# Convergence of population processes with small and frequent mutations to the canonical equation of adaptive dynamics

Nicolas Champagnat, Vincent Hass

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Nicolas Champagnat, Vincent Hass. Convergence of population processes with small and frequent mutations to the canonical equation of adaptive dynamics. 2023. hal-04034027v3

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# Convergence of population processes with small and frequent mutations to the canonical equation of adaptive dynamics

Nicolas Champagnat, Vincent Hass

► To cite this version:

Nicolas Champagnat, Vincent Hass. Convergence of population processes with small and frequent mutations to the canonical equation of adaptive dynamics. The Annals of Applied Probability, 2025, 35 (1), pp.1-63. 10.1214/24-AAP2103 . hal-04034027v4

HAL Id: hal-04034027

https://hal.science/hal-04034027v4

Submitted on 7 Jul 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Convergence of individual-based models with small and frequent mutations to the canonical equation of adaptive dynamics

NICOLAS CHAMPAGNAT*, VINCENT HASS*<fnref n="1" />

MAY 4, 2023

* Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France

**Abstract.** In this article, a stochastic individual-based model describing Darwinian evolution of asexual, phenotypic trait-structured population, is studied. We consider a large population with constant population size characterised by a resampling rate modeling competition pressure driving selection and a mutation rate where mutations occur during life. In this model, the population state at fixed time is given as a measure on the space of phenotypes and the evolution of the population is described by a continuous time, measure-valued MARKOV process. We investigate the asymptotic behavior of the system, where mutations are frequent, in the double simultaneous limit of large population ($K \to +\infty$) and small mutational effects ($\sigma \to 0$) proving convergence to an ODE known as the canonical equation of adaptive dynamics. This result holds only for a certain range of $\sigma$ parameters (as a function of $K$) which must be small enough but not too small either. The canonical equation describes the evolution in time of the dominant trait in the population driven by a fitness gradient. This result is based on an slow-fast asymptotic analysis. We use an averaging method, inspired by KURTZ [49], which exploits a martingale approach and compactness-uniqueness arguments. The contribution of the fast component, which converges to the centered FLEMING-VIOT process, is obtained by averaging according to its invariant measure, recently characterised in [16].

**Keywords.** Adaptive dynamics, Canonical equation, Individual-based model, Measure-valued MARKOV process, Slow-fast asymptotic analysis, Averaging method, Centered FLEMING-VIOT process.

**MSC subject classification.** Primary 60B10, 60G44, 60G57, 92D10, 92D25, 92D40; Secondary 60F10, 60G10, 60J35, 60J60, 60J68.

## 1 Introduction

In this article we study, at the individual level and in the interplay between ecology and Darwinian evolution, a population model, structured by a 1-dimensional quantitative phenotypic

<footnote n="1">¹Corresponding author at: IECL - Site de Nancy Faculté des Sciences et Technologies Campus, Boulevard des Aiguillettes 54506 Vandœuvre-lès-Nancy, Cedex, France.

*E-mail addresses:* nicolas.champagnat@inria.fr (Nicolas Champagnat), vincent.hass@inria.fr (Vincent Hass).</footnote>
# Convergence of population processes with small and frequent mutations to the canonical equation of adaptive dynamics

NICOLAS CHAMPAGNAT*, VINCENT HASS†

JULY 7, 2024

* Université de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France

† Université de Franche-Comté, IUT Nord Franche-Comté, F-90000 Belfort, France

**Abstract.** In this article, a stochastic individual-based model describing Darwinian evolution of asexual, phenotypic trait-structured population, is studied. We consider a large population with constant population size characterised by a resampling rate modeling competition pressure driving selection and a mutation rate where mutations occur during life. In this model, the population state at fixed time is given as a measure on the space of phenotypes and the evolution of the population is described by a continuous time, measure-valued MARKOV process. We investigate the asymptotic behaviour of the system, where mutations are frequent, in the double simultaneous limit of large population ($K \to +\infty$) and small mutational effects ($\sigma_K \to 0$) proving convergence to an ODE known as the canonical equation of adaptive dynamics. This result holds only for a certain range of $\sigma_K$ parameters (as a function of $K$) which must be small enough but not too small either. The canonical equation describes the evolution in time of the dominant trait in the population driven by a fitness gradient. This result is based on a slow-fast asymptotic analysis. We use an averaging method, inspired by KURTZ [57], which exploits a martingale approach and compactness-uniqueness arguments. The contribution of the fast component, which converges to the centered FLEMING-VIOT process, is obtained by averaging according to its invariant measure, recently characterised in [18].

**Keywords.** Adaptive dynamics, Canonical equation, Individual-based model, Measure-valued MARKOV process, Slow-fast asymptotic analysis, Averaging method, Centered FLEMING-VIOT process.

**MSC subject classification.** Primary 60B10, 60G44, 60G57, 92D10, 92D25, 92D40; Secondary 60F10, 60G10, 60J35, 60J60, 60J68.

# 1 Introduction

In this article we study, at the individual level and in the interplay between ecology and Darwinian evolution, a population model, structured by a 1-dimensional quantitative phenotypic

<footnote n="1">¹Corresponding author at: IUT Nord Franche-Comté, Département Génie Civil et Construction Durable, 19 Avenue du Maréchal Juin, 90 016 Belfort, Cedex, France.</footnote>

<footnote>*E-mail addresses:* nicolas.champagnat@inria.fr (Nicolas Champagnat), vincent.hass@univ-fcomte.fr (Vincent Hass).</footnote>
trait. The Darwinian evolution is based on three basic mechanisms. Firstly, *heredity* which allows the transmission of the individual phenotypic characteristics from one generation to another. Secondly, a source of variation in the individual phenotypic characteristics: in our case it is only *mutations*. Finally, a *selection* mechanism which can result from interaction between individuals in the population such as competition. Our model is an *individual-based model* (in short, IBM) which involves a finite and asexual population with constant population size in which each individual's birth, death and mutation events are described. IBMs were first introduced in ecology as a tool to describe local interactions or complex phenomena at the individual level [65, 11, 12, 29, 45, 24]. Ecological studies using IBMs are mainly numerical and the models are mostly posed in discrete space as systems of interacting particles [65] and more rarely in continuous space [11, 12, 29]. Many IBMs (with non-constant population size) have been proposed and studied in the context of Darwinian evolutionary by the biology community [59, 27, 35] and the mathematical community [14, 15, 5]. Others are dispersal models in spatially structured populations where the trait is viewed as a spatial location and mutations as dispersal [11, 12, 54, 40]. Other models, structured in age, were developed in [19] and studied mathematically in [66, 73], or structured in age and traits in [36, 58, 62].

We consider an IBM with fixed population size, so that births and deaths occur simultaneously in so-called *resampling* (or swap phenomenon) events. The mutation and resampling rate of an individual depends on its phenotype. When a mutation occurs, the new mutant trait is close to its parent's one yielding a slow variation of the trait. In population genetics, the WRIGHT-FISHER model (and its extensions with selection, mutation or immigration), CANNINGS model or the MORAN model are interested in the evolution of allele frequencies according to various mechanisms [32]. In [37, 23, 22, 31], the authors construct the FLEMING-VIOT process as a scaling limit of large population from the MORAN process. Several extensions exist, including frequency-dependent selection, recombination, other reproduction mechanisms [33, 44]. In particular, the last article provides a bridge between population genetics and eco-evolutionary models. Others, based on KERMACK-MCKENDRICK's model, are interested in epidemiological questions [61, 63].

The aim of this article is to describe the evolutionary dynamics of the dominant trait, at the population level, on a long time scale, as a solution to an ordinary differential equation (in short, ODE) called *Canonical Equation of the Adaptive Dynamics* (in short, CEAD). Canonical equations are well-known tools in evolutionary biology, used to predict the evolutionary fate of ecological communities. More precisely, such equations describe the evolution of dominant traits in a biological population as driven by mutations and a fitness gradient which describes the strength of selection that pushes the population to locally increase its fitness [76, 53, 72, 26].

Fitness measures the selective value of a given individual in a given environment including the population under consideration itself. This individual can be any (fictitious) mutant individual that can be born in the population at the time under consideration. The way to construct a fitness landscape depends on the ecological context [60]. For continuous time homogeneous MARKOV models as studied below, it is the instantaneous growth rate (birth rate minus death rate) of the individual considered in the environment considered. If we further assume that the population constituting the environment is in a stationary state, then we can consider that the fitness of a given mutant individual in this population governs the possibility of invasion of the descendants of this mutant in the population.

The canonical equation has been studied and derived in different contexts such as game
trait. The Darwinian evolution is based on three basic mechanisms. Firstly, *heredity* which allows the transmission of the individual phenotypic characteristics from one generation to another. Secondly, a source of variation in the individual phenotypic characteristics: in our case it is only *mutations*. Finally, a *selection* mechanism which can result from interaction between individuals in the population such as competition. Our model is an *individual-based model* (in short, IBM) which involves a finite and asexual population with constant population size in which each individual's birth, death and mutation events are described. IBMs were first introduced in ecology as a tool to describe local interactions or complex phenomena at the individual level [73, 12, 13, 33, 52, 28]. Ecological studies using IBMs are mainly numerical and the models are mostly posed in discrete space as systems of interacting particles [73] and more rarely in continuous space [12, 13, 33]. Many IBMs (with non-constant population size) have been proposed and studied in the context of Darwinian evolution by the biology community [68, 31, 41] and the mathematical community [16, 17, 5]. Others are dispersal models in spatially structured populations where the trait is viewed as a spatial location and mutations as dispersal [12, 13, 62, 46]. Other models, structured in age, were developed in [22] and studied mathematically in [74, 82], or structured in age and traits in [42, 66, 67].

We consider an IBM with fixed population size, so that births and deaths occur simultaneously in so-called *resampling* (or swap phenomenon) events. The mutation and resampling rate of an individual depends on its phenotype. When a mutation occurs, the new mutant trait is close to its parent's one yielding a slow variation of the trait. In population genetics, the WRIGHT-FISHER model (and its extensions with selection, mutation or immigration), CANNINGS model or the MORAN model are interested in the evolution of allele frequencies according to various mechanisms [38]. In [43, 27, 26, 37], the authors construct the FLEMING-VIOT process as a scaling limit of large population from the MORAN process. Several extensions exist, including frequency-dependent selection, recombination, other reproduction mechanisms [39, 51]. In particular, the last article provides a bridge between population genetics and eco-evolutionary models. Others, based on KERMACK-MCKENDRICK's model, are interested in epidemiological questions [70, 71].

The aim of this article is to describe the evolutionary dynamics of the dominant trait, at the population level, on a long time scale, as a solution to an ordinary differential equation (in short, ODE) called *Canonical Equation of the Adaptive Dynamics* (in short, CEAD). Canonical equations are well-known tools in evolutionary biology, used to predict the evolutionary fate of ecological communities. More precisely, such equations describe the evolution of dominant traits in a biological population as driven by mutations and a fitness gradient which describes the strength of selection that pushes the population to locally increase its fitness [85, 61, 81, 30].

Fitness measures the selective value of a given individual in a given environment including the population under consideration itself. This individual can be any (fictitious) mutant individual that can be born in the population at the time under consideration. The way to construct a fitness landscape depends on the ecological context [69]. For continuous time homogeneous MARKOV models as studied below, it is the instantaneous growth rate (birth rate minus death rate) of the individual considered in the environment considered. If we further assume that the population constituting the environment is in a stationary state, then we can consider that the fitness of a given mutant individual in this population governs the possibility of invasion of the descendants of this mutant in the population.

The canonical equation has been studied and derived in different contexts such as game theory [54] and quantitative genetics [61]. In the branch of evolutionary biology called *adapt-
theory [47] and quantitative genetics [53]. In the branch of evolutionary biology called *adaptive dynamics* [48, 64, 57], the CEAD has been introduced heuristically in [26]. The theory of adaptive dynamics studies the links between ecology and Darwinian evolution, more precisely, it investigates the effects of the ecological aspects of population dynamics on the evolutionary process, and so describes the population dynamics on the phenotypic level instead of the genotypic level. The theory of adaptive dynamics is based on biological assumptions of *rare* and *small mutations* and of *large population* under which the CEAD was proposed.

Two mathematical approaches were developed to give a proper mathematical justification of this theory: a *deterministic* one, and a *stochastic* one. All these approaches are based on the use of IBMs and different combinations of the three previous biological assumptions and consider a parameter scaling under which the population distribution over the trait space concentrates to Dirac masses, i.e. to subpopulations in which all the individuals have the same trait. The canonical equation corresponds to the motion of Dirac masses. Let us introduce three parameters corresponding to the different biological assumptions: $p$ for the mutation probability, $\sigma$ for the mutation size and $K$ for the population size.

In the deterministic approach, [40] first establishes that, under the asymptotic of large population ($K \to +\infty$), the IBM converges in law to a deterministic process which is a weak solution of a partial differential equation. Then by adding the small mutation assumption ($\sigma \to 0$) and an appropriate time scaling $1/\sigma$, [28, 69, 56] establish the convergence to a version of the canonical equation different from the one of [26] and described by a HAMILTON-JACOBI equation with constraint.

The stochastic approach was developed in [14, 18]. In [14] it is proved that the IBM converges, for finite dimensional distributions, under the double simultaneous asymptotic of large population ($K \to +\infty$) and rare mutation ($p \to 0$) to a stochastic process: the TSS for *Trait Substitution Sequence* introduced in [59, Section 6.4]. This is a pure jump Markov process in the trait space where the population is at all times monomorphic and where the jumps describe the invasion and then the fixation of a mutant $y$ in a monomorphic resident population of trait $x$. By adding the small mutation assumption ($\sigma \to 0$) to the TSS, its convergence to the canonical equation proposed in [26] is established in [18]. The time scale involved in observing the canonical equation phenomenon is $1/K\sigma^2p$. Extensions of these results were obtained in [17] for chemostat models, in [30, 58] for age-structured populations models, in [55] for spatial-structured populations models and [2] which studies the simultaneous application of the three limits ($K \to +\infty, p \to 0, \sigma \to 0$) in order to determine precisely the range of parameters leading to the canonical equation.

Despite their success, the proposed approaches are criticised by biologists [74, 70]. Among the biological assumptions of adaptive dynamics, the assumption of rare mutations is the most critised as unrealistic [74]. The rate of molecular mutation is relatively well known and generally involves several nucleotide substitutions per generation. The adaptive dynamics response is based on the fact that only non-synonymous mutations (changing phenotypes) producing viable individuals should be taken into account [1] [59, Section 6.4]. Since only a small fraction of DNA codes for proteins and many mutations produce non-functional proteins, and thus non-viable individuals, it is not unreasonable to assume that mutations are rare, but probably not as rare as assumed in the stochastic approach [14, 18, 2]. Another criticism of stochastic approaches is that the phenomenon of the canonical equation takes place on a too long evolutionary time scale. In order to solve these problems, we propose to apply a double simultaneous asymptotic of small
tive dynamics [55, 72, 65], the CEAD has been introduced heuristically in [30]. The theory of adaptive dynamics studies the links between ecology and Darwinian evolution, more precisely, it investigates the effects of the ecological aspects of population dynamics on the evolutionary process, and so describes the population dynamics on the phenotypic level instead of the genotypic level. The theory of adaptive dynamics is based on biological assumptions of rare and small mutations and of large population under which the CEAD was proposed.

Two mathematical approaches were developed to give a proper mathematical justification of this theory: a deterministic one, and a stochastic one. All these approaches are based on the use of IBMs and different combinations of the three previous biological assumptions and consider a parameter scaling under which the population distribution over the trait space concentrates to Dirac masses, i.e. to subpopulations in which all the individuals have the same trait. The canonical equation corresponds to the motion of Dirac masses. Let us introduce three parameters corresponding to the different biological assumptions: $p$ for the mutation probability, $\sigma$ for the mutation size and $K$ for the population size. In the sequel, we denote $p_K := p(K)$ and $\sigma_K := \sigma(K)$ to emphasise the dependence of $K$.

In the deterministic approach, [46] first establishes that, under the asymptotic of large population ($K \to +\infty$), the IBM converges in law to a deterministic process which is a weak solution of a partial differential equation. Then by adding the small mutation assumption ($\sigma \to 0$) and an appropriate time scaling $1/\sigma$, [32, 78, 64] establish the convergence to a version of the canonical equation different from the one of [30] and described by a HAMILTON-JACOBI equation with constraint.

The stochastic approach was developed in [16, 20]. In [16] it is proved that the IBM converges, for finite dimensional distributions, under the double simultaneous asymptotic of large population ($K \to +\infty$) and rare mutation ($p_K \to 0$) to a stochastic process: the TSS for Trait Substitution Sequence introduced in [68, Section 6.4]. This is a pure jump Markov process in the trait space where the population is at all times monomorphic and where the jumps describe the invasion and then the fixation of a mutant $y$ in a monomorphic resident population of trait $x$. By adding the small mutation assumption ($\sigma \to 0$) to the TSS, its convergence to the canonical equation proposed in [30] is established in [20]. The time scale involved in observing the canonical equation phenomenon is $1/K\sigma_K^2 p_K$. Extensions of these results were obtained in [19] for chemostat models, in [35, 66] for age-structured populations models, in [63] for spatial-structured populations models and [2] which studies the simultaneous application of the three limits ($K \to +\infty$, $p_K \to 0$, $\sigma_K \to 0$) in order to determine precisely the range of parameters leading to the canonical equation. Note that another approach, studied in [23, 36], consider models of large population ($K \to +\infty$) scaling with rare-but-not-too-rare mutations (with power-law mutation rates $p_K := K^{-\alpha} \to 0$). This specific mutational scale implies that negligible subpopulations of size $K^\beta$, $\beta \in (0,1)$ may have a strong contribution to evolution. A similar scaling was also studied in [34, 15, 21, 11] with fixed mutation size. Building on these works, [76] recently proposed a derivation of a CEAD assuming small mutations.

Despite their success, the proposed approaches are criticised by biologists [83, 79]. Among the biological assumptions of adaptive dynamics, the assumption of rare mutations is the most critised as unrealistic [83]. The rate of molecular mutation is relatively well known and generally involves several nucleotide substitutions per generation. The adaptive dynamics response is based on the fact that only non-synonymous mutations (changing phenotypes) producing viable individuals should be taken into account [1] [68, Section 6.4]. Since only a small fraction of DNA codes for proteins and many mutations produce non-functional proteins, and thus non-viable
mutations ($\sigma \to 0$) and large population ($K \to +\infty$), but frequent mutations ($p \equiv 1$). After conveniently scaling the population state, this leads to a slow-fast dynamics where the phenomenon of the canonical equation is visible on a time scale $1/K\sigma^2$. So, there is some consistency between the previous stochastic approaches developed in [18, 2] and ours: it is exactly the same canonical equation. However in our situation, the CEAD is visible on a shorter evolutionary scale than theirs, and therefore is biologically more reasonable.

As the mutation size parameter is small, the population distribution tends to be close to a Dirac mass. Our aim is to describe the evolution of the support of this Dirac mass on a large time scale. The mean trait appears as the natural slow component. It will be proved to act on the time scale $1/K\sigma^2$. In our case, the fast component acts on the time scale $K$ and is given by the dynamics of the centered and dilated distribution of traits corresponding to a discrete version of the centered FLEMING-VIOT process [16]. Since the centered FLEMING-VIOT process is ergodic [16], we expect the centered and dilated distribution of traits to stabilise on the slow time scale, hence the distribution of traits should stabilise to a Dirac mass. Therefore, the dynamics of the dominant trait corresponds to that of the mean trait.

The reason for not considering the same IBM as in [14, 18, 2] is because it involves three time scales, a slow one corresponding to the dynamics of the mean trait acting on the evolutionary time scale $1/K\sigma^2$, a fast one corresponding to the dynamics of the centered, dilated distribution of traits acting on the evolutionary time scale $K$ and a very fast one corresponding to the population size dynamics acting on the ecological time scale 1. For simplicity, we focus here on a model with constant population size to reduce the number of time scales to two. We expect our results to extend to general IBMs as in [14, 18, 2] and we leave this for future works.

To prove convergence in the framework of slow-fast dynamics (also called stochastic singular perturbations), different techniques can be used.

