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

Nicolas Champagnat Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy,
France; E-mail<EMAIL_ADDRESS>Sylvie Méléard Ecole Polytechnique
et Institut Universitaire de France, CNRS, Institut polytechnique de Paris,
route de Saclay, 91128 Palaiseau Cedex-France; E-mail:
<EMAIL_ADDRESS>Sepideh Mirrahimi  Institut Montpelliérain
Alexander Grothendieck, Univ. Montpellier, CNRS, Montpellier, France; E-mail:
<EMAIL_ADDRESS>Viet Chi Tran LAMA, Univ Gustave Eiffel,
Univ Paris Est Creteil, CNRS, F-77454 Marne-la-Vallée, France; E-mail:
<EMAIL_ADDRESS>
###### 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
mutation 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 result 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 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 proposed 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 trait
$x\in\mathbb{T}$, 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 $x\in\mathbb{T}$, 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 by $G(h)$ the mutation kernel. Assuming
that the population density solves the PDE

$\begin{cases}\varepsilon\partial_{t}u_{\varepsilon}(t,x)=u_{\varepsilon}(t,x)\left(b(x)-d(x)\right)+\int_{\mathbb{T}}\frac{1}{\varepsilon}G\left(\frac{x-y}{\varepsilon}\right)p(y)u_{\varepsilon}(t,y)\mathrm{d}y&(t,x)\in\mathbb{R}_{+}\times\mathbb{T}\\\
u_{\varepsilon}(t,0)=\exp\left(\frac{\beta_{0}(x)}{\varepsilon}\right),&x\in\mathbb{T}\end{cases}$
(1.1)

in the limit $\varepsilon\to 0$ of small mutations and large time, and
applying the Hof-Cole transformation

$\beta_{\varepsilon}(t,x)=\varepsilon\log
u_{\varepsilon}(t,x),\quad\text{or}\quad
u_{\varepsilon}(t,x)=\exp\left(\frac{\beta_{\varepsilon}(t,x)}{\varepsilon}\right),$
(1.2)

it is proved in [2] (in a slightly different setting, considering
$x\in\mathbb{R}$ and taking into account a competition term) that
$\beta_{\varepsilon}$ converges to the unique solution $\beta$ of the
Hamilton-Jacobi equation

$\begin{cases}\frac{\partial}{\partial
t}\beta(t,x)=b(x)-d(x)+p(x)\int_{\mathbb{R}}G(h)e^{h\partial_{x}\beta(t,x)}dh,&(t,x)\in\mathbb{R}_{+}\times\mathbb{T}\\\
\beta(0,x)=\beta_{0}(x),&x\in\mathbb{T}\end{cases}$ (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 $K^{\beta}$ and
characterize the asymptotic dynamics of the exponent $\beta$. In particular,
local extinction is possible when the exponent $\beta$ 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 $\delta_{K}$, where $K$ is a scaling parameter such that the population
is of the order of $K^{\widetilde{\beta}^{K}(t,x)}$, assuming frequent and
small mutations. In the individual-based model, individuals with trait
$x\in\mathbb{T}$ 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 $\log KG(\log K\cdot)$. Mutation steps are
of the order of $1/\log K$ and the discretization step $\delta_{K}$ is assumed
much smaller than $1/\log K$. 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 of $\widetilde{\beta}^{K}$, 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. However, 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 interpretation 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 population of individuals characterized by a quantitative
phenotypic or genetic trait $x\in\mathbb{T}$. Starting from a finite
population whose initial size is parameterized by $K\in\mathbb{N}$, our goal
is to recover, in the limit $K\to+\infty$, an evolutionary dynamics described
by the Hamilton-Jacobi partial differential equation (1.3). For this, we
consider a discretization of the trait space $\mathbb{T}$ with step
$\delta_{K}\to 0$. For the sake of simplicity, we will consider in what
follows that $1/\delta_{K}\in\mathbb{N}$. Then, the population is composed of
individuals with traits belonging to the discrete space

${\cal
X}_{K}:=\left\\{i\delta_{K}:i\in\\{0,1,\cdots,{1\over\delta_{K}}-1\\}\right\\},$

embedded with the torus distance: $\forall x,y\in[0,1)$,

$\displaystyle\rho(x,y)$ $\displaystyle=\min\left\\{|x^{\prime}-y^{\prime}|,\
x^{\prime}=x\text{ mod }1,\ y^{\prime}=y\text{ mod }1\right\\}$
$\displaystyle=\min\big{(}|x-y|,1-|x-y|\big{)}.$

It’s enough to define $\rho(x,y)$ for $x,y\in\mathbb{T}$ by considering their
representative in $[0,1)$.

The number of individuals with trait $i\delta_{K}$ is described by the
stochastic process $(N_{i}^{K}(t),t\geq 0)$. The total population size at time
$t$ is then given by

$N^{K}(t)=\sum_{i=0}^{1/\delta_{K}-1}N_{i}^{K}(t).$

An individual with trait $x\in{\cal X}_{K}$

  * •

gives birth to a new individual with the same trait $x$ at rate $b(x)$;

  * •

dies at rate $d(x)$;

  * •

gives birth to a mutant individual with trait $y\in{\cal X}_{K}$ at rate

$p(x)\delta_{K}\log K\,G\left((\overline{y-x})\,\log K\right),$ (2.1)

where $\overline{z}$ is the unique real number of $[-1/2,1/2)$ that is equal
to $z$ modulo 1, i.e. $\overline{z}=z-\lfloor z+1/2\rfloor$, where
$\lfloor\cdot\rfloor$ is the integer part function.

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

######  Assumption 2.1.

1\. We assume that $b$, $d$ and $p$ are nonnegative Lipschitz continuous
functions defined on $\mathbb{T}$, such that for all $x\in\mathbb{T}$,

$b(x)>d(x)\hbox{ and }p(x)>0.$ (2.2)

This means that the birth-death process for each trait is super-critical. In
the sequel, we denote by $\bar{b}$, $\bar{p}$ and $\bar{d}$ the upper bounds
of these functions on $\mathbb{T}$, by $\,\underline{p}>0$ the lower bound of
$p$, and by $\|b\|_{\text{Lip}}$, $\|d\|_{\text{Lip}}$ and
$\|p\|_{\text{Lip}}$ their Lipschitz norm.

2\. The function $G$ defined on $\mathbb{R}$ is nonnegative, continuous,
satisfies $\int_{\mathbb{R}}G(y)\,dy=1$ and has finite exponential moments of
any order. Moreover, we assume that there exists $R>0$ such that $G$ is
nonincreasing on $[R,+\infty)$ and nondecreasing on $(-\infty,-R]$.
An example of function $G$ satisfying Assumption 2 is given by the Gaussian
kernel $G(h)=\frac{1}{\sqrt{2\pi}\sigma}e^{-h^{2}/2\sigma^{2}}$.

3\. There exists a constant $a_{1}>0$ such that, for all $K\in\mathbb{N}$ and
all $i\in\\{0,1,\cdots,{1\over\delta_{K}}-1\\}$,

$N^{K}_{i}(0)\geq K^{a_{1}}.$ (2.3)

4\. There exists $a_{2}<a_{1}$ such that

$K^{-a_{2}/4}\ll\delta_{K}\ll\frac{1}{\log K}\quad\mbox{ as }\quad
K\rightarrow+\infty.$ (2.4)

Point 4 above implies that

$h_{K}:=\delta_{K}\log K\ll 1.$ (2.5)

Therefore, the interpretation of (2.1) is that the rate at which $x$ gives
birth to a mutant individual is close to $p(x)$. Indeed, for an individual
with trait $x_{K}=i_{K}\delta_{K}$ with $i_{K}=\lfloor x/\delta_{K}\rfloor$
and $x\in\mathbb{T}$ fixed,

$\displaystyle\lim_{K\rightarrow+\infty}p(x_{K})\sum_{j=0}^{{1\over\delta_{K}}-1}h_{K}G\left(\overline{(i_{K}-j)\delta_{K}}\,\log
K\right)$ $\displaystyle=\lim_{K\rightarrow+\infty}p(x_{K})\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}G\left(h_{K}\ell\right)$
$\displaystyle=p(x)\int_{\mathbb{R}}G(y)\,dy=p(x),$ (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 trait $y$ of the offspring has the
distribution $G$ scaled by a factor $1/\log K$ 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
$K^{\alpha_{i}}$ for some $\alpha_{i}>0$. Note that in the case where
$p(x)=0\,$ for all $x\in\mathbb{T}$, the process $N^{K}_{i}(t)$ is a super-
critical one-dimensional branching process, hence
$\mathbb{E}(N^{K}_{i}(t))=\mathbb{E}(N^{K}_{i}(0))e^{(b(i\delta_{K})-d(i\delta_{K}))t}$.
Therefore, if the initial condition is of order $K^{\alpha_{i}}$, then
$\mathbb{E}[N^{K}_{i}(t\log K)]\sim
K^{\alpha_{i}+(b(i\delta_{K})-d(i\delta_{K}))t}$. This suggests to study

$\beta_{i}^{K}(t)=\frac{\log(N_{i}^{K}(t\log K))}{\log(K)},$ (2.7)

with the convention that $\beta^{K}_{i}(t)=0\,$ if $\,N^{K}_{i}(t\log K)=0$.
So the sub-population of trait $i\delta_{K}$ at time $t\log K$ has size
$N^{K}_{i}(t\log K)=K^{\beta^{K}_{i}(t)}$.

We make the following assumption on the initial condition $\beta^{K}_{i}(0)$.

######  Assumption 2.2.

Assume that there exists a constant $A>0$ such that

$\lim_{K\rightarrow+\infty}\mathbb{P}\Big{(}\sup_{i\neq
j}\frac{|\beta^{K}_{i}(0)-\beta^{K}_{j}(0)|}{\rho(i\,\delta_{K},j\,\delta_{K})}>A\Big{)}=0.$
(2.8)

Notations: (i) We shall use the following Riemann approximation repeatedly in
the proofs: for all $\alpha>0$,

$\lim_{K\rightarrow+\infty}\sum_{j=0}^{{1\over\delta_{K}}-1}h_{K}G\left(\overline{(i_{K}-j)\delta_{K}}\,\log
K\right)e^{\alpha\,\log K\,\rho(j\delta_{K},i_{K}\delta_{K})}\\\
=\lim_{K\rightarrow+\infty}\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}G\left(h_{K}\ell\right)e^{\alpha\,{h_{K}}\,|{\ell}|}=\int_{\mathbb{R}}e^{\alpha|y|}G(y)\,dy,$
(2.9)

and thus, there exists a constant $\overline{G}(\alpha)$ depending only on
$\alpha>0$ such that

$\sup_{K\geq
1}\,\sum_{j=0}^{{1\over\delta_{K}}-1}h_{K}G\left(\overline{(i_{K}-j)\delta_{K}}\,\log
K\right)e^{L\,\log K\,\rho(j\delta_{K},i_{K}\delta_{K})}=\ :\
\overline{G}(\alpha)<+\infty.$ (2.10)

(ii) In what follows and for any function $f$ on
$\\{0,1,\cdots,{1\over\delta_{K}}-1\\}$, we will use the notation

$\Delta_{K}f_{i}=\frac{f_{i+1}-f_{i}}{\delta_{K}},$

with the convention that $f_{1/\delta_{K}}=f_{0}$.

