -C, Equation ( 6 ) has a unique solution n K ∈ C 1 ( R + , e ℓ 1 A ( Z )) , where e ℓ A 1 ( Z ) := { v ∈ R Z , X i ∈ Z e − C A | i h K | | v i | < + ∞} , (48) and C A := 2 A + 1 , introduced in Proposition 4.1. See Appendix B for the proof . The next proposition shows the converg ence of the Hopf -Cole transformation u K to the viscosity solution of the Hamil ton-J acobi equation, under Assumptions A -C. Proposition 4.3. Under Assump tions A -C, ( e u K ) K converges locally uniformly to the unique Lipschitz-continuous viscosity solution of the Hamilton-J acobi equation:          ∂ t u ( t , x ) = b ( x ) − d ( x ) + p R R G ( y ) e ∇ u ( t ,x ) .y d y , ( t , x ) ∈ (0 , + ∞ ) × R , u (0 , . ) = u 0 ( . ) . (49) See Appendix B for the proof . 21 4.3 Conv ergence result in the supercritical regime Theorem 4.4. Under Assumptions A -C, the sequence of stochastic processes ( e β K ) K converges in probability locally uniformly to the unique Lipschitz-continuous viscosity solution of the equation          ∂ t u ( t , x ) = b ( x ) − d ( x ) + p R R G ( y ) e ∇ u ( t ,x ) .y d y , ( t , x ) ∈ (0 , + ∞ ) × R , u (0 , . ) = u 0 ( . ) . This theorem gives a probabilistic deriva tion of the classical Hamilton-J acobi equation in a uniformly supercritical dynamics. Proposition 4.5. Let T > 0 . F or any ( t , i ) ∈ [0 , T ] × Z , we have E N K i ( t log K ) − n K i ( t log K ) n K i ( t log K ) ! 2 ≤ C ( T ) K a/ 2 , (50) where C ( T ) is a positive constant increasing on T , but independent on K . Moreover , for all η > 0 and T , D > 0 , we have P       sup { ( t ,i ) ∈ [0 ,T ] × Z / i δ K ∈ [ − D ,D ] }     N K i ( t log K ) − n K i ( t log K ) n K i ( t log K )     > η       ≤ C ( T , D ) η 2 δ K K a/ 2 , (51) where C ( T , D ) is a positive constant depending on T , D and increasing on T , but independent on K . Hence, on a logarithmic time scale, N K / n K converges in probability to 1 locally uniformly in time and space. Proof. Let T > 0 . For all ( t , i ) ∈ [0 , T ] × Z , we define S K i ( t ) = N K i ( t ) − n K i ( t ) n K i ( t ) . Using Itô ’ s lemma and the semi-marting ale decom position ( 3 ), which is w ell-defined by ( 41 ), w e ha ve for all 0 < s < t S K i ( t ) = S K i ( s ) + Z t s X l ∈ Z ph K G ( l h K ) n K l + i ( τ ) n K i ( τ ) ( S K l + i ( τ ) − S K i ( τ )) d τ + Z t s d M K i ( τ ) n K i ( τ ) . Then  S K i ( t )  2 =  S K i ( s )  2 + Z t s X l ∈ Z ph K G ( l h K ) n K l + i ( τ ) n K i ( τ ) 2 S K i ( τ )( S K l + i ( τ ) − S K i ( τ )) d s + Z t s d ⟨ M K i ⟩ τ  n K i ( τ )  2 + e M K i ( t ) − e M K i ( s ) , (52) 22 where t 7→ e M K i is a martingale, which is a consequence of ( 41 ). W e define, Y K i ( t ) = E   S K i ( t )  2  , which is also w ell-defined by ( 41 ). By Y oung’ s inequality , we obtain that Y K i ( t ) ≤ Y K i ( s ) + Z t s X l ∈ Z ph K G ( l h K ) n K l + i ( τ ) n K