Filling the gap between individual-based evolutionary models and Hamilton-Jacobi equations

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Filling the gap between individual-based evolutionary models and Hamilton-Jacobi equations

Nicolas Champagnat, Sylvie Méléard, Sepideh Mirrahimi, Viet-Chi Tran

To cite this version:

Nicolas Champagnat, Sylvie Méléard, Sepideh Mirrahimi, Viet-Chi Tran. Filling the gap between individual-based evolutionary models and Hamilton-Jacobi equations. 2022. �hal-03656608v2�

Filling the gap between individual-based evolutionary models and Hamilton-Jacobi equations

Nicolas Champagnat^* Sylvie M´el´eard^„ Sepideh Mirrahimi Viet Chi Tran ^§ June 15, 2022

Abstract

We consider a stochastic model for the evolution of a discrete population structured by a trait with values on a finite grid of the torus, and with mutation and selection. Traits are vertically inherited unless a mutation occurs, and influence the birth and death rates.

We focus on a parameter scaling where population is large, individual mutations are small but not rare, and the grid mesh for the trait values is much smaller than the size of mu- tation steps. When considering the evolution of the population in a long time scale, the contribution of small sub-populations may strongly influence the dynamics. Our main re- sult quantifies the asymptotic dynamics of sub-population sizes on a logarithmic scale. We establish that under the parameter scaling the logarithm of the stochastic population size process, conveniently normalized, converges to the unique viscosity solution of a Hamilton- Jacobi equation. Such Hamilton-Jacobi equations have already been derived from parabolic integro-differential equations and have been widely developed in the study of adaptation of quantitative traits. Our work provides a justification of this framework directly from a stochastic individual based model, leading to a better understanding of the results obtained within this approach. The proof makes use of almost sure maximum principles and careful controls of the martingale parts.

Keywords: stochastic birth death models, large population approximation, selection, mutation, viscosity solution, maximum principle.

MSC 2000 subject classification: 92D25, 92D15, 60J80, 60F99, 35F21.

1 Introduction and presentation of the model

Long-term evolutionary dynamics of biological populations may be strongly influenced by small populations and local extinction in some areas of the phenotypical trait space. Survival of small

*Universit´e de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France; E-mail: [email protected]

„Ecole Polytechnique et Institut Universitaire de France, CNRS, Institut polytechnique de Paris, route de Saclay, 91128 Palaiseau Cedex-France; E-mail: [email protected]

…Institut Montpelli´erain Alexander Grothendieck, Univ. Montpellier, CNRS, Montpellier, France; E-mail:

[email protected]

§LAMA, Univ Gustave Eiffel, Univ Paris Est Creteil, CNRS, F-77454 Marne-la-Vall´ee, France; E-mail:

[email protected]

populations in very large populations is crucial when evolution proceeds by selective sweeps [20] or for the evolution of antibiotic resistance for bacteria [24, 27]. For example, in bacterial populations involving horizontal transfer, it was shown in [3, 6] that the individual-based long- term dynamics is very sensitive to random survival of very small populations, which may either drive the population to evolutionary suicide or to cyclic dynamics. In such context, the bacterial population is very large making the tracking of small populations very challenging.

From a point of view of mathematical modeling, one wishes to consider large population scalings allowing for survival of much smaller populations. Two approaches emerged: a purely deterministic one, based on partial differential equations (PDE), and a stochastic one, based on birth and death processes (so-called individual-based models in biology). Both approaches describes exponentially small populations sizes and characterize the dynamics of the exponents.

An analytical approach allowing to deal with negligible but non-extinct populations was pro- posed in [13] and then widely developed (see for instance [25, 2, 21]) for the asymptotic study of parabolic integro-differential selection-mutation models. Let us present it in a setting close to [2]. We consider a population whose individuals are differentiated by a traitx∈T, the torus of dimension 1, identified below with the interval [0,1). The trait can vary from an individual to the other. The evolution of the population is driven by two effects: mutation of the traits, and selection as the reproductive and survival abilities of an individual depend on its trait x.

For an individual of trait x ∈ T, let us denote by b(x) (resp. d(x) and p(x)) the clonal birth rate (resp. the death rate and the birth rate with mutation), and byG(h) the mutation kernel.

Assuming that the population density solves the PDE (ε∂_tu_ε(t, x) =u_ε(t, x) (b(x)−d(x)) +R

T 1

εG ^x−y_ε

p(y)u_ε(t, y)dy (t, x)∈R+×T u_ε(t,0) = exp_β

0(x) ε

, x∈T

(1.1) in the limitε→0 of small mutations and large time, and applying the Hof-Cole transformation

β_ε(t, x) =εlogu_ε(t, x), or u_ε(t, x) = exp

βε(t, x) ε

, (1.2)

it is proved in [2] (in a slightly different setting, considering x ∈ R and taking into account a competition term) thatβ_ε converges to the unique solutionβ of the Hamilton-Jacobi equation