Firstly, the method of the perturbed test function initially proposed by PAPANICOLAU, STROOCK and VARADHAN in [67] identifies the generator of the limit process with a martingale approach whose idea is the following. If we consider a family of stochastic processes $((X^\epsilon, Y^\epsilon))_{\epsilon>0}$ of generator $L^\epsilon$ where $X^\epsilon$ is the slow component and $Y^\epsilon$ is the fast component in the form $Y^\epsilon(t) = Y(t/\epsilon)$ where $Y$ is a MARKOV process, the slow-fast problem consists in identifying the limit process of $X^\epsilon$ using ergodicity properties of the fast dynamics. Assuming that the family $(X^\epsilon)_{\epsilon>0}$ is tight, we consider $X$ a limiting value. We can expect to characterise $X$ with a martingale problem derived from the martingale problem of $(X^\epsilon, Y^\epsilon)$ provided that the solution to this problem is unique. We would then obtain the convergence of $X^\epsilon$ to $X$ in law. However, for multiscale singular problems, the convergence of $L^\epsilon\varphi((X_t^\epsilon, Y_t^\epsilon))$ when $\epsilon \to 0$, for a test function $\varphi$ depending only on the $X$ component, does not (in general) take place because it contains diverging terms in $\epsilon$. To overcome this difficulty, firstly the idea is to decompose the generator $L^\epsilon$ in the following form $L^\epsilon = \frac{1}{\epsilon}L_1 + L_0$ where $L_1$ is the infinitesimal generator of the MARKOV process $Y$ in the variable $y$ and $L_0$ the operator of the slow component depending on slow and fast variables. Secondly, the idea is to perturb the initial test function $\varphi(x)$ into a test function $\varphi^\epsilon(x, y) := \varphi(x) + \epsilon\varphi_1(x, y)$ such that

$$L^\epsilon \varphi^\epsilon = \left[ \frac{1}{\epsilon} L_1 + L_0 \right] (\varphi + \epsilon \varphi_1) = (L_1 \varphi_1 + L_0 \varphi - \bar{L}_0 \varphi) + \bar{L}_0 \varphi + \epsilon L_0 \varphi_1 \quad (1)$$
individuals, it is not unreasonable to assume that mutations are rare, but probably not as rare
as assumed in the stochastic approach [16, 20, 2]. Another criticism of stochastic approaches
is that the phenomenon of the canonical equation takes place on a too long evolutionary time
scale. In order to solve these problems, we propose to apply a double simultaneous asymptotic
of small mutations ($\sigma_K \to 0$) and large population ($K \to +\infty$), but frequent mutations ($p \equiv 1$).
After conveniently scaling the population state, this leads to a slow-fast dynamics where the
phenomenon of the canonical equation is visible on a time scale $1/K\sigma_K^2$. So, there is some cons-
sistency between the previous stochastic approaches developed in [20, 2] and ours: it is exactly
the same canonical equation. However in our situation, the CEAD is visible on a shorter evolu-
tionary scale than theirs, and therefore is biologically more reasonable. Note however that the
restrictions on $\sigma_K$ in [2] are in a completely different regime than ours ($\sigma_K \gg K^{-1/2}$, as opposed
to $\sigma_K \ll K^{-2}$ here). Hence it is possible to choose $p_K$ in [2] in such a way that the CEAD
appears on a faster time scale than in our work, where the fastest scale is in $K^3$.

As the mutation size parameter is small, the population distribution tends to be close to a
DIRAC mass. Our aim is to describe the evolution of the support of this DIRAC mass on a large
time scale. The mean trait appears as the natural slow component. It will be proved to act on
the time scale $1/K\sigma_K^2$. In our case, the fast component acts on the time scale $K$ and is given
by the dynamics of the centered and dilated distribution of traits corresponding to a discrete
version of the centered FLEMING-VIOT process [18]. Since the centered FLEMING-VIOT process
is ergodic [18], we expect the centered and dilated distribution of traits to stabilise on the slow
time scale, hence the distribution of traits should stabilise to a DIRAC mass. Therefore, the
dynamics of the dominant trait corresponds to that of the mean trait.

The reason for not considering the same IBM as in [16, 20, 2] is because it involves three time
scales, a slow one corresponding to the dynamics of the mean trait acting on the evolutionary
time scale $1/K\sigma_K^2$, a fast one corresponding to the dynamics of the centered, dilated distribu-
tion of traits acting on the evolutionary time scale $K$ and a very fast one corresponding to the
population size dynamics acting on the ecological time scale 1. For simplicity, we focus here on
a model with constant population size to reduce the number of time scales to two. We expect
our results to extend to general IBMs as in [16, 20, 2] and we leave this for future works.

To prove convergence in the framework of slow-fast dynamics (also called *stochastic singular perturbations*), different techniques can be used.

Firstly, the method of the *perturbed test function* initially proposed by PAPANICOLAU,
STROOCK and VARADHAN in [75] identifies the generator of the limit process with a mar-
tingale approach whose idea is the following. If we consider a family of stochastic processes
((X^ε, Y^ε))_{ε>0} of generator L^ε where X^ε is the *slow component* and Y^ε is the *fast component*
in the form Y^ε(t) = Y (t/ε) where Y is a MARKOV process, the slow-fast problem consists in
identifying the limit process of X^ε using ergodicity properties of the fast dynamics. Assuming
that the family (X^ε)_{ε>0} is tight, we consider X a limiting value. We can expect to characterise
X with a martingale problem derived from the martingale problem of (X^ε, Y^ε) provided that
the solution to this problem is unique. We would then obtain the convergence of X^ε to X in law.
However, for multiscale singular problems, the convergence of L^εϕ((X_t^ε, Y_t^ε)) when ε → 0, for a
test function ϕ depending only on the X component, does not (in general) take place because
it contains diverging terms in ε. To overcome this difficulty, firstly the idea is to decompose the
generator L^ε in the following form L^ε = (1/ε)L_1 + L_0 where L_1 is the infinitesimal generator of the
because of $L_1\varphi = 0$ and where $\overline{L}_0$ is the operator of the limit slow component *averaged* by the unique invariant probability measure of the limit fast component $Y$. Provided that $\varphi_1$ solves the POISSON equation (with respect to $L_1$ and the variable $y$)

$$L_1\varphi_1 + L_0\varphi - \overline{L}_0\varphi = 0, \qquad (2)$$

we deduce that $L^\varepsilon\varphi^\varepsilon = \overline{L}_0\varphi + O(\varepsilon)$. Then, $\varphi^\varepsilon(x, y) \to \varphi(x)$ and $L^\varepsilon\varphi^\varepsilon(x, y) \to \overline{L}_0\varphi(x)$ when $\varepsilon \to 0$ as expected.

The perturbed test function method has been extended in [10, 50, 51] and in the books [38, Section 6.3], [52]. Many other references apply the perturbed test function strategy in various settings: in finance [39] in transport problems as in [67] or in [21] where the tightness of the fast component is established using its occupation measure. Similar methods are used in homogenisation [67, 7, 68] (see also [6] which exploits spectral and semi-group properties in addition), and in stochastic stability and control [52].

Finally, an important method is the stochastic averaging, in a generic framework, that was proposed by KURTZ in [49]. Many approaches [3, 62, 46, 20, 43, 13, 4, 8] including ours, are based on it. The main idea consists in exploiting the occupation measure $\Gamma^\varepsilon$ of the fast component $Y^\varepsilon$ which is formally defined for all $t \ge 0$, for any Borelian $B$ by

$$\Gamma^\varepsilon ([0, t] \times B) = \int_0^t 1_B (Y_s^\varepsilon) \, ds,$$

and to establish the convergence of the couple $(X^\varepsilon, \Gamma^\varepsilon)$ when $\varepsilon \to 0$ using *compactness-uniqueness* arguments. Proceeding in this way allows us to escape the difficulties created by the fluctuations of the fast component and avoids to obtain tightness result of the “slow-fast” couple $(X^\varepsilon, Y^\varepsilon)$. The proof can be divided into four steps.

The first one consists in establishing uniform tightness of the sequence of laws of the couple $(X^\varepsilon, \Gamma^\varepsilon)$. The second one consists in establishing a martingale problem for any limiting value of the family of laws of the couple. The third one is based on establishing the uniqueness of the limiting value of $\Gamma^\varepsilon$ expressed in terms of the limit $X$ of $X^\varepsilon$ which is given. The characterisation of the limiting value $\Gamma$ of $\Gamma^\varepsilon$ is usually based on ergodicity arguments. Finally, the last step is to establish the uniqueness of the martingale problem for the slow component limit $X$ when $\Gamma$ is given as above.

The KURTZ approach seems to be better adapted to our situation than the perturbed test function method. Indeed, the generator $L_1$ being that of a measure-valued diffusion, it is thus delicate to find the good class of test function $\varphi_1$ satisfying (2) and the computations are difficult because of the moment terms (see Section 2.3). In addition, the remainder terms generated by (1) are difficult to control and the fast component $Y^\varepsilon$ does not take the form $Y(t/\varepsilon)$. However, KURTZ’s approach can neither be implemented directly because of difficulties inherent to our model described in Section 2. Therefore, our result required a complete reworking of the classical arguments. In particular, in our case, we don’t have uniform moment estimates but only a uniform control, in probability, of the second moment of the fast variable up to a stopping time $\tau^K$ where the diameter of the support of the fast component becomes large.

This paper is organised as follows. In Section 2, we define our trait structured IBM, state the central theorem about the CEAD characterising the limit model which consists of an ODE ruling the dynamics of the limit slow component. We establish in this section the sketch and outline of
MARKOV process $Y$ in the variable $y$ and $L_0$ the operator of the slow component depending on slow and fast variables. Secondly, the idea is to perturb the initial test function $\varphi(x)$ into a test function $\varphi^\varepsilon(x, y) := \varphi(x) + \varepsilon\varphi_1(x, y)$ such that

$$L^\varepsilon \varphi^\varepsilon = \left[ \frac{1}{\varepsilon} L_1 + L_0 \right] (\varphi + \varepsilon \varphi_1) = (L_1 \varphi_1 + L_0 \varphi - \bar{L}_0 \varphi) + \bar{L}_0 \varphi + \varepsilon L_0 \varphi_1 \quad (1)$$

because of $L_1\varphi = 0$ and where $\bar{L}_0$ is the operator of the limit slow component averaged by the unique invariant probability measure of the limit fast component $Y$. Provided that $\varphi_1$ solves the POISSON equation (with respect to $L_1$ and the variable $y$)

$$L_1 \varphi_1 + L_0 \varphi - \bar{L}_0 \varphi = 0, \quad (2)$$

we deduce that $L^\varepsilon \varphi^\varepsilon = \bar{L}_0 \varphi + O(\varepsilon)$. Then, $\varphi^\varepsilon(x, y) \to \varphi(x)$ and $L^\varepsilon \varphi^\varepsilon(x, y) \to \bar{L}_0 \varphi(x)$ when $\varepsilon \to 0$ as expected.

The perturbed test function method has been extended in [10, 58, 59] and in the books [44, Section 6.3], [60]. Many other references apply the perturbed test function strategy in various settings: in finance [45] in transport problems as in [75] or in [25] where the tightness of the fast component is established using its occupation measure. Similar methods are used in homogenisation [75, 7, 77] (see also [6] which exploits spectral and semi-group properties in addition), and in stochastic stability and control [60].

Finally, an important method is the stochastic averaging, in a generic framework, that was proposed by KURTZ in [57]. Many approaches [3, 67, 53, 24, 49, 14, 4, 8] including ours, are based on it. The main idea consists in exploiting the occupation measure $\Gamma^\varepsilon$ of the fast component $Y^\varepsilon$ which is formally defined for all $t \ge 0$, for any Borelian $B$ by

$$\Gamma^\varepsilon ([0, t] \times B) = \int_0^t 1_B (Y_s^\varepsilon) \, ds,$$

and to establish the convergence of the couple $(X^\varepsilon, \Gamma^\varepsilon)$ when $\varepsilon \to 0$ using compactness-uniqueness arguments. Proceeding in this way allows us to escape the difficulties created by the fluctuations of the fast component and avoids to obtain tightness result of the “slow-fast” couple $(X^\varepsilon, Y^\varepsilon)$. The proof can be divided into four steps.

The first one consists in establishing uniform tightness of the sequence of laws of the couple $(X^\varepsilon, \Gamma^\varepsilon)$. The second one consists in establishing a martingale problem for any limiting value of the family of laws of the couple. The third one is based on establishing the uniqueness of the limiting value of $\Gamma^\varepsilon$ expressed in terms of the limit $X$ of $X^\varepsilon$ which is given. The characterisation of the limiting value $\Gamma$ of $\Gamma^\varepsilon$ is usually based on ergodicity arguments. Finally, the last step is to establish the uniqueness of the martingale problem for the slow component limit $X$ when $\Gamma$ is given as above.

The KURTZ approach seems to be better adapted to our situation than the perturbed test function method. Indeed, it is delicate to find the good class of test function $\varphi_1$ satisfying (2) and the computations are difficult because of the moment terms (see Section 2.3). In addition, the remainder terms generated by (1) are difficult to control and the fast component $Y^\varepsilon$ does not take the form $Y(t/\varepsilon)$. However, KURTZ’s approach can neither be implemented directly because of difficulties inherent to our model described in Section 2. Therefore, our result required a complete reworking of the classical arguments. In particular, in our case, we do not have uniform moment estimates but only a uniform control, in probability, of the second moment of the fast variable up
the main proof and the difficulties encountered. We give also some prospects that let us believe that we can improve our main result by relaxing some assumptions. The rest of this paper is devoted to prove the central theorem of Section 2. In Section 3 some approximations of the infinitesimal generator of the slow-fast process are proved. In Section 4, we establish estimates of moments, in particular of the sixth and second order moment up to time $\tilde{\tau}^K$. We also prove the convergence of $\tilde{\tau}^K$ to $+\infty$ when $K \to +\infty$ exploiting coupling arguments between the moment of order 2 and a certain biased random walk for which large deviations estimates are established. In Section 5, we prove the tightness of the couple “slow-occupation measure fast” in the torus case. In Section 6, we establish, in the torus case, the uniqueness of the limit occupation measure which is described by the unique invariant probability measure of the centered FLEMING-VIOT process. This result is used in Section 7 to characterise the limit slow component as the unique solution of the CEAD with values in the torus. Thanks to the non-explosion of this ODE and choosing the torus large enough we are able to conclude the proof of our main result given in Section 2.

# 2 A trait structured IBM

Let us describe the microscopic model which models population evolution, in a Darwinian sense, at the individual level.

## 2.1 Parameters and assumptions of the model

We consider a discrete population of constant size in continuous time in which the survival and reproductive capacity of each individual is characterised by a continuous quantitative phenotypic trait $x \in \mathbb{R}$, i.e. an overall characteristic subject to selection such as body size at maturity. The individuals reproduce asexually during their lives, i.e. the reproduction scheme is assumed clonal simultaneously with death of another individual, with frequency-dependent rates. Each birth of an individual occurs simultaneously with the death of another individual in the population. A mutation occurs during life at the individual level at rate 1: in this sense mutations are considered frequent. Each mutation has an amplitude of the order of magnitude $\sigma \in (0, +\infty)$. Small $\sigma$ means mutations have a small phenotypic effect, i.e. evolution acts slowly on the individual phenotypic characteristics.

We are interested in approximating the long-term dynamics of a large population. We assume that the number of individuals alive at each time $t \ge 0$ is always equal to $K$. Let us denote by $x_1(t), \dots, x_K(t)$, the phenotypic trait values of these individuals at time $t$. The state of the population at time $t$, can be described by the finite point measure on $\mathbb{R}$ rescaled by $K$

$$ \nu_t^{K,\sigma} := \frac{1}{K} \sum_{i=1}^{K} \delta_{x_i(t)} $$

where $\delta_x$ is the Dirac measure at $x$.

For all $x, y \in \mathbb{R}$, let us introduce the following biological parameters:
to a stopping time $\tilde{\tau}^K$ where the diameter of the support of the fast component becomes large.

This paper is organised as follows. In Section 2, we define our trait structured IBM, state the central theorem about the CEAD characterising the limit model which consists of an ODE ruling the dynamics of the limit slow component. We establish in this section the sketch and outline of the main proof and the difficulties encountered. We give also some prospects that let us believe that we can improve our main result by relaxing some assumptions. The rest of this paper is devoted to prove the central theorem of Section 2. In Section 3 some approximations of the infinitesimal generator of the slow-fast process are proved. In Section 4, we establish estimates of moments, in particular of the sixth and second order moment up to time $\tilde{\tau}^K$. We also prove the convergence of $\tilde{\tau}^K$ to $+\infty$ when $K \to +\infty$ exploiting coupling arguments between the moment of order 2 and a certain biased random walk for which large deviations estimates are established. In Section 5, we prove the tightness of the couple “slow-occupation measure fast” in the torus case. In Section 6, we establish, in the torus case, the uniqueness of the limit occupation measure which is described by the unique invariant probability measure of the centered FLEMING-VIOT process. This result is used in Section 7 to characterise the limit slow component as the unique solution of the CEAD with values in the torus. Thanks to the non-explosion of this ODE and choosing the torus large enough we are able to conclude the proof of our main result given in Section 2.

# 2 A trait structured IBM

Let us describe the microscopic model which models population evolution, in a Darwinian sense, at the individual level.

## 2.1 Parameters and assumptions of the model

We consider a discrete population of constant size in continuous time in which the survival and reproductive capacity of each individual is characterised by a continuous quantitative phenotypic trait $x \in \mathbb{R}$, i.e. an overall characteristic subject to selection such as body size at maturity. The individuals reproduce asexually during their lives, i.e. the reproduction scheme is assumed clonal simultaneously with death of another individual, with frequency-dependent rates. Each birth of an individual occurs simultaneously with the death of another individual in the population.

We are interested in approximating the long-term dynamics of a large population. We assume that the number of individuals alive at each time $t \ge 0$ is always equal to $K$. Let us denote by $x_1(t), \dots, x_K(t)$, the phenotypic trait values of these individuals at time $t$. A mutation occurs during life at the individual level at rate 1: in this sense mutations are considered *frequent*. Each mutation has an amplitude of the order of magnitude $\sigma_K \in (0, +\infty)$. Small $\sigma_K$ means mutations have a small phenotypic effect, i.e. evolution acts slowly on the individual phenotypic characteristics. The state of the population at time $t$, can be described by the finite point measure on $\mathbb{R}$ rescaled by $K$

$$ \nu_t^K := \frac{1}{K} \sum_{i=1}^{K} \delta_{x_i(t)} $$

where $\delta_x$ is the Dirac measure at $x$.
* $b(x, y) \in \mathbb{R}_+$ is the resampling rate, i.e. the rate of simultaneous birth of an individual holding trait $y$ and death of an individual holding trait $x$ and it can be interpreted as modeling a competitive pressure driving selection.

* $\theta(x) \in \mathbb{R}_+$ is the rate of mutation of an individual holding trait $x$.

* $m(x, dh)$ is the mutation law of the scaled mutation step $h$, born from an individual with trait $x$, where the mutant trait is given by $y := x + \sigma h \in \mathbb{R}$. It is a probability measure on $\mathbb{R}$.

Let us also introduce the following notations, used throughout this paper:

* $\beta(x) := \frac{\theta(x)}{b(x,x)}$, the ratio between the mutation rate and the resampling rate in a monomorphic population with trait $x$, which can be interpreted (in the scaling limit considered below) as the mean number of mutations between two resampling events.

* Fit$(y, x) := b(x, y) - b(y, x)$ is the adaptive value or fitness of a mutant individual with trait $y$ in the population of $(K - 1)$ individuals of trait $x$. The fitness can be interpreted as the initial growth rate of a mutant individual $y$ in a resident monomorphic population with trait $x$.

Indeed, the total birth rate of a mutant individual $y$ in this population is $b(x, y)(K - 1)$ and the total death rate of a mutant individual $y$ in this population is $b(y, x)(K - 1)$. Hence, the (initial) growth rate of the mutant population is Fit$(y, x)(K - 1)$.

Let $\mathcal{M}_1(\mathbb{R})$ be the set of probability measures on $\mathbb{R}$, endowed with the topology of weak convergence making it a Polish space [9]. For a measurable real bounded function $f$, and a measure $\nu \in \mathcal{M}_1(\mathbb{R})$, we denote $\langle f, \nu \rangle := \int_{\mathbb{R}} f(x)\nu(dx)$. We denote by id the identity function. We denote also $\mathbb{N} := \{0, 1, 2, \dots\}$ and $\mathbb{N}^* := \mathbb{N} \setminus \{0\}$. If $\mathcal{I}$ is an interval of $\mathbb{R}$, then for all $\ell \in \mathbb{N}$, $p \in \mathbb{N}^*$, we denote by $\mathcal{C}^\ell(\mathcal{I}^p, \mathbb{R})$ the space of functions of class $\mathcal{C}^\ell$ from $\mathcal{I}^p$ to $\mathbb{R}$ and by $\mathcal{C}_b^\ell(\mathcal{I}^p, \mathbb{R})$ the space of functions of class $\mathcal{C}^\ell(\mathcal{I}^p, \mathbb{R})$ with bounded derivatives. Finally, we define for all $K \in \mathbb{N}^*$,

$$ \mathcal{M}_{1,K}(\mathbb{R}) := \left\{ \frac{1}{K} \sum_{i=1}^{K} \delta_{x_i} \mid (x_i)_{1 \le i \le K} \in \mathbb{R}^K \right\}. $$

the set of probability measures on $\mathbb{R}$ of $K$ atoms of mass $1/K$.