i ( τ ) ( Y K l + i ( τ ) − Y K i ( τ )) d τ + Z t s d E ( ⟨ M K i ⟩ τ )  n K i ( τ )  2 . Furthermore, from ( 25 ) the spatial Lipschitz bound of u K , is global on [0 , T ] × Z , and T 7→ C Li p ( T ) is increasing. W e obtain Z t s d E ( ⟨ M K i ⟩ τ )  n K i ( τ )  2 = Z t s 1 n K i ( τ )  b ( i δ K ) + d ( i δ K ) + X l ∈ Z ph K G ( l h K ) n K l + i ( τ ) n K i ( τ ) d τ  ≤ Z t s C ( T / log K ) d τ n K i ( τ ) , where T 7→ C ( T ) is increasing. Then by ( 39 ), we deduce that d Y K i ( t ) d t ≤ X l ∈ Z ph K G ( l h K ) n K l + i ( t ) n K i ( t ) ( Y K l + i ( t ) − Y K i ( t )) + C ( T / log K ) K a . (53) Therefore, by the maximum principle (see Appendix B for the proof), and ( 38 ), we deduce ( 50 ). Moreover , let η > 0 , we hav e by the submartingale inequality and from ( 52 ), that for all i ∈ Z P ( sup t ∈ [0 ,T ] | e M K i ( t ) | > η ) ≤ 1 η E ( | e M K i ( T ) | ) ≤ 1 η  E  ( S K i ( T )) 2  + E  ( S K i (0)) 2  + Z T 0 X l ∈ Z ph K G ( l h K ) n K l + i ( s ) n K i ( s ) 2 E  | S K i ( s )( S K l + i ( s ) − S K i ( s )) |  d s + Z T 0 d E ( ⟨ M K i ⟩ s )  n K i ( s )  2  . Moreover , we hav e X l ∈ Z ph K G ( l h K ) n K l + i ( s ) n K i ( s ) E  2 | S K i ( s )( S K l + i ( s ) − S K i ( s )) |  ≤ p X l ∈ Z h K G ( l h K ) e C Li p ( T / log K ) | l h K | (3 Y K i ( s ) + Y K l + i ( s )) . (54) 23 Thus, from ( 50 ) we ded uce that P ( sup t ∈ [0 ,T ] | e M K i ( t ) | > η ) ≤ C ( T / log K ) η K a/ 2 . (55) Therefore, from ( 52 ) and ( 55 ), w e obtain for K larg e enough that P ( sup t ∈ [0 ,T ] ( S K i ( t )) 2 > η ) ≤ P  ( S K i (0)) 2 > η / 4  + P  sup t ∈ [0 ,T ] | e M K i ( t ) | > η / 4  + P  Z T 0 X l ∈ Z ph K G ( l h K ) n K l + i ( s ) n K i ( s ) 2 | S K i ( s )( S K l + i ( s ) − S K i ( s )) | d s > η / 4  ≤ C ( T / log K ) η K a/ 2 . (56) This implies ( 51 ), and hence N K / n K conv erges in probability to 1 , on a logarithmic time scale. W e conclude the proof of Proposition 4.5. 4.4 Proof of Theorem 4.4 Let us prov e the conv ergence toward the Hamil ton-J acobi equation. By similar computa- tions as in the proof of Theorem 3.1, and by ( 51 ), we ded uce that P       sup { ( t ,x ) ∈ [0 ,T ] × [ − D ,D ] }    e β K ( t , x ) − e u K ( t , x )    > η  ≤ P       sup { ( t ,i ) ∈ [0 ,T ] × Z / i δ K ∈ [ − 2 D , 2 D ] }    β K i ( t ) − u K i ( t )    > η  = P       sup ( t ,i δ K ) ∈ [0 ,T ] × [ − 2 D , 2 D ] 1 log K     log  1 + N K i ( t log K ) − n K i ( t log K ) n K i ( t log K )      > η  ≤ ( 1 η 2 log 2 K + 4) C ( T , 2 D ) δ K K a/ 2 . Moreover , from Proposition 4.3, u K conv erges to the viscosity solution of ( 49 ), w e con- clude the proof of Theorem 4.4. 