(∂

∂tβ(t, x) =b(x)−d(x) +p(x)R

RG(h)e^h∂xβ(t,x)dh, (t, x)∈R+×T

β(0, x) =β0(x), x∈T

(1.3) Scaling limits of individual-based models on a discrete trait space with rare mutations and large population, and allowing to deal with negligible populations and local extinction, were proposed in [14, 5, 10, 11, 4]. These references focus on population sizes of the order of K^β and characterize the asymptotic dynamics of the exponent β. In particular, local extinction is possible when the exponent β hits 0. The fact that the trait space is discrete allows to describe separately the dynamics of each small sub-population. However, this makes the detailed description of the asymptotic dynamics very complicated (see [10, 11]). Note that mutations are assumed individually rare in these references, but they are more frequent than in the scaling limits of adaptive dynamics (see e.g. [22, 12, 7, 9]), where negligible populations either fixate or go extinct fast, due to the fact that the populational mutation rate vanishes here. Scaling limits with non-vanishing populational mutation rates partly solve the criticisms raised by biologists [28]

concerning the too slow evolutionary speed in adaptive dynamics, particularly in microorganism populations. Biological criticisms were also raised for the analytical approach [26], because of the so-called tail problem: exponentially small populations, which may actually be extinct, can have a strong influence on the future evolutionary dynamics of the population. In particular, evolutionary branching is too fast. Modification of the Hamilton-Jacobi equation were proposed in [26, 23, 16] to solve this problem, but we believe that an individual-based approach is crucial to provide a more realistic and biologically relevant solution to the tail problem.

The purpose of our work is to provide a stochastic individual-based justification of Hamilton- Jacobi equations. To our knowledge, this is the first proof of this kind in the literature. We follow an individual-based approach, assuming a continuous trait space with a vanishing discretization step δK, where K is a scaling parameter such that the population is of the order of K^βeK(t,x), assuming frequent and small mutations. In the individual-based model, individuals with trait x ∈ T give birth to a clone at rate b(x), die at rate d(x) or give birth to a mutant at rate p(x). Mutant traits are drawn according to a discretization of the distribution logKG(logK·).

Mutation steps are of the order of 1/logK and the discretization step δ_K is assumed much smaller than 1/logK. In this first work, we focus on the understanding of the relevant scales allowing to capture the limiting Hamilton-Jacobi dynamics. Thus, we consider a simplified model where the birth rate b is assumed larger than the death rate d, making the stochastic process super-critical, and the trait space has no boundary. Generalization is a work in progress.

The proof of our main result makes use of uniform Lipschitz bounds on the finite variation part ofβe^K, obtained using an almost sure maximum principle and careful bounds for the martingale part. The identification of the limit is done by checking that it is almost surely viscosity solution of (2.12). We describe the model and state our main result in Section 2. The proof is divided into two main steps—proof of tightness and identification of the limit—which are detailed in Sections 3 and 4, respectively.

Note that the Hopf-Cole transformation (1.2) is reminiscent of large deviations scalings. How- ever, our scaling is more of a law of large numbers type. As far as we know, the large deviations interpretation of the Hamilton-Jacobi equation can be done through a Feynman-Kac interpreta- tion of the PDE (1.1) [8]. However, the stochastic process involved in the Feynman-Kac formula does not seem to be directly related to the biological population process, even though some works suggest that it may be related to the ancestral trait process of living individuals [15].

2 Model and main result

2.1 The model

We consider a super-critical stochastic birth-death-mutation model describing an asexual popu- lation of individuals characterized by a quantitative phenotypic or genetic traitx∈T. Starting from a finite population whose initial size is parameterized by K ∈ N, our goal is to recover, in the limit K→ +∞, an evolutionary dynamics described by the Hamilton-Jacobi partial dif- ferential equation (1.3). For this, we consider a discretization of the trait space T with step δK →0. For the sake of simplicity, we will consider in what follows that 1/δK ∈N. Then, the population is composed of individuals with traits belonging to the discrete space

X_K :=

iδ_K :i∈ {0,1,· · ·, 1 δK

−1}

,

embedded with the torus distance: ∀x, y∈[0,1), ρ(x, y) = min

|x^′−y^′|, x^′ =x mod 1, y^′ =y mod 1

= min |x−y|,1− |x−y|

.

It’s enough to defineρ(x, y) for x, y∈Tby considering their representative in [0,1).

The number of individuals with traitiδK is described by the stochastic process (N_i^K(t), t≥0).

The total population size at timet is then given by N^K(t) =

1/δK−1

X

i=0

N_i^K(t).

An individual with traitx∈ X_K

ˆ gives birth to a new individual with the same trait x at rateb(x);

ˆ dies at rated(x);

ˆ gives birth to a mutant individual with traity∈ X_K at rate

p(x)δ_KlogK G((y−x) logK), (2.1) where z is the unique real number of [−1/2,1/2) that is equal to z modulo 1, i.e. z = z− ⌊z+ 1/2⌋, where⌊·⌋ is the integer part function.

In the rest of the paper, the following assumptions are made:

Assumption 2.1. 1. We assume thatb,dandpare nonnegative Lipschitz continuous functions defined on T, such that for all x∈T,

b(x)> d(x) andp(x)>0. (2.2) This means that the birth-death process for each trait is super-critical. In the sequel, we denote by ¯b, p¯and d¯the upper bounds of these functions on T, by p >0 the lower bound of p, and by

∥b∥_Lip, ∥d∥_Lip and∥p∥_Lip their Lipschitz norm.