**Assumptions.** Let us denote by (A) the following two assumptions:

(A1) The maps $b : \mathbb{R}^2 \to \mathbb{R}_+$, and $\theta : \mathbb{R} \to \mathbb{R}_+$ are respectively in $\mathcal{C}_b^2(\mathbb{R}^2, \mathbb{R})$ and $\mathcal{C}_b^2(\mathbb{R}, \mathbb{R})$ and there exists $0 < \underline{b}, \bar{b}, \underline{\theta}, \bar{\theta} < \infty$ such that:

$$ \underline{b} \le b(\cdot, \cdot) \le \bar{b}, \quad \text{and} \quad \underline{\theta} \le \theta(\cdot) \le \bar{\theta}. $$

(A2) (a) There exists $A_m \in (0, +\infty)$ such that the law $m(x, dh)$ is absolutely continuous with respect to the LEBESGUE measure on $\mathbb{R}$ with density $m(x, h)$ centered and satisfies

$$ \forall x \in \mathbb{R}, \forall |h| \ge A_m, \quad m(x, h) = 0. $$
For all $x, y \in \mathbb{R}$, let us introduce the following biological parameters:

* $b(x, y) \in \mathbb{R}_+$ is the resampling rate, i.e. the rate of simultaneous birth of an individual holding trait $y$ and death of an individual holding trait $x$ and it can be interpreted as modeling a competitive pressure driving selection.

* $\theta(x) \in \mathbb{R}_+$ is the rate of mutation of an individual holding trait $x$.

* $m(x, dh)$ is the mutation law of the scaled mutation step $h$, born from an individual with trait $x$, where the mutant trait is given by $y := x + \sigma_K h \in \mathbb{R}$. It is a probability measure on $\mathbb{R}$.

Let us also introduce the following notations, used throughout this paper:

* $\beta(x) := \frac{\theta(x)}{b(x,x)}$, the ratio between the mutation rate and the resampling rate in a monomorphic population with trait $x$, which can be interpreted (in the scaling limit considered below) as the mean number of mutations between two resampling events.

* Fit$(y, x) := b(x, y) - b(y, x)$ is the adaptive value or fitness of a mutant individual with trait $y$ in the population of $(K-1)$ individuals of trait $x$. The fitness can be interpreted as the initial growth rate of a mutant individual $y$ in a resident monomorphic population with trait $x$.
  Indeed, the total birth rate of a mutant individual $y$ in this population is $b(x, y)(K-1)$ and the total death rate of a mutant individual $y$ in this population is $b(y, x)(K-1)$. Hence, the (initial) growth rate of the mutant population is Fit$(y, x)(K-1)$.

Let $\mathcal{M}_1(\mathbb{R})$ be the set of probability measures on $\mathbb{R}$, endowed with the topology of weak convergence making it a Polish space [9]. For a measurable real bounded function $f$, and a measure $\nu \in \mathcal{M}_1(\mathbb{R})$, we denote $\langle f, \nu \rangle := \int_{\mathbb{R}} f(x)\nu(dx)$. We denote by id the identity function. We denote also $\mathbb{N} := \{0, 1, 2, \dots\}$ and $\mathbb{N}^* := \mathbb{N} \setminus \{0\}$. If $\mathcal{I}$ is an interval of $\mathbb{R}$, then for all $\ell \in \mathbb{N}$, $q \in \mathbb{N}^*$, we denote by $\mathcal{C}^\ell(\mathcal{I}^q, \mathbb{R})$ the space of functions of class $\mathcal{C}^\ell$ from $\mathcal{I}^q$ to $\mathbb{R}$ and by $\mathcal{C}_b^\ell(\mathcal{I}^q, \mathbb{R})$ the space of functions of class $\mathcal{C}^\ell(\mathcal{I}^q, \mathbb{R})$ with bounded derivatives. Finally, we define for all $K \in \mathbb{N}^*$,

$$ \mathcal{M}_{1,K}(\mathbb{R}) := \left\{ \frac{1}{K} \sum_{i=1}^{K} \delta_{x_i} \mid (x_i)_{1 \le i \le K} \in \mathbb{R}^{K} \right\} $$

the set of probability measures on $\mathbb{R}$ of $K$ atoms of mass $1/K$.

**Assumptions.** Let us denote by (A) the following two assumptions:

(A1) The maps $b: \mathbb{R}^2 \to \mathbb{R}_+$, and $\theta: \mathbb{R} \to \mathbb{R}_+$ are respectively in $\mathcal{C}_b^2(\mathbb{R}^2, \mathbb{R})$ and $\mathcal{C}_b^2(\mathbb{R}, \mathbb{R})$ and there exists $0 < \underline{b}, \bar{b}, \underline{\theta}, \bar{\theta} < \infty$ such that:

$$ \underline{b} \le b(\cdot, \cdot) \le \bar{b}, \quad \text{and} \quad \underline{\theta} \le \theta(\cdot) \le \bar{\theta}. $$

(A2) (a) There exists $A_m \in (0, +\infty)$ such that the law $m(x, dh)$ is absolutely continuous with respect to the LEBESGUE measure on $\mathbb{R}$ with density $m(x, h)$ centered and satisfies

$$ \forall x \in \mathbb{R}, \forall |h| \ge A_m, \quad m(x, h) = 0. $$
(b) For all $\alpha \in \mathbb{R}$, for all $\ell \in \mathbb{N}$, we denote by

$$m_{\ell}(\alpha) := \int_{\mathbb{R}} |h|^{\ell} m(\alpha, h) dh$$

the $\ell$th moment of the mutation law. We assume for $\ell \in \{2, 4, 6\}$ that $m_\ell$ is LIPSCHITZ and there exists $\underline{m}_\ell, \overline{m}_\ell \in (0, +\infty)$ such that for all $\alpha \in \mathbb{R}$, $\underline{m}_\ell \le m_\ell(\alpha) \le \overline{m}_\ell$.

Let us now give the infinitesimal generator $L^{K,\sigma}$ of the $M_{1,K}(\mathbb{R})$-valued MARKOV process $\nu^{K,\sigma} := (\nu_t^{K,\sigma})_{t \ge 0}$ describing the ecological dynamics of the population with resampling. The generator $L^{K,\sigma}$ is defined for any bounded measurable map $\phi$ from $M_{1,K}(\mathbb{R})$ to $\mathbb{R}$, by

$$L^{K,\sigma}\phi(\nu) = K \int_{\mathbb{R}} \nu(dx) \int_{\mathbb{R}} \nu(dy)b(x,y) \left[ \phi\left(\nu - \frac{\delta_x}{K} + \frac{\delta_y}{K}\right) - \phi(\nu) \right] \\ + K \int_{\mathbb{R}} \theta(x)\nu(dx) \int_{\mathbb{R}} m(x,h) \left[ \phi\left(\nu - \frac{\delta_x}{K} + \frac{\delta_{x+\sigma h}}{K}\right) - \phi(\nu) \right] dh. \quad (3)$$

The first term describes the resampling event of one individual by another and the second term describes the effect of mutations over the lifetime of individuals.

## 2.2 Main result

The main result of this article is the following theorem. For all $\nu \in M_1(\mathbb{R})$, let us denote by Diam (Supp $\nu$) the diameter of the support of $\nu$.

**Theorem 2.1.** Assume (A) and for all $K \in \mathbb{N}^*$, $\sigma \in (0, +\infty)$, for some $x_0 \in \mathbb{R}$, for all $\ell \in \{1, 2, 3\}$,

$$\langle \mathrm{id}, \nu_0^{K,\sigma} \rangle := x_0, \quad \text{and} \quad \langle (\mathrm{id} - x_0)^{2\ell}, \nu_0^{K,\sigma} \rangle \le C_{2\ell}^* K^\ell \sigma^{2\ell} \quad (4)$$

for some $C_{2\ell}^* > 0$. Assume that $K \to +\infty$, $\sigma \to 0$ and $\sigma = \sigma(K)$ such that

$$\exists \varepsilon > 0, \quad \forall C > 0, \quad K^{-C \log(K)} \ll \sigma \ll K^{-(2+\varepsilon)}. \quad (5)$$

Then, for the SKOROHOD topology, the sequence of the mean trait processes $((\langle \mathrm{id}, \nu_{t/K\sigma^2}^{K,\sigma} \rangle)_{t \ge 0})_{K \in \mathbb{N}^*}$ converges in law in $\mathbb{D}(\mathbb{R}_+, \mathbb{R})$ when $K \to +\infty$, $\sigma \to 0$, to the unique deterministic process $(\zeta_t)_{t \ge 0} \in \mathscr{C}^0(\mathbb{R}_+, \mathbb{R})$ with initial condition $\zeta_0 := x_0$, which is solution to the Canonical Equation of Adaptive Dynamics (CEAD):

$$\dot{y} = \partial_1 \mathrm{Fit}(y, y) \times \beta(y)m_2(y).$$

In addition, we have the following support concentration property : for all $T > 0$,

$$\sup_{t \in [0, T]} \mathrm{Diam} \left( \mathrm{Supp} \, \nu_{t/K\sigma^2}^{K,\sigma} \right) \le \frac{1}{K} \quad (6)$$

holds with probability which tends to 1 when $K \to +\infty$.
(b) For all $\alpha \in \mathbb{R}$, for all $\ell \in \mathbb{N}$, we denote by

$$m_{\ell}(\alpha) := \int_{\mathbb{R}} |h|^{\ell} m(\alpha, h) \, dh$$

the $\ell^{th}$ moment of the mutation law. We assume for $\ell \in \{2, 4, 6\}$ that $m_{\ell}$ is LIPSCHITZ and there exists $\underline{m}_{\ell}, \bar{m}_{\ell} \in (0, +\infty)$ such that for all $\alpha \in \mathbb{R}$, $\underline{m}_{\ell} \le m_{\ell}(\alpha) \le \bar{m}_{\ell}$.

Let us now give the infinitesimal generator $L^K$ of the $M_{1,K}(\mathbb{R})$-valued MARKOV process $\nu^K := (\nu_t^K)_{t \ge 0}$ describing the ecological dynamics of the population with resampling. The generator $L^K$ is defined for any bounded measurable map $\phi$ from $M_{1,K}(\mathbb{R})$ to $\mathbb{R}$, by

$$L^K \phi(\nu) = K \int_{\mathbb{R}} \nu(dx) \int_{\mathbb{R}} \nu(dy) b(x, y) \left[ \phi \left( \nu - \frac{\delta_x}{K} + \frac{\delta_y}{K} \right) - \phi(\nu) \right] \\ + K \int_{\mathbb{R}} \theta(x) \nu(dx) \int_{\mathbb{R}} m(x, h) \left[ \phi \left( \nu - \frac{\delta_x}{K} + \frac{\delta_{x+\sigma_K h}}{K} \right) - \phi(\nu) \right] dh. \quad (3)$$

The first term describes the resampling event of one individual by another and the second term describes the effect of mutations over the lifetime of individuals.

## 2.2 Main result

The main result of this article is the following theorem. For all $\nu \in M_1(\mathbb{R})$, let us denote by Diam (Supp $\nu$) the diameter of the support of $\nu$.

**Theorem 2.1.** Assume (A) and for all $K \in \mathbb{N}^*$, $\sigma_K \in (0, +\infty)$, for some $x_0 \in \mathbb{R}$, for all $\ell \in \{1, 2, 3\}$,

$$\langle \mathrm{id}, \nu_0^K \rangle := x_0, \quad \text{and} \quad \langle (\mathrm{id} - x_0)^{2\ell}, \nu_0^K \rangle \le C_{2\ell}^* K^\ell \sigma_K^{2\ell} \quad (4)$$

for some $C_{2\ell}^* > 0$. Assume that $K \to +\infty$, $\sigma_K \to 0$ such that

$$\exists \varepsilon > 0, \quad \forall C > 0, \quad K^{-C \log(K)} \ll \sigma_K \ll K^{-(2+\varepsilon)}. \quad (5)$$

Then, for the SKOROHOD topology, the sequence of the mean trait processes

$$\left( \left( \left\langle \mathrm{id}, \nu_{t/K \sigma_K^2}^K \right\rangle \right)_{t \ge 0} \right)_{K \in \mathbb{N}^*}$$

converges in law in $D(\mathbb{R}_+, \mathbb{R})$ when $K \to +\infty$, $\sigma_K \to 0$, to the unique deterministic process $(\zeta_t)_{t \ge 0} \in C^0(\mathbb{R}_+, \mathbb{R})$ with initial condition $\zeta_0 := x_0$, which is solution to the Canonical Equation of Adaptive Dynamics (CEAD):

$$y = \partial_1 \mathrm{Fit}(y, y) \times \frac{\beta(y) m_2(y)}{2}.$$

In addition, we have the following support concentration property : for all $T > 0$,

$$\sup_{t \in [0, T]} \mathrm{Diam} \left( \mathrm{Supp} \, \nu_{t/K \sigma_K^2}^K \right) \le \frac{1}{K} \quad (6)$$

holds with probability which tends to 1 when $K \to +\infty$.
The canonical equation is composed of two terms. The first term: the fitness gradient, describes the strength of selection that pushes the population to increase its adaptive value locally. The second term describes the effect of mutations. Note that (6) describes, in some sense, when $K \to +\infty$, the convergence of the population distribution $(\nu_{t/K\sigma^2}^{K,\sigma})$ to a Dirac mass.

**Remark 2.2.** (1) We conjecture that the assumption $\sigma \ll K^{-(2+\epsilon)}$ given by (5) is too restrictive and that it can be weakened by $\sigma \ll K^{-(\frac{3}{2}+\epsilon)}$. This conjecture is supported by the discussion in Section 2.3.4 below.

(2) We actually prove Theorem 2.1 (see Section 4.4.3) under a slightly weaker assumption than the left bound of (5): for a universal constant $C_0$ depending only on $b, \theta, m$ and not on $K$ and $\sigma$,

$$ \exists \epsilon > 0, \quad K^{-\epsilon^2 C_0 \log(K)} \ll \sigma \ll K^{-(2+\epsilon)}. $$

(3) The Assumption (A2)(a) is only used in Section 4.4 and could be weakened with some tail bounds (typically exponential) on $m(x, \cdot)$.

The aim of the rest of the paper is to prove Theorem 2.1. Since $\sigma = \sigma(K)$, we will use the notation $\nu^K$ to designate $\nu^{K,\sigma}$.

## 2.3 Sketch and outline of the proof

Note that, the convergence result of Theorem 2.1 takes place on the time scale $1/K\sigma^2$. The idea of the proof is based on slow-fast asymptotic analysis techniques developed by KURTZ [49]. Let us begin by introducing the slow and fast dynamics involved in our model, then the difficulties encountered and finally the outline of the proof.

### 2.3.1 Slow-fast asymptotic analysis

Let $a, \alpha \in \mathbb{R}$ and $\mathcal{B}(\mathbb{R})$ the BOREL $\sigma$-field on $\mathbb{R}$. Let us define respectively by $\tau_\alpha$ and $h_a$, the translation operator of vector $\alpha$ and the homothety of ratio $a$. For all $x \in \mathbb{R}$, for all $A \in \mathcal{B}(\mathbb{R})$,

$$ (h_a \circ \tau_\alpha) \sharp \delta_x(A) = \delta_{a(x+\alpha)}(A) \quad \text{and} \quad (\tau_\alpha \circ h_a) \sharp \delta_x(A) = \delta_{ax+\alpha}(A). $$

Note that for all $K \in \mathbb{N}^*$, any $\nu \in \mathcal{M}_{1,K}(\mathbb{R})$ has all its moments finite. We define for all $K \in \mathbb{N}^*$, $\mathcal{M}_{1,K}^c(\mathbb{R})$ the set of centered probability measures of $\mathcal{M}_{1,K}(\mathbb{R})$. From the population process $(\nu_{t/K\sigma^2}^K)_{t \ge 0}$, we introduce two evolutionary dynamics:

* The slow dynamics, with value in $\mathbb{R}$, corresponds to that of the mean trait defined by

$$ z_t^K := \langle \mathrm{id}, \nu_{t/K\sigma^2}^K \rangle. $$

* The fast dynamics, with value in $\mathcal{M}_{1,K}^c(\mathbb{R})$, corresponds to that of the centered and dilated distribution of traits defined by

$$ \mu_t^K := \left(h_{\frac{1}{\sigma\sqrt{K}}} \circ \tau_{-z_t^K}\right) \sharp \nu_{t/K\sigma^2}^K = \frac{1}{K} \sum_{i=1}^K \delta_{\frac{1}{\sigma\sqrt{K}}(x_i(t)-z_t^K)}. \quad (7) $$

Note that $\nu_t^K = (\tau_{z_{tK\sigma^2}}^K \circ h_{\sigma\sqrt{K}}) \sharp \mu_{tK\sigma^2}^K$
The canonical equation is composed of two terms. The first term: the fitness gradient, describes the strength of selection that pushes the population to increase its adaptive value locally. The second term describes the effect of mutations. Note that (6) describes, in some sense, when $K \to +\infty$, the convergence of the population distribution $(\nu_{t/K\sigma_K^2}^K)$ to a Dirac mass.

**Remark 2.2.** (i) We conjecture that the assumption $\sigma_K \ll K^{-(2+\varepsilon)}$ given by (5) is too restrictive and that it can be weakened by $\sigma_K \ll K^{-(\frac{3}{2}+\varepsilon)}$. This conjecture is supported by the discussion in Section 2.3.5 below.

(ii) We actually prove Theorem 2.1 (see Section 4.4.3) under a slightly weaker assumption than the left bound of (5): for a universal constant $C_0$ depending only on $b, \theta, m$ and not on $K$,

$$ \exists \varepsilon > 0, \quad K^{-\varepsilon^2 C_0 \log(K)} \ll \sigma_K \ll K^{-(2+\varepsilon)}. $$

(iii) Assumption (A2)(a) is only used in Section 4.4 and could be weakened with some tail bounds (typically exponential) on $m(x, \cdot)$.

The aim of the rest of the paper is to prove Theorem 2.1.

## 2.3 Sketch and outline of the proof

Note that, the convergence result of Theorem 2.1 takes place on the time scale $1/K\sigma_K^2$. The idea of the proof is based on slow-fast asymptotic analysis techniques developed by KURTZ [57]. Let us begin by introducing the slow and fast dynamics involved in our model, then the difficulties encountered and finally the outline of the proof.

### 2.3.1 Slow-fast asymptotic analysis

Let $a, \alpha \in \mathbb{R}$ and $\mathcal{B}(\mathbb{R})$ the BOREL $\sigma$-field on $\mathbb{R}$. Let us define respectively by $\tau_\alpha$ and $h_a$, the translation operator of vector $\alpha$ and the homothety of ratio $a$. For all $x \in \mathbb{R}$, for all $A \in \mathcal{B}(\mathbb{R})$,

$$ (h_a \circ \tau_\alpha) \sharp \delta_x(A) = \delta_{a(x+\alpha)}(A) \quad \text{and} \quad (\tau_\alpha \circ h_a) \sharp \delta_x(A) = \delta_{ax+\alpha}(A). $$

Note that for all $K \in \mathbb{N}^*$, any $\nu \in \mathcal{M}_{1,K}(\mathbb{R})$ has all its moments finite. We define for all $K \in \mathbb{N}^*$, $\mathcal{M}_{1,K}^c(\mathbb{R})$ the set of centered probability measures of $\mathcal{M}_{1,K}(\mathbb{R})$. From the population process $(\nu_{t/K\sigma_K^2}^K)_{t \ge 0}$, we introduce two evolutionary dynamics:

* The slow dynamics, with value in $\mathbb{R}$, corresponds to that of the mean trait defined by

$$ z_t^K := \langle \mathrm{id}, \nu_{t/K\sigma_K^2}^K \rangle. $$

* The fast dynamics, with value in $\mathcal{M}_{1,K}^c(\mathbb{R})$, corresponds to that of the centered and dilated distribution of traits defined by

$$ \mu_t^K := \left( h_{\frac{1}{\sigma_K \sqrt{K}}} \circ \tau_{-z_t^K} \right) \sharp \nu_{t/K\sigma_K^2}^K = \frac{1}{K} \sum_{i=1}^K \delta_{\frac{1}{\sigma_K \sqrt{K}}} (x_i(t) - z_t^K). \quad (7) $$

$$ \text{Note that } \nu_t^K = \left( \tau_{z_{t/K\sigma_K^2}}^K \circ h_{\sigma_K \sqrt{K}} \right) \sharp \mu_{t/K\sigma_K^2}^K. $$
For all $\ell \in \mathbb{N}$ and $\nu \in \mathcal{M}_1(\mathbb{R})$, we denote by $M_\ell(\nu) := \langle |\mathrm{id}|^\ell, \nu \rangle$ the $\ell^{\mathrm{th}}$ moment of the measure $\nu$. Let us consider for all $\ell \in \mathbb{N}^*$,

$$ \mathcal{M}_1^{c,\ell}(\mathbb{R}) := \left\{ \mu \in \mathcal{M}_1(\mathbb{R}) \mid M_\ell(\mu) < \infty, \langle \mathrm{id}, \mu \rangle = 0 \right\} $$

and the convention $M_\ell(\mu) = +\infty$ if $\mu \notin \mathcal{M}_1^{c,\ell}(\mathbb{R})$. Let us denote by $\mathcal{B}(E, \mathbb{R})$ the set of BOREL functions on the metric space $E$.