24 Appendix A Proof of the maximum principle for ( 30 ) : Recall that K is fixed in this proof . Let T > 0 . W e recall that Y K i ( t ) = E  ( N K i ( t ) − n K i ( t )) 2 n K i ( t )  . W e start by bounding the growth of Y K i when | i | → + ∞ . W e hav e, Y K i ( t ) ≤ 2 E  ( N K i ( t )) 2  n K i ( t ) + 2 n K i ( t ) . From ( 18 ), w e have E  ( N K i ( t )) 2  ≤ C ( T , K ) , and n K i ( t ) ≤ p C ( T , K ) , ∀ t ∈ [0 , T ] . Furthermore, from ( 24 ) we have that n K i ( t ) ≥ e − L | i h K | + A 4 log K + A 5 t . Therefore, we deduce that Y K i ( t ) ≤ e C ( T , K )( e L | i h K | + 1) . (57) Our goal is to deduce from ( 30 ) tha t ( Y K i ( t ) − C ( T / log K ) t ) e − ( α + o (1)) t − C ≤ 0 , (58) where C is given by ( 16 ). T o this end, let us define for all ( t , i ) ∈ [0 , T ] × Z , and for some positive constan t D to be chosen latter e Y K i ( t ) :=   Y K i ( t ) − C ( T / log K ) t  e − ( α + o (1)) t − C  e − ( L +1) | l h K | e − D t . By contradiction, w e assume that M = sup ( t ,i ) ∈ [0 ,T ] × Z e Y K i ( t ) > 0 . (59) 25 From ( 57 ), e Y K i ( t ) v anishes as | i | → + ∞ . Let ( t K , i K ) be a maximum point of f Y K in [0 , T ] × Z . By ( 38 ) we ha ve that t K > 0 . Thus, it foll ows from ( 30 ) at ( t K , i K ) that d d t e Y K i K ( t K ) ≤ − D e Y K i K ( t K ) + p X l ∈ Z h K G ( l h K ) n K l + i K ( t K ) n K i K ( t K ) ( e ( L +1)( | ( l + i K ) h K |−| i K h K | ) e Y K l + i K ( t K ) − e Y K i K ( t K )) ≤ − D M + p X l ∈ Z h K G ( l h K ) n K l + i K ( t K ) n K i K ( t K ) ( e ( L +1) | l h K | M − M ) ≤       − D + p X l ∈ Z h K G ( l h K )( e ( L +1+ C Li p ( T / log K )) | l h K |       M . Then, for D > p P l ∈ Z h K G ( l h K )( e ( L +1+ C Li p ( T / log K )) | l h K | , w e have d d t e Y K i K ( t K ) < 0 . This is in contradiction with the f act that t K > 0 . Hence ( 58 ) is prov ed. Appendix B Proof of Proposition 4.2 The proof follows the classical Picard iteration method. The novel ty here is that the initial condition is not integrable. W e introduce a new functional space in which existence and uniqueness hold. For more details about the existence and uniqueness of such a discrete system, we ref er to Appendix A in [ 12 ]. W e define B K = ( a K i ,j ) i ,j ∈ Z the infinite real matrix where a K i ,j = p h K G (( j − i ) h K ) , D K the infinite diagonal matrix whose diag onal elements are b ( i δ K ) − d ( i δ K ) for all i ∈ Z , and we define the infinite vector n K = ( n K i ) i ∈ Z . W e write Equation ( 6 ) as d d t n K ( t ) = ( D K + B K ) n K ( t ) . (60) Let T > 0 . W e consider the following closed subset of C ([0 , T ] , e ℓ 1 A ( Z )) : H :=  n ∈ C ([0 , T ] , e ℓ 1 A ( Z )) , ∥ n ( t ) ∥ e ℓ 1 A ( Z ) ≤ C ( t ) ∥ n (0) ∥ e ℓ 1 A ( Z ) , ∀ t ∈ [0 , T ]  , (61) where ∥ n ∥ L ∞ ([0 ,T ] , e ℓ 1 A ( Z )) := sup t ∈ [0 ,T ] X i ∈ Z e − C A | i h K | | n i ( t ) | , 26 and C ( t ) := 1 + e ( b + d +2 p ) t . Note that C ([0 , T ] , e ℓ 1 A ( Z )) endowed with the norm ∥ . ∥ L ∞ ([0 ,T ] , e ℓ 1 A ( Z )) , is a Banach space. Let us now define the mapping Φ : H 