2. The function G defined on R is nonnegative, continuous, satisfies R

RG(y)dy = 1 and has finite exponential moments of any order. Moreover, we assume that there existsR >0such that G is nonincreasing on [R,+∞) and nondecreasing on (−∞,−R].

An example of function G satisfying Assumption 2 is given by the Gaussian kernel G(h) =

√1

2πσe^−h2/2σ2.

3. There exists a constant a1 >0 such that, for all K ∈N and all i∈ {0,1,· · ·,_δ¹

K −1},

N_i^K(0)≥K^a1. (2.3)

4. There exists a₂ < a₁ such that

K^−a2/4 ≪δ_K ≪ 1

logK as K →+∞. (2.4)

Point 4 above implies that

hK :=δKlogK≪1. (2.5)

Therefore, the interpretation of (2.1) is that the rate at whichxgives birth to a mutant individual is close to p(x). Indeed, for an individual with trait xK =iKδK with iK =⌊x/δ_K⌋ and x ∈T fixed,

K→+∞lim p(x_K)

1 δK−1

X

j=0

h_KG

(i_K−j)δ_K logK

= lim

K→+∞p(x_K)

1/δK−1−⌊1/2δ_K⌋

X

ℓ=−⌊1/2δ_K⌋

h_KG(h_Kℓ)

=p(x) Z

R

G(y)dy=p(x), (2.6) where we used Assumption 2.1.2 to control the tails of the Riemann sum.

Therefore, the individual mutation rate is order 1 and mutation steps are small: conditionally on being a mutant, the traityof the offspring has the distributionGscaled by a factor 1/logK representing the order of magnitude of the mutation steps. Note also that (2.5) means that the mesh size is much smaller than the mutation step.

Our goal is to study the asymptotic behavior (when K tends to infinity) of the population sizes N_i^K when N_i^K(0) is of the order of K^αi for some α_i > 0. Note that in the case where p(x) = 0 for allx∈T, the processN_i^K(t) is a super-critical one-dimensional branching process, hence E(N_i^K(t)) =E(N_i^K(0))e^{(b(iδK)−d(iδK))t}. Therefore, if the initial condition is of orderK^αi, thenE[N_i^K(tlogK)]∼K^{αi+(b(iδK)−d(iδK))t}. This suggests to study

β_i^K(t) = log(N_i^K(tlogK))

log(K) , (2.7)

with the convention that β_i^K(t) = 0 if N_i^K(tlogK) = 0. So the sub-population of trait iδK at timetlogK has sizeN_i^K(tlogK) =K^βiK(t).

We make the following assumption on the initial condition β_i^K(0).

Assumption 2.2. Assume that there exists a constant A >0 such that

K→+∞lim P

sup

i̸=j

|β_i^K(0)−β_j^K(0)|

ρ(i δ_K, j δ_K) > A

= 0. (2.8)

Notations: (i) We shall use the following Riemann approximation repeatedly in the proofs:

for all α >0,

K→+∞lim

1 δK−1

X

j=0

h_KG

(i_K−j)δ_K logK

e^{αlogK ρ(jδK,iKδK)}

= lim

K→+∞

1/δK−1−⌊1/2δK⌋

X

ℓ=−⌊1/2δ_K⌋

h_KG(h_Kℓ)e^{α hK|ℓ|}= Z

R

e^α|y|G(y)dy, (2.9)

and thus, there exists a constantG(α) depending only onα >0 such that

sup

K≥1

1 δK−1

X

j=0

h_KG

(i_K−j)δ_K logK

e^{LlogK ρ(jδK,iKδK)} = : G(α)<+∞. (2.10)

(ii) In what follows and for any functionf on {0,1,· · · ,_δ¹

K −1}, we will use the notation

∆_Kf_i = f_i+1−f_i δ_K , with the convention that f_1/δK =f0.

2.2 The main result - Sketch of the proof

Since we are interested in the convergence of the quantitiesβ_i^K to a continuous function defined on the trait space T, when K → +∞ and δK → 0, we introduce the following interpolation of theβ_i^K’s: for allx∈Tand K≥1, leti be such thatx∈[iδK,(i+ 1)δK), and define

βe_t^K(x) =β_i^K(t)

1− x δ_K +i

+β_i+1^K (t) x

δ_K −i

, (2.11)

with the convention thatβ_1/δ^K

K(t) =β₀^K(t). The sequence of processes (βe_t^K, t∈[0, T])_K belongs to D([0, T],C(T,R)), where C(T,R) is endowed with the topology of uniform convergence and D([0, T],C(T,R)) is the Skorohod space of c`adl`ag paths with the associated Skorokhod topology.

Let us state our main theorem.

Theorem 2.3. Let T > 0. Under the Assumptions 2.1 and 2.2, and assuming that βe₀^K(·) converges in probability for the topology of uniform convergence on C(T,R) to a deterministic functionβ0(·)∈ C(T,R), the sequence(βe^K)K converges in probability inD([0, T],C(T,R))to the unique Lipschitz viscosity solution of the Hamilton-Jacobi equation

(∂

∂tβ(t, x) =b(x)−d(x) +p(x)R

RG(h)e^h∂xβ(t,x)dh, (t, x)∈R+×T

β(0, x) =β₀(x), x∈T. (2.12)

The proof of Theorem 2.3 will be classically obtained in two steps: tightness and identification of the limiting values. Therefore we will prove the two next results, respectively in Sections 3 and 4.