In the following, for all $K \in \mathbb{N}^*$, we study the couple of processes $((z_t^K, \mu_t^K))_{t \ge 0}$. Denoting $\mathcal{L}^K$ the infinitesimal generator of the process $((z_t^K, \mu_t^K))_{t \ge 0}$, computed in Proposition 3.1, we will check the following assertions, proved respectively in Propositions 3.2 and 3.3.

(1) If we consider a function $f : (z, \mu) \mapsto f(z) \in \mathcal{C}_b^2(\mathbb{R}, \mathbb{R})$, the generator $\mathcal{L}^K$ satisfies the following decomposition

$$ \mathcal{L}^K f(z, \mu) = \mathcal{L}_{\text{SLOW}} f(z, \mu) + O\left(\frac{1}{K^2} + \frac{M_2(\mu)}{K} + \sigma \sqrt{K} M_3(\mu)\right) \quad (8) $$

where the operator $\mathcal{L}_{\text{SLOW}}$, of the limit slow component, is defined from $\mathcal{C}_b^1(\mathbb{R}, \mathbb{R})$ to $\mathcal{B}(\mathbb{R} \times \mathcal{M}_1^{c,2}(\mathbb{R}), \mathbb{R})$ by

$$ \mathcal{L}_{\text{SLOW}} f(z, \mu) = f'(z) M_2(\mu) \partial_1 \text{Fit}(z, z). \quad (9) $$

(2) If we consider a function $F_\varphi : (z, \mu) \mapsto F_\varphi(\mu) := F(\langle \varphi, \mu \rangle)$ with $F \in \mathcal{C}^3(\mathbb{R}, \mathbb{R})$ and $\varphi \in \mathcal{C}_b^3(\mathbb{R}, \mathbb{R})$, the generator $\mathcal{L}^K$ satisfies the following decomposition

$$ \mathcal{L}^K F_\varphi(z, \mu) = \frac{\theta(z) m_2(z)}{K^2 \sigma^2} \left[ \mathcal{L}_{\text{FVc}}^{\lambda(z)} F_\varphi(z, \mu) + O\left(\frac{1}{\sqrt{K}} + \sigma K^{\frac{3}{2}} M_2(\mu) + \frac{M_3(\mu)}{K}\right) \right] \quad (10) $$

where $\mathcal{L}_{\text{FVc}}^{\lambda(z)}$ is the generator of the centered FLEMING-VIOT process with resampling rate $\lambda(z) := \frac{b(z,z)}{\theta(z)m_2(z)}$, as studied in [16] and whose definition is recalled below. For all $F \in \mathcal{C}^2(\mathbb{R}, \mathbb{R})$ and $g \in \mathcal{C}_b^2(\mathbb{R}, \mathbb{R})$,

$$ \mathcal{L}_{\text{FVc}}^{\lambda} F_g(\mu) := F'(\langle g, \mu \rangle) \left( \left\langle \frac{g''}{2}, \mu \right\rangle + \lambda \left[ \langle g'', \mu \rangle M_2(\mu) - 2 \langle g' \times \mathrm{id}, \mu \rangle \right] \right) \\ + \lambda F''(\langle g, \mu \rangle) \left[ \langle g^2, \mu \rangle - \langle g, \mu \rangle^2 + \langle g', \mu \rangle^2 M_2(\mu) - 2 \langle g', \mu \rangle \langle g \times \mathrm{id}, \mu \rangle \right]. \quad (11) $$

From (10) and (8), note that the fast component $(\mu_{K\sigma^2t}^K)_{t \ge 0} = \left( \left( h_{\frac{1}{\sigma\sqrt{K}}} \circ \tau_{-\langle \mathrm{id}, \nu_t^K \rangle} \right) \sharp \nu_t^K \right)_{t \ge 0}$ moves on the evolutionary time scale $K$ whereas the slow component $(z_{K\sigma^2t}^K)_{t \ge 0} = (\langle \mathrm{id}, \nu_t^K \rangle)_{t \ge 0}$ moves over a much longer evolutionary time scale $1/K\sigma^2$. The different time scales involved in this model can be represented as in Figure 2.1. It follows that the fast component will be the first to stabilise in its equilibrium state. Note that the operator of the slow component (9) depends on the fast variable through the second-order moment $M_2(\mu)$. This is a standard difficulty in slow-fast analysis, usually solved by assuming that once the fast component is stabilised in its
For all $\ell \in \mathbb{N}$ and $\nu \in \mathcal{M}_1(\mathbb{R})$, we denote by $M_\ell(\nu) := \langle |\mathrm{id}|^\ell, \nu \rangle$ the $\ell^{\mathrm{th}}$ moment of the measure $\nu$. Let us consider for all $\ell \in \mathbb{N}^*$,

$$ \mathcal{M}_1^{c,\ell}(\mathbb{R}) := \left\{ \mu \in \mathcal{M}_1(\mathbb{R}) \mid M_\ell(\mu) < \infty, \langle \mathrm{id}, \mu \rangle = 0 \right\} $$

and the convention $M_\ell(\mu) = +\infty$ if $\mu \notin \mathcal{M}_1^{c,\ell}(\mathbb{R})$. Let us denote by $\mathcal{B}(E, \mathbb{R})$ the set of BOREL functions on the metric space $E$.

In the following, for all $K \in \mathbb{N}^*$, we study the couple of processes $((z_t^K, \mu_t^K))_{t \ge 0}$. Denoting $\mathcal{L}^K$ the infinitesimal generator of the process $((z_t^K, \mu_t^K))_{t \ge 0}$, computed in Proposition 3.1, we will check the following assertions, proved respectively in Propositions 3.2 and 3.3.

(i) If we consider a function $f : (z, \mu) \mapsto f(z) \in \mathcal{C}_b^2(\mathbb{R}, \mathbb{R})$, the generator $\mathcal{L}^K$ satisfies the following decomposition

$$ \mathcal{L}^K f(z, \mu) = \mathcal{L}_{\text{SLOW}} f(z, \mu) + O\left(\frac{1}{K^2} + \frac{M_2(\mu)}{K} + \sigma_K \sqrt{K} M_3(\mu)\right) \quad (8) $$

where the operator $\mathcal{L}_{\text{SLOW}}$, of the limit slow component, is defined from $\mathcal{C}_b^1(\mathbb{R}, \mathbb{R})$ to $\mathcal{B}(\mathbb{R} \times \mathcal{M}_1^{c,2}(\mathbb{R}), \mathbb{R})$ by

$$ \mathcal{L}_{\text{SLOW}} f(z, \mu) = f'(z) M_2(\mu) \partial_1 \text{Fit}(z, z). \quad (9) $$

(ii) If we consider a function $F_\varphi : (z, \mu) \mapsto F_\varphi(\mu) := F(\langle \varphi, \mu \rangle)$ with $F \in \mathcal{C}^3(\mathbb{R}, \mathbb{R})$ and $\varphi \in \mathcal{C}_b^3(\mathbb{R}, \mathbb{R})$, the generator $\mathcal{L}^K$ satisfies the following decomposition

$$ \mathcal{L}^K F_\varphi(z, \mu) = \frac{\theta(z) m_2(z)}{K^2 \sigma_K^2} \left[ \mathcal{L}_{\text{FVC}}^{\lambda(z)} F_\varphi(z, \mu) + O\left(\frac{1}{\sqrt{K}} + \sigma_K K^{\frac{3}{2}} M_2(\mu) + \frac{M_3(\mu)}{K}\right) \right] \quad (10) $$

where $\mathcal{L}_{\text{FVC}}^{\lambda(z)}$ is the generator of the centered FLEMING-VIOT process with resampling rate $\lambda(z) := \frac{b(z,z)}{\theta(z)m_2(z)}$, as studied in [18] and whose definition is recalled below. For all $F \in \mathcal{C}^2(\mathbb{R}, \mathbb{R})$ and $g \in \mathcal{C}_b^2(\mathbb{R}, \mathbb{R})$,

$$ \mathcal{L}_{\text{FVC}}^{\lambda} F_g(\mu) := F'\left(\langle g, \mu \rangle\right) \left( \left\langle \frac{g''}{2}, \mu \right\rangle + \lambda \left[ \langle g'', \mu \rangle M_2(\mu) - 2 \langle g' \times \mathrm{id}, \mu \rangle \right] \right) \\ + \lambda F''\left(\langle g, \mu \rangle\right) \left[ \langle g^2, \mu \rangle - \langle g, \mu \rangle^2 + \langle g', \mu \rangle^2 M_2(\mu) - 2 \langle g', \mu \rangle \langle g \times \mathrm{id}, \mu \rangle \right]. \quad (11) $$

From (10) and (8), note that the fast component $(\mu_{K\sigma_K^2 t}^K)_{t \ge 0} = \left( \left( h_{\frac{1}{\sigma_K \sqrt{K}}} \circ \tau_{-\langle \mathrm{id}, \nu_t^K \rangle} \right) \sharp \nu_t^K \right)_{t \ge 0}$ moves on the evolutionary time scale $K$ whereas the slow component $(z_{K\sigma_K^2 t}^K)_{t \ge 0} = (\langle \mathrm{id}, \nu_t^K \rangle)_{t \ge 0}$ moves over a much longer evolutionary time scale $1/K\sigma_K^2$. The different time scales involved in this model can be represented as in Figure 2.1. It follows that the fast component will be the first to stabilise in its equilibrium state. Note that the operator of the slow component (9) depends on the fast variable through the second-order moment $M_2(\mu)$. This is a standard difficulty in slow-fast analysis, usually solved by assuming that once the fast component is stabilised in its
<figure><img src="image_2.png" /><figcaption>Figure 2.1: Different time scales involved in our individual-based model expected in the original time scale of the process $(\nu_t^K)_{t \ge 0}$.</figcaption></figure>

equilibrium state, $M_2(\mu)$ can be characterised in terms of the slow component leading to an autonomous slow dynamics in the limit $K \to +\infty$.

As the fast limit component is driven by a centered FLEMING-VIOT process, we expect that it will inherit ergodicity properties as stated in [16, Section 4]. This reference establishes the existence of a unique invariant probability measure $\pi^\lambda$ for the centered FLEMING-VIOT process and characterises it. In particular, we expect that the averaging principle applied to the drift coefficient in (9) will lead to the averaged drift

$$ \partial_1 \text{Fit}(z_t, z_t) \int_{M_1(\mathbb{R})} M_2(\mu) \pi^{\lambda(z_t)}(d\mu) = \frac{\partial_1 \text{Fit}(z_t, z_t)}{\lambda(z_t)} = \partial_1 \text{Fit}(z_t, z_t) \beta(z_t) m_2(z_t) $$

where $(z_t)_{t \ge 0}$ is the limit slow component (see [16, Corollary 4.16] for the computation of the integral). Formally, by replacing $M_2(\mu)$ by its mean value $\int_{M_1(\mathbb{R})} M_2(\mu) \pi^{\lambda(z_t)}(d\mu)$ in (9), we obtain that (see details in Section 7)

$$ \mathcal{L}_{\text{CEAD}} f(z, \mu) = f'(z) \partial_1 \text{Fit}(z, z) \beta(z) m_2(z) $$

and we recognise the generator of the announced CEAD.

### 2.3.2 Difficulties encountered

The slow-fast analysis relies on a stochastic averaging result that exploits tightness arguments. The classical approach to prove tightness of sequences of laws [5, 40] requires to have uniform pathwise estimates on the moments of the process. In our situation, we note the presence of moments in (8) and (10). However, we could not establish control in expectation of $\sup_{0 \le t \le T} M_2(\mu_t^K)$ for all $T \ge 0$ because of the long time scale which does not allow to exploit the martingale problem associated to the decomposition of $M_2(\mu_t^K)$. This is an important difference, for example with [62] which studies similar forms of processes (individual-based models). Instead, we will first use fine pathwise estimates and expectation bounds on $M_2(\mu_t^K)$ up to the stopping time $\tau^K$ defined for all $K \in \mathbb{N}^*$ by

$$ \tau^K := \check{\tau}^K \wedge \hat{\tau}^K. \qquad (12) $$

where

$$ \hat{\tau}^K := \inf \left\{ t \ge 0 \mid M_2(\mu_t^K) \ge K^\epsilon \right\} \qquad (13) $$
<figure><img src="image_2.png" /><figcaption>Figure 2.1: Different time scales involved in our individual-based model expected in the original time scale of the process $(\nu_t^K)_{t \ge 0}$.</figcaption></figure>

equilibrium state, $M_2(\mu)$ can be characterised in terms of the slow component leading to an autonomous slow dynamics in the limit $K \to +\infty$.

As the fast limit component is driven by a centered FLEMING-VIOT process, we expect that it will inherit ergodicity properties as stated in [18, Section 4]. This reference establishes the existence of a unique invariant probability measure $\pi^\lambda$ for the centered FLEMING-VIOT process and characterises it. In particular, we expect that the averaging principle applied to the drift coefficient in (9) will lead to the averaged drift

$$ \partial_1 \text{Fit}(z_t, z_t) \int_{M_1(\mathbb{R})} M_2(\mu) \pi^{\lambda(z_t)}(d\mu) = \frac{\partial_1 \text{Fit}(z_t, z_t)}{2\lambda(z_t)} = \partial_1 \text{Fit}(z_t, z_t) \frac{\beta(z_t) m_2(z_t)}{2} $$

where $(z_t)_{t \ge 0}$ is the limit slow component (see [18, Corollary 4.16] for the computation of the integral). Formally, by replacing $M_2(\mu)$ by its mean value $\int_{M_1(\mathbb{R})} M_2(\mu) \pi^{\lambda(z_t)}(d\mu)$ in (9), we obtain that (see details in Section 7)

$$ \mathcal{L}_{\text{CEAD}} f(z, \mu) = f'(z) \partial_1 \text{Fit}(z, z) \frac{\beta(z) m_2(z)}{2} $$

and we recognise the generator of the announced CEAD.

### 2.3.2 An example

The goal of this section is to illustrate numerically the phenomenon of the CEAD. We have performed numerical simulations of the process $\nu^K$ for a simple model from the biological literature. The simulation is based on the IBMPopSim package developed by GIORGI ET AL in [50].

The biological model is adapted from KISDI [56] for which the trait space is $\mathcal{X} := [0, 4]$ (and not $\mathbb{R}$ as previously) and the parameters are:

$$ \theta(x) := 1, \quad c(x, y) := 1 - \frac{1}{1 + 1.2 \exp(-4(x - y))}, \quad b(x, y) := (4 - y) \times c(x, y) \quad (12) $$

and $m(x, dh)$ is a Gaussian distribution $N(x, \sigma_K^2)$ conditional on the mutant trait remaining in $\mathcal{X}$.

Let us recall that $b(x, y)$ is a resampling rate, i.e. the rate of simultaneous death of an individual with trait $x$ and birth of an individual with trait $y$ and it can be interpreted as
with $\varepsilon > 0$ given in Theorem 2.1 and where

$$ \check{\tau}^K := \inf \left\{ t \ge 0 \mid \mathrm{Diam} \left( \mathrm{Supp} \, \mu_t^K \right) > \frac{1}{\sigma K^{\frac{3+\varepsilon}{2}}} \right\} \quad (14) $$

which allows to control the diameter of the support of the centered and dilated distribution of the traits $\mu^K$. An important difficulty arises in Lemma 3.6 from the presence of third order moment in the DOOB semi-martingale decomposition of $M_2(\mu_t^K)$:

$$ M_2(\mu_t^K) = M_2(\mu_0^K) - \frac{1}{K^2\sigma^2} \int_0^t \left\{ 2b(z_s^K, z_s^K) M_2(\mu_s^K) - \theta(z_s^K) m_2(z_s^K) \right\} ds \\ + \frac{1}{K^2\sigma^2} \int_0^t O\left(\frac{1}{K} + \sigma K M_2(\mu_s^K) + \sigma K^{\frac{3}{2}} M_3(\mu_s^K)\right) ds + M_t^{K,P_{\mathrm{id}2,1}} $$

where $(M_t^{K,P_{\mathrm{id}2,1}})_{t \ge 0}$ is a local-martingale. This leads to introduce the stopping time $\check{\tau}^K$ which guarantees that the error term in $M_3(\mu_t^K)$ remains under control. This leads to a new difficulty: establishing that $\check{\tau}^K$ tends to $+\infty$ in probability when $K \to +\infty$. To prove this, we use pathwise estimates on $M_2(\mu_t^K)$ up to time $\check{\tau}^K$ to estimate the different transitions of $M_2(\mu_t^K)$ between thresholds of the form $3^{\ell-1}K^{\frac{\varepsilon}{2}}$ and $3^{\ell+1}K^{\frac{\varepsilon}{2}}$ and to construct a coupling between these transitions and biased random walks. This allows us to use large deviations results on random walks and estimates on the exit from an attracting domain (see e.g. [41, 25]) to prove that the stopping time $\hat{\tau}^K$, defined by (13), converges to $+\infty$ in probability when $K \to +\infty$. This in turn implies that $\check{\tau}^K \to +\infty$ in probability when $K \to +\infty$ (see Section 2.3.4 for a discussion of the method used for this step).

The implementation of the slow-fast method of KURTZ [49] is done in two steps. To establish the tightness of the sequence of laws of the slow component, we exploit criteria developed by ETHIER-KURTZ [34, Theorems 3.9.1 and 3.9.4] by restricting ourselves to the torus case. This strategy allows to overcome the difficulty related to the verification of the compact containment condition, on the real line, of the slow component stopped at the stopping time $\tau^K$. The second step consists in characterising each accumulation point of the sequence of laws of the slow dynamics and the occupation measure of the fast dynamics. For this, we need to check that any measure $\gamma$ in $M_1(M_1(\mathbb{R}))$ satisfying

$$ \int_{M_1(\mathbb{R})} \mathcal{L}_{FVc}^{\lambda} \phi(\mu) \gamma(d\mu) = 0 \quad (15) $$

for a certain class $\mathcal{F}$ of functions $\phi$ must be $\pi^{\lambda}$. Equation (10) suggests to take $\mathcal{F}$ as the set of functions of the form $F_{\phi}(\mu) := F(\langle\phi, \mu\rangle)$. However, the last property seems hard to prove for this choice of $\mathcal{F}$. Instead, we adapt a result of DAWSON [22, Theorem 2.7.1] that applies to the set $\mathcal{F}$ of so-called *polynomial functions* of the form

$$ P_{f,n}(\mu) := \int_{\mathbb{R}} \cdots \int_{\mathbb{R}} f(x_1, \cdots, x_n) \mu(dx_1) \cdots \mu(dx_n). $$

However the duality property used by DAWSON in [22] does not hold in our case. In [16], it is proved only a weak duality relation involving stopping times. This difficulty will be solved by proving that the measure $\gamma$ in (15) gives mass only to measures having its first five moments finite. This will allow us to characterise the limit fast component and hence the limit slow component on the torus, as solution to an ODE. Since this ODE is non-explosive, we choose the torus large enough to conclude the proof on the real line.
modelling a competitive pressure driving selection. Here, it is composed of two terms which both have a biological interpretation. Note that the first term $c(x, y)$ depends only on $x - y$ and tends to 0 when $x - y \to +\infty$ and to 1 when $x - y \to -\infty$. So, this function models an *asymmetric competition* in favour of high traits: $x$ feels little competition from a smaller trait $y$, but strong competition from a higher trait $y$. Since the second term $4 - y$ only depends on the reproducing trait and is favourable to small traits, evolution is expected to select an optimal trait.

We consider the previous model with three scaling parameters: $K$ (for the population size) and $\sigma_K$ (for the size of mutation) as in Section 2.1, and a parameter $p_K$ giving the probability of mutation at each birth event. This amouts to replace in Section 2.1 the mutation kernel $m(x, h)dh$ by $p_K m(x, h)dh + (1-p_K)\delta_x(dh)$. We performed simulations of KISDI's model, starting with a monomorphic initial condition of trait 1 and varying the parameters $K, p_K$ and $\sigma_K$. Some of these simulations are presented in the following figures and show a wide variety of evolutionary behaviours.

Note that, when $K$ and $\sigma_K$ are not too large and $p_K$ not too small, the population evolves according to a relatively stable general scenario as in Figures 2.3, 2.4 and 2.5: it initially remains concentrated around the trait value equal to 1 that progressively moves towards a trait $x^*$ close to 3.5 corresponding to the optimal trait value of this population, in the sense that it is best adapted to survive and reproduce. Once this point is reached, the population stabilises in its steady state. In other words, Figures 2.4 and 2.5 show the long-term behaviour of the mean trait of the population, i.e. the one of the slow component. Figure 2.6 shows the behaviour of the fast component by zooming in the initial dynamics of Figure 2.5. Note that Figure 2.2 illustrates the approach developed by [20].