###  2.2 The main result - Sketch of the proof

Since we are interested in the convergence of the quantities $\beta_{i}^{K}$
to a continuous function defined on the trait space $\mathbb{T}$, when
$K\to+\infty$ and $\delta_{K}\to 0$, we introduce the following interpolation
of the $\beta_{i}^{K}$’s: for all $x\in\mathbb{T}$ and $K\geq 1$, let $i$ be
such that $x\in[i\delta_{K},(i+1)\delta_{K})$, and define

$\widetilde{\beta}^{K}_{t}(x)=\beta_{i}^{K}(t)\Big{(}1-\frac{x}{\delta_{K}}+i\Big{)}+\beta_{i+1}^{K}(t)\Big{(}\frac{x}{\delta_{K}}-i\Big{)},$
(2.11)

with the convention that $\beta^{K}_{1/\delta_{K}}(t)=\beta^{K}_{0}(t)$. The
sequence of processes $(\widetilde{\beta}^{K}_{t},t\in[0,T])_{K}$ belongs to
$\mathbb{D}([0,T],\mathcal{C}(\mathbb{T},\mathbb{R}))$, where
$\mathcal{C}(\mathbb{T},\mathbb{R})$ is endowed with the topology of uniform
convergence and $\mathbb{D}([0,T],\mathcal{C}(\mathbb{T},\mathbb{R}))$ is the
Skorohod space of càdlàg 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
$\widetilde{\beta}^{K}_{0}(\cdot)$ converges in probability for the topology
of uniform convergence on $\mathcal{C}(\mathbb{T},\mathbb{R})$ to a
deterministic function
$\beta_{0}(\cdot)\in\mathcal{C}(\mathbb{T},\mathbb{R})$, the sequence
$(\widetilde{\beta}^{K})_{K}$ converges in probability in
$\mathbb{D}([0,T],\mathcal{C}(\mathbb{T},\mathbb{R}))$ to the unique Lipschitz
viscosity solution of the Hamilton-Jacobi equation

$\begin{cases}\frac{\partial}{\partial
t}\beta(t,x)=b(x)-d(x)+p(x)\int_{\mathbb{R}}G(h)e^{h\partial_{x}\beta(t,x)}dh,&(t,x)\in\mathbb{R}_{+}\times\mathbb{T}\\\
\beta(0,x)=\beta_{0}(x),&x\in\mathbb{T}.\end{cases}$ (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 $(\tilde{\beta}^{K}_{t},t\in[0,T])_{K}$ is relatively
compact in ${\cal P}(\mathbb{D}([0,T],\mathcal{C}(\mathbb{T},\mathbb{R})))$.

######  Theorem 2.5.

The limiting values $\beta$ of $(\tilde{\beta}^{K}_{t},t\in[0,T])_{K}$ are
characterized as the unique viscosity solution of the Hamilton-Jacobi equation

$\frac{\partial}{\partial
t}\beta(t,x)=b(x)-d(x)+p(x)\int_{\mathbb{R}}G(h)e^{h\partial_{x}\beta(t,x)}dh.$

The proofs of these two results require to control the increments of the
functions $\widetilde{\beta}^{K}_{t}(.)$. This functions can also be written
as

$\widetilde{\beta}^{K}_{t}(x)=(x-i\delta_{K})\Delta_{K}\beta^{K}_{i}(t)+\beta^{K}_{i}(t).$

From this expression, we observe that two technical steps are required: to
estimate uniformly $\beta_{i}^{K}(t)$ and to control uniformly $\
\Delta_{K}\beta^{K}_{i}(t)=\frac{\beta_{i+1}^{K}(t)-\beta_{i}^{K}(t)}{\delta_{K}}\
$ (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
$\beta_{i}^{K},\ i=0,\cdots,1/\delta_{K}-1$, which can easily be deduced from
the Doob-Meyer decomposition of the semi-martingales $N^{K}_{i},\
i=0,\cdots,1/\delta_{K}-1$.

We have

$\beta_{i}^{K}(t)=M^{K}_{i}(t)+A^{K}_{i}(t)$ (2.13)

with

$\displaystyle A_{i}^{K}(t)$ $\displaystyle=\beta_{i}^{K}(0)+{1\over\log
K}\int_{0}^{t\log K}\Big{(}b(i\delta_{K})N^{K}_{i}(s)\log\left(1+{1\over
N^{K}_{i}(s)}\right)$ (2.14) $\displaystyle\hskip
142.26378pt+d(i\delta_{K})N^{K}_{i}(s)\log\left(1-{1\over
N^{K}_{i}(s)}\right)\Big{)}ds$ $\displaystyle+{1\over\log
K}\sum_{\ell=-\lfloor 1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}\,p((i+\ell)\delta_{K})G(h_{K}\ell)\int_{0}^{t\log
K}N^{K}_{i+\ell}(s)\log\left(1+{1\over N^{K}_{i}(s)}\right)ds,$

with the conventions that, when the index
$j\notin\\{0,\dots,1/\delta_{K}-1\\}$,

$\displaystyle N^{K}_{j}=N^{K}_{j-\lfloor
j\delta_{K}\rfloor/\delta_{K}}\quad\text{and}\quad p(j\delta_{K})=p((j-\lfloor
j\delta_{K}\rfloor/\delta_{K})\delta_{K}),\qquad\mbox{ when }j\geq
1/\delta_{K},$ (2.15) $\displaystyle
N^{K}_{j}=N^{K}_{j+\lceil|j|\delta_{K}\rceil/\delta_{K}}\quad\text{and}\quad
p(j\delta_{K})=p((j+\lceil|j|\delta_{K}\rceil/\delta_{K})\delta_{K}),\qquad\mbox{
when }j<0.$ (2.16)

The process $M^{K}_{i}$ is a local martingale with quadratic variation

$\displaystyle\langle M^{K}_{i}\rangle_{t}$
$\displaystyle={1\over\log^{2}K}\int_{0}^{t\log
K}\Big{(}b(i\delta_{K})N^{K}_{i}(s)\log^{2}\Big{(}1+{1\over
N^{K}_{i}(s)}\Big{)}$ (2.17)
$\displaystyle\qquad\qquad\qquad+d(i\delta_{K})N^{K}_{i}(s)\log^{2}\Big{(}1-{1\over
N^{K}_{i}(s)}\Big{)}\Big{)}ds$
$\displaystyle+{1\over\log^{2}K}\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((i+\ell)\delta_{K})G(h_{K}\ell)\int_{0}^{t\log
K}N^{K}_{i+\ell}(s)\log^{2}\left(1+{1\over N^{K}_{i}(s)}\right)ds.$

In Section 3, we will prove technical uniform estimates on the martingale part
$\Delta_{K}M^{K}_{i}(t)$ and the finite variation part
$\Delta_{K}A^{K}_{i}(t)$ of the processes $\Delta_{K}\beta^{K}_{i}(t)$. In
that aim, let us introduce sequences of stopping times playing an important
role in the proofs.

Let $a\in(a_{2},a_{1})$ be fixed during the rest of the proof, $a_{1}$ being
defined in (2.3) and $a_{2}$ in(2.4). For any $K$, we define

$\tau^{\prime}_{K}=\inf\Big{\\{}t\geq 0,\exists
i\in\\{0,1,\cdots,{1\over\delta_{K}}-1\\};N^{K}_{i}(t\log K)<K^{a}\Big{\\}}.$
(2.18)

Recall Assumption 2.3 on the initial condition. It implies that
$N^{K}_{i}(t\log K)\geq 1$ for all $t\leq\tau^{\prime}_{K}$.

For all $L>0$, we also define

$\tau_{K}=\tau_{K}(L)=\inf\left\\{t\geq 0:\exists
i\in\\{0,1,\cdots,{1\over\delta_{K}}-1\\},\,|\beta^{K}_{i+1}(t)-\beta^{K}_{i}(t)|>L\delta_{K}\right\\},$
(2.19)

with the usual convention that $\beta^{K}_{1/\delta_{K}}=\beta^{K}_{0}$. It is
easy to check that

$\tau_{K}(L)=\inf\left\\{t\geq 0:\exists
i,j\in\\{0,1,\cdots,{1\over\delta_{K}}-1\\},\,|\beta^{K}_{i}(t)-\beta^{K}_{j}(t)|>L\rho(i\delta_{K},j\delta_{K})\right\\}$

and

$\tau_{K}(L)=\inf\left\\{t\geq 0:\exists
x,y\in\mathbb{T},\,|\widetilde{\beta}^{K}_{t}(x)-\widetilde{\beta}^{K}_{t}(y)|>L\
\rho(x,y)\right\\}.$ (2.20)

We will study the processes until the stopping time

$\theta_{K}(L)=\tau_{K}(L)\wedge\tau^{\prime}_{K}.$ (2.21)

Before the stopping time $\theta_{K}(L)$, the functions
$\widetilde{\beta}^{K}_{t}$ are Lipschitz and the population size of each
trait is controlled by $K^{a}$, by definition. For each $L$ fixed, we will
provide uniform estimates on the martingale parts $M^{K}_{i}(t)$ and
$\Delta_{K}M^{K}_{i}(t)$ and the finite variation parts $A^{K}_{i}(t)$ and
$\Delta_{K}A^{K}_{i}(t)$ of the processes $\beta^{K}_{i}(t)$ and
$\Delta_{K}\beta^{K}_{i}(t)$, before the stopping time $\theta_{K}(L)$. This
will allow us to prove the next lemma:

######  Lemma 2.6.

For all $T>0$, there exists $L_{0}>0$ large enough, such that

$\lim_{K\rightarrow+\infty}\mathbb{P}(\theta_{K}(L_{0})>T)=1.$

An expression for $L_{0}$ 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 set $\mathbb{F}$ of functions $F_{f}$, for
$f\in\mathcal{C}(\mathbb{T},\mathbb{R})$, defined on
$\mathcal{C}(\mathbb{T},\mathbb{R})$ by

$\forall g\in\mathcal{C}(\mathbb{T},\mathbb{R}),\
F_{f}(g)=\int_{\mathbb{T}}f(x)g(x)dx.$

We have to prove that $\mathbb{F}$ satisfies the required properties:

(i) For each $\varepsilon>0$, there exists a compact set
$C_{\varepsilon}\subset\mathcal{C}(\mathbb{T},\mathbb{R})$ such that

$\forall K,\
\mathbb{P}\left(\widetilde{\beta}^{K}\in\mathbb{D}([0,T],C_{\varepsilon})\right)>1-\varepsilon.$

(ii) For each $f\in\mathcal{C}(\mathbb{T},\mathbb{R})$, the sequence of laws
of real-valued processes

$X^{K}_{f}(\cdot)=\int_{\mathbb{T}}\widetilde{\beta}^{K}(.,x)f(x)dx$ (3.1)

is tight.

Point (i) is the hard part of the proof. Using Ascoli’s characterization of
compact subsets of $\mathcal{C}(\mathbb{T},\mathbb{R})$, we need to obtain
estimates related to equi-boundedness and to equi-continuity for the processes
$\widetilde{\beta}^{K}_{t}(.)$. The proof relies on Lipschitz estimates (in
$x$) of the functions $\widetilde{\beta}^{K}_{t}$. In Section 3.1, we show
that the martingale part of $\beta^{K}_{i}$ remains small with our rescaling
and in Section 3.2, we apply the maximum principle almost surely on the finite
variation part of $\beta^{K}_{i}$. 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
$\widetilde{\beta}^{K}$. 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
$t\leq\tau_{K}(L)$ and all $i,j\leq 1/\delta_{K}-1$,

$\frac{N^{K}_{j}(t\log K)}{N^{K}_{i}(t\log K)}=\exp(\log
K(\beta_{j}(t)-\beta_{i}(t))\leq e^{L\,\rho(j\delta_{K},i\delta_{K})\,\log
K}.$ (3.2)

######  Lemma 3.1.

For all $T>0$, there exists a constant $C$ independent of $K$, $T$, $L$ and
$i$ such that, almost surely, for all $t\leq T$ and all
$i\in\\{0,\ldots,1/\delta_{K}-1\\}$,

$\langle
M^{K}_{i}\rangle_{t\wedge\theta_{K}(L)}\leq\left(\frac{C\overline{G}(L)}{K^{a}\log
K}\right)t.$ (3.3)

In particular,

$\mathbb{E}\Big{(}\sup_{t\leq T\wedge\theta_{K}(L)}\sup_{i}\langle
M^{K}_{i}\rangle_{t}\Big{)}\leq{C\overline{G}(L)T\over K^{a}\log K}$ (3.4)

and for all $A>0$,

$\mathbb{P}\Big{(}\sup_{t\leq T\wedge\theta_{K}(L)}\sup_{i}|M^{K}_{i}(t)|\geq
A\Big{)}\leq\frac{C\overline{G}(L)T}{A^{2}\delta_{K}K^{a}\log K}.$ (3.5)