Theorem 2.4. The sequence of laws of( ˜β_t^K, t∈[0, T])_Kis relatively compact inP(D([0, T],C(T,R))).

Theorem 2.5. The limiting values β of ( ˜β_t^K, t ∈[0, T])_K are characterized as the unique vis- cosity solution of the Hamilton-Jacobi equation

∂

∂tβ(t, x) =b(x)−d(x) +p(x) Z

R

G(h)e^h∂xβ(t,x)dh.

The proofs of these two results require to control the increments of the functionsβe_t^K(.). This functions can also be written as

βe_t^K(x) = (x−iδK)∆Kβ_i^K(t) +β_i^K(t).

From this expression, we observe that two technical steps are required: to estimate uniformly β_i^K(t) and to control uniformly ∆Kβ_i^K(t) = ^β

K

i+1(t)−β_i^K(t)

δK (the second estimate being the harder part, it is a major difficulty and constitutes the technical interest of the paper). Such estimates are also obtained in the deterministic derivation of Hamilton-Jacobi equations of type (2.12) from parabolic integro-differential equations using the maximum principle and the Bernstein method which consists in applying again the maximum principle to the equation satisfied by the increments (see [2]). Here, since we have stochastic processes we cannot apply the Bernstein method directly. Using the Doob-Meyer decomposition, the stochastic processes can be separated into a finite variation part and a martingale part. We show indeed that the martingale part remains small with our rescaling and we apply the maximum principle almost surely on the finite variation part.

Let us detail now the semimartingale Doob-Meyer decomposition of the processes β_i^K, i = 0,· · · ,1/δ_K−1, which can easily be deduced from the Doob-Meyer decomposition of the semi- martingalesN_i^K, i= 0,· · · ,1/δ_K−1.

We have

β_i^K(t) =M_i^K(t) +A^K_i (t) (2.13) with

A^K_i (t) =β^K_i (0) + 1 logK

Z tlogK 0

b(iδ_K)N_i^K(s) log

1 + 1 N_i^K(s)

(2.14) +d(iδK)N_i^K(s) log

1− 1

N_i^K(s)

ds

+ 1

logK

1/δ_K−1−⌊1/2δ_K⌋

X

ℓ=−⌊1/2δ_K⌋

h_Kp((i+ℓ)δ_K)G(h_Kℓ)

Z tlogK 0

N_i+ℓ^K (s) log

1 + 1

N_i^K(s)

ds,

with the conventions that, when the index j /∈ {0, . . . ,1/δ_K−1}, N_j^K =N_j−⌊jδ^K

K⌋/δK and p(jδ_K) =p((j− ⌊jδ_K⌋/δ_K)δ_K), whenj ≥1/δ_K, (2.15) N_j^K =N_j+⌈|j|δ^K

K⌉/δ_K and p(jδ_K) =p((j+⌈|j|δ_K⌉/δ_K)δ_K), when j <0. (2.16) The process M_i^K is a local martingale with quadratic variation

⟨M_i^K⟩_t= 1 log²K

Z tlogK 0

b(iδK)N_i^K(s) log²

1 + 1

N_i^K(s)

(2.17) +d(iδ_K)N_i^K(s) log²

1− 1

N_i^K(s)

ds

+ 1

log²K

1/δK−1−⌊1/2δ_K⌋

X

ℓ=−⌊1/2δ_K⌋

hKp((i+ℓ)δK)G(hKℓ)

Z tlogK 0

N_i+ℓ^K (s) log²

1 + 1

N_i^K(s)

ds.

In Section 3, we will prove technical uniform estimates on the martingale part ∆_KM_i^K(t) and the finite variation part ∆_KA^K_i (t) of the processes ∆_Kβ_i^K(t). In that aim, let us introduce sequences of stopping times playing an important role in the proofs.

Leta∈(a2, a1) be fixed during the rest of the proof, a1 being defined in (2.3) anda2 in(2.4).

For anyK, we define τ_K^′ = inf

n

t≥0,∃i∈ {0,1,· · · , 1 δK

−1};N_i^K(tlogK)< K^a o

. (2.18)

Recall Assumption 2.3 on the initial condition. It implies thatN_i^K(tlogK)≥1 for allt≤τ_K^′ . For allL >0, we also define

τK =τK(L) = inf

t≥0 :∃i∈ {0,1,· · · , 1 δK

−1},|β_i+1^K (t)−β^K_i (t)|> LδK

, (2.19) with the usual convention that β_1/δ^K

K =β^K₀ . It is easy to check that τ_K(L) = inf

t≥0 :∃i, j∈ {0,1,· · · , 1

δ_K −1},|β_i^K(t)−β_j^K(t)|> Lρ(iδ_K, jδ_K)

and

τ_K(L) = infn

t≥0 :∃x, y∈T,|eβ^K_t (x)−βe_t^K(y)|> L ρ(x, y)o

. (2.20)

We will study the processes until the stopping time

θ_K(L) =τ_K(L)∧τ_K^′ . (2.21)

Before the stopping timeθ_K(L), the functionsβe_t^K are Lipschitz and the population size of each trait is controlled by K^a, by definition. For each Lfixed, we will provide uniform estimates on the martingale partsM_i^K(t) and ∆KM_i^K(t) and the finite variation parts A^K_i (t) and ∆KA^K_i (t) of the processes β_i^K(t) and ∆_Kβ_i^K(t), before the stopping time θ_K(L). This will allow us to prove the next lemma:

Lemma 2.6. For all T >0, there exists L₀ >0 large enough, such that

K→+∞lim P(θ_K(L0)> T) = 1.