<figure><img src="image_3.png" /><figcaption>Figure 2.2: Numerical simulation of the trait distribution of the microscopic model with parameters (12) and $K = 300$, $p_K = 0.00003$, $\sigma_K = 0.05$.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 2.3: Numerical simulation of the trait distribution of the microscopic model with parameters (12) and $K = 300$, $p_K = 1$, $\sigma_K = 0.05$.</figcaption></figure>
### 2.3.3 Outline of the proof

In Section 3, we begin by giving an approximation of the infinitesimal generator of the slow-fast process $((z_t^K, \mu_t^K))_{t \ge 0}$ for a class of test functions on $\mathbb{R} \times \mathcal{M}_{1,K}^c(\mathbb{R})$, large enough to be convergence determining and we characterise martingales associated to our process. We give also martingale problems of the slow-fast process for polynomials in $\mu$. In Section 4, we prove some moment estimates. In Section 4.1, we give estimates of the moment of order 6. In Section 4.2, we establish some inequalities on $M_2(\mu_t^K)$ and we control its bracket that we will use in Section 4.3 to prove $\mathbb{P}$-a.s. that $M_2(\mu_t^K)$ takes superlinear (of the order of $K^{C \log(K)}$) long time before hitting $K^\epsilon$. The fact that $\tau^K \to +\infty$ in probability when $K \to +\infty$ is proved in Section 4.3. The rest of the proof deals with the compactness-uniqueness argument associated to our slow-fast problem. Firstly, in Section 5, we establish uniform tightness of the sequence of laws of $(z^K, \Gamma^K)$ stopped at time $\check{\tau}^K$ in the torus case where $\Gamma^K$ designates the occupation measure of the process $\mu^K$. In Section 6, we identify and characterise in a unique way the limiting distribution of the fast component on the torus. In Section 7.1, we proceed similarly for the slow component. Thanks to the uniqueness on the torus of the limit slow component, we deduce in Section 7.2 the announced result of Theorem 2.1.

### 2.3.4 Prospects

In the first part of this section, we explain the origin of exponent 2 in the inequality $\sigma \ll K^{-(2+\epsilon)}$ of Assumption (5). In the second part, it is explained how Assumption (5) may be improved into the assumption $\sigma \ll K^{-(\frac{3}{2}+\epsilon)}$ by using estimates for moments of order $2\ell$. Finally, in the last part of this section, we explain the difficulties in obtaining these moment estimates. Note that the assumption $\sigma \ll K^{-\frac{3}{2}}$ is the best we can expect because of the error term $\sigma K^{\frac{3}{2}}$ in (10).

(a) **Origin of the exponent 2 in Assumption (5).** On the one hand, as for all $t \ge 0$ and $K \in \mathbb{N}^*$, $\text{Diam}(\text{Supp } \mu_t^K)^2 \le (2 \max \{|x| \mid x \in \text{Supp } \mu_t^K\})^2 \le 4KM_2(\mu_t^K)$, we deduce that

$$ \begin{aligned} \mathbb{P}(\check{\tau}^K < \hat{\tau}^K \wedge T) &\le \mathbb{P}(\exists t < T \wedge \hat{\tau}^K, \text{Diam}(\text{Supp } \mu_t^K) > \frac{1}{\sigma K^{\frac{3+\epsilon}{2}}}) \\ &\le \mathbb{P}(\exists t < T \wedge \hat{\tau}^K, M_2(\mu_t^K) > \frac{1}{4\sigma^2 K^{4+\epsilon}}) \end{aligned} $$

On the other hand, note that thanks to the definition (13) of the stopping time $\hat{\tau}^K$,

$$ \mathbb{P}(\exists t < T \wedge \hat{\tau}^K, M_2(\mu_t^K) > K^\epsilon) = 0. $$

Hence, Assumption (5) implies that $\mathbb{P}(\check{\tau}^K < \hat{\tau}^K \wedge T) = 0$ for $K$ large enough. Using this relation, we prove in Section 4.3 that $\mathbb{P}(\tau^K < T) \le \mathbb{P}(\hat{\tau}^K \le T \wedge \check{\tau}^K)$. Hence, to establish that $\tau^K \to +\infty$ when $K \to +\infty$ in probability, we need to prove that

$$ \lim_{K \to +\infty} \mathbb{P}(\hat{\tau}^K \le T \wedge \check{\tau}^K) = \lim_{K \to +\infty} \mathbb{P}\left(\sup_{0 \le t \le T \wedge \check{\tau}^K} M_2(\mu_t^K) \ge K^\epsilon\right) = 0 $$

as established in Section 4.4.
<figure><img src="image_5.png" /><figcaption>Figure 2.4: Numerical simulation of the trait distribution of the microscopic model with parameters (12) and $K = 300$, $p_K = 1$, $\sigma_K = 0.001$.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 2.5: Numerical simulation of the trait distribution of the microscopic model with parameters (12) and $K = 300$, $p_K = 1$, $\sigma_K = 0.0005$.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 2.6: Numerical simulation of the trait distribution of the microscopic model with parameters (12) and $K = 300$, $p_K = 1$, $\sigma_K = 0.0005$.</figcaption></figure>
(b) Improvement of Assumption (5) with moments of order $2\ell$. Assume now that $\sigma \ll K^{-\left(\frac{3}{2}+\varepsilon\right)}$. Then, there exists $\ell \in \mathbb{N}^*$, $\tilde{\varepsilon} > 0$ such that

$$ \forall C > 0, \quad K^{-C \log(K)} \ll \sigma \ll K^{-\left(\frac{3}{2} + \frac{1}{2\ell} + \frac{\ell+1}{2\ell} \tilde{\varepsilon}\right)} \quad (16) $$

and we consider the stopping times

$$ \hat{\tau}_{\text{bis}}^{K} := \inf \left\{ t \ge 0 \mid M_2 (\mu_t^K) \ge K^{\tilde{\varepsilon}} \right\} \quad \text{and} \quad \check{\tau}_{\text{bis}}^{K} := \inf \left\{ t \ge 0 \mid \text{Diam} (\text{Supp } \mu_t^K) > \frac{1}{\sigma K^{\frac{3+\tilde{\varepsilon}}{2}}} \right\}. $$

Using now that for all $t \ge 0$, for all $K \in \mathbb{N}^*$, for all $\ell \in \mathbb{N}^*$, $\text{Diam} (\text{Supp } \mu_t^K)^{2\ell} \le 2\ell KM_{2\ell} (\mu_t^K)$, we deduce that

$$ \mathbb{P} \left( \exists t < T \wedge \hat{\tau}_{\text{bis}}^K, \text{Diam} (\text{Supp } \mu_t^K) > \frac{1}{\sigma K^{\frac{3+\tilde{\varepsilon}}{2}}} \right) \\ \le \mathbb{P} \left( \exists t < T \wedge \hat{\tau}_{\text{bis}}^K, M_{2\ell} (\mu_t^K) > \frac{1}{2^\ell \sigma^{2\ell} K^{3\ell+1+\ell\tilde{\varepsilon}}} \right), $$

which is zero for $K$ large enough by (16). As previously, if we can prove for all $T \ge 0$ that

$$ \lim_{K \to +\infty} \mathbb{P} \left( \hat{\tau}_{\text{bis}}^{K} \le T \wedge \check{\tau}_{\text{bis}}^{K} \right) = \lim_{K \to +\infty} \mathbb{P} \left( \sup_{0 \le t \le T \wedge \check{\tau}_{\text{bis}}^{K}} M_{2\ell} (\mu_t^K) \ge K^{\tilde{\varepsilon}} \right) = 0, \quad (17) $$

then, we can conclude as before.

(c) **Difficulties encountered.** The main difficulty consists in proving (17). For this we could seek for an extension of Lemma 4.1 to the framework of moments of order $2\ell$. Lemma 4.1 relies on the inequality for all $t \ge 0$, $K$ large enough,

$$ \begin{aligned} & \frac{3}{4} M_6 (\mu_{t \wedge \check{\tau}^K}) + 3 M_4 (\mu_{t \wedge \check{\tau}^K}^K) M_2 (\mu_{t \wedge \check{\tau}^K}^K) + \frac{3}{2} M_2^3 (\mu_{t \wedge \check{\tau}^K}^K) \\ & \le \frac{3}{4} M_6 (\mu_0^K) + 7 M_4 (\mu_0^K) M_2 (\mu_0^K) + \frac{3}{2} M_2^3 (\mu_{t \wedge \check{\tau}^K}) \\ & \quad - \frac{C_1}{K^2 \sigma^2} \int_0^{t \wedge \check{\tau}^K} \left( \frac{3}{4} M_6 (\mu_{t \wedge \check{\tau}^K}) + 3 M_4 (\mu_s^K) M_2 (\mu_s^K) + \frac{3}{2} M_2^3 (\mu_s^K) - C_2 \right) ds + \text{Mart}_{t \wedge \check{\tau}^K} \end{aligned} $$

for some constants $C_1, C_2 > 0$ and where $(\text{Mart}_{t \wedge \check{\tau}^K})_{t \ge 0}$ is a martingale. However a similar calculation does not seem to be successful for moments of order 8 or more.

## 3 Infinitesimal generator approximation

Let us begin this section by giving the generator of the couple of process $(z^K, \mu^K)$. Then, we give an approximation of the previous formula for a class of test functions which is convergence determining. Finally, we give an extension to this result in the case of polynomials in $\mu$.
### 2.3.3 Difficulties encountered

The slow-fast analysis relies on a stochastic averaging result that exploits tightness arguments. The classical approach to prove tightness of sequences of laws [5, 46] requires to have uniform pathwise estimates on the moments of the process. In our situation, we note the presence of moments in (8) and (10). However, we could not establish control in expectation of $sup_{0 \le t \le T} M_2(\mu_t^K)$ for all $T \ge 0$ because of the long time scale which does not allow to exploit the martingale problem associated to the decomposition of $M_2(\mu_t^K)$. This is an important difference, for example with [67] which studies similar forms of processes (individual-based models). Instead, we will first use fine pathwise estimates and expectation bounds on $M_2(\mu_t^K)$ up to the stopping time $\tau^K$ defined for all $K \in \mathbb{N}^*$ by

$$ \tau^K := \check{\tau}^K \wedge \hat{\tau}^K. \qquad (13) $$

where

$$ \hat{\tau}^K := \inf \left\{ t \ge 0 \mid M_2(\mu_t^K) \ge K^\epsilon \right\} \qquad (14) $$

with $\epsilon > 0$ given in Theorem 2.1 and where

$$ \check{\tau}^K := \inf \left\{ t \ge 0 \mid \mathrm{Diam} \left( \mathrm{Supp} \, \mu_t^K \right) > \frac{1}{\sigma_K K^{\frac{3+\epsilon}{2}}} \right\} \qquad (15) $$

which allows to control the diameter of the support of the centered and dilated distribution of the traits $\mu^K$. An important difficulty arises in Lemma 3.6 from the presence of third order moment in the DOOB semi-martingale decomposition of $M_2(\mu_t^K)$:

$$ M_2(\mu_t^K) = M_2(\mu_0^K) - \frac{1}{K^2 \sigma_K^2} \int_0^t \left\{ 2b(z_s^K, z_s^K) M_2(\mu_s^K) - \theta(z_s^K) m_2(z_s^K) \right\} ds \\ + \frac{1}{K^2 \sigma_K^2} \int_0^t O \left( \frac{1}{K} + \sigma_K K M_2(\mu_s^K) + \sigma_K K^{\frac{3}{2}} M_3(\mu_s^K) \right) ds + M_t^{K, P_{\mathrm{id}2,1}} $$

where $(M_t^{K, P_{\mathrm{id}2,1}})_{t \ge 0}$ is a local-martingale. This leads to introduce the stopping time $\check{\tau}^K$ which guarantees that the error term in $M_3(\mu_t^K)$ remains under control. This leads to a new difficulty: establishing that $\check{\tau}^K$ tends to $+\infty$ in probability when $K \to +\infty$. To prove this, we use pathwise estimates on $M_2(\mu_t^K)$ up to time $\check{\tau}^K$ to estimate the different transitions of $M_2(\mu_t^K)$ between thresholds of the form $3^{\ell-1} K^{\frac{\epsilon}{2}}$ and $3^{\ell+1} K^{\frac{\epsilon}{2}}$ and to construct a coupling between these transitions and biased random walks. This allows us to use large deviations results on random walks and estimates on the exit from an attracting domain (see e.g. [47, 29]) to prove that the stopping time $\hat{\tau}^K$, defined by (14), converges to $+\infty$ in probability when $K \to +\infty$. This in turn implies that $\check{\tau}^K \to +\infty$ in probability when $K \to +\infty$ (see Section 2.3.5 for a discussion of the method used for this step).

The implementation of the slow-fast method of KURTZ [57] is done in two steps. To establish the tightness of the sequence of laws of the slow component, we exploit criteria developed by ETHIER-KURTZ [40, Theorems 3.9.1 and 3.9.4] by restricting ourselves to the torus case. This strategy allows to overcome the difficulty related to the verification of the compact containment condition, on the real line, of the slow component stopped at the stopping time $\tau^K$. The second
## 3.1 Generators and martingales

For all $K \in \mathbb{N}^*$, let us introduce the filtration $(\mathcal{F}_t^K)_{t \ge 0}$ defined by $\mathcal{F}_t^K := \sigma(z_s^K, \mu_s^K \mid s \le t)$.

**Proposition 3.1.** The infinitesimal generator of the $\mathbb{R} \otimes \mathcal{M}_{1,K}^c(\mathbb{R})$-valued MARKOV process $(z^K, \mu^K)$, defined for all bounded measurable function $\Phi$ from $\mathbb{R} \times \mathcal{M}_{1,K}^c(\mathbb{R})$ to $\mathbb{R}$, is given by

$$ \mathcal{L}^K \Phi (z, \mu) = \frac{K}{K \sigma^2} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) b \left( \sigma \sqrt{K} x + z, \sigma \sqrt{K} y + z \right) \\ \times \left[ \Phi \left( z + \frac{\sigma \sqrt{K}}{K} (y-x), \tau_{-\frac{y-x}{K}} \sharp \left[ \mu - \frac{\delta_x}{K} + \frac{\delta_y}{K} \right] \right) - \Phi(z, \mu) \right] \\ + \frac{K}{K \sigma^2} \int_{\mathbb{R}} \mu(dx) \theta \left( \sigma \sqrt{K} x + z \right) \int_{\mathbb{R}} m \left( \sigma \sqrt{K} x + z, h \right) \\ \times \left[ \Phi \left( z + \frac{\sigma \sqrt{K} h}{K^{\frac{3}{2}}}, \tau_{-\frac{h}{K^{3/2}}} \sharp \left[ \mu - \frac{\delta_x}{K} + \frac{\delta_{x+\frac{h}{\sqrt{K}}}}{K} \right] \right) - \Phi(z, \mu) \right] dh $$

Moreover, for all $K \in \mathbb{N}^*$, for all bounded measurable function $\Phi$ from $\mathbb{R} \times \mathcal{M}_{1,K}^c(\mathbb{R})$ to $\mathbb{R}$, the following process $(M_t^{K,\Phi})_{t \ge 0}$ defined by

$$ M_t^{K,\Phi} := \Phi(z_t^K, \mu_t^K) - \Phi(z_0^K, \mu_0^K) - \int_0^t \mathcal{L}^K \Phi(z_s^K, \mu_s^K) ds \quad (18) $$

is a $(\mathcal{F}_t^K)_{t \ge 0}$-martingale, square integrable, with quadratic variation:

$$ \langle M_t^{K,\Phi} \rangle_t = \frac{K}{K\sigma^2} \int_0^t ds \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) b \left( \sigma\sqrt{K}x + z, \sigma\sqrt{K}y + z \right) \\ \times \left[ \Phi \left( z + \frac{\sigma\sqrt{K}}{K}(y-x), \tau_{-\frac{y-x}{K}} \sharp \left[ \mu - \frac{\delta_x}{K} + \frac{\delta_y}{K} \right] \right) - \Phi(z, \mu) \right]^2 \\ + \frac{K}{K\sigma^2} \int_0^t ds \int_{\mathbb{R}} \mu(dx) \theta \left( \sigma\sqrt{K}x + z \right) \int_{\mathbb{R}} m \left( \sigma\sqrt{K}x + z, h \right) \\ \times \left[ \Phi \left( z + \frac{\sigma\sqrt{K}h}{K^{\frac{3}{2}}}, \tau_{-\frac{h}{K^{3/2}}} \sharp \left[ \mu - \frac{\delta_x}{K} + \frac{\delta_{x+\frac{h}{\sqrt{K}}}}{K} \right] \right) - \Phi(z, \mu) \right]^2 dh $$

Note that the factor $1/K\sigma^2$ corresponds to the time scaling of $\nu_t$ used to define $z_t$ and $\mu_t$.

**Proof. Step 1. About the generators.** On the one hand, from (3), note that for all $\Phi : \mathbb{R} \times \mathcal{M}_{1,K}^c(\mathbb{R}, \mathbb{R}) \to \mathbb{R}$ and $\phi : \mathcal{M}_{1,K}(\mathbb{R}) \to \mathbb{R}$ such that $\phi(\nu) = \Phi(\langle \mathrm{id}, \nu \rangle, (h_{\frac{1}{\sigma\sqrt{K}}} \circ \tau_{-\langle \mathrm{id}, \nu \rangle} \sharp \nu)$,

$$ \mathcal{L}^K \Phi (z, \mu) = \frac{1}{K \sigma^2} \times L^K, \sigma \phi \left( (\tau_z \circ h_{\sigma \sqrt{K}}) \sharp \mu \right). \quad (19) $$
step consists in characterising each accumulation point of the sequence of laws of the slow dynamics and the occupation measure of the fast dynamics. For this, we need to check that any measure $\gamma$ in $M_1(M_1(\mathbb{R}))$ satisfying

$$ \int_{M_1(\mathbb{R})} \mathcal{L}_{FVc}^{\lambda} \phi(\mu) \gamma(d\mu) = 0 \qquad (16) $$

for a certain class $\mathcal{F}$ of functions $\phi$ must be $\pi^{\lambda}$. Equation (10) suggests to take $\mathcal{F}$ as the set of functions of the form $F_{\phi}(\mu) := F(\langle\phi, \mu\rangle)$. However, the last property seems hard to prove for this choice of $\mathcal{F}$. Instead, we adapt a result of DAWSON [26, Theorem 2.7.1] that applies to the set $\mathcal{F}$ of so-called polynomial functions of the form

$$ P_{f,n}(\mu) := \int_{\mathbb{R}} \cdots \int_{\mathbb{R}} f(x_1, \cdots, x_n) \mu(dx_1) \cdots \mu(dx_n). $$

However the duality property used by DAWSON in [26] does not hold in our case. In [18], it is proved only a weak duality relation involving stopping times. This difficulty will be solved by proving that the measure $\gamma$ in (16) gives mass only to measures having its first four moments finite. This will allow us to characterise the limit fast component and hence the limit slow component on the torus, as solution to an ODE. Since this ODE is non-explosive, we choose the torus large enough to conclude the proof on the real line.

### 2.3.4 Outline of the proof

In Section 3, we begin by giving in Proposition 3.1 an approximation of the infinitesimal generator of the slow-fast process $((z_t^K, \mu_t^K))_{t \ge 0}$ for a class of test functions on $\mathbb{R} \times M_{1,K}^c(\mathbb{R})$, large enough to be convergence determining and we characterise martingales associated to our process. We give also martingale problems of the slow-fast process for polynomials in $\mu$ where the main results are given in Lemma 3.4 for bounded test functions and Lemmas 3.6 and 3.7 for unbounded test functions. In Section 4, we prove some moment estimates. In Section 4.1, we give estimates of the moment of order 6. The most important result is given by Lemma 4.2 because consequences are useful for all results in Section 4 as well as in Theorem 5.1. In Section 4.2, we establish some inequalities on $M_2(\mu_t^K)$ and we control its bracket that we will use in Section 4.3 to prove $\mathbb{P}$-a.s. that $M_2(\mu_t^K)$ takes superlinear (of the order of $K^{c\log(K)}$) long time before hitting $K^{\epsilon}$. The fact that $\tau^K \to +\infty$ in probability when $K \to +\infty$ is proved in Section 4.3 using the technical Lemma 4.9. Section 4.4 is dedicated to prove this lemma by constructing a coupling between the moment of order 2 and a biased random walk. The rest of the proof deals with the compactness-uniqueness argument associated to our slow-fast problem. Firstly, in Section 5, we establish uniform tightness of the sequence of laws of $((z^K, \Gamma^K))_{K \in \mathbb{N}^*}$ stopped at time $\tau^K$ in the torus case where $\Gamma^K$ designates the occupation measure of the process $\mu^K$. The main result of section is given by Theorem 5.1. In Proposition 6.3 of Section 6, we identify and characterise in a unique way the limiting distribution of the fast component on the torus. Section 6.1 is devoted to proving this result by exploiting the key Lemma 6.4 whose proof is given in Section 6.2. In Lemma 7.1 of Section 7.1, we proceed similarly for the slow component. Thanks to the uniqueness on the torus of the limit slow component, we deduce in Section 7.2 the announced result of Theorem 2.1.
On the other hand, note that for all $\nu \in \mathcal{M}_{1,K}(\mathbb{R})$, $u, v \in \mathbb{R}$, $\psi: \mathbb{R} \to \mathbb{R}$ and $\Psi: \mathcal{M}_{1,K}(\mathbb{R}) \to \mathbb{R}$ measurable bounded functions, we have

$$ 
\begin{aligned}
& \int_{\mathbb{R}} \psi(x) \Psi \left( \nu + \frac{\delta_{x+\sigma\sqrt{K}u}}{K} - \frac{\delta_v}{K} \right) \nu(dx) \\
& = \int_{\mathbb{R}} \psi \left( \sigma\sqrt{K}y + \langle \mathrm{id}, \nu \rangle \right) \Psi \left( \left( \tau_{\langle \mathrm{id}, \nu \rangle} \circ h_{\sigma\sqrt{K}} \right) \sharp \left[ \mu + \frac{\delta_{y+u}}{K} - \frac{\delta_v}{K} \right] \right) \mu(dy).
\end{aligned} 
\quad (20) $$

Finally, by noting that for all $u \in \mathbb{R}$ the mean trait of $(\tau_{\langle \mathrm{id}, \nu \rangle} \circ h_{\sigma\sqrt{K}}) \sharp [\mu - \frac{\delta_x}{K} + \frac{\delta_u}{K}]$ is $\langle \mathrm{id}, \nu \rangle + \frac{\sigma\sqrt{K}}{K}(u-x)$ and the centered and dilated distribution of traits of $(\tau_{\langle \mathrm{id}, \nu \rangle} \circ h_{\sigma\sqrt{K}}) \sharp [\mu - \frac{\delta_x}{K} + \frac{\delta_u}{K}]$ is $\tau_{-\frac{u-x}{K}} \sharp [\mu - \frac{\delta_x}{K} + \frac{\delta_u}{K}]$, the announced result follows from (19) and (20).