###### Proof.

It follows from (2.17) that

$\langle
M^{K}_{i}\rangle_{t\wedge\theta_{K}}\leq\frac{C(\bar{b}+\bar{d})}{\log^{2}K}\int_{0}^{(t\wedge\theta_{K})\log
K}\frac{ds}{N^{K}_{i}(s)}\\\
+\frac{\bar{p}}{\log^{2}K}\int_{0}^{(t\wedge\theta_{K})\log
K}\sum_{\ell=-\lfloor 1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}G(h_{K}\ell)\frac{N^{K}_{i+\ell}(s)}{(N^{K}_{i}(s))^{2}}\,ds.$

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

$\displaystyle\mathbb{P}\Big{(}\sup_{t\leq
T\wedge\theta_{K}}\sup_{i}|M^{K}_{i}(t)|\geq A\Big{)}$
$\displaystyle\leq\sum_{i=0}^{1/\delta_{K}-1}\mathbb{P}\Big{(}\sup_{t\leq
T\wedge\theta_{K}}|M^{K}_{i}(t)|\geq A\Big{)}$ $\displaystyle\leq$
$\displaystyle\frac{1}{A^{2}}\sum_{i=0}^{1/\delta_{K}-1}\mathbb{E}\Big{(}\sup_{t\leq
T\wedge\theta_{K}}|M^{K}_{i}(t)|^{2}\Big{)},$

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 $t\leq T$. Then, for any $\varepsilon>0$ and any
$i\in\\{0,\ldots,1/\delta_{K}-1\\}$

$\mathbb{P}\Big{(}\sup_{s\leq
t\wedge\theta_{K}(L)}|\Delta_{K}M^{K}_{i}(s)|>\varepsilon\Big{)}\leq{C\over\varepsilon}\sqrt{\frac{\overline{G}(L)t}{\delta_{K}^{2}K^{a}\log
K}},$ (3.6)

where the constant $\overline{G}(L)$ is defined in (2.10).

###### Proof.

Let $\varepsilon>0$. By the submartingale maximal lemma, we have that

$\mathbb{P}\Big{(}\sup_{s\leq
t}|\Delta_{K}M^{K}_{i}(s)|>\varepsilon\Big{)}\leq{1\over\varepsilon}\mathbb{E}(|\Delta_{K}M^{K}_{i}(t)|)\leq{1\over\varepsilon}\mathbb{E}\big{(}|\Delta_{K}M^{K}_{i}(t)|^{2}\big{)}^{1/2}.$

Now, Lemma 3.1 yields

$\displaystyle\langle\Delta_{K}M^{K}_{i}\rangle_{t\wedge\theta_{K}}\leq\frac{2}{\delta_{K}^{2}}(\langle
M^{K}_{i+1}\rangle_{t\wedge\theta_{K}}+\langle
M^{K}_{i}\rangle_{t\wedge\theta_{K}})\leq\frac{4C\overline{G}(L)t}{\delta_{K}^{2}K^{a}\log
K}.$

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 $T>0$ and $\varepsilon_{K}=\delta_{K}^{-1}(K^{a}\log K)^{-1/4}$. We define
the event

$\Omega_{K}(L)=\left\\{\sup_{0\leq i\leq 1/\delta_{K}-1,\ t\leq
T\wedge\theta_{K}(L)}|\Delta_{K}M^{K}_{i}(t)|\leq\varepsilon_{K}\right\\}.$

Then there exists a constant $C>0$ such that

$\mathbb{P}(\Omega_{K}^{c}(L))\leq\frac{C\sqrt{\overline{G}(L)T}}{\delta_{K}\varepsilon_{K}\delta_{K}\sqrt{K^{a}\log
K}}=\frac{C\sqrt{\overline{G}(L)T}}{\delta_{K}(K^{a}\log K)^{1/4}}.$

Note that, by (2.4), $\delta_{K}^{4}K^{a}\log K$ tends to infinity, so
$\mathbb{P}(\Omega_{K}(L))$ tends to $1$, as $K$ goes to infinity. In
addition, since $|j-i|\delta_{K}\leq 1$ for all $0\leq i,j\leq
1/\delta_{K}-1$, we also have that

$\Omega_{K}(L)\subset\left\\{\sup_{0\leq i,j\leq 1/\delta_{K}-1,\ t\leq
T\wedge\theta_{K}(L)}\left|M^{K}_{i}(t)-M^{K}_{j}(t)\right|\leq\varepsilon_{K}\right\\},$

so it also follows from the last corollary that

$\mathbb{P}\left(\sup_{0\leq i,j\leq 1/\delta_{K}-1,\ t\leq
T\wedge\theta_{K}(L)}\left|M^{K}_{i}(t)-M^{K}_{j}(t)\right|>\varepsilon_{K}\right)\leq
C\sqrt{\overline{G}(L)T}\varepsilon_{K}.$ (3.7)

From now on, we will work on the probability subspace $\Omega_{K}(L)$.

###  3.2 Control of the finite variation part

Let us now focus on the finite variation part $A^{K}_{i}$. We will prove that

######  Proposition 3.4.

Let $T>0$. Then, there exists a constant $C_{1}$ such that for $K$ large
enough, for all $t\leq T$ and all $i\in\\{0,\ldots,1/\delta_{K}-1\\}$ the
following inequality holds almost surely on $\Omega_{K}(L)$:

$|A^{K}_{i}(t\wedge\theta_{K}(L))|\leq\max_{0\leq j\leq
1/\delta_{K}-1}\beta^{K}_{i}(0)+C_{1}t.$ (3.8)

###### Proof.

We only provide the proof of the upper bound on
$A_{i}^{K}(t\wedge\theta_{K})$. The lower bound can be obtained following
similar arguments.

Let $t$ and $s$ be less than $T$ such that $s<t$. Using that $\log(1+x)\leq x$
and (3.2), we have

$A_{i}^{K}(t\wedge\theta_{K})-A_{i}^{K}(s\wedge\theta_{K})\\\ \begin{aligned}
&={1\over\log K}\int_{(s\wedge\theta_{K})\log K}^{(t\wedge\theta_{K})\log
K}\Big{(}b(i\delta_{K})N^{K}_{i}(u)\log\left(1+{1\over
N^{K}_{i}(u)}\right)+d(i\delta_{K})N^{K}_{i}(u)\log\left(1-{1\over
N^{K}_{i}(u)}\right)\Big{)}du\\\ &+{1\over\log K}\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((i+\ell)\delta_{K})G(h_{K}\ell)\int_{(s\wedge\theta_{K})\log
K}^{(t\wedge\theta_{K})\log K}N^{K}_{i+\ell}(u)\log\left(1+{1\over
N^{K}_{i}(u)}\right)du\\\ &\leq C(\bar{b}+\bar{d})(t-s)+{1\over\log
K}\sum_{\ell=-\lfloor 1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((i+\ell)\delta_{K})G(h_{K}\ell)\int_{(s\wedge\theta_{K})\log
K}^{(t\wedge\theta_{K})\log K}\frac{N^{K}_{i+\ell}(u)}{N^{K}_{i}(u)}du\\\
&\leq C(\bar{b}+\bar{d})(t-s)\\\ &\quad+{1\over\log K}\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((i+\ell)\delta_{K})G(h_{K}\ell)\int_{(s\wedge\theta_{K})\log
K}^{(t\wedge\theta_{K})\log K}\exp(\log
K(\beta^{K}_{i+\ell}(u)-\beta^{K}_{i}(u))du.\end{aligned}$

Recall that on $\Omega_{K}(L)$,

$\beta^{K}_{j}(u)-\beta^{K}_{i}(u)=A_{j}^{K}(u)-A_{i}^{K}(u)+M_{j}^{K}(u)-M_{i}^{K}(u)\leq
A_{j}^{K}(u)-A_{i}^{K}(u)+\varepsilon_{K}.$

Thus we obtain that, on the event $\Omega_{K}(L)$,

$A_{i}^{K}(t\wedge\theta_{K})-A_{i}^{K}(s\wedge\theta_{K})\leq
C(\bar{b}+\bar{d})(t-s)+{1\over\log K}\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((i+\ell)\delta_{K})G(h_{K}\ell)\\\
\int_{(s\wedge\theta_{K})\log K}^{(t\wedge\theta_{K})\log
K}e^{\varepsilon_{K}\log K}\exp(\log K(A^{K}_{i+\ell}(u)-A^{K}_{i}(u)))du.$

We deduce that, almost surely on $\Omega_{K}(L)$ and for all
$t\leq\theta_{K}(L)$,

$\frac{dA^{K}_{i}(t)}{dt}\leq C(\bar{b}+\bar{d})\\\ +{1\over\log
K}\sum_{\ell=-\lfloor 1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((i+\ell)\delta_{K})G(h_{K}\ell)e^{\varepsilon_{K}\log
K}\exp(\log K(A^{K}_{i+\ell}(t)-A^{K}_{i}(t)))$ (3.9)

Defining
$\widetilde{A}^{K}_{i}(t)=A^{K}_{i}(t)-2C(\bar{b}+\bar{d})t-2\bar{p}t$, we
deduce that for any $t\leq\theta_{K}$,

${d\widetilde{A}^{K}_{i}(t)\over dt}<\bar{p}e^{\varepsilon_{K}\log
K}\sum_{\ell=-\lfloor 1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}G(h_{K}\ell)\exp(\log
K(\widetilde{A}^{K}_{i+\ell}(t)-\widetilde{A}^{K}_{i}(t))-2\bar{p}.$ (3.10)

Let us introduce

$(i_{K},t_{K})=(i_{K}(\omega),t_{K}(\omega))=\text{argmax}_{{i\in\\{0,\cdots,{1\over\delta_{K}}-1\\},t\in[0,\theta_{K}(\omega)]}}\widetilde{A}^{K}_{i}(t).$

We can prove that $t_{K}=0.$ Indeed, if conversely we assume that $t_{K}>0$,
then the right term of (3.10), for $K$ large enough, is non positive for
$i=i_{K}$ and $t=t_{K}$ and then the left term is negative, contradicting the
fact that $\widetilde{A}^{K}_{i_{K}}(t)$ is maximal for $t=t_{K}$. Hence, we
have proved that for $K$ large enough, almost surely on the event
$\Omega_{K}$, for all $t\leq\theta_{K}$ and $0\leq i\leq 1/\delta_{K}-1$,

$\displaystyle
A^{K}_{i}(t)=\widetilde{A}^{K}_{i}(t)+2C(\bar{b}+\bar{d})t+2\bar{p}t\leq$
$\displaystyle\max_{0\leq j\leq
1/\delta_{K}}\widetilde{A}^{K}_{j}(0)+2C(\bar{b}+\bar{d})t+2\bar{p}t$
$\displaystyle=$ $\displaystyle\max_{0\leq j\leq
1/\delta_{K}}\beta^{K}_{j}(0)+2C(\bar{b}+\bar{d})t+2\bar{p}t.$

∎

The last result has a consequence that will be useful to prove the tightness
of $\widetilde{\beta}^{K}$ in Section 3.4.