An expression for L₀ will be given in (3.18) in the proof.

3 Proof of the tightness

We will use the criterion of Theorem 3.1 in Jakubowski [18]: let us consider the setFof functions Ff, forf ∈ C(T,R), defined on C(T,R) by

∀g∈ C(T,R), F_f(g) = Z

T

f(x)g(x)dx.

We have to prove thatF satisfies the required properties:

(i) For each ε >0, there exists a compact set C_ε⊂ C(T,R) such that

∀K, P

βe^K ∈D([0, T], C_ε)

>1−ε.

(ii) For eachf ∈ C(T,R), the sequence of laws of real-valued processes X_f^K(·) =

Z

T

βe^K(., x)f(x)dx (3.1)

is tight.

Point (i) is the hard part of the proof. Using Ascoli’s characterization of compact subsets of C(T,R), we need to obtain estimates related to equi-boundedness and to equi-continuity for the processes βe_t^K(.). The proof relies on Lipschitz estimates (in x) of the functions βe_t^K. In Section 3.1, we show that the martingale part ofβ_i^K remains small with our rescaling and in Section 3.2, we apply the maximum principle almost surely on the finite variation part ofβ_i^K. This allows us to prove Lemma 2.6 in Section 3.3. The proof of the tightness is ended in Section 3.4.

3.1 Control of the martingale part

Our first estimate will be useful to prove the tightness of the laws of βe^K. In the sequel, C denotes a constant not depending on any parameter and that may change from line to line.

The following estimate will be used repeatedly: by (2.20), for all t ≤ τ_K(L) and all i, j ≤ 1/δK−1,

N_j^K(tlogK)

N_i^K(tlogK) = exp(logK(βj(t)−βi(t))≤e^{L ρ(jδK,iδK) logK}. (3.2) Lemma 3.1. For all T >0, there exists a constantC independent ofK, T, L andi such that, almost surely, for all t≤T and all i∈ {0, . . . ,1/δK−1},

⟨M_i^K⟩_t∧θK(L)≤

CG(L) K^alogK

t. (3.3)

In particular,

E

sup

t≤T∧θ_K(L)

sup

i

⟨M_i^K⟩_t

≤ CG(L)T

K^alogK (3.4)

and for all A >0, P

sup

t≤T∧θ_K(L)

sup

i

|M_i^K(t)| ≥A

≤ CG(L)T

A²δKK^alogK. (3.5) Proof. It follows from (2.17) that

⟨M_i^K⟩_t∧θK ≤ C(¯b+ ¯d) log²K

Z (t∧θ_K) logK 0

ds N_i^K(s) + p¯

log²K

Z (t∧θK) logK 0

1/δK−1−⌊1/2δ_K⌋

X

ℓ=−⌊1/2δ_K⌋

hKG(hKℓ) N_i+ℓ^K (s) (N_i^K(s))² ds.

Therefore, using (2.18) and (3.2), (3.3) follows. Then P

sup

t≤T∧θ_K

sup

i

|M_i^K(t)| ≥A

≤

1/δK−1

X

i=0

P

sup

t≤T∧θ_K

|M_i^K(t)| ≥A

≤ 1 A²

1/δ_K−1

X

i=0

E

sup

t≤T∧θ_K

|M_i^K(t)|² ,

so we obtain (3.5) using Doob’s inequality and (3.3).

The second step is a technical lemma that we state now.

Lemma 3.2. Let t≤T. Then, for any ε >0 and any i∈ {0, . . . ,1/δ_K−1}

P

sup

s≤t∧θ_K(L)

|∆_KM_i^K(s)|> ε

≤ C ε

s

G(L)t

δ_K² K^alogK, (3.6) where the constant G(L) is defined in (2.10).

Proof. Let ε >0. By the submartingale maximal lemma, we have that P

sup

s≤t

|∆_KM_i^K(s)|> ε

≤ 1

εE(|∆_KM_i^K(t)|)≤ 1

εE |∆_KM_i^K(t)|²1/2

.

Now, Lemma 3.1 yields

⟨∆_KM_i^K⟩_t∧θK ≤ 2

δ_K² (⟨M_i+1^K ⟩_t∧θK+⟨M_i^K⟩_t∧θK)≤ 4CG(L)t δ_K²K^alogK. Hence (3.6) and thus Lemma 3.2 are proved.

Using a similar argument as in the proof of Lemma 3.1, we have the following result.

Corollary 3.3. Let T >0 andε_K =δ_K⁻¹(K^alogK)^−1/4. We define the event Ω_K(L) =

(

sup

0≤i≤1/δ_K−1, t≤T∧θ_K(L)

|∆_KM_i^K(t)| ≤ε_K )

.