**Step 2. About the martingales.** The martingale property follows from classical arguments since $\Phi$ and $\mathcal{L}^K\Phi$ are bounded [34, Chapter 4]. To compute the bracket, we proceed according to the following classical method (see e.g. [40]). We apply (18) replacing $\Phi$ by $\Phi^2$ to obtain the martingale $M^{K,\Phi^2}$. Then, we apply the $\mathrm{IT}\hat{\mathrm{O}}$ formula to compute $\Phi^2(z_t^K, \mu_t^K)$ from (18). We deduce that

$$ \mathrm{Mart}_t := \Phi^2(z_t^K, \mu_t^K) - \Phi^2(z_0^K, \mu_0^K) - 2 \int_0^t \mathcal{L}^K \Phi(z_s^K, \mu_s^K) \mathrm{d}s - \langle M^{K,\Phi} \rangle_t $$

is a martingale. Hence, the martingale $M_t^{K,\Phi^2} - \mathrm{Mart}_t$ has finite variation, so it is null [42, Theorem 4.1], leading to

$$ \langle M^{K,\Phi} \rangle_t = \int_0^t \left[ \mathcal{L}^K \Phi^2(z_s^K, \mu_s^K) - 2\Phi(z_s^K, \mu_s^K) \mathcal{L}^K \Phi(z_s^K, \mu_s^K) \right] \mathrm{d}s $$

then the announced conclusion. $\square$

## 3.2 Asymptotic expansions

Recall that, a set $S \subset \mathcal{C}_b^0(\mathbb{R}, \mathbb{R})$ is called $\mathcal{M}_{1,K}^c(\mathbb{R})$-convergence determining if whenever $(P_n)_{n \in \mathbb{N}} \in \mathcal{M}_{1,K}^c(\mathbb{R})^N$, $P \in \mathcal{M}_{1,K}^c(\mathbb{R})$ and $\lim_{n \to +\infty} \langle f, P_n \rangle = \langle f, P \rangle$ for all $f \in S$, we have that $(P_n)_{n \in \mathbb{N}}$ converges weakly to $P$ [34, Chapter 3, Section 4, p.112]. As developed in [22, Theorem 3.2.6], the class of functions on $\mathcal{M}_{1,K}^c(\mathbb{R})$,

$$ \mathbb{F}_b^2 := \left\{ F_\varphi \mid F_\varphi(\mu) := F(\langle \varphi, \mu \rangle), F \in \mathcal{C}^2(\mathbb{R}, \mathbb{R}), \varphi \in \mathcal{C}_b^2(\mathbb{R}, \mathbb{R}) \right\}. $$

is $\mathcal{M}_{1,K}^c(\mathbb{R})$-convergence determining.

In the following sections, we prove the assertions (1) and (2) of Section 2.3.1.

### 3.2.1 Slow component

**Proposition 3.2.** For all $f \in \mathcal{C}_b^2(\mathbb{R}, \mathbb{R})$, understood as a function of $(z, \mu)$ which depends only on $z$, the infinitesimal generator $\mathcal{L}^K$ defined in Proposition 3.1, satisfies the following decomposition

$$ \mathcal{L}^K f(z, \mu) = \mathcal{L}_{\mathrm{SLOW}} f(z, \mu) + O \left( \frac{1}{K^2} + \frac{M_2(\mu)}{K} + \sigma \sqrt{K} M_3(\mu) \right) $$
### 2.3.5 Prospects

In the first part of this section, we explain the origin of exponent 2 in the inequality $\sigma_K \ll K^{-(2+\varepsilon)}$ of Assumption (5). In the second part, it is explained how Assumption (5) may be improved into the assumption $\sigma_K \ll K^{-(\frac{3}{2}+\varepsilon)}$ by using estimates for moments of order $2\ell$. Finally, in the last part of this section, we explain the difficulties in obtaining these moment estimates. Note that the assumption $\sigma_K \ll K^{-\frac{3}{2}}$ is the best we can expect because of the error term $\sigma_K K^{\frac{3}{2}}$ in (10).

(a) Origin of the exponent 2 in Assumption (5). On the one hand, as for all $t \ge 0$ and $K \in \mathbb{N}^*$, $\text{Diam}(\text{Supp } \mu_t^K)^2 \le (2\max\{|x| \mid x \in \text{Supp } \mu_t^K\})^2 \le 4KM_2(\mu_t^K)$, we deduce that

$$ 
\begin{aligned}
\mathbb{P}(\check{\tau}^K < \hat{\tau}^K \wedge T) &\le \mathbb{P}\left(\exists t < T \wedge \hat{\tau}^K, \text{Diam}(\text{Supp } \mu_t^K) > \frac{1}{\sigma_K K^{\frac{3+\varepsilon}{2}}}\right) \\
&\le \mathbb{P}\left(\exists t < T \wedge \hat{\tau}^K, M_2(\mu_t^K) > \frac{1}{4\sigma_K^2 K^{4+\varepsilon}}\right)
\end{aligned}
$$

On the other hand, note that thanks to the definition (14) of the stopping time $\hat{\tau}^K$,

$$ \mathbb{P}(\exists t < T \wedge \hat{\tau}^K, M_2(\mu_t^K) > K^{\varepsilon}) = 0. $$

Hence, Assumption (5) implies that $\mathbb{P}(\check{\tau}^K < \hat{\tau}^K \wedge T) = 0$ for $K$ large enough. Using this relation, we prove in Section 4.3 that $\mathbb{P}(\tau^K < T) \le \mathbb{P}(\hat{\tau}^K \le T \wedge \check{\tau}^K)$. Hence, to establish that $\tau^K \to +\infty$ when $K \to +\infty$ in probability, we need to prove that

$$ \lim_{K \to +\infty} \mathbb{P}(\hat{\tau}^K \le T \wedge \check{\tau}^K) = \lim_{K \to +\infty} \mathbb{P}\left(\sup_{0 \le t \le T \wedge \check{\tau}^K} M_2(\mu_t^K) \ge K^{\varepsilon}\right) = 0 $$

as established in Section 4.4.

(b) Improvement of Assumption (5) with moments of order $2\ell$. Assume now that $\sigma_K \ll K^{-(\frac{3}{2}+\varepsilon)}$. Then, there exists $\ell \in \mathbb{N}^*$, $\tilde{\varepsilon} > 0$ such that

$$ \forall C > 0, \quad K^{-C \log(K)} \ll \sigma_K \ll K^{-\left(\frac{3}{2} + \frac{1}{2\ell} + \frac{\ell+1}{2\ell} \tilde{\varepsilon}\right)} \quad (17) $$

and we consider the stopping times

$$ \hat{\tau}_{\text{bis}}^K := \inf \left\{ t \ge 0 \mid M_2(\mu_t^K) \ge K^{\tilde{\varepsilon}} \right\} $$

and

$$ \check{\tau}_{\text{bis}}^K := \inf \left\{ t \ge 0 \mid \text{Diam}(\text{Supp } \mu_t^K) > \frac{1}{\sigma_K K^{\frac{3+\tilde{\varepsilon}}{2}}} \right\}. $$

Using now that for all $t \ge 0$, for all $K \in \mathbb{N}^*$, for all $\ell \in \mathbb{N}^*$, $\text{Diam}(\text{Supp } \mu_t^K)^{2\ell} \le 2^{2\ell} KM_{2\ell}(\mu_t^K)$, we deduce that

$$ 
\begin{aligned}
\mathbb{P}\left(\exists t < T \wedge \hat{\tau}_{\text{bis}}^K, \text{Diam}(\text{Supp } \mu_t^K) > \frac{1}{\sigma_K K^{\frac{3+\tilde{\varepsilon}}{2}}}\right) & \\
&\le \mathbb{P}\left(\exists t < T \wedge \hat{\tau}_{\text{bis}}^K, M_{2\ell}(\mu_t^K) > \frac{1}{2^{2\ell} \sigma_K^{2\ell} K^{3\ell+1+\ell\tilde{\varepsilon}}}\right),
\end{aligned}
$$
where the operator $\mathcal{L}_{\text{SLOW}}$ is given by (9).

*Proof.* From Proposition 3.1 for the choice of test functions $\Phi(z, \mu) := f(z)$ we obtain that
$$ \mathcal{L}^K f(z, \mu) = (\mathbf{A})^K + (\mathbf{B})^K \text{ where} $$

$$ (\mathbf{A})^K := \frac{K}{K\sigma^2} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) b\left(\sigma\sqrt{K}x + z, \sigma\sqrt{K}y + z\right) \left[f\left(z + \frac{\sigma\sqrt{K}}{K}(y-x)\right) - f(z)\right], $$

$$ (\mathbf{B})^K := \frac{K}{K\sigma^2} \int_{\mathbb{R}} \mu(dx) \theta\left(\sigma\sqrt{K}x + z\right) \int_{\mathbb{R}} m\left(\sigma\sqrt{K}x + z, h\right) \left[f\left(z + \frac{\sigma\sqrt{K}}{K^{\frac{3}{2}}}h\right) - f(z)\right] dh. $$

Now, we want to decompose and study $(\mathbf{A})^K$. Denoting $(\mathbf{C})^K_{x,y}(z) := f\left(z + \frac{\sigma\sqrt{K}}{K}(y-x)\right) - f(z)$, from TAYLOR'S formula, note that $(\mathbf{A})^K = (\mathbf{A})^K_1 + (\mathbf{A})^K_2 + (\mathbf{A})^K_3$ where

$$ (\mathbf{A})_1^K := \frac{K}{K\sigma^2} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) b(z, z) (\mathbf{C})_{x,y}^K(z), $$

$$ (\mathbf{A})_2^K := \frac{K}{K\sigma^2} \times \sigma\sqrt{K} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) \left(x\partial_1 b(z, z) + y\partial_2 b(z, z)\right) (\mathbf{C})_{x,y}^K(z), $$

$$ (\mathbf{A})_3^K := \frac{K}{K\sigma^2} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) \left[ b\left(\sigma\sqrt{K}x + z, \sigma\sqrt{K}y + z\right) - b(z, z), \right. \\ \left. - \sigma\sqrt{K} \left(x\partial_1 b(z, z) + y\partial_2 b(z, z)\right) \right] (\mathbf{C})_{x,y}^K(z) $$

Using that $(\mathbf{C})^K_{x,y}(z) = f'(z) \frac{\sigma\sqrt{K}}{K}(x-y) + O\left(\frac{\sigma^2}{K}|y-x|^2\right)$, we deduce that

$$ (\mathbf{A})_1^K = O\left(\frac{M_2(\mu)}{K}\right), $$

$$ (\mathbf{A})_2^K = f'(z) (\partial_2 b(z, z) - \partial_1 b(z, z)) M_2(\mu) + O\left(\frac{\sigma\sqrt{K} M_3(\mu)}{K}\right), $$

and noting that $|(\mathbf{C})^K_{x,y}| \le C_1 \frac{\sigma\sqrt{K}}{K} |y-x|$ for some constant $C_1 > 0$ and

$$ \left| b\left(\sigma\sqrt{K}x + z, \sigma\sqrt{K}y + z\right) - b(z, z) - \sigma\sqrt{K} \left(x\partial_1 b(z, z) + y\partial_2 b(z, z)\right) \right| \le C_2 \sigma^2 K (x^2 + y^2) $$

for some constant $C_2 > 0$, we deduce that $(\mathbf{A})^K_3 = O(\sigma\sqrt{K}M_3(\mu))$. Therefore,

$$ (\mathbf{A})^K = f'(z) (\partial_2 b(z, z) - \partial_1 b(z, z)) M_2(\mu) + O\left(\frac{M_2(\mu)}{K} + \sigma\sqrt{K} M_3(\mu)\right). $$

As previously, we obtain that

$$ (\mathbf{B})^K = O\left(\frac{\overline{\theta m_2}}{K^2}\right). $$

and the announced result follows. $\square$
which is zero for $K$ large enough by (17). As previously, if we can prove for all $T \ge 0$ that

$$ \lim_{K \to +\infty} \mathbb{P}(\hat{\tau}_{\text{bis}}^K \le T \wedge \check{\tau}_{\text{bis}}^K) = \lim_{K \to +\infty} \mathbb{P}\left(\sup_{0 \le t \le T \wedge \check{\tau}_{\text{bis}}^K} M_{2\ell}(\mu_t^K) \ge K^{\tilde{\epsilon}}\right) = 0, \quad (18) $$

then, we can conclude as before.

(c) *Difficulties encountered.* The main difficulty consists in proving (18). For this we could seek for an extension of Lemma 4.2 to the framework of moments of order $2\ell$. Lemma 4.2 relies on the inequality for all $t \ge 0$, $K$ large enough,

$$ \begin{aligned} & \frac{3}{4} M_6 (\mu_{t \wedge \check{\tau}^K}) + 3 M_4 (\mu_{t \wedge \check{\tau}^K}^K) M_2 (\mu_{t \wedge \check{\tau}^K}^K) + \frac{3}{2} M_2^3 (\mu_{t \wedge \check{\tau}^K}^K) \\ & \le \frac{3}{4} M_6 (\mu_0^K) + 7 M_4 (\mu_0^K) M_2 (\mu_0^K) + \frac{3}{2} M_2^3 (\mu_{t \wedge \check{\tau}^K}) \\ & \quad - \frac{C_1}{K^2 \sigma_K^2} \int_0^{t \wedge \check{\tau}^K} \left( \frac{3}{4} M_6 (\mu_{t \wedge \check{\tau}^K}) + 3 M_4 (\mu_s^K) M_2 (\mu_s^K) + \frac{3}{2} M_2^3 (\mu_s^K) - C_2 \right) ds + \text{Mart}_{t \wedge \check{\tau}^K} \end{aligned} $$

for some constants $C_1, C_2 > 0$ and where $(\text{Mart}_{t \wedge \check{\tau}^K})_{t \ge 0}$ is a martingale. However a similar calculation does not seem to be successful for moments of order 8 or more.

# 3 Infinitesimal generator approximation

Let us begin this section by giving the generator of the couple of process $(z^K, \mu^K)$. Then, we give an approximation of the previous formula for a class of test functions which is convergence determining. Finally, we give an extension to this result in the case of polynomials in $\mu$.

## 3.1 Generators and martingales

For all $K \in \mathbb{N}^*$, let us introduce the filtration $(\mathcal{F}_t^K)_{t \ge 0}$ defined by $\mathcal{F}_t^K := \sigma(z_s^K, \mu_s^K \mid s \le t)$.

**Proposition 3.1.** The infinitesimal generator of the $\mathbb{R} \times \mathcal{M}_{1,K}^c(\mathbb{R})$-valued MARKOV process $(z^K, \mu^K)$, defined for all bounded measurable function $\Phi$ from $\mathbb{R} \times \mathcal{M}_{1,K}^c(\mathbb{R})$ to $\mathbb{R}$, is given by

$$ \begin{aligned} \mathcal{L}^K \Phi (z, \mu) = & \frac{K}{K \sigma_K^2} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) b \left( \sigma_K \sqrt{K} x + z, \sigma_K \sqrt{K} y + z \right) \\ & \times \left[ \Phi \left( z + \frac{\sigma_K \sqrt{K}}{K} (y-x), \tau_{-\frac{y-x}{K}} \sharp \left[ \mu - \frac{\delta_x}{K} + \frac{\delta_y}{K} \right] \right) - \Phi(z, \mu) \right] \\ & + \frac{K}{K \sigma_K^2} \int_{\mathbb{R}} \mu(dx) \theta \left( \sigma_K \sqrt{K} x + z \right) \int_{\mathbb{R}} m \left( \sigma_K \sqrt{K} x + z, h \right) \\ & \times \left[ \Phi \left( z + \frac{\sigma_K \sqrt{K} h}{K^{\frac{3}{2}}}, \tau_{-\frac{h}{K^{3/2}}} \sharp \left[ \mu - \frac{\delta_x}{K} + \frac{\delta_{x+\frac{h}{\sqrt{K}}}}{K} \right] \right) - \Phi(z, \mu) \right] dh. \end{aligned} $$

Moreover, for all $K \in \mathbb{N}^*$, for all bounded measurable function $\Phi$ from $\mathbb{R} \times \mathcal{M}_{1,K}^c(\mathbb{R})$ to $\mathbb{R}$, the following process $(M_t^{K,\Phi})_{t \ge 0}$ defined by

$$ M_t^{K,\Phi} := \Phi(z_t^K, \mu_t^K) - \Phi(z_0^K, \mu_0^K) - \int_0^t \mathcal{L}^K \Phi(z_s^K, \mu_s^K) ds \quad (19) $$
### 3.2.2 Fast component

**Proposition 3.3.** For all $F \in \mathcal{C}^3(\mathbb{R}, \mathbb{R})$ and $\varphi \in \mathcal{C}_b^3(\mathbb{R}, \mathbb{R})$, the infinitesimal generator $\mathcal{L}^K$ defined in Proposition 3.1, satisfies the following decomposition

$$ \mathcal{L}^K F_{\varphi}(z, \mu) = \frac{\theta(z)m_2(z)}{K^2\sigma^2} \left[ \mathcal{L}_{\text{FVc}}^{\lambda(z)} F_{\varphi}(z, \mu) + O\left(\frac{1}{\sqrt{K}} + \sigma K^{\frac{3}{2}} M_2(\mu) + \frac{M_3(\mu)}{K}\right) \right] $$

where the operator $\mathcal{L}_{\text{FVc}}^{\lambda(z)}$ is given by (11) and $F_{\varphi}$ is understood as a function of $(z, \mu)$ which depends only on $\mu$.