7→ H n 7→ Φ ( n ) , where Φ ( n ) is the unique solution of          d d t Φ ( n )( t ) = ( D K + B K ) n ( t ) Φ ( n )(0) = n K (0) . Note that from ( 35 ) there exists a positive constant C ( K ) such that ∥ n K (0) ∥ e ℓ 1 A ( Z ) ≤ C ( K ) . Then, the proof follows by simple computations proving that , for a small time T > 0 , Φ ( H ) ⊂ H , and it is a contraction in H with the norm ∥ . ∥ L ∞ ([0 ,T ] , e ℓ 1 A ( Z )) . W e conclude the existence and uniqueness until this small time T , and by classical iteration arguments, we concl ude the existence and uniqueness of the global solution. Proof of Proposition 4.3 The proof of this resul t follows similar arguments to the proof of Theorem 3.3. Note that from ( 35 ), and by adapting the proof of the comparison principle in Proposition 3.4 in [ 12 ], we obtain for some positive constant C , that u K i ( t ) ≤ A | i δ K | + B + C t . Moreover , from ( 39 ), we ha ve u K i ( t ) ≥ a. W e deduce that, u K is locall y uniformly bounded. The spatial Lipschitz bound is given by Proposition 4.2 in [ 12 ], and the Lipschitz bound in time can be deduced as in the proof of Theorem 3.3. Then, by the Arzelà–Ascoli theorem, we conclude the compactness of ( e u K ) K , and classical arguments yield the converg ence along subsequences to the Hamilton-J acobi equation. The uniqueness of such Hamilton-J acobi equations with an initial condition of linear growth is well-known (see for instance [ 2 ]). Hence, the local uniform converg ence of the full sequence ( e u K ) K to the unique viscosity solution of ( 49 ). Proof of the maximum principle of ( 53 ) The only change from the proof giv en in Appendix A is the beha vior of Y K at infin- ity in space. In this case, it is easy to see from ( 41 ) that max  n K i ( t ) , E  ( N K i ( t )) 2   ≤ C ( T , K ) e C A | i h K | . Moreov er , since n K i ( t ) ≥ K a , we deduce the exponential growth of Y K for 27 fixed K , i,e Y K i ( t ) ≤ C ( T , K ) e C | i h K | , where C is a positive constant. Then, the remainder of the proof is exactly as in Appendix A. Acknowledg ements: The a uthor thanks Nicolas Champagnat and Sylvie Méléard for their comments and careful reading. This work is funded by the European Union (ERC, SINGER, 101054787). V iews and opinions expressed are howev er those of the author only and do not necessarily reflect those of the E uropean Union or the European Research Council. Neither the European Union nor the granting authority can be hel d responsible for them. This wor k has also been supported by the Chair "Modélisation Mathématique et Biodiversité " of V eolia En vironnement -École P olytechnique-Museum Na tional d’Histoire Naturelle-Fonda tion X. 28 Ref erences [1] V . Bansay e and S. Méléard. S tochastic Models for S tructured P opulations , volume 1.4. 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