Then there exists a constant C >0 such that

P(Ω^c_K(L))≤ C q

G(L)T δ_Kε_Kδ_K√

K^alogK = C q

G(L)T δ_K(K^alogK)^1/4.

Note that, by (2.4),δ_K⁴ K^alogKtends to infinity, soP(Ω_K(L)) tends to 1, asKgoes to infinity.

In addition, since |j−i|δ_K ≤1 for all 0≤i, j≤1/δK−1, we also have that Ω_K(L)⊂

(

sup

0≤i,j≤1/δ_K−1, t≤T∧θ_K(L)

M_i^K(t)−M_j^K(t) ≤ε_K

) ,

so it also follows from the last corollary that

P sup

0≤i,j≤1/δ_K−1, t≤T∧θ_K(L)

M_i^K(t)−M_j^K(t) > ε_K

!

≤C q

G(L)T ε_K. (3.7) From now on, we will work on the probability subspace ΩK(L).

3.2 Control of the finite variation part

Let us now focus on the finite variation partA^K_i . We will prove that

Proposition 3.4. Let T >0. Then, there exists a constant C₁ such that for K large enough, for allt≤T and alli∈ {0, . . . ,1/δ_K−1}the following inequality holds almost surely onΩ_K(L):

|A^K_i (t∧θK(L))| ≤ max

0≤j≤1/δ_K−1β_i^K(0) +C1t. (3.8)

Proof. We only provide the proof of the upper bound on A^K_i (t∧θ_K). The lower bound can be obtained following similar arguments.

Lett ands be less thanT such that s < t. Using that log(1 +x)≤x and (3.2), we have A^K_i (t∧θK)−A^K_i (s∧θK)

= 1

logK

Z (t∧θ_K) logK (s∧θ_K) logK

b(iδ_K)N_i^K(u) log

1 + 1

N_i^K(u)

+d(iδ_K)N_i^K(u) log

1− 1

N_i^K(u)

du

+ 1

logK

1/δK−1−⌊1/2δ_K⌋

X

ℓ=−⌊1/2δK⌋

hKp((i+ℓ)δK)G(hKℓ)

Z (t∧θ_K) logK (s∧θK) logK

N_i+ℓ^K (u) log

1 + 1

N_i^K(u)

du

≤C(¯b+ ¯d)(t−s) + 1 logK

1/δ_K−1−⌊1/2δ_K⌋

X

ℓ=−⌊1/2δ_K⌋

h_Kp((i+ℓ)δ_K)G(h_Kℓ)

Z (t∧θ_K) logK (s∧θK) logK

N_i+ℓ^K (u) N_i^K(u) du

≤C(¯b+ ¯d)(t−s)

+ 1

logK

1/δK−1−⌊1/2δ_K⌋

X

ℓ=−⌊1/2δK⌋

hKp((i+ℓ)δK)G(hKℓ)

Z (t∧θ_K) logK (s∧θ_K) logK

exp(logK(β^K_i+ℓ(u)−β_i^K(u))du.

Recall that on ΩK(L),

β_j^K(u)−β_i^K(u) =A^K_j (u)−A^K_i (u) +M_j^K(u)−M_i^K(u)≤A^K_j (u)−A^K_i (u) +ε_K. Thus we obtain that, on the event ΩK(L),

A^K_i (t∧θK)−A^K_i (s∧θK)≤C(¯b+ ¯d)(t−s) + 1 logK

1/δK−1−⌊1/2δ_K⌋

X

ℓ=−⌊1/2δK⌋

hKp((i+ℓ)δK)G(hKℓ) Z (t∧θ_K) logK

(s∧θK) logK

e^εKlogKexp(logK(A^K_i+ℓ(u)−A^K_i (u)))du.

We deduce that, almost surely on Ω_K(L) and for allt≤θ_K(L), dA^K_i (t)

dt ≤C(¯b+ ¯d)

+ 1

logK

1/δK−1−⌊1/2δK⌋

X

ℓ=−⌊1/2δ_K⌋

h_Kp((i+ℓ)δ_K)G(h_Kℓ)e^εKlogKexp(logK(A^K_i+ℓ(t)−A^K_i (t))) (3.9)

DefiningAe^K_i (t) =A^K_i (t)−2C(¯b+ ¯d)t−2¯pt, we deduce that for anyt≤θK, dAe^K_i (t)

dt <pe¯ ^εKlogK

1/δK−1−⌊1/2δ_K⌋

X

ℓ=−⌊1/2δ_K⌋

h_KG(h_Kℓ) exp(logK(Ae^K_i+ℓ(t)−Ae^K_i (t))−2¯p. (3.10)

Let us introduce

(i_K, t_K) = (i_K(ω), t_K(ω)) = argmax_{i∈{0,···,} ¹

δK−1},t∈[0,θK(ω)]Ae^K_i (t).

We can prove that tK = 0. Indeed, if conversely we assume that tK > 0, then the right term of (3.10), for K large enough, is non positive for i= iK and t =tK and then the left term is negative, contradicting the fact thatAe^K_i

K(t) is maximal fort=t_K. Hence, we have proved that forK large enough, almost surely on the event ΩK, for all t≤θK and 0≤i≤1/δK−1,

A^K_i (t) =Ae^K_i (t) + 2C(¯b+ ¯d)t+ 2¯pt≤ max

0≤j≤1/δK

Ae^K_j (0) + 2C(¯b+ ¯d)t+ 2¯pt

= max

0≤j≤1/δ_Kβ_j^K(0) + 2C(¯b+ ¯d)t+ 2¯pt.