*Proof.* Noting that

$$ \begin{aligned} (\mathbf{D})_{x,y}^K &:= \left\langle \varphi, \tau_{-\frac{y-x}{K}} \sharp \left[ \mu - \frac{\delta_x}{K} + \frac{\delta_y}{K} \right] \right\rangle - \langle \varphi, \mu \rangle \\ &= \left\langle \varphi \circ \tau_{-\frac{y-x}{K}}, \mu \right\rangle + \frac{1}{K} \left\{ \varphi\left(y - \frac{y-x}{K}\right) - \varphi\left(x - \frac{y-x}{K}\right) \right\} - \langle \varphi, \mu \rangle, \end{aligned} $$

setting $(\mathbf{C})_{x,y}^K := F(\langle \varphi, \mu \rangle + (\mathbf{D})_{x,y}^K) - F(\langle \varphi, \mu \rangle)$ and from Proposition 3.1 we obtain that $\mathcal{L}^K F_{\varphi}(z, \mu) = (\mathbf{A})^K + (\mathbf{B})^K$ where

$$ \begin{aligned} (\mathbf{A})^K &:= \frac{K}{K\sigma^2} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy)b\left(\sigma\sqrt{K}x + z, \sigma\sqrt{K}y + z\right) (\mathbf{C})_{x,y}^K, \\ (\mathbf{B})^K &:= \frac{K}{K\sigma^2} \int_{\mathbb{R}} \mu(dx)\theta\left(\sigma\sqrt{K}x + z\right) \int_{\mathbb{R}} m\left(\sigma\sqrt{K}x + z, h\right) (\mathbf{C})_{x,x+h/\sqrt{K}}^K dh. \end{aligned} $$

**Step 1. Decomposition and study of $(\mathbf{A})^K$.** From TAYLOR'S formula, we obtain that $(\mathbf{A})^K = (\mathbf{A})_1^K + (\mathbf{A})_2^K$ where

$$ \begin{aligned} (\mathbf{A})_1^K &:= \frac{K}{K\sigma^2} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) b(z, z) (\mathbf{C})_{x,y}^K, \\ (\mathbf{A})_2^K &:= \frac{K}{K\sigma^2} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) \left[ b\left(\sigma\sqrt{K}x + z, \sigma\sqrt{K}y + z\right) - b(z, z) \right] (\mathbf{C})_{x,y}^K. \end{aligned} $$

Denoting $(\mathbf{E})_{x,y}^K := \frac{1}{K}(\varphi(y) - \varphi(x) - (y-x)\langle \varphi', \mu \rangle)$ and from TAYLOR'S formula again, we obtain that $(\mathbf{A})_1^K = (\mathbf{A})_{11}^K + (\mathbf{A})_{12}^K + (\mathbf{A})_{13}^K + (\mathbf{A})_{14}^K$ where

$$ \begin{aligned} (\mathbf{A})_{11}^K &:= \frac{K}{K\sigma^2} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) b(z, z) F'\left(\langle \varphi, \mu \rangle\right) (\mathbf{D})_{x,y}^K, \\ (\mathbf{A})_{12}^K &:= \frac{K}{2K\sigma^2} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) b(z, z) F''\left(\langle \varphi, \mu \rangle\right) \left[(\mathbf{E})_{x,y}^K\right]^2, \\ (\mathbf{A})_{13}^K &:= \frac{K}{2K\sigma^2} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) b(z, z) F''\left(\langle \varphi, \mu \rangle\right) \left[[\mathbf{D}]_{x,y}^K\right]^2 - \left[(\mathbf{E})_{x,y}^K\right]^2, \\ (\mathbf{A})_{14}^K &:= \frac{K}{K\sigma^2} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) b(z, z) \left( (\mathbf{C})_{x,y}^K - F'\left(\langle \varphi, \mu \rangle\right) (\mathbf{D})_{x,y}^K - \frac{F''\left(\langle \varphi, \mu \rangle\right)}{2} \left[(\mathbf{D})_{x,y}^K\right]^2 \right). \end{aligned} $$
is a $(\mathcal{F}_t^K)_{t \ge 0}$-martingale, square integrable, with quadratic variation:

$$
\begin{align*}
\langle M_t^{K, \Phi} \rangle_t ={}& \frac{K}{K \sigma_K^2} \int_0^t ds \int_{\mathbb{R}} \mu_s^K(dx) \int_{\mathbb{R}} \mu_s^K(dy) b \left( \sigma_K \sqrt{K} x + z_s^K, \sigma_K \sqrt{K} y + z_s^K \right) \\
& \times \left[ \Phi \left( z_s^K + \frac{\sigma_K \sqrt{K}}{K} (y-x), \tau_{-\frac{y-x}{K}} \sharp \left[ \mu_s^K - \frac{\delta_x}{K} + \frac{\delta_y}{K} \right] \right) - \Phi \left( z_s^K, \mu_s^K \right) \right]^2 \\
& + \frac{K}{K \sigma_K^2} \int_0^t ds \int_{\mathbb{R}} \mu_s^K(dx) \theta \left( \sigma_K \sqrt{K} x + z_s^K \right) \int_{\mathbb{R}} m \left( \sigma_K \sqrt{K} x + z_s^K, h \right) \\
& \times \left[ \Phi \left( \mu_s^K + \frac{\sigma_K \sqrt{K} h}{K^{\frac{3}{2}}}, \tau_{-\frac{h}{K^{3/2}}} \sharp \left[ \mu_s^K - \frac{\delta_x}{K} + \frac{\delta_{x+\frac{h}{\sqrt{K}}}}{K} \right] \right) - \Phi \left( \mu_s^K, \mu_s^K \right) \right]^2 dh.
\end{align*}
$$

Note that the factor $1/K\sigma_K^2$ corresponds to the time scaling of $\nu_t^K$ used to define $z_t^K$ and $\mu_t^K$.

*Proof. Step 1. About the generators.* On the one hand, from (3), note that for all $\Phi : \mathbb{R} \times \mathcal{M}_{1,K}^c(\mathbb{R}, \mathbb{R}) \to \mathbb{R}$ and $\phi : \mathcal{M}_{1,K}(\mathbb{R}) \to \mathbb{R}$ such that $\phi(\nu) = \Phi(\langle \mathrm{id}, \nu \rangle, \left(h_{\frac{1}{\sigma_K \sqrt{K}}} \circ \tau_{-\langle \mathrm{id}, \nu \rangle} \right) \sharp \nu)$,

$$
\mathcal{L}^K \Phi (z, \mu) = \frac{1}{K \sigma_K^2} \times L^K \phi \left( (\tau_z \circ h_{\sigma_K \sqrt{K}}) \sharp \mu \right). \quad (20)
$$

On the other hand, note that for all $\nu \in \mathcal{M}_{1,K}(\mathbb{R})$, $u, v \in \mathbb{R}$, $\psi : \mathbb{R} \to \mathbb{R}$ and $\Psi : \mathcal{M}_{1,K}(\mathbb{R}) \to \mathbb{R}$ measurable bounded functions, we have

$$
\begin{align}
\int_{\mathbb{R}} \psi(x) \Psi \left( \nu + \frac{\delta_{x+\sigma_K \sqrt{K} u}}{K} - \frac{\delta_v}{K} \right) \nu(dx) \tag{21} \\
&= \int_{\mathbb{R}} \psi \left( \sigma_K \sqrt{K} y + \langle \mathrm{id}, \nu \rangle \right) \Psi \left( \left( \tau_{\langle \mathrm{id}, \nu \rangle} \circ h_{\sigma_K \sqrt{K}} \right) \sharp \left[ \mu + \frac{\delta_{y+u}}{K} - \frac{\delta_v}{K} \right] \right) \mu(dy). \nonumber
\end{align}
$$

Finally, by noting that for all $u \in \mathbb{R}$ the mean trait of $(\tau_{\langle \mathrm{id}, \nu \rangle} \circ h_{\sigma_K \sqrt{K}}) \sharp [\mu - \frac{\delta_x}{K} + \frac{\delta_u}{K}]$ is

$$
\langle \mathrm{id}, \nu \rangle + \frac{\sigma_K \sqrt{K}}{K} (u - x)
$$

and the centered and dilated distribution of traits of $(\tau_{\langle \mathrm{id}, \nu \rangle} \circ h_{\sigma_K \sqrt{K}}) \sharp [\mu - \frac{\delta_x}{K} + \frac{\delta_u}{K}]$ is

$$
\tau_{-\frac{u-x}{K}} \sharp \left[ \mu - \frac{\delta_x}{K} + \frac{\delta_u}{K} \right],
$$

the announced result follows from (20) and (21).

*Step 2. About the martingales.* The martingale property follows from classical arguments since $\Phi$ and $\mathcal{L}^K \Phi$ are bounded [40, Chapter 4]. To compute the bracket, we proceed according to the following classical method (see e.g. [46]). We apply (19) replacing $\Phi$ by $\Phi^2$ to obtain the martingale $M^{K,\Phi^2}$. Then, we apply the $\mathrm{ItA'}$ formula to compute $\Phi^2(z_t^K, \mu_t^K)$ from (19). We deduce that

$$
\mathrm{Mart}_t := \Phi^2(z_t^K, \mu_t^K) - \Phi^2(z_0^K, \mu_0^K) - 2 \int_0^t \Phi(z_s^K, \mu_s^K) \mathcal{L}^K \Phi(z_s^K, \mu_s^K) ds - \langle M^{K, \Phi} \rangle_t
$$
From TAYLOR'S formula again and the centered condition of $\mu$, we obtain that

$$ (A)_{11}^K = \frac{K b(z, z)}{K \sigma^2} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) F'((\langle \varphi, \mu \rangle)) \left[ (E)_{x,y}^K - \frac{1}{K^2}(y-x)(\varphi'(y) - \varphi'(x)) + \frac{(y-x)^2 \langle \varphi'', \mu \rangle}{2K^2} + O\left(\frac{|y-x|^2 + |y-x|^3}{K^3}\right) \right] \\ = \frac{b(z, z)}{K^2 \sigma^2} F'((\langle \varphi, \mu \rangle)) [-2 \langle \varphi' \times \mathrm{id}, \mu \rangle + \langle \varphi'', \mu \rangle M_2(\mu)] + \frac{1}{K^2 \sigma^2} \times O\left(\frac{M_2(\mu) + M_3(\mu)}{K}\right) $$

In a straightforward way, we obtain that

$$ (A)_{12}^K = \frac{b(z, z)}{K^2 \sigma^2} F''((\langle \varphi, \mu \rangle)) [\langle \varphi^2, \mu \rangle - \langle \varphi, \mu \rangle^2 + M_2(\mu) \langle \varphi', \mu \rangle^2 - 2 \langle \varphi \times \mathrm{id}, \mu \rangle \langle \varphi', \mu \rangle] $$

As $\varphi$ is bounded and so $F$ too, we deduce that there exists two constants $C_1, C_2 > 0$ such that

$$ \left| \left[ (D)_{x,y}^K \right]^2 - \left[ (E)_{x,y}^K \right]^2 \right| = \left| (D)_{x,y}^K - (E)_{x,y}^K \right| \left| (D)_{x,y}^K + (E)_{x,y}^K \right| \le C_1 \frac{|y-x| + (y-x)^2}{K^2} \left( \frac{1}{K} + \frac{|y-x|}{K} \right), \\ \left| (C)_{x,y}^K - F'((\langle \varphi, \mu \rangle)) (D)_{x,y}^K - \frac{F''((\langle \varphi, \mu \rangle))}{2} \left[ (D)_{x,y}^K \right]^2 \right| \le C_2 \frac{1 + |y-x|^3}{K^3}. $$

Hence,

$$ (A)_{13}^K = \frac{1}{K^2 \sigma^2} \times O\left(\frac{1 + M_3(\mu)}{K}\right) \quad \text{and} \quad (A)_{14}^K = \frac{1}{K^2 \sigma^2} \times O\left(\frac{1 + M_3(\mu)}{K}\right). $$

Finally, as $\left| b\left(\sigma\sqrt{K}x + z, \sigma\sqrt{K}y + z\right) - b(z, z) \right| \le C_3\sigma\sqrt{K}(|x| + |y|)$ and $\left| (C)_{x,y}^K \right| \le C_4 \frac{1+|y-x|}{K}$ for some constants $C_3, C_4 > 0$, we deduce that

$$ (A)_{2}^{K} = \frac{1}{K^{2}\sigma^{2}} \times O\left(\sigma K^{\frac{3}{2}} [M_{1}(\mu) + M_{2}(\mu)]\right). $$

Therefore, using HÖLDER'S inequalities to bound $M_1(\mu) \le \sqrt{M_2(\mu)} \le 1 + M_2(\mu)$ and $M_2(\mu) \le M_3^{2/3}(\mu) \le 1 + M_3(\mu)$ we have that

$$ (A)^K = \frac{b(z, z)}{K^2 \sigma^2} F'((\langle \varphi, \mu \rangle)) [-2 \langle \varphi' \times \mathrm{id}, \mu \rangle + \langle \varphi'', \mu \rangle M_2(\mu)] \\ + \frac{b(z, z)}{K^2 \sigma^2} F''((\langle \varphi, \mu \rangle)) [\langle \varphi^2, \mu \rangle - \langle \varphi, \mu \rangle^2 + M_2(\mu) \langle \varphi', \mu \rangle^2 - 2 \langle \varphi \times \mathrm{id}, \mu \rangle \langle \varphi', \mu \rangle] \\ + \frac{1}{K^2 \sigma^2} \times O\left(\frac{1}{K} + \sigma K^{\frac{3}{2}} [1 + M_2(\mu)] + \frac{M_3(\mu)}{K}\right) $$

**Step 2. Conclusion.** In similar way to Step 1, we obtain that

$$ (B)^K = \frac{\theta(z)m_2(z)}{K^2 \sigma^2} F'((\langle \varphi, \mu \rangle)) \langle \frac{\varphi''}{2}, \mu \rangle + \frac{1}{K^2 \sigma^2} \times O\left(\frac{1}{\sqrt{K}} + \sigma K M_1(\mu)\right). $$

Since, by Assumption (5), $\sigma K^{\frac{3}{2}} \ll \frac{1}{\sqrt{K}}$, the announced result follows. □
is a martingale. Hence, the martingale $M_t^{K, \Phi^2}$ – Mart$_t$ has finite variation, so it is null [48, Theorem 4.1], leading to

$$ \langle M^{K, \Phi} \rangle_t = \int_0^t \left[ \mathcal{L}^K \Phi^2 (z_s^K, \mu_s^K) - 2\Phi (z_s^K, \mu_s^K) \mathcal{L}^K \Phi (z_s^K, \mu_s^K) \right] ds $$

then the announced conclusion.

□

## 3.2 Asymptotic expansions

Recall that, a set $S \subset \mathscr{C}_b^0(\mathbb{R}, \mathbb{R})$ is called $\mathcal{M}_{1,K}^c(\mathbb{R})$ – convergence determining if whenever $(P_n)_{n \in \mathbb{N}} \in \mathcal{M}_{1,K}^c(\mathbb{R})^\mathbb{N}, P \in \mathcal{M}_{1,K}^c(\mathbb{R})$ and

$$ \lim_{n \to +\infty} \langle f, P_n \rangle = \langle f, P \rangle $$

for all $f \in S$, we have that $(P_n)_{n \in \mathbb{N}}$ converges weakly to $P$ [40, Chapter 3, Section 4, p.112]. As developed in [26, Theorem 3.2.6], the class of functions on $\mathcal{M}_{1,K}^c(\mathbb{R})$,

$$ \mathbb{F}_b^2 := \left\{ F_\varphi \mid F_\varphi(\mu) := F(\langle \varphi, \mu \rangle), F \in \mathscr{C}^2(\mathbb{R}, \mathbb{R}), \varphi \in \mathscr{C}_b^2(\mathbb{R}, \mathbb{R}) \right\} $$

is $\mathcal{M}_{1,K}^c(\mathbb{R})$-convergence determining.

In the following sections, we prove the assertions (i) and (ii) of Section 2.3.1.

### 3.2.1 Slow component

**Proposition 3.2.** For all $f \in \mathscr{C}_b^2(\mathbb{R}, \mathbb{R})$, understood as a function of $(z, \mu)$ which depends only on $z$, the infinitesimal generator $\mathcal{L}^K$ defined in Proposition 3.1, satisfies the following decomposition

$$ \mathcal{L}^K f (z, \mu) = \mathcal{L}_{\text{SLOW}} f (z, \mu) + O \left( \frac{1}{K^2} + \frac{M_2(\mu)}{K} + \sigma_K \sqrt{K} M_3(\mu) \right) $$

where the operator $\mathcal{L}_{\text{SLOW}}$ is given by (9).

*Proof.* From Proposition 3.1 for the choice of test functions $\Phi(z, \mu) := f(z)$ we obtain that $\mathcal{L}^K f(z, \mu) = (\mathbf{A})^K + (\mathbf{B})^K$ where

$$ \begin{aligned} (\mathbf{A})^K := \frac{K}{K \sigma_K^2} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) b \left( \sigma_K \sqrt{K} x + z, \sigma_K \sqrt{K} y + z \right), & \\ & \times \left[ f \left( z + \frac{\sigma_K \sqrt{K}}{K} (y-x) \right) - f(z) \right] & \\ (\mathbf{B})^K := \frac{K}{K \sigma_K^2} \int_{\mathbb{R}} \mu(dx) \theta \left( \sigma_K \sqrt{K} x + z \right) \int_{\mathbb{R}} m \left( \sigma_K \sqrt{K} x + z, h \right) & \\ & \times \left[ f \left( z + \frac{\sigma_K \sqrt{K}}{K^{\frac{3}{2}}} h \right) - f(z) \right] dh. & \end{aligned} $$
## 3.3 Generators and martingales in the case of polynomials in $\mu$

### 3.3.1 For bounded test functions

In this section, we extend in Lemma 3.4 the result of Proposition 3.1 to test functions of the form

$$ P_{f,n}(\mu) := \langle f, \mu^n \rangle := \int_{\mathbb{R}} \dots \int_{\mathbb{R}} f(x_1, \dots, x_n) \mu(dx_1) \dots \mu(dx_n) \quad (21) $$

with $n \in \mathbb{N}^*$, $\mu \in \mathcal{M}_{1,K}^c(\mathbb{R})$, $f \in \mathcal{C}_b^3(\mathbb{R}^n, \mathbb{R})$ and where $\mu^n$ is the $n$-fold product measure of $\mu$. In Lemmas 3.6 and 3.7 we extend this result to the case of specific unbounded test functions.

For all $n \in \mathbb{N}^*$, we denote by $\mathbf{1} \in \mathbb{R}^n$, the vector whose coordinates are all 1 and by $\Delta$ the Laplacian operator on $\mathbb{R}^n$. Let us introduce, for all $n \in \mathbb{N}^*$, $\lambda > 0$ and $f \in \mathcal{C}_b^2(\mathbb{R}^n, \mathbb{R})$, the operator $B_\lambda^{(n)}$ defined by

$$ B_{\lambda}^{(n)} f(x) := \frac{1}{2} \Delta f(x) - 2\lambda (\nabla f(x) \cdot \mathbf{1})(x \cdot \mathbf{1}), \quad x \in \mathbb{R}^n. \quad (22) $$

Let us consider for all $n \in \mathbb{N}^*$ for all $i, j \in \{1, \dots, n\}$,

* $\Phi_{i,j} : \mathcal{C}_b^2(\mathbb{R}^n, \mathbb{R}) \to \mathcal{C}_b^2(\mathbb{R}^{n-1}, \mathbb{R})$, with $i \neq j$, is the function obtained from $f$ by inserting the variable $x_i$ between $x_{j-1}$ and $x_j$ when $i < j$ and by inserting the variable $x_{i-1}$ between $x_{j-1}$ and $x_j$ when $i > j$:

$$ \begin{aligned} \Phi_{i,j} f(x_1, \dots, x_{n-1}) &= f(x_1, \dots, x_{j-1}, x_i, x_j, x_{j+1}, \dots, x_{n-1}) & i < j \\ \Phi_{i,j} f(x_1, \dots, x_{n-1}) &= f(x_1, \dots, x_{j-1}, x_{i-1}, x_j, x_{j+1}, \dots, x_{n-1}) & i > j. \end{aligned} \quad (23) $$

* $K_{i,j} : \mathcal{C}_b^2(\mathbb{R}^n, \mathbb{R}) \to \mathcal{C}^2(\mathbb{R}^{n+1}, \mathbb{R})$ is defined as

$$ K_{i,j} f(x_1, \dots, x_n, x_{n+1}) := \partial_{ij}^2 f(x_1, \dots, x_n) x_{n+1}^2. \quad (24) $$

We recall from [16, Definition 2.8] that the extended generator (in the sense of DYNKIN, see (26) below) $\mathcal{L}_{FVc}^\lambda$ of the centered FLEMING-VIOT process with resampling rate $\lambda$ is defined for any $n \in \mathbb{N}^*$, for any test functions $f \in \mathcal{C}_b^2(\mathbb{R}^n, \mathbb{R})$ by

$$ \mathcal{L}_{FVc}^{\lambda} P_{f,n}(\mu) = \langle B_{\lambda}^{(n)} f, \mu^n \rangle + \lambda \sum_{\substack{i,j=1 \\ i \neq j}}^{n} \left[ \langle \Phi_{i,j} f, \mu^{n-1} \rangle - \langle f, \mu^n \rangle \right] + \lambda \sum_{i,j=1}^{n} \langle K_{i,j} f, \mu^{n+1} \rangle. \quad (25) $$

From [16, Definition 2.8] and denoting

$$ \tilde{\Omega} := \left\{ X \in \mathcal{C}^0([0, +\infty), \mathcal{M}_1^{c,2}(\mathbb{R})) \mid \forall T > 0, \sup_{0 \le t \le T} M_2(X_t) < \infty \right\}, $$

we recall that a probability measure $P_\mu$ on $\tilde{\Omega}$ is said to solve the centered FLEMING-VIOT martingale problem for polynomials with initial condition $\mu \in \mathcal{M}_1^{c,2}(\mathbb{R})$, if the canonical process $(X_t)_{t \ge 0}$ on $\tilde{\Omega}$ satisfies $P_\mu(X_0 = \mu) = 1$ and, for all $n \in \mathbb{N}^*$ and $f \in \mathcal{C}_b^2(\mathbb{R}^n, \mathbb{R})$,

$$ \widehat{M}_t^{P_{f,n}} := P_{f,n}(X_t) - P_{f,n}(X_0) - \int_0^t \mathcal{L}_{FVc}^{\lambda} P_{f,n}(X_s) ds \quad (26) $$

is a $P_\mu$-martingale.
Now, we want to decompose and study $(\mathbf{A})^K$. Denoting $(\mathbf{C})_{x,y}^K(z) := f\left(z + \frac{\sigma_K \sqrt{K}}{K}(y-x)\right)$
$f(z)$, from TAYLOR'S formula, note that $(\mathbf{A})^K = (\mathbf{A})_1^K + (\mathbf{A})_2^K + (\mathbf{A})_3^K$ where

$$
\begin{align*}
(\mathbf{A})_1^K &:= \frac{K}{K\sigma_K^2} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) b(z, z) (\mathbf{C})_{x,y}^K(z), \\
(\mathbf{A})_2^K &:= \frac{K}{K\sigma_K^2} \times \sigma_K \sqrt{K} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) (x\partial_1 b(z, z) + y\partial_2 b(z, z)) (\mathbf{C})_{x,y}^K(z), \\
(\mathbf{A})_3^K &:= \frac{K}{K\sigma_K^2} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) \left[ b\left(\sigma_K \sqrt{K}x + z, \sigma_K \sqrt{K}y + z\right) - b(z, z) \right. \\
&\qquad \left. - \sigma_K \sqrt{K} \left(x\partial_1 b(z, z) + y\partial_2 b(z, z)\right) \right] (\mathbf{C})_{x,y}^K(z).
\end{align*}
$$

Using that $(\mathbf{C})_{x,y}^K(z) = f'(z) \frac{\sigma_K \sqrt{K}}{K} (x-y) + O\left(\frac{\sigma_K^2}{K} |y-x|^2\right)$, we deduce that

$$
(\mathbf{A})_1^K = O\left(\frac{M_2(\mu)}{K}\right), \quad (\mathbf{A})_2^K = f'(z) (\partial_2 b(z, z) - \partial_1 b(z, z)) M_2(\mu) + O\left(\frac{\sigma_K \sqrt{K} M_3(\mu)}{K}\right),
$$

and noting that $|(\mathbf{C})_{x,y}^K| \le C_1 \frac{\sigma_K \sqrt{K}}{K} |y-x|$ for some constant $C_1 > 0$ and

$$
\left| b \left( \sigma_K \sqrt{K} x + z, \sigma_K \sqrt{K} y + z \right) - b(z, z) - \sigma_K \sqrt{K} \left( x \partial_1 b(z, z) + y \partial_2 b(z, z) \right) \right| \le C_2 \sigma_K^2 K (x^2 + y^2)
$$

for some constant $C_2 > 0$, we deduce that $(\mathbf{A})_3^K = O(\sigma_K \sqrt{K} M_3(\mu))$. Therefore,

$$
(\mathbf{A})^K = f'(z) (\partial_2 b(z, z) - \partial_1 b(z, z)) M_2(\mu) + O\left(\frac{M_2(\mu)}{K} + \sigma_K \sqrt{K} M_3(\mu)\right).
$$

As previously, we obtain that

$$
(\mathbf{B})^K = O\left(\frac{\bar{\theta}\bar{m}_2}{K^2}\right).
$$

and the announced result follows.

<figure><img src="image_8.png" /></figure>

3.2.2 Fast component

**Proposition 3.3.** For all $F \in \mathscr{C}^3(\mathbb{R}, \mathbb{R})$ and $\varphi \in \mathscr{C}_b^3(\mathbb{R}, \mathbb{R})$, the infinitesimal generator $\mathcal{L}^K$
*defined in Proposition 3.1, satisfies the following decomposition*

$$
\mathcal{L}^K F_\varphi (z, \mu) = \frac{\theta(z) m_2(z)}{K^2 \sigma_K^2} \left[ \mathcal{L}_{\text{FVc}}^{\lambda(z)} F_\varphi (z, \mu) + O \left( \frac{1}{\sqrt{K}} + \sigma_K K^{\frac{3}{2}} M_2 (\mu) + \frac{M_3 (\mu)}{K} \right) \right]
$$

where the operator $\mathcal{L}_{\text{FVc}}^{\lambda(z)}$ is given by (11) and $F_{\varphi}$ is understood as a function of $(z, \mu)$ which
depends only on $\mu$.
**Lemma 3.4.** The infinitesimal generator $\mathcal{L}^K$ of the $\mathbb{R} \otimes \mathcal{M}_{1,K}^c(\mathbb{R})$-valued MARKOV process $(z_t^K, \mu_t^K)_{t \ge 0}$ given by Proposition 3.1, satisfies for all $n \in \mathbb{N}^*$ and $f \in \mathcal{C}_b^3(\mathbb{R}^n, \mathbb{R})$, the following relations:

$$ \mathcal{L}^K P_{f,n}(z, \mu) = \frac{\theta(z)m_2(z)}{K^2\sigma^2} \mathcal{L}_{FVc}^{\lambda(z)} P_{f,n}(\mu) + \frac{1}{K^2\sigma^2} \times O\left(\frac{1}{\sqrt{K}} + \sigma K^{\frac{3}{2}}M_2(\mu) + \frac{M_3(\mu)}{K}\right) $$

where $\lambda(z) := \frac{b(z,z)}{\theta(z)m_2(z)}$. Moreover, for all $K, n \in \mathbb{N}^*$ and test functions $f \in \mathcal{C}_b^3(\mathbb{R}^n, \mathbb{R})$, the process $(M_t^{K,P_{f,n}})_{t \ge 0}$ defined by

$$ M_t^{K,P_{f,n}} := P_{f,n}(\mu_t^K) - P_{f,n}(\mu_0^K) - \int_0^t \mathcal{L}^K P_{f,n}(z_s^K, \mu_s^K) ds $$

is a square integrale martingale started at 0.

The proof of this result is given in Section 3.4.

### 3.3.2 For some unbounded test functions

The following results are particular extensions of Lemma 3.4 when the test functions are no longer bounded but the processes are stopped at time $\tilde{\tau}^K$. Then, we give in Lemmas 3.6 and 3.7 DOOB's semi-martingale decomposition of moments of order 2 and those whose degree is 6.

**Proposition 3.5.** For all $n \in \mathbb{N}^*$, for all $f \in \mathcal{C}^3(\mathbb{R}^n, \mathbb{R})$, the process $(M_{t \wedge \tilde{\tau}^K}^{K,P_{f,n}})_{t \ge 0}$ defined by

$$ \begin{aligned} M_{t \wedge \tilde{\tau}^K}^{K,P_{f,n}} := & P_{f,n}(\mu_{t \wedge \tilde{\tau}^K}^K) - P_{f,n}(\mu_0^K) - \frac{K}{K\sigma^2} \int_0^{t \wedge \tilde{\tau}^K} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) b\left(\sigma\sqrt{K}x + z_s^K, \sigma\sqrt{K}x + z_s^K\right) \\ & \times \left[P_{f \circ \tau_{-\frac{y-x}{K}}, n}\left(\mu - \frac{\delta_x}{K} + \frac{\delta_y}{K}\right) - P_{f,n}(\mu)\right] ds \\ & + \frac{K}{K\sigma^2} \int_0^{t \wedge \tilde{\tau}^K} \int_{\mathbb{R}} \mu(dx) \theta\left(\sigma\sqrt{K}x + z_s^K\right) \int_{\mathbb{R}} m\left(\sigma\sqrt{K}x + z_s^K, h\right) dh \\ & \times \left[P_{f \circ \tau_{-\frac{h}{K^2}}, n}\left(\mu - \frac{\delta_x}{K} + \frac{\delta_{x+\frac{h}{\sqrt{K}}}}{K}\right) - P_{f,n}(\mu)\right] ds \end{aligned} $$

is a bounded martingale.

*Proof.* Thanks to the stopping time $\tilde{\tau}^K$, $(M_{t \wedge \tilde{\tau}^K}^{K,P_{f,n}})_{t \ge 0}$ is bounded. Hence the martingality of this process follows from Lemma 3.4. $\square$

(a) Moment of order 2. The following result will be useful in Section 4.

**Lemma 3.6.** For all $K \in \mathbb{N}^*$, the process $(M_{t \wedge \tilde{\tau}^K}^{K,P_{id}^2,1})_{t \ge 0}$ defined in Proposition 3.5 satisfies for all $t \ge 0$

$$ \begin{aligned} M_{t \wedge \tilde{\tau}^K}^{K,P_{id}^2,1} = & M_2(\mu_{t \wedge \tilde{\tau}^K}^K) - M_2(\mu_0^K) + \frac{1}{K^2\sigma^2} \int_0^{t \wedge \tilde{\tau}^K} \left\{ 2b(z_s^K, z_s^K) M_2(\mu_s^K) - \theta(z_s^K) m_2(z_s^K) \right\} ds \\ & + \frac{1}{K^2\sigma^2} \int_0^{t \wedge \tilde{\tau}^K} O\left(\frac{1}{K} + \sigma\sqrt{K}M_1(\mu_s^K) + \sigma K^{\frac{3}{2}}M_3(\mu_s^K)\right) ds \end{aligned} $$
Proof. Noting that

$$
\begin{align*}
(\mathbf{D})_{x,y}^K &:= \left\langle \varphi, \tau_{-\frac{y-x}{K}} \sharp \left[ \mu - \frac{\delta_x}{K} + \frac{\delta_y}{K} \right] \right\rangle - \langle \varphi, \mu \rangle \\
&= \left\langle \varphi \circ \tau_{-\frac{y-x}{K}}, \mu \right\rangle + \frac{1}{K} \left\{ \varphi \left(y - \frac{y-x}{K}\right) - \varphi \left(x - \frac{y-x}{K}\right) \right\} - \langle \varphi, \mu \rangle,
\end{align*}
$$

setting $(\mathbf{C})_{x,y}^K := F(\langle\varphi, \mu\rangle + (\mathbf{D})_{x,y}^K) - F(\langle\varphi, \mu\rangle)$ and from Proposition 3.1 we obtain that
$\mathcal{L}^K F_\varphi(z, \mu) = (\mathbf{A})^K + (\mathbf{B})^K$ where

$$
\begin{align*}
(\mathbf{A})^K &:= \frac{K}{K \sigma_K^2} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) b \left( \sigma_K \sqrt{K} x + z, \sigma_K \sqrt{K} y + z \right) (\mathbf{C})_{x,y}^K, \\
(\mathbf{B})^K &:= \frac{K}{K \sigma_K^2} \int_{\mathbb{R}} \mu(dx) \theta \left( \sigma_K \sqrt{K} x + z \right) \int_{\mathbb{R}} m \left( \sigma_K \sqrt{K} x + z, h \right) (\mathbf{C})_{x,x+h/\sqrt{K}}^K \, dh.
\end{align*}
$$

Step 1. Decomposition and study of $(\mathbf{A})^K$. From TAYLOR’s formula, we obtain that $(\mathbf{A})^K = (\mathbf{A})_1^K + (\mathbf{A})_2^K$ where

$$
\begin{align*}
(\mathbf{A})_1^K &:= \frac{K}{K \sigma_K^2} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) b(z, z) (\mathbf{C})_{x,y}^K, \\
(\mathbf{A})_2^K &:= \frac{K}{K \sigma_K^2} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) \left[ b\left(\sigma_K \sqrt{K} x + z, \sigma_K \sqrt{K} y + z\right) - b(z, z) \right] (\mathbf{C})_{x,y}^K.
\end{align*}
$$

Denoting $(\mathbf{E})_{x,y}^K := \frac{1}{K} (\varphi(y) - \varphi(x) - (y-x) \langle \varphi', \mu \rangle)$ and from TAYLOR’s formula again, we obtain that $(\mathbf{A})_1^K = (\mathbf{A})_{11}^K + (\mathbf{A})_{12}^K + (\mathbf{A})_{13}^K + (\mathbf{A})_{14}^K$ where

$$
\begin{align*}
(\mathbf{A})_{11}^K &:= \frac{K}{K \sigma_K^2} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) b(z, z) F'\left(\langle \varphi, \mu \rangle\right) (\mathbf{D})_{x,y}^K, \\
(\mathbf{A})_{12}^K &:= \frac{K}{2K \sigma_K^2} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) b(z, z) F''\left(\langle \varphi, \mu \rangle\right) \left[(\mathbf{E})_{x,y}^K\right]^2, \\
(\mathbf{A})_{13}^K &:= \frac{K}{2K \sigma_K^2} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) b(z, z) F''\left(\langle \varphi, \mu \rangle\right) \left[\left[(\mathbf{D})_{x,y}^K\right]^2 - \left[(\mathbf{E})_{x,y}^K\right]^2\right], \\
(\mathbf{A})_{14}^K &:= \frac{K}{K \sigma_K^2} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) b(z, z) \\
&\quad \times \left( (\mathbf{C})_{x,y}^K - F'\left(\langle \varphi, \mu \rangle\right) (\mathbf{D})_{x,y}^K - \frac{F''\left(\langle \varphi, \mu \rangle\right)}{2} \left[(\mathbf{D})_{x,y}^K\right]^2 \right)
\end{align*}
$$

From TAYLOR’s formula again and the centered condition of $\mu$, we obtain that

$$
\begin{align*}
(\mathbf{A})_{11}^K &= \frac{K b(z, z)}{K \sigma_K^2} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) F'\left(\langle \varphi, \mu \rangle\right) \left[ (\mathbf{E})_{x,y}^K - \frac{1}{K^2}(y-x)(\varphi'(y) - \varphi'(x)) \right. \\
&\qquad \left. + \frac{(y-x)^2 \langle \varphi'', \mu \rangle}{2K^2} + O\left( \frac{|y-x|^2 + |y-x|^3}{K^3} \right) \right] \\
&= \frac{b(z, z)}{K^2 \sigma_K^2} F'\left(\langle \varphi, \mu \rangle\right) \left[ -2 \langle \varphi' \times \mathrm{id}, \mu \rangle + \langle \varphi'', \mu \rangle M_2(\mu) \right] + \frac{1}{K^2 \sigma_K^2} \times O\left( \frac{M_2(\mu) + M_3(\mu)}{K} \right)
\end{align*}
$$
is a square integrable martingale with quadratic variation:

$$
\begin{equation}
\begin{split}
\langle M^{K, P_{\mathrm{id}2,1}} \rangle_{t \wedge \hat{\tau}^K} ={}& \frac{2}{K^2 \sigma^2} \int_0^{t \wedge \hat{\tau}^K} b(z_s^K, z_s^K) \{M_4(\mu_s^K) - M_2^2(\mu_s^K)\} \mathrm{d}s \\
& + \frac{1}{K^2 \sigma^2} \int_0^{t \wedge \hat{\tau}^K} O\left(\frac{\sigma \sqrt{K}}{K^2} + \frac{M_2(\mu_s^K)}{K} + \frac{\sigma \sqrt{K} M_3(\mu_s^K)}{K} + \sigma \sqrt{K} M_5(\mu_s^K)\right) \mathrm{d}s.
\end{split}
\tag{27}
\end{equation}
$$

Proof. **Step 1. Approximation of the Doob decomposition.** Note that for all $x, y \in \mathbb{R}$,

$$
P_{\mathrm{id}2 \circ \tau_{-\frac{y-x}{K}, 1}}\left(\mu-\frac{\delta_{x}}{K}+\frac{\delta_{y}}{K}\right)-P_{\mathrm{id}2, 1}\left(\mu\right)=\frac{1}{K}\left(y^{2}-x^{2}\right)-\frac{1}{K^{2}}\left(-x+y\right)^{2}.
$$

From TAYLOR'S formula, it follows that

$$
\begin{align*}
& \frac{K}{K\sigma^2} \int_0^t \mathrm{d}s \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) b\left(\sigma\sqrt{K}x + z_s^K, \sigma\sqrt{K}y + z_s^K\right) \\
& \qquad \times \left[ P_{\mathrm{id}2\circ\tau_{-\frac{y-x}{K},1}}\left(\mu - \frac{\delta_x}{K} + \frac{\delta_y}{K}\right) - P_{\mathrm{id}2,1}(\mu) \right] \\
&= \frac{1}{K^2\sigma^2} \int_0^t \left[ -2b(z_s^K, z_s^K) M_2(\mu_s^K) + O\left(\sigma K^{\frac{3}{2}} M_3(\mu_s^K)\right) \right] \mathrm{d}s
\end{align*}
$$

and

$$
\begin{align*}
& \frac{K}{K\sigma^2} \int_0^t \mathrm{d}s \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} m\left(\sigma\sqrt{K}x + z_s^K, h\right) \theta\left(\sigma\sqrt{K}x + z_s^K\right) \\
& \qquad \times \left[ P_{\mathrm{id}2\circ\tau_{-\frac{h}{K^{3/2}},1}}\left(\mu - \frac{\delta_x}{K} + \frac{\delta_{x+\frac{h}{\sqrt{K}}}}{K}\right) - P_{\mathrm{id}2,1}(\mu) \right] \mathrm{d}h \\
&= \frac{1}{K^2\sigma^2} \int_0^t \left[ \theta(z_s^K) m_2(z_s^K) + O\left(\frac{1}{K} + \sigma\sqrt{K}M_1(\mu_s^K)\right) \right] \mathrm{d}s.
\end{align*}
$$

We deduce the first announced result.

**Step 2. Quadratic variation.** In similar way to the proof of Proposition 3.1, we have

$$
\begin{align*}
\langle M^{K, P_{\mathrm{id}2,1}} \rangle_t ={}& \frac{K}{K\sigma^2} \int_0^t \mathrm{d}s \int_{\mathbb{R}} \mu_s^K(dx) \int_{\mathbb{R}} \mu_s^K(dy) b\left(\sigma\sqrt{K}x + z_s^K, \sigma\sqrt{K}y + z_s^K\right) \\
& \times \left[ \frac{1}{K}(y^2 - x^2) - \frac{1}{K^2}(-x+y)^2 \right]^2 \\
& + \frac{K}{K\sigma^2} \int_0^t \mathrm{d}s \int_{\mathbb{R}} \mu_s^K(dx) \int_{\mathbb{R}} m\left(\sigma\sqrt{K}x + z_s^K, h\right) \theta\left(\sigma\sqrt{K}x + z_s^K\right) \left[ \frac{h^2}{K^2} - \frac{h^2}{K^3} + \frac{2xh}{K^{\frac{3}{2}}} \right]^2 \mathrm{d}h.
\end{align*}
$$

The announced result follows from TAYLOR'S formula and straightforward computations. $\square$

(b) Moments of degree 6. Let us denote for all $\ell \in \mathbb{N}$, $\tilde{M}_{\ell}(\mu) := \langle \mathrm{id}^{\ell}, \mu \rangle$.
In a straightforward way, we obtain that

$$ (\mathbf{A})_{12}^K = \frac{b(z, z)}{K^2 \sigma_K^2} F''(\langle\varphi, \mu\rangle) [\langle\varphi^2, \mu\rangle - \langle\varphi, \mu\rangle^2 + M_2(\mu) \langle\varphi', \mu\rangle^2 - 2 \langle\varphi \times \mathrm{id}, \mu\rangle \langle\varphi', \mu\rangle]. $$

As $\varphi$ is bounded and so $F$ too, we deduce that there exists two constants $C_1, C_2 > 0$ such that

$$ \begin{aligned} |[(\mathbf{D})_{x,y}^K]^2 - [(\mathbf{E})_{x,y}^K]^2| &= |(\mathbf{D})_{x,y}^K - (\mathbf{E})_{x,y}^K| |(\mathbf{D})_{x,y}^K + (\mathbf{E})_{x,y}^K| \\ &\le C_1 \frac{|y-x| + (y-x)^2}{K^2} \left( \frac{1}{K} + \frac{|y-x|}{K} \right), \end{aligned} $$

$$ |(\mathbf{C})_{x,y}^K - F'((\langle\varphi, \mu\rangle)(\mathbf{D})_{x,y}^K) - \frac{F''((\langle\varphi, \mu\rangle))}{2} [(\mathbf{D})_{x,y}^K]^2| \le C_2 \frac{1 + |y-x|^3}{K^3}. $$

Hence,

$$ (\mathbf{A})_{13}^K = \frac{1}{K^2 \sigma_K^2} \times O \left( \frac{1 + M_3(\mu)}{K} \right) \quad \text{and} \quad (\mathbf{A})_{14}^K = \frac{1}{K^2 \sigma_K^2} \times O \left( \frac{1 + M_3(\mu)}{K} \right). $$

Finally, as $|b(\sigma_K \sqrt{K}x + z, \sigma_K \sqrt{K}y + z) - b(z, z)| \le C_3 \sigma_K \sqrt{K} (|x| + |y|)$ and $|(\mathbf{C})_{x,y}^K| \le C_4 \frac{1+|y-x|}{K}$ for some constants $C_3, C_4 > 0$, we deduce that

$$ (\mathbf{A})_2^K = \frac{1}{K^2 \sigma_K^2} \times O \left( \sigma_K K^{\frac{3}{2}} [M_1(\mu) + M_2(\mu)] \right). $$

Therefore, using Hölder's inequalities to bound $M_1(\mu) \le \sqrt{M_2(\mu)} \le 1 + M_2(\mu)$ and $M_2(\mu) \le M_3^{2/3}(\mu) \le 1 + M_3(\mu)$ we have that

$$ \begin{aligned} (\mathbf{A})^K = & \frac{b(z, z)}{K^2 \sigma_K^2} F'((\langle\varphi, \mu\rangle)) [-2 \langle\varphi' \times \mathrm{id}, \mu\rangle + \langle\varphi'', \mu\rangle M_2(\mu)] \\ & + \frac{b(z, z)}{K^2 \sigma_K^2} F''((\langle\varphi, \mu\rangle)) [\langle\varphi^2, \mu\rangle - \langle\varphi, \mu\rangle^2 + M_2(\mu) \langle\varphi', \mu\