######  Corollary 3.5.

For all $T>0$, there exists $C(T)$ such that,

$\lim_{K\to+\infty}\mathbb{P}\left(\sup_{0\leq i\leq
1/\delta_{K}-1}\sup_{t\in[0,T\wedge\theta_{K}(L)]}\beta^{K}_{i}(t)\geq
C(T)\right)=0.$

###### Proof.

We use the semimartingale decomposition (2.13) of $\beta^{K}_{i}$, Proposition
3.4, (3.5) with $A=1$ and Corollary 3.3 to deduce that, for all $t\leq T$ and
$K$ large enough,

$|\beta^{K}_{i}(t\wedge\theta_{K})|\leq\sup_{0\leq j\leq
1/\delta_{K}-1}|\beta^{K}_{j}(0)|+C_{1}T+1$

with probability at least $1-\frac{C\overline{G}(L)T}{\delta_{K}K^{a}\log
K}-\frac{C\sqrt{\overline{G}(L)T}}{\delta_{K}(K^{a}\log K)^{1/4}}$. Since
$\widetilde{\beta}^{K}_{i}(0)$ converges in probability to $\beta_{0}$,
$\mathbb{P}(\sup_{i}\beta^{K}_{i}(0)\geq\|\beta_{0}\|_{\infty}+1)$ converges
to $0$ when $K$ goes to $+\infty$. Hence the result follows with
$C(T)=\|\beta_{0}\|_{\infty}+C_{1}T+2$. ∎

###  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 $T>0$,

$\lim_{K\to+\infty}\mathbb{P}(\tau^{\prime}_{K}\geq T)=1.$

###### 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 $b(i\delta_{K})$, death rate
$d(i\delta_{K})$ and initial condition $K^{a+\varepsilon}$ for
$\varepsilon=a_{1}-a>0$. In addition, the processes $(Z_{i}^{K}(t))_{t}$ for
$0\leq i\leq{1\over\delta_{K}}-1$ are independent. Let us define

$\theta^{\prime\prime}_{K}=\inf\Big{\\{}t\geq 0,\exists
i\in\\{0,1,\cdots,{1\over\delta_{K}}-1\\};Z^{K}_{i}(t\log K)<K^{a}\Big{\\}}.$

In order to prove that $\ \lim_{K\to\infty}\mathbb{P}(\tau^{\prime}_{K}>T)=1$,
it is enough to prove that

$\ \lim_{K\to\infty}\mathbb{P}(\theta^{\prime\prime}_{K}=+\infty)=1.$

We have

$\displaystyle\mathbb{P}(\theta^{\prime\prime}_{K}=+\infty)$ $\displaystyle=$
$\displaystyle\mathbb{P}\Big{(}\forall
i\in\\{0,1,\cdots,{1\over\delta_{K}}-1\\},\ \forall t\geq 0\ Z^{K}_{i}(t\log
K)>K^{a}\Big{)}$ $\displaystyle=$
$\displaystyle\prod_{i=0}^{{1\over\delta_{K}}-1}\mathbb{P}\Big{(}\inf_{t\geq
0}Z^{K}_{i}(t\log K)>K^{a}\Big{)}.$

Fix $i\in\\{0,\ldots,{1\over\delta_{K}}-1\\}$. It is usual to prove (by time
change) that the probability $\mathbb{P}\big{(}\inf_{t\geq 0}Z^{K}_{i}(t\log
K)>K^{a}\big{)}$ is equal to the probability for a random walk
$M({b(i\delta_{K})\over b(i\delta_{K})+d(i\delta_{K})},{d(i\delta_{K})\over
b(i\delta_{K})+d(i\delta_{K})})$ on $\mathbb{Z}_{+}$ (adding $+1$ with
probability ${b(i\delta_{K})\over b(i\delta_{K})+d(i\delta_{K})}$ and $-1$
with probability ${d(i\delta_{K})\over b(i\delta_{K})+d(i\delta_{K})}$) with
initial value $K^{a+\varepsilon}$ never to attain $K^{a}$. This quantity is
well known and equal to

$1-\Big{(}{d(i\delta_{K})\over
b(i\delta_{K})}\Big{)}^{K^{a+\varepsilon}-K^{a}}.$

Since $\alpha=\max_{x\in\mathbb{T}}d(x)/b(x)<1$, it follows from (2.4) that

$\displaystyle\mathbb{P}(\theta^{\prime\prime}_{K}=+\infty)$
$\displaystyle\geq$
$\displaystyle\exp\bigg{(}{1\over\delta_{K}}\log\big{(}1-\alpha^{K^{a+\varepsilon}-K^{a}}\big{)}\bigg{)}$
$\displaystyle\sim$
$\displaystyle\exp\bigg{(}-{1\over\delta_{K}}\alpha^{K^{a+\varepsilon}-K^{a}}\bigg{)}$
$\displaystyle\gg$ $\displaystyle
1-K^{{a_{2}}/4}\alpha^{K^{a+\varepsilon}-K^{a}},$

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 $L_{0}$ in the
definition (2.19) of $\tau_{K}(L)$ such that

$\lim_{K\rightarrow+\infty}\mathbb{P}(\tau_{K}(L_{0})>T)=1.$

###### Proof of Proposition 3.7.

For $i\in\\{0,\ldots,1/\delta_{K}-1\\}$, let us consider the increments

$\Delta_{K}\beta^{K}_{i}(t\wedge\theta_{K})=\frac{\beta_{i+1}^{K}(t\wedge\theta_{K})-\beta_{i}^{K}(t\wedge\theta_{K})}{\delta_{K}}=\Delta_{K}M^{K}_{i}(t\wedge\theta_{K})+\Delta_{K}A^{K}_{i}(t\wedge\theta_{K}).$

Our aim is to prove that
$\Delta_{K}\beta^{K}_{i}(t)-\Delta_{K}\beta^{K}_{i}(s)$ is close to its finite
variation part for large $K$ and to apply to the latter an almost sure maximum
principle.

Let us introduce

$g^{K}_{i}(t)=\Delta_{K}A^{K}_{i}(t\wedge\theta_{K})+\frac{\|p\|_{\text{Lip}}}{\underline{p}}A^{K}_{i+1}(t\wedge\theta_{K}),$
(3.11)

Using (2.14), we obtain for $K$ large enough and $0\leq s<t\leq T$,

$\displaystyle g^{K}_{i}(t)-g^{K}_{i}(s)$
$\displaystyle=\frac{1}{h_{K}}\int_{(s\wedge\theta_{K})\log
K}^{(t\wedge\theta_{K})\log
K}\left[b((i+1)\delta_{K})N^{K}_{i+1}(u)\log\left(1+{1\over
N^{K}_{i+1}(u)}\right)-b(i\delta_{K})N^{K}_{i}(u)\log\left(1+{1\over
N^{K}_{i}(u)}\right)\right]du$
$\displaystyle+\frac{1}{h_{K}}\int_{(s\wedge\theta_{K})\log
K}^{(t\wedge\theta_{K})\log
K}\left[d((i+1)\delta_{K})N^{K}_{i+1}(u)\log\left(1-{1\over
N^{K}_{i+1}(u)}\right)-d(i\delta_{K})N^{K}_{i}(u)\log\left(1-{1\over
N^{K}_{i}(u)}\right)\right]du$
$\displaystyle+\frac{1}{h_{K}}\int_{(s\wedge\theta_{K})\log
K}^{(t\wedge\theta_{K})\log K}\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}G(h_{K}\ell)$ $\displaystyle\hskip
28.45274pt\left[p((i+1+\ell)\delta_{K})N^{K}_{i+1+\ell}(u)\log\left(1+{1\over
N^{K}_{i+1}(u)}\right)-p((i+\ell)\delta_{K})N^{K}_{i+\ell}(u)\log\left(1+{1\over
N^{K}_{i}(u)}\right))\right]du$
$\displaystyle+\frac{\|p\|_{\text{Lip}}}{\underline{p}\log
K}\int_{(s\wedge\theta_{K})\log K}^{(t\wedge\theta_{K})\log
K}\left[b((i+1)\delta_{K})N^{K}_{i+1}(u)\log\left(1+{1\over
N^{K}_{i+1}(u)}\right)\right.$ $\displaystyle\hskip
113.81102pt+\left.d((i+1)\delta_{K})N^{K}_{i+1}(u)\log\left(1-{1\over
N^{K}_{i+1}(u)}\right)\right]du$
$\displaystyle+\frac{\|p\|_{\text{Lip}}}{\underline{p}\log
K}\int_{(s\wedge\theta_{K})\log K}^{(t\wedge\theta_{K})\log
K}\sum_{\ell=-\lfloor 1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}G(h_{K}\ell)p((i+1+\ell)\delta_{K})N^{K}_{i+1+\ell}(u)\log\left(1+{1\over
N^{K}_{i+1}(u)}\right)du$
$\displaystyle\leq\frac{C(\bar{b}+\bar{d})}{h_{K}}\int_{(s\wedge\theta_{K})\log
K}^{(t\wedge\theta_{K})\log
K}\left[\frac{1}{N^{K}_{i+1}(u)}+\frac{1}{N^{K}_{i}(u)}\right]du+(C_{1}(\bar{b}+\bar{d})+\|b\|_{\text{Lip}}+\|d\|_{\text{Lip}})(t\wedge\theta_{K}-s\wedge\theta_{K})$
$\displaystyle+\frac{1}{h_{K}}\int_{(s\wedge\theta_{K})\log
K}^{(t\wedge\theta_{K})\log K}\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((\ell+i)\delta_{K})G(h_{K}\ell)\left[N_{\ell+i+1}^{K}(u)\log\left(1+\frac{1}{N_{i+1}^{K}(u)}\right)\right.$
$\displaystyle\hskip
170.71652pt\left.-N_{\ell+i}^{K}(u)\log\left(1+\frac{1}{N_{i}^{K}(u)}\right)\right]du$
$\displaystyle+\frac{3\|p\|_{\text{Lip}}}{\underline{p}\log
K}\int_{(s\wedge\theta_{K})\log K}^{(t\wedge\theta_{K})\log
K}\sum_{\ell=-\lfloor 1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((\ell+i)\delta_{K})G(h_{K}\ell)N_{\ell+i+1}^{K}(u)\log\left(1+\frac{1}{N_{i+1}^{K}(u)}\right)du,$
(3.12)

where we used that for all $x$ such that $|x|\leq 1/2$ we have

$\displaystyle\left|\frac{1}{x}\log(1+x)\right|\leq C,$ (3.13)
$\displaystyle\left|\frac{1}{x}\log(1+x)-\frac{1}{y}\log(1+y)\right|\leq
C(|x|+|y|).$ (3.14)

and the fact that (recalling the convention (2.15) and that $p$ is periodic)

$\displaystyle\left|p((\ell+i+1)\delta_{K})-p((\ell+i)\delta_{K})\right|=$
$\displaystyle
p((\ell+i)\delta_{K})\frac{\left|p((\ell+i+1)\delta_{K})-p((\ell+i)\delta_{K})\right|}{p((\ell+i)\delta_{K})}$
$\displaystyle\leq$
$\displaystyle\frac{\|p\|_{\text{Lip}}\delta_{K}}{\underline{p}}p((\ell+i)\delta_{K})$
(3.15)

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

$\frac{\|p\|_{\text{Lip}}\delta_{K}}{\underline{p}}\leq 1,$

so that

$p((\ell+i+1)\delta_{K})\leq 2p((\ell+i)\delta_{K}).$

Next, notice that for all $x^{\prime},y^{\prime},x,y$ such that $|x|,|y|\leq
1/2$,

$\displaystyle{1\over y^{\prime}}\log(1+{y})-{1\over
x^{\prime}}\log(1+{x})\leq{y\over y^{\prime}}-{x\over
x^{\prime}}+C\Big{(}{y^{2}\over y^{\prime}}+{x^{2}\over x^{\prime}}\Big{)}.$
(3.16)

Using this inequality and (3.2), we have

$\displaystyle\frac{1}{h_{K}}$ $\displaystyle\int_{(s\wedge\theta_{K})\log
K}^{(t\wedge\theta_{K})\log K}\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((\ell+i)\delta_{K})G(h_{K}\ell)\left[N_{\ell+i+1}^{K}(u)\log\left(1+\frac{1}{N_{i+1}^{K}(u)}\right)\right.$
$\displaystyle\hskip
170.71652pt\left.-N_{\ell+i}^{K}(u)\log\left(1+\frac{1}{N_{i}^{K}(u)}\right)\right]du$
$\displaystyle\leq\frac{1}{h_{K}}\int_{(s\wedge\theta_{K})\log
K}^{(t\wedge\theta_{K})\log K}\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((\ell+i)\delta_{K})G(h_{K}\ell)\left[\frac{N_{\ell+i+1}^{K}(u)}{N_{i+1}^{K}(u)}-\frac{N_{\ell+i}^{K}(u)}{N_{i}^{K}(u)}\right]du$
$\displaystyle+\frac{C\bar{p}}{h_{K}}\int_{(s\wedge\theta_{K})\log
K}^{(t\wedge\theta_{K})\log K}\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}G(h_{K}\ell)e^{Lh_{K}|\ell|}\left(\frac{1}{N_{i+1}^{K}(u)}+\frac{1}{N_{i}^{K}(u)}\right)du.$

Therefore, using (2.10) and the definition of $\tau^{\prime}_{K}$, we have
proved that

$\displaystyle g^{K}_{i}(t)-$ $\displaystyle
g^{K}_{i}(s)\leq\left(\frac{C\bar{G}(L)\log K}{K^{a}\ h_{K}}+C\right)(t-s)$
$\displaystyle+\frac{1}{h_{K}}\int_{(s\wedge\theta_{K})\log
K}^{(t\wedge\theta_{K})\log K}\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((\ell+i)\delta_{K})G(h_{K}\ell)\left[\frac{N_{\ell+i+1}^{K}(u)}{N_{i+1}^{K}(u)}-\frac{N_{\ell+i}^{K}(u)}{N_{i}^{K}(u)}\right]du$
$\displaystyle+\frac{{3}\|p\|_{\text{Lip}}}{\underline{p}\log
K}\int_{(s\wedge\theta_{K})\log K}^{(t\wedge\theta_{K})\log
K}\sum_{\ell=-\lfloor 1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((\ell+i)\delta_{K})G(h_{K}\ell)\frac{N_{\ell+i+1}^{K}(u)}{N_{i+1}^{K}(u)}du.$

Using that for any real numbers $\lambda,\alpha$, $e^{\lambda}\leq
e^{\alpha}+e^{\lambda}(\lambda-\alpha)$ and then (2.13), we have

$\displaystyle\frac{1}{h_{K}}\int_{(s\wedge\theta_{K})\log
K}^{(t\wedge\theta_{K})\log K}\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((\ell+i)\delta_{K})G(h_{K}\ell)\left[\frac{N_{\ell+i+1}^{K}(u)}{N_{i+1}^{K}(u)}-\frac{N_{\ell+i}^{K}(u)}{N_{i}^{K}(u)}\right]du$
$\displaystyle\leq\frac{\log K}{h_{K}}\int_{(s\wedge\theta_{K})\log
K}^{(t\wedge\theta_{K})\log K}\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((\ell+i)\delta_{K})G(h_{K}\ell)$
$\displaystyle\qquad\qquad\left(\beta^{K}_{\ell+i+1}\big{(}\frac{u}{\log
K}\big{)}-\beta^{K}_{\ell+i}\big{(}\frac{u}{\log
K}\big{)}-(\beta^{K}_{i+1}\big{(}\frac{u}{\log
K}\big{)}-\beta^{K}_{i}\big{(}\frac{u}{\log
K}\big{)})\right)\frac{N^{K}_{\ell+i+1}(u)}{N^{K}_{i+1}(u)}du$
$\displaystyle\leq\int_{(s\wedge\theta_{K})\log K}^{(t\wedge\theta_{K})\log
K}\sum_{\ell=-\lfloor 1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((\ell+i)\delta_{K})G(h_{K}\ell)$
$\displaystyle\qquad\qquad\left(\Delta_{K}\beta^{K}_{\ell+i}\big{(}\frac{u}{\log
K}\big{)}-\Delta_{K}\beta^{K}_{i}\big{(}\frac{u}{\log
K}\big{)}\right)\frac{N^{K}_{\ell+i+1}(u)}{N^{K}_{i+1}(u)}du$
$\displaystyle\leq\int_{(s\wedge\theta_{K})\log K}^{(t\wedge\theta_{K})\log
K}\sum_{\ell=-\lfloor 1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((\ell+i)\delta_{K})G(h_{K}\ell)$
$\displaystyle\qquad\left(\Delta_{K}M^{K}_{\ell+i}\big{(}\frac{u}{\log
K}\big{)}-\Delta_{K}M^{K}_{i}\big{(}\frac{u}{\log
K}\big{)}+\Delta_{K}A^{K}_{\ell+i}\big{(}\frac{u}{\log
K}\big{)}-\Delta_{K}A^{K}_{i}\big{(}\frac{u}{\log
K}\big{)}\right)\frac{N^{K}_{\ell+i+1}(u)}{N^{K}_{i+1}(u)}du.$

Thus, using (2.4) and (3.11), we deduce that

$g^{K}_{i}(t)-g^{K}_{i}(s)\\\ -\int_{(s\wedge\theta_{K})\log
K}^{(t\wedge\theta_{K})\log K}\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((\ell+i)\delta_{K})G(h_{K}\ell)\left(g_{\ell+i}\big{(}\frac{u}{\log
K}\big{)}-g_{i}\big{(}\frac{u}{\log
K}\big{)}\right)\frac{N^{K}_{\ell+i+1}(u)}{N^{K}_{i+1}(u)}du\\\
\begin{aligned} &\leq
C_{0}(K,L)(t-s)+\frac{{3}\|p\|_{\text{Lip}}}{\underline{p}\log
K}\int_{(s\wedge\theta_{K})\log K}^{(t\wedge\theta_{K})\log
K}\sum_{\ell=-\lfloor 1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((\ell+i)\delta_{K})G(h_{K}\ell)\frac{N_{\ell+i+1}^{K}(u)}{N_{i+1}^{K}(u)}du\\\
&-\frac{\|p\|_{\text{Lip}}}{\underline{p}}\int_{(s\wedge\theta_{K})\log
K}^{(t\wedge\theta_{K})\log K}\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((\ell+i)\delta_{K})G(h_{K}\ell)\\\ &\hskip
199.16928pt\left(\beta_{\ell+i+1}\big{(}\frac{u}{\log
K}\big{)}-\beta_{i+1}\big{(}\frac{u}{\log
K}\big{)}\right)\frac{N^{K}_{\ell+i+1}(u)}{N^{K}_{i+1}(u)}du\\\
&+\frac{\|p\|_{\text{Lip}}}{\underline{p}}\int_{(s\wedge\theta_{K})\log
K}^{(t\wedge\theta_{K})\log K}\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((\ell+i)\delta_{K})G(h_{K}\ell)\\\ &\hskip
199.16928pt\left(M_{\ell+i+1}\big{(}\frac{u}{\log
K}\big{)}-M_{i+1}\big{(}\frac{u}{\log
K}\big{)}\right)\frac{N^{K}_{\ell+i+1}(u)}{N^{K}_{i+1}(u)}du\\\
&+\int_{(s\wedge\theta_{K})\log K}^{(t\wedge\theta_{K})\log
K}\sum_{\ell=-\lfloor 1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((\ell+i)\delta_{K})G(h_{K}\ell)\\\ &\hskip
199.16928pt\left(\Delta_{K}M^{K}_{\ell+i}\big{(}\frac{u}{\log
K}\big{)}-\Delta_{K}M^{K}_{i}\big{(}\frac{u}{\log
K}\big{)}\right)\frac{N^{K}_{\ell+i+1}(u)}{N^{K}_{i+1}(u)}du,\end{aligned}$

where

$C_{0}(K,L)=C+\frac{C\bar{G}(L)}{\delta_{K}K^{a}}.$

Now, using Corollary 3.3 and (3.2), on the event $\Omega_{K}(L)$,

$\displaystyle\int_{(s\wedge\theta_{K})\log K}^{(t\wedge\theta_{K})\log
K}\sum_{\ell=-\lfloor 1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((\ell+i)\delta_{K})G(h_{K}\ell)$
$\displaystyle\hskip
199.16928pt\left|\Delta_{K}M^{K}_{\ell+i}\big{(}\frac{u}{\log
K}\big{)}-\Delta_{K}M^{K}_{i}\big{(}\frac{u}{\log
K}\big{)}\right|\frac{N^{K}_{\ell+i+1}(u)}{N^{K}_{i+1}(u)}du$
$\displaystyle\leq 2\bar{p}\log
K(t-s)\varepsilon_{K}\sum_{\ell\in\mathbb{Z}}h_{K}G(k_{k}\ell)e^{Lh_{K}|\ell|}\leq
2\bar{p}\log K\varepsilon_{K}\bar{G}(L)(t-s).$

Similarly, using (3.7), on the event $\Omega_{K}(L)$,

$\int_{(s\wedge\theta_{K})\log K}^{(t\wedge\theta_{K})\log
K}\sum_{\ell=-\lfloor 1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((\ell+i)\delta_{K})G(h_{K}\ell)\\\
\times\left|M_{\ell+i+1}\big{(}\frac{u}{\log
K}\big{)}-M_{i+1}\big{(}\frac{u}{\log
K}\big{)}\right|\frac{N^{K}_{\ell+i+1}(u)}{N^{K}_{i+1}(u)}du\leq\bar{p}\log
K\varepsilon_{K}\bar{G}(L)(t-s).$

To conclude, we use the inequality $e^{x}(3-x)\leq e^{2}$ for all
$x\in\mathbb{R}$ to deduce that

$\frac{N^{K}_{\ell+i+1}(u)}{N^{K}_{i+1}(u)}\left[{3}-\log
K\left(\beta_{\ell+i+1}\big{(}\frac{u}{\log
K}\big{)}-\beta_{i+1}\big{(}\frac{u}{\log K}\big{)}\right)\right]\leq{e^{2}}.$

Combining the four previous inequalities, we deduce that

$g^{K}_{i}(t)-g^{K}_{i}(s)\leq C(K,L)(t-s)\\\ +\bar{p}\log
K\int_{(s\wedge\theta_{K})}^{(t\wedge\theta_{K})}\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}G(h_{K}\ell)\left[g^{K}_{\ell+i}(v)-g^{K}_{i}(v)\right]^{+}\,e^{h_{K}L|\ell|}dv,$

where $[x]^{+}=x\vee 0$ is the positive part of $x$ and

$C(K,L)=C+\frac{C\bar{G}(L)}{\delta_{K}K^{a}}+\left(2+\frac{\|p\|_{\text{Lip}}}{\underline{p}}\right)\bar{p}\log
K\varepsilon_{K}\bar{G}(L).$

Thus

${dg^{K}_{i}(t)\over dt}\leq C(K,L)+\bar{p}\log K\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}G(h_{K}\ell)\left[g^{K}_{\ell+i}(t)-g^{K}_{i}(t)\right]^{+}\,e^{h_{K}L|\ell|}.$

We will now use the maximum principle for $\omega\in\Omega_{K}(L)$ fixed.
Defining $\widetilde{g}^{K}_{i}(t)=g^{K}_{i}(t)-2C(K,L)t$, we deduce that for
any $t\leq\theta_{K}$,

${d\widetilde{g}^{K}_{i}(t)\over dt}<\bar{p}\log K\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}G(h_{K}\ell)\left[\widetilde{g}^{K}_{\ell+i}(t)-\widetilde{g}^{K}_{i}(t)\right]^{+}\,e^{h_{K}L|\ell|}.$
(3.17)

Let us introduce

$(i_{K},t_{K})=(i_{K}(\omega),t_{K}(\omega))=\text{argmax}_{{i\in\\{0,\cdots,{1\over\delta_{K}}-1\\},t\in[0,\theta_{K}(\omega)]}}\widetilde{g}^{K}_{i}(t)$

and let us prove that

$t_{K}=0.$

By contradiction, if we assume that $t_{K}>0$, then the right term of (3.17)
is non-positive for $i=i_{K}$ and then the left term is negative,
contradicting the fact that $\widetilde{g}^{K}_{i_{K}}(t)$ is maximal for
$t=t_{K}$. Hence, we have proved that, almost surely on the event
$\Omega_{K}(L)$, for all $t\leq\theta_{K}$ and $0\leq i\leq 1/\delta_{K}-1$,

$g^{K}_{i}(t)=\widetilde{g}^{K}_{i}(t)+2C(K,L)t\leq\max_{0\leq j\leq
1/\delta_{K}-1}\widetilde{g}^{K}_{j}(0)+2C(K,L)t=\max_{0\leq j\leq
1/\delta_{K}-1}g^{K}_{j}(0)+2C(K,L)t,$

so that, by Proposition 3.4,

$\displaystyle\Delta_{K}\beta_{i}^{K}(t\wedge\theta_{K})$
$\displaystyle=g_{i}^{K}(t)-\frac{\|p\|_{\text{Lip}}}{\underline{p}}A_{i+1}^{K}(t\wedge\theta_{K})+\Delta_{K}M_{i}^{K}(t\wedge\theta_{K})$
$\displaystyle\leq\max_{0\leq j\leq
1/\delta_{K}}g^{K}_{j}(0)+2C(K,L)t+\frac{\|p\|_{\text{Lip}}}{\underline{p}}\left(\max_{0\leq
j\leq 1/\delta_{K}-1}\beta^{K}_{i}(0)+C_{1}t\right)+\varepsilon_{K}.$

A similar argument applied to
$(\beta^{K}_{i}(t\wedge\theta_{K})-\beta^{K}_{i-1}(t\wedge\theta_{K}))/\delta_{K}$
gives the converse inequality, so we finally obtain that there exists a
constant $C$ independent of $K$, $i$, $t$ and $L$ such that, almost surely on
the event $\Omega_{K}(L)$, for all $i\in\\{0,\ldots,1/\delta_{K}-1\\}$ and
$t\leq T$,

$|\Delta_{K}\beta_{i}^{K}(t\wedge\theta_{K})|\leq C\left[\max_{0\leq j\leq
1/\delta_{K}-1}\left(|\Delta_{K}\beta^{K}_{i}(0)|+\beta^{K}_{i}(0)\right)+1+T+\bar{G}(L)T\left(\frac{1}{\delta_{K}K^{a}}+\varepsilon_{K}\log
K\right)\right],$

Finally, defining $\widetilde{\Omega}_{K}$ as the event of probability
converging to $1$ where $\max_{0\leq j\leq
1/\delta_{K}-1}|\Delta_{K}\beta^{K}_{i}(0)|+\beta^{K}_{i}(0)\leq
A+\|\beta_{0}\|_{\infty}+1$, where the constant $A$ comes from Assumption 2.2,
on the event $\Omega_{K}(L)\cap\widetilde{\Omega}_{K}$, we have

$|\Delta_{K}\beta_{i}^{K}(t\wedge\theta_{K})|\leq
C\left[A+\|\beta_{0}\|_{\infty}+2+T+\bar{G}(L)T\left(\frac{1}{\delta_{K}K^{a}}+\varepsilon_{K}\log
K\right)\right].$

To conclude the proof of Proposition 3.7, we first fix $T>0$, set

$L_{0}=C(A+\|\beta_{0}\|_{\infty}+3+T)$ (3.18)

and choose $K$ large enough such that
$C\,T\,\bar{G}(L)\left(\frac{1}{\delta_{K}K^{a}}+\varepsilon_{K}\log
K\right)<1$. Then

$\mathbb{P}(\tau_{K}(L_{0})>T)\geq\mathbb{P}(\Omega_{K}(L_{0})\cap\widetilde{\Omega}_{K})\xrightarrow[K\to\infty]{}1.$

Hence Proposition 3.7 is proved. ∎

###  3.4 Proof of Theorem 2.4: tightness of $\tilde{\beta}^{K}$

Our goal is to prove that the sequence of laws of
$(\tilde{\beta}^{K}_{t},t\in[0,T])_{K}$ is tight in ${\cal
P}(\mathbb{D}([0,T],\mathcal{C}(\mathbb{T},\mathbb{R})))$. We will see in
Corollary 3.8 that it is actually $C$-tight.

Let us recall that the random functions $\
\tilde{\beta}^{K}\in\mathbb{D}([0,T],\mathcal{C}(\mathbb{T},\mathbb{R}))$ is
defined in (2.11) as follows. For all $x\in\mathbb{T}$, let
$i\in\\{0,\ldots,1/\delta_{K}-1\\}$ be such that
$x\in[i\delta_{K},(i+1)\delta_{K})$, and set

$\widetilde{\beta}^{K}_{t}(x):=\widetilde{\beta}^{K}(t,x)=\beta_{i}^{K}(t)\Big{(}1-\frac{x}{\delta_{K}}+i\Big{)}+\beta_{i+1}^{K}(t)\Big{(}\frac{x}{\delta_{K}}-i\Big{)},$

where, by convention, $\beta^{K}_{1/\delta_{K}}(t)=\beta^{K}_{0}(t)$.

Let us recall that the proof of Theorem 2.4 is based on the criterion of
Jakubovski [18] recalled in Section 3. Our goal is to prove Conditions (i) and
(ii) therein.

Let us first prove (i). By Ascoli’s Theorem, we know that a compact set
$K_{\varepsilon}$ is a set of equi-continuous and equi-bounded functions. By
Corollary 3.5 and Proposition 3.7, we have, on an event of probability
converging to $1$ when $K$ tends to infinity, that, for all $x\in\mathbb{T}$
and all $t\in[0,T]$,

$\widetilde{\beta}^{K}_{t}(x)=(x-i\delta_{K})\Delta_{K}\beta^{K}_{i}(t)+\beta_{i}^{K}(t)\leq
L\delta_{K}+C(T),$

so the sequence $(\widetilde{\beta}^{K}_{t},t\in[0,T])_{K}$ is equi-bounded.
Furthermore, recall that, by (2.20), for $x,y\in\mathbb{T}$,

$\left|\widetilde{\beta}^{K}_{t}(x)-\widetilde{\beta}^{K}_{t}(y)\right|=\rho(x,y)\sup_{0\leq
j\leq 1/\delta_{K}-1}\left|\Delta_{K}\beta^{K}_{j}(t)\right|\leq L\rho(x,y).$

We deduce that the sequence is equi-continuous and (i) is proved.

Let us now prove (ii), i.e. that for all
$f\in\mathcal{C}(\mathbb{T},\mathbb{R})$, the sequence of laws of the real-
valued processes

$\displaystyle X^{K}(t)$
$\displaystyle=\int_{\mathbb{T}}\widetilde{\beta}^{K}(.,x)f(x)dx$
$\displaystyle=\sum_{i=0}^{1/\delta_{K}-1}\bigg{[}\beta^{K}_{i}(t)\int_{i\delta_{K}}^{(i+1)\delta_{K}}\Big{(}1+i-{x\over\delta_{K}}\Big{)}f(x)dx+\beta^{K}_{i+1}(t)\int_{i\delta_{K}}^{(i+1)\delta_{K}}\Big{(}{x\over\delta_{K}}-i\Big{)}f(x)dx\bigg{]}$

is tight. Recalling that $\beta_{i}^{K}(t)=A^{K}_{i}(t)+M^{K}_{i}(t)$, the
process $X^{K}_{f}$ is a local semi-martingale with Doob-Meyer decomposition
$X^{K}_{f}=A^{K}_{f}+M^{K}_{f}$, where

$A^{K}_{f}(t)=\sum_{i}\bigg{[}A^{K}_{i}(t)\int_{i\delta_{K}}^{(i+1)\delta_{K}}(1+i-{x\over\delta_{K}})f(x)dx+A^{K}_{i+1}(t)\int_{i\delta_{K}}^{(i+1)\delta_{K}}({x\over\delta_{K}}-i)f(x)dx\bigg{]}$

and $M^{K}_{f}$ is defined similarly using $M^{K}_{i}(t)$ instead of
$A^{K}_{i}(t)$.

We use Aldous and Rebolledo criteria (see for example Joffe-Métivier [19]) to
prove the tightnes of the sequence $(X^{K}_{f})$. Let $S$ be a stopping times
for the filtration of the underlying Poisson point measures, a.s. in $[0,T]$.
We need to estimate for $\alpha>0$, the quantity
$\mathbb{P}(|A^{K}_{f}((S+\alpha)\wedge T)-A^{K}_{f}(S)|>\eta)$ for $\eta>0$.
From (2.14), we deduce

$\displaystyle A^{K}_{f}((S+\alpha)\wedge T)-A^{K}_{f}(S)=$
$\displaystyle\sum_{i}\left\\{\Bigg{(}\int_{i\delta_{K}}^{(i+1)\delta_{K}}(1+i-{x\over\delta_{K}})f(x)dx\right)\bigg{[}{1\over\log
K}\int_{S\log K}^{((S+\alpha)\wedge T)\log
K}\Bigg{(}b(i\delta_{K})N^{K}_{i}(s)\log\left(1+{1\over N^{K}_{i}(s)}\right)$
$\displaystyle\hskip 199.16928pt+d(i\delta_{K})N^{K}_{i}(s)\log\left(1-{1\over
N^{K}_{i}(s)}\right)\Bigg{)}ds$ $\displaystyle+{{1\over\log
K}}\,\sum_{\ell=-\lfloor 1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((\ell+i)\delta_{K})G(h_{K}\ell)$
$\displaystyle\hskip 199.16928pt\times\int_{S\log K}^{((S+\alpha)\wedge T)\log
K}N^{K}_{\ell+i}(s)\log\left(1+{1\over N^{K}_{i}(s)}\right)ds\Bigg{]}$
$\displaystyle+\left(\int_{i\delta_{K}}^{(i+1)\delta_{K}}({x\over\delta_{K}}-i)f(x)dx\right)\Bigg{[}{1\over\log
K}\int_{S\log K}^{((S+\alpha)\wedge T)\log
K}\Bigg{(}b((i+1)\delta_{K})N^{K}_{i+1}(s)\log\left(1+{1\over
N^{K}_{i+1}(s)}\right)$ $\displaystyle\hskip
199.16928pt+d((i+1)\delta_{K})N^{K}_{i+1}(s)\log\left(1-{1\over
N^{K}_{i+1}(s)}\right)\Bigg{)}ds$ $\displaystyle+{{1\over\log
K}}\sum_{\ell=-\lfloor 1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((\ell+i+1)\delta_{K})G(h_{K}\ell)$
$\displaystyle\hskip 199.16928pt\times\int_{S\log K}^{((S+\alpha)\wedge T)\log
K}N^{K}_{\ell+i+1}(s)\log\left(1+{1\over
N^{K}_{i+1}(s)}\right)ds\Bigg{]}\Bigg{\\}}.$

Using (3.13) and the definition of $\theta_{K}$, proceeding as in the proof of
Proposition 3.7 we have

$\mathbb{E}(|A^{K}_{f}((S+\alpha)\wedge\theta_{K}\wedge
T)-A^{K}_{f}(S\wedge\theta_{K})|)\\\ \begin{aligned} &\leq{C\over\log
K}\sum_{i=0}^{1/\delta_{K}-1}\int_{i\delta_{K}}^{(i+1)\delta_{K}}|f(x)|dx\Bigg{\\{}2(\bar{b}+\bar{d})\,\alpha\log
K\\\ &\qquad+\sum_{\ell=-\lfloor 1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((\ell+i)\delta_{K})G(h_{K}\ell)\mathbb{E}\Big{(}\int_{(S\wedge\theta_{K})\log
K}^{((S+\alpha)\wedge\theta_{K}\wedge T)\log K}{N^{K}_{\ell+i}(s)\over
N^{K}_{i}(s)}ds\Big{)}\\\ &\qquad+\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((\ell+i+1)\delta_{K})G(h_{K}\ell)\mathbb{E}\Big{(}\int_{(S\wedge\theta_{K})\log
K}^{((S+\alpha)\wedge\theta_{K}\wedge T)\log K}{N^{K}_{\ell+i+1}(s)\over
N^{K}_{i+1}(s)}ds\Big{)}\Bigg{\\}}\\\ &\leq\alpha
C\left[2(\bar{b}+\bar{d})+2\bar{p}\overline{G}(L)\right]\|f\|_{\infty}.\end{aligned}$

By Proposition 3.7, $\theta_{K}>T$ with probability converging to 1, so we
deduce from Markov’s inequality that, for all $\varepsilon>0$ and $\eta>0$,
there exists $\alpha$ such that,

$\limsup_{K\to+\infty}\sup_{S}\mathbb{P}(|A^{K}_{f}((S+\alpha)\wedge
T)-A^{K}_{f}(S)|>\eta)\leq\varepsilon,$

where the supremum is taken over all stopping times $S\leq T$. This is Aldous
criterion for $A^{K}_{f}(t)$.

It remains to to prove a similar property replacing $A^{K}_{f}$ by $\langle
M^{K}_{f}\rangle$. This can be done similarly using (2.17). Computations are
actually simpler by Lemma 3.1. $\Box$

Hence we have proved Theorem 2.4.We prove in the next corollary that the
sequence of laws of $(\tilde{\beta}^{K}_{t},t\in[0,T])_{K}$ is actually
$C$-tight.

######  Corollary 3.8.

The sequence of laws of $(\widetilde{\beta}^{K}_{t},t\in[0,T])_{K}$ is C-tight
in ${\cal P}(\mathbb{D}([0,T],\mathcal{C}(\mathbb{T},\mathbb{R})))$. In
addition, for all $T>0$, given a value of $L$ as in Proposition 3.7, for any
$\beta$ distributed as a limiting value of the laws of
$(\widetilde{\beta}^{K}_{t},t\in[0,T])_{K}$, we have almost surely

$\sup_{t\in[0,T]}\sup_{x,y\in\mathbb{T}\ s.t.\ x\neq
y}\frac{|\beta(t,x)-\beta(t,y)|}{\rho(x,y)}\leq L.$

###### Proof.

Since $|\beta_{i}^{K}(t)-\beta_{i}^{K}(t-)|\leq C/\log K$ for any $K$, $i$ and
$t$, we have

$\lim_{K}\mathbb{P}(\sup_{t\leq
T}\|\widetilde{\beta}^{K}_{t}-\widetilde{\beta}^{K}_{t-}\|_{\infty}>\varepsilon)=0.$

Then, we deduce from Proposition 3.26 in Jacod-Shiryaev p.351 [17] that, for
all $f\in\mathcal{C}(\mathbb{T},\mathbb{R})$, the sequence of laws of
$X_{f}^{K}$ defined in (3.1) is $C$-tight. We proceed by contradiction to
deduce that $\widetilde{\beta}^{K}$ is also $C$-tight: if this is not true,
there exists an event $\Omega_{1}$ of positive probability such that, for all
$\omega\in\Omega_{1}$, there exists $t_{0}(\omega)$, $\alpha(\omega)$ and a
ball $B(\omega)\subset\mathbb{T}$ of positive radius such that, for all $x\in
B$,

$|\widetilde{\beta}^{K}_{t_{0}}(x)-\widetilde{\beta}^{K}_{t_{0}-}(x)|>\alpha.$
(3.19)

Therefore, there exists non-random $\alpha>0$ and $\varepsilon>0$ and
$i\in\\{0,1,\ldots,\lfloor 1/\varepsilon\rfloor-1\\}$ and an event
$\Omega_{2}\subset\Omega_{1}$ of positive probability such that (3.19) holds
true for all $x\in[i\varepsilon,(i+1)\varepsilon]$ and for this non-random
$\alpha$. Now, we define $f_{i}\in\mathcal{C}(\mathbb{T},\mathbb{R})$ with
support in $[i\varepsilon,(i+1)\varepsilon]$ and positive on
$(i\varepsilon,(i+1)\varepsilon)$. Then, for all $\omega\in\Omega_{2}$,

$\liminf_{K\to+\infty}|X_{f_{i}}^{K}(t_{0})-X_{f_{i}}^{K}(t_{0}-)|\geq\alpha\inf_{x\in[(i+1/3)\varepsilon,(i+2/3)\varepsilon]}|f_{i}(x)|>0.$

This is a contradiction with the $C$-tightness of $X^{K}_{f_{i}}$.

We now prove the Lipschitz estimate for $\beta$. Using the Skorohod
representation theorem, we can construct copies $\hat{\beta}^{K}$ of
$\tilde{\beta}^{K}$ and $\hat{\beta}$ of $\beta$ such that $\hat{\beta}^{K}$
converges (up to a subsequence) almost surely for the $L^{\infty}$ norm on
$[0,T]$ to $\hat{\beta}$. We then define

$\hat{\tau}^{K}=\inf_{i\neq
j\in\\{0,\ldots,1/\delta_{K}-1\\}}\inf\left\\{t\geq
0:\frac{|\hat{\beta}^{K}(t,i\delta_{K})-\hat{\beta}^{K}(t,j\delta_{K})|}{\rho(i\delta_{K},j\delta_{K})}>L\right\\}.$

Then $\hat{\tau}_{K}$ is distributed as $\tau_{K}$ in (2.19). It then follows
from Propostion 3.7 that $\hat{\tau}_{K}>T$ with probability converging to 1.
Hence, for all $x\neq y\in\mathbb{T}$, almost surely

$|\hat{\beta}(t,x)-\hat{\beta}(t,y)|=\lim_{K\to+\infty}|\hat{\beta}^{K}(t,x)-\hat{\beta}^{K}(t,y)|\leq
L\,\rho(x,y).$

By continuity of $\hat{\beta}$ we deduce that this property holds, almost
surely, for all $x,y\in\mathbb{T}$. ∎

##  4 Identification of the limit as a viscosity solution of a Hamilton-Jacobi
equation

In the previous section, we obtained the tightness of the laws of
$(\widetilde{\beta}^{K}_{t},t\in[0,T])_{K}$ for all $T>0$. Hence, the sequence
of laws of $(\widetilde{\beta}^{K}_{t},t\in[0,+\infty))_{K}$ admits at least
one limiting value. Our aim is now to identify the limiting path as the unique
viscosity solution of the Hamilton-Jacobi equation

$\frac{\partial}{\partial
t}\beta(t,x)=b(x)-d(x)+p(x)\int_{\mathbb{R}}G(h)e^{h\partial_{x}\beta(t,x)}dh.$

Let $\beta$ be distributed as a limiting value of the laws of
$(\tilde{\beta}^{K}_{t},t\in[0,+\infty))_{K}$. By Corollary 3.8, $\beta$
belongs to $C([0,+\infty)\times\mathbb{T},\mathbb{R})$. In the sequel and with
an abuse of notation, we denote again by
$(\tilde{\beta}^{K}_{t},t\in[0,+\infty))_{K}$ the subsequence that converges
in distribution to $\beta$.

It also follows from Lemma 3.1 that $\widetilde{A}^{K}-\widetilde{\beta}^{K}$
converges in law, and thus in probability, to $0$. Therefore
$(\widetilde{A}^{K}-\widetilde{\beta}^{K},\widetilde{\beta}^{K})$ converges in
law to $(0,\beta)$ and thus $(\widetilde{A}^{K},\widetilde{\beta}^{K})$
converges in law to $(\beta,\beta)$. Using Skorokhod’s representation theorem,
there exist a new probability space, still denoted (by abuse of notation) by
$(\Omega,\mathcal{A},\mathbb{P})$, and random variables still denoted by
$\widetilde{A}^{K},\widetilde{\beta}^{K},\widetilde{M}^{K}$ and $\beta$ on
this space, such that $(\widetilde{A}^{K},\widetilde{\beta}^{K})$ converges
almost surely to $(\beta,\beta)$. Let us denote by $\widetilde{\Omega}_{0}$
the event where the convergence holds.

We also define for the value $L_{0}$ defined in Theorem 2.6

$\widetilde{\Omega}_{K}=\left\\{\omega\in\Omega\ :\
\sup_{x\in\mathbb{T}}\,\sup_{t\in[0,T]}\,|\widetilde{M}^{K}(t,x)|\leq\varepsilon^{\prime}_{K},\
\|\widetilde{\beta}^{K}\|_{\text{Lip}}\leq L_{0}\right\\},$

where $\varepsilon^{\prime}_{K}=\delta_{K}^{-1/2}K^{-a/2}$ converges to 0 by
(2.4). It follows from Lemma 3.1 and Proposition 3.7 that
$\mathbb{P}(\widetilde{\Omega}_{K})\to 1$ when $K\to+\infty$. Hence, the set

$\Omega_{0}:=\widetilde{\Omega}_{0}\cap\limsup_{K\to+\infty}\widetilde{\Omega}_{K}$

has probability 1.

To prove that $\beta$ is a viscosity sub-solution of Equation (2.12), we work
$\omega$ by $\omega$ in $\Omega_{0}$. Let $\omega\in\Omega_{0}$ and $T>0$ and
consider a continuous function $\varphi:[0,T]\times\mathbb{T}$ (depending on
$\omega$) such that $\beta(\omega)-\varphi$ attains a strict global maximum on
$[0,T]\times\mathbb{T}$ at the point $(\bar{t}(\omega),\bar{x}(\omega))$ such
that $\bar{t}(\omega)>0$. We will prove that

$\frac{\partial}{\partial
t}\varphi(\bar{t},\bar{x})\leq{b(\bar{x})-d(\bar{x})}+p(x)\int_{\mathbb{R}}G(h)e^{h\partial_{x}\varphi(\bar{t},\bar{x})}dh.$

Since $\widetilde{A}^{K}(\omega)$ converges in
$L^{\infty}([0,T]\times\mathbb{T})$ to $\beta$, there exists for $K$ large
enough a local maximum of $\widetilde{A}^{K}(\omega)-\varphi$ on
$[0,T]\times\mathbb{T}$ at a point $(t_{K}(\omega),x_{K}(\omega))$ such that
$(t_{K}(\omega),x_{K}(\omega))\to(\bar{t}(\omega),\bar{x}(\omega))$ as
$K\to\infty$. Assume $K$ is large enough so that $t_{K}(\omega)>0$. From now
on, we will omit the dependencies with respect to $\omega\in\Omega_{0}$ to
avoid heavy notation.

Defining $i_{K}\in\\{0,\ldots,1/\delta_{K}-1\\}$ such that
$i_{K}\delta_{K}\leq{x_{K}}<(i_{K}+1)\delta_{K}$, we have

$\displaystyle\frac{\partial}{\partial
t}\widetilde{A}^{K}(t_{K},x_{K})=(1-\frac{x_{K}}{\delta_{K}}+i_{K})\frac{d}{dt}A_{i_{K}}^{K}(t_{K})+(\frac{x_{K}}{\delta_{K}}-i_{K})\frac{d}{dt}A_{i_{K}+1}^{K}(t_{K})$
$\displaystyle=(1-\frac{x_{K}}{\delta_{K}}+i_{K})N_{i_{K}}^{K}(t_{K})\Big{(}b(i_{K}\delta_{K})\log\big{(}1+\frac{1}{N_{i_{K}}^{K}(s)}\big{)}+d(i_{K}\delta_{K})\log\big{(}1-\frac{1}{N_{i_{K}}^{K}(s)}\big{)}\Big{)}$
$\displaystyle+(\frac{x_{K}}{\delta_{K}}-i_{K})N_{i_{K}+1}^{K}(t_{K})\Big{(}b\big{(}(i_{K}+1)\delta_{K}\big{)}\log\big{(}1+\frac{1}{N_{i_{K}+1}^{K}(s)}\big{)}+d\big{(}(i_{K}+1)\delta_{K}\big{)}\log\big{(}1-\frac{1}{N_{i_{K}+1}^{K}(s)}\big{)}\Big{)}$
$\displaystyle+(1-\frac{x_{K}}{\delta_{K}}+i_{K})\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((\ell+i_{K})\delta_{K})G(h_{K}\ell)N_{\ell+i_{K}}^{K}(t_{K})\log\big{(}1+\frac{1}{N_{i_{K}}^{K}(t_{K})}\big{)}$
$\displaystyle+(\frac{x_{K}}{\delta_{K}}-i_{K})\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((\ell+i_{K}+1)\delta_{K})G(h_{K}\ell)N_{\ell+i_{K}+1}^{K}(t_{K})\log\big{(}1+\frac{1}{N_{i_{K}+1}^{K}(t_{K})}\big{)}.$

Using that forall $x\geq-1/2$,

$b\log(1+x)+d\log(1-x)\leq(b-d)x.$ (4.1)

and using Lemma 3.6, we deduce that

$\displaystyle\frac{\partial}{\partial
t}\widetilde{A}^{K}(t_{K},x_{K})\leq(1-\frac{x_{K}}{\delta_{K}}+i_{K})\big{(}b(i_{K}\delta_{K})-d(i_{K}\delta_{K})\big{)}(1+\frac{C}{K^{a}})$
$\displaystyle+(\frac{x_{K}}{\delta_{K}}-i_{K})\big{(}b((i_{K}+1)\delta_{K})-d((i_{K}+1)\delta_{K})\big{)}(1+\frac{C}{K^{a}})$
$\displaystyle+(1-\frac{x_{K}}{\delta_{K}}+i_{K})\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((\ell+i_{K})\delta_{K})G(h_{K}\ell)e^{\log(K)(\beta_{\ell+i_{K}}^{K}(t_{K})-\beta_{i_{K}}^{K}(t_{K}))}(1+\frac{C}{K^{a}})$
$\displaystyle+(\frac{x_{K}}{\delta_{K}}-i_{K})\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}p((\ell+i_{K}+1)\delta_{K})G(h_{K}\ell)e^{\log(K)(\beta_{\ell+i_{K}+1}^{K}(t_{K})-\beta_{i_{K}+1}^{K}(t_{K}))}(1+\frac{C}{K^{a}})$

We next use the fact that $\mu$, $b$ and $d$ are $C^{1}$ functions in
$\mathbb{T}$ to obtain, modifying the constant $C$ if necessary,

$\displaystyle\frac{\partial}{\partial
t}\widetilde{A}^{K}(t_{K},x_{K})\leq\big{(}b(x_{K})-d(x_{K})+C\delta_{K})(1+\frac{C}{K^{a}})$
$\displaystyle+(1-\frac{x_{K}}{\delta_{K}}+i_{K})\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}(p(x_{K})+C\delta_{K}(|\ell|+1))G(h_{K}\ell)e^{\log(K)(\beta_{\ell+i_{K}}^{K}(t_{K})-\beta_{i_{K}}^{K}(t_{K}))}(1+\frac{C}{K^{a}})$
$\displaystyle+(\frac{x_{K}}{\delta_{K}}-i_{K})\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}(p(x_{K})+C\delta_{K}(|\ell|+1))G(h_{K}\ell)e^{\log(K)(\beta_{\ell+i_{K}+1}^{K}(t_{K})-\beta_{i_{K}+1}^{K}(t_{K}))}(1+\frac{C}{K^{a}})$
(4.2)

Since $(t_{K},x_{K})$ is a maximum point of $\widetilde{A}^{K}-\varphi$, we
deduce that

$\displaystyle\beta_{j}^{K}(t_{K})-\beta_{i_{K}}^{K}(t_{K})$
$\displaystyle=\widetilde{\beta}^{K}(t_{K},j\delta_{K})-\widetilde{\beta}^{K}(t_{K},x_{K})-\left(\beta_{i_{K}}^{K}(t_{K})-\widetilde{\beta}^{K}(t_{K},x_{K})\right)$
$\displaystyle\leq\varphi(t_{K},j\delta_{K})-\varphi(t_{K},x_{K})+\widetilde{M}^{K}(t_{K},j\delta_{K})-\widetilde{M}^{K}(t_{K},x_{K})-\left(\beta_{i_{K}}^{K}(t_{K})-\widetilde{\beta}^{K}(t_{K},x_{K})\right)$

and similarly for $\beta_{j+1}^{K}(t_{K})-\beta_{i_{K}+1}^{K}(t_{K})$.
In addition,

$\varphi(t_{K},j\delta_{K})-\varphi(t_{K},x_{K})\leq(j-i_{K})\delta_{K}\partial_{x}\varphi(t_{K},x_{K})+O(|x_{K}-i_{K}\delta_{K}|)+O(|j-i_{K}|^{2}\delta_{K}^{2}).$

Therefore, since $\omega\in\limsup\widetilde{\Omega}_{K}$, there exists a
subsequence in $K$ (still denoted $K$) along which

$\displaystyle\beta_{j}^{K}(t_{K})-\beta_{i_{K}}^{K}(t_{K})$
$\displaystyle\leq(j-i_{K})\delta_{K}\partial_{x}\varphi(x_{K})+C\left(|x_{K}-i_{K}\delta_{K}|+|j-i_{K}|^{2}\delta_{K}^{2}+\varepsilon_{K}\right)$
$\displaystyle\leq(j-i_{K})\delta_{K}\partial_{x}\varphi(x_{K})+C(|j-i_{K}|^{2}\delta_{K}^{2}+\delta_{K}+\varepsilon_{K})$

and

$\beta_{j+1}^{K}(t_{K})-\beta_{i_{K}+1}^{K}(t_{K})\leq(j-i_{K})\delta_{K}\partial_{x}\varphi(t_{K},x_{K})+C(|j-i_{K}|^{2}\delta_{K}^{2}+\delta_{K}+\varepsilon_{K}).$

Combining these inequalities with (4.2), we obtain

$\frac{\partial}{\partial t}\varphi(t_{K},x_{K})\leq\frac{\partial}{\partial
t}\widetilde{A}^{K}(t_{K},x_{K})\leq\big{(}b(x_{K})-d(x_{K})+C\delta_{K})(1+\frac{C}{K^{a}})\\\
+\sum_{\ell=-\lfloor 1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}(p(x_{K})+C\delta_{K}(|\ell|+1))G(h_{K}\ell)e^{h_{K}\ell\partial_{x}\varphi(t_{K},x_{K})+C\frac{(h_{K}\ell)^{2}}{\log
K}+o(1)}(1+\frac{C}{K^{a}})$ (4.3)

Since $p(x_{K})\to p(\bar{x})$ when $K\to+\infty$, to prove the convergence of
the sum in the right-hand side of (4.3), it is sufficient to study the
convergence of

$\displaystyle S=$ $\displaystyle\sum_{\ell=-\lfloor
1/2\delta_{K}\rfloor}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}G\big{(}h_{K}\ell\big{)}e^{h_{K}\ell\partial_{x}\varphi(t_{K},x_{K})+C\frac{(h_{K}\ell)^{2}}{\log
K}}.$

Recall from Assumption 2.1 that $G$ is continuous and that there exists $R>0$
such that $G$ is nonincreasing on $[R,+\infty)$ and nondecreasing on
$(-\infty,-R]$. We first notice that

$S_{0}=\sum_{\ell=\lfloor-R/h_{K}\rfloor}^{\lfloor
R/h_{K}\rfloor}h_{K}G\big{(}h_{K}\ell\big{)}e^{h_{K}\ell\partial_{x}\varphi(t_{K},x_{K})+C\frac{(h_{K}\ell)^{2}}{\log
K}}$

is a Riemann sum which converges to
$\int_{-R}^{R}G(y)e^{y\partial_{x}\varphi(\bar{t},\bar{x})}dy$. Hence we only
have to deals with the remainder $S-S_{0}$. We detail the analysis for

$S_{+}=\sum_{\ell=\lfloor R/h_{K}\rfloor+1}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}G\big{(}h_{K}\ell\big{)}e^{h_{K}\ell\partial_{x}\varphi(t_{K},x_{K})+C\frac{(h_{K}\ell)^{2}}{\log
K}}.$

A similar computation applies to the lower tail.

For all $\varepsilon>0$, there exists $K_{0}$ such that for $K\geq K_{0}$,
$|\partial_{x}\varphi(t_{K},x_{K})-\partial_{x}\varphi(\bar{t},\bar{x})|\leq\varepsilon$.
Hence, setting $a=\partial_{x}\varphi(\bar{t},\bar{x})$ and recalling that $G$
is nonincreasing on $[R,+\infty)$, for $K$ large enough,

$\displaystyle S_{+}$ $\displaystyle\leq\sum_{\ell=\lfloor
R/h_{K}\rfloor+1}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}h_{K}G\big{(}h_{K}\ell\big{)}e^{ah_{K}\ell+C\frac{(h_{K}\ell)^{2}}{\log
K}+\varepsilon h_{K}|\ell|}$ $\displaystyle\leq\sum_{\ell=\lfloor
R/h_{K}\rfloor+1}^{1/\delta_{K}-1-\lfloor
1/2\delta_{K}\rfloor}\int_{h_{K}(\ell-1)}^{h_{K}\ell}G\big{(}h_{K}\ell\big{)}e^{ah_{K}\ell+C\frac{(h_{K}\ell)^{2}}{\log
K}+\varepsilon h_{K}|\ell|}\ dy$
$\displaystyle\leq\int_{R}^{h_{K}/2\delta_{K}}G(y)e^{ay+|a|h_{K}+C\frac{y^{2}+h_{K}(2|y|+h_{K}))}{\log
K}+\varepsilon(|y|+h_{K})}\ dy$ $\displaystyle\leq\int_{R}^{\log
K}G(y)e^{ay+C\frac{y^{2}+2h_{K}|y|}{\log K}+\varepsilon|y|}(1+\varepsilon)\
dy.$

Observing that, for $|y|\leq(\log K)^{1/3}$, $\frac{y^{2}+2h_{K}|y|}{\log
K}\leq(\log K)^{-1/3}+2\delta_{K}(\log K)^{1/3}\to 0$ when $K\to+\infty$ and
that, for $|y|\leq\log K$, $\frac{y^{2}+2h_{K}|y|}{\log K}\leq y+2h_{K}$, we
can decompose the domain integration as $[R,\log K]=[R,(\log
K)^{1/3})\cup[(\log K)^{1/3},\log K]$ to deduce that, for $K$ large enough,

$\displaystyle S_{+}$ $\displaystyle\leq(1+2\varepsilon)\int_{R}^{(\log
K)^{1/3}}G(y)e^{ay+\varepsilon|y|}dy+(1+2\epsilon)\int_{(\log K)^{1/3}}^{\log
K}G(y)e^{(a+C)y+\varepsilon|y|}dy$
$\displaystyle\leq(1+2\varepsilon)\int_{R}^{+\infty}G(y)e^{ay+\varepsilon|y|}dy+\frac{1+2\epsilon}{e^{(\log
K)^{1/3}}}\int_{\mathbb{R}}G(y)e^{(a+C+2)|y|}dy.$

Now, by dominated convergence,

$\int_{R}^{+\infty}G(y)e^{ay+\varepsilon|y|}dy\xrightarrow[\varepsilon\to
0]{}\int_{R}^{+\infty}G(y)e^{ay}dy.$

To conclude, we have proved that

$\limsup_{K\to+\infty}S\leq\int_{\mathbb{R}}G(y)e^{y\partial_{x}\varphi(\bar{t},\bar{x})}dy.$

Therefore,

$\frac{\partial}{\partial t}\varphi(\bar{t},\bar{x})\leq
b(\bar{x})-d(\bar{x})+p(\bar{x})\int_{\mathbb{R}}G(h)e^{h\partial_{x}\varphi(\bar{t},\bar{x})}dh.$

We conclude that $\beta$ is a viscosity sub-solution of (2.12) in
$(0,T]\times\mathbb{T}$.

Following similar arguments, we can prove that $\beta$ is a viscosity super-
solution, and hence a viscosity solution of (2.12) in $(0,T]\times\mathbb{T}$.
The result then follows from uniqueness of a Lipschitz viscosity solution of
(2.12) [1].

Acknowledgements: This work has been supported by the Chair “Modélisation
Mathématique et Biodiversité” of Veolia Environnement-Ecole Polytechnique-
Museum National d’Histoire Naturelle-Fondation X. S.M. was partially supported
by the ANR project DEEV ANR-20-CE40-0011-01. V.C.T. also acknowledges support
from Labex Bézout (ANR-10-LABX-58).

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