The last result has a consequence that will be useful to prove the tightness ofβe^Kin Section 3.4.

Corollary 3.5. For all T >0, there existsC(T) such that,

K→+∞lim P sup

0≤i≤1/δK−1

sup

t∈[0,T∧θK(L)]

β_i^K(t)≥C(T)

!

= 0.

Proof. We use the semimartingale decomposition (2.13) ofβ_i^K, Proposition 3.4, (3.5) withA= 1 and Corollary 3.3 to deduce that, for all t≤T and K large enough,

|β_i^K(t∧θ_K)| ≤ sup

0≤j≤1/δK−1

|β_j^K(0)|+C₁T+ 1

with probability at least 1−_δ^CG(L)T

KK^alogK − ^C

√

G(L)T

δK(K^alogK)^1/4. Sinceβe_i^K(0) converges in probability toβ₀,P(sup_iβ_i^K(0)≥ ∥β₀∥_∞+ 1) converges to 0 when K goes to +∞. Hence the result follows withC(T) =∥β₀∥_∞+C₁T + 2.

3.3 Proof of Lemma 2.6

The proof of Lemma 2.6 results from the following two results:

Lemma 3.6. Under Assumption 2.1.1 and 2.1.3, for all T >0,

K→+∞lim P(τ_K^′ ≥T) = 1.

Proof of Lemma 3.6. By a coupling procedure, it is easy to prove that for each i, the process (N_i^K(t))t is pathwisely bounded below by a branching process (Z_i^K(t))t with birth rate b(iδ_K), death rate d(iδ_K) and initial condition K^a+ε for ε = a₁−a > 0. In addition, the processes (Z_i^K(t))t for 0≤i≤ _δ¹

K −1 are independent. Let us define θ^′′_K = inf

n

t≥0,∃i∈ {0,1,· · ·, 1

δ_K −1};Z_i^K(tlogK)< K^a o

.

In order to prove that limK→∞P(τ_K^′ > T) = 1, it is enough to prove that

K→∞lim P(θ^′′_K = +∞) = 1.

We have

P(θ_K^′′ = +∞) = P

∀i∈ {0,1,· · · , 1 δK

−1}, ∀t≥0 Z_i^K(tlogK)> K^a

=

1 δK−1

Y

i=0

P

t≥0infZ_i^K(tlogK)> K^a .

Fixi∈ {0, . . . ,_δ¹

K−1}. It is usual to prove (by time change) that the probabilityP inft≥0Z_i^K(tlogK)>

K^a

is equal to the probability for a random walkM(_b(iδ^b(iδK)

K)+d(iδK),_b(iδ^d(iδK)

K)+d(iδK)) onZ+ (adding +1 with probability _b(iδ^b(iδK)

K)+d(iδK) and−1 with probability _b(iδ^d(iδK)

K)+d(iδK)) with initial valueK^a+ε never to attain K^a. This quantity is well known and equal to

1−d(iδK) b(iδ_K)

K^a+ε−K^a

.

Since α= maxx∈Td(x)/b(x)<1, it follows from (2.4) that P(θ_K^′′ = +∞) ≥ exp

1

δ_K log 1−α^Ka+ε−Ka

∼ exp

− 1 δK

α^Ka+ε−Ka

≫ 1−K^a2/4α^Ka+ε−Ka, which tends to 1 when K tends to infinity.

Proposition 3.7. Under the Assumptions 2.1 and 2.2, for all T > 0, there exists L₀ in the definition (2.19) of τ_K(L) such that

K→+∞lim P(τ_K(L₀)> T) = 1.

Proof of Proposition 3.7. Fori∈ {0, . . . ,1/δK−1}, let us consider the increments

∆_Kβ^K_i (t∧θ_K) = β_i+1^K (t∧θK)−β_i^K(t∧θK)

δ_K = ∆_KM_i^K(t∧θ_K) + ∆_KA^K_i (t∧θ_K).

Our aim is to prove that ∆Kβ_i^K(t)−∆Kβ_i^K(s) is close to its finite variation part for large K and to apply to the latter an almost sure maximum principle.

Let us introduce

g^K_i (t) = ∆KA^K_i (t∧θK) +∥p∥_Lip

p A^K_i+1(t∧θK), (3.11) Using (2.14), we obtain forK large enough and 0≤s < t≤T,

g^K_i (t)−g_i^K(s)

= 1 hK

Z (t∧θK) logK

(s∧θ_K) logK

b((i+ 1)δ_K)N_i+1^K (u) log

1 + 1

N_i+1^K (u)

−b(iδ_K)N_i^K(u) log

1 + 1 N_i^K(u)

du

+ 1 h_K

Z (t∧θK) logK

(s∧θK) logK

d((i+ 1)δK)N_i+1^K (u) log

1− 1

N_i+1^K (u)

−d(iδK)N_i^K(u) log

1− 1 N_i^K(u)

du

+ 1 hK

Z (t∧θ_K) logK

(s∧θK) logK

1/δK−1−⌊1/2δK⌋

X

ℓ=−⌊1/2δK⌋

hKG(hKℓ)

p((i+ 1 +ℓ)δK)N_i+1+ℓ^K (u) log

1 + 1

N_i+1^K (u)

−p((i+ℓ)δK)N_i+ℓ^K (u) log

1 + 1 N_i^K(u)

)

du

+ ∥p∥_Lip plogK

Z (t∧θK) logK

(s∧θK) logK

b((i+ 1)δ_K)N_i+1^K (u) log

1 + 1

N_i+1^K (u)

+d((i+ 1)δ_K)N_i+1^K (u) log

1− 1

N_i+1^K (u)

du

+ ∥p∥Lip

plogK

Z (t∧θK) logK

(s∧θK) logK

1/δK−1−⌊1/2δK⌋

X

ℓ=−⌊1/2δK⌋

hKG(hKℓ)p((i+ 1 +ℓ)δK)N_i+1+ℓ^K (u) log

1 + 1

N_i+1^K (u)

du

≤C(¯b+ ¯d) hK

Z (t∧θK) logK

(s∧θK) logK

1

N_i+1^K (u)+ 1 N_i^K(u)

du+ (C1(¯b+ ¯d) +∥b∥Lip+∥d∥Lip)(t∧θK−s∧θK)

+ 1 hK

Z (t∧θK) logK

(s∧θ_K) logK

1/δ_K−1−⌊1/2δK⌋

X

ℓ=−⌊1/2δ_K⌋

h_Kp((ℓ+i)δ_K)G(h_Kℓ)

N_ℓ+i+1^K (u) log

1 + 1

N_i+1^K (u)

−N_ℓ+i^K (u) log

1 + 1

N_i^K(u)

du

+3∥p∥Lip

plogK

Z (t∧θK) logK

(s∧θK) logK

1/δK−1−⌊1/2δK⌋

X

ℓ=−⌊1/2δK⌋

hKp((ℓ+i)δK)G(hKℓ)N_ℓ+i+1^K (u) log

1 + 1

N_i+1^K (u)

du, (3.12)

where we used that for allx such that |x| ≤1/2 we have

1

xlog(1 +x)

≤C, (3.13)

1

xlog(1 +x)− 1

ylog(1 +y)

≤C(|x|+|y|). (3.14)

and the fact that (recalling the convention (2.15) and that pis periodic)

|p((ℓ+i+ 1)δ_K)−p((ℓ+i)δ_K)|=p((ℓ+i)δ_K)|p((ℓ+i+ 1)δ_K)−p((ℓ+i)δ_K)|

p((ℓ+i)δ_K)

≤∥p∥_LipδK

p p((ℓ+i)δK) (3.15)

in the last inequality. Note also that, to obtain this last inequality we have takenK large enough such that

∥p∥_LipδK

p ≤1, so that

p((ℓ+i+ 1)δK)≤2p((ℓ+i)δK).

Next, notice that for all x^′, y^′, x, y such that|x|,|y| ≤1/2, 1

y^′ log(1 +y)− 1

x^′log(1 +x)≤ y y^′ − x

x^′ +C y²

y^′ +x² x^′

. (3.16)

Using this inequality and (3.2), we have 1

h_K

Z (t∧θK) logK (s∧θ_K) logK

1/δK−1−⌊1/2δ_K⌋

X

ℓ=−⌊1/2δ_K⌋

hKp((ℓ+i)δK)G(hKℓ)

"

N_ℓ+i+1^K (u) log 1 + 1 N_i+1^K (u)

!

−N_ℓ+i^K (u) log

1 + 1

N_i^K(u)

du

≤ 1 hK

Z (t∧θ_K) logK (s∧θK) logK

1/δK−1−⌊1/2δK⌋

X

ℓ=−⌊1/2δ_K⌋

h_Kp((ℓ+i)δ_K)G(h_Kℓ)

"

N_ℓ+i+1^K (u)

N_i+1^K (u) −N_ℓ+i^K (u) N_i^K(u)

# du

+Cp¯ h_K

Z (t∧θ_K) logK (s∧θ_K) logK

1/δK−1−⌊1/2δK⌋

X

ℓ=−⌊1/2δ_K⌋

h_KG(h_Kℓ)e^LhK|ℓ| 1

N_i+1^K (u) + 1 N_i^K(u)

! du.

Therefore, using (2.10) and the definition ofτ_K^′ , we have proved that g_i^K(t)−g_i^K(s)≤

CG(L) log¯ K K^a hK

+C

(t−s) + 1

hK

Z (t∧θ_K) logK (s∧θK) logK

1/δ_K−1−⌊1/2δ_K⌋

X

ℓ=−⌊1/2δK⌋

h_Kp((ℓ+i)δ_K)G(h_Kℓ)

"

N_ℓ+i+1^K (u)

N_i+1^K (u) −N_ℓ+i^K (u) N_i^K(u)

# du

+3∥p∥_Lip plogK

Z (t∧θ_K) logK (s∧θK) logK

1/δK−1−⌊1/2δK⌋

X

ℓ=−⌊1/2δ_K⌋

h_Kp((ℓ+i)δ_K)G(h_Kℓ)N_ℓ+i+1^K (u) N_i+1^K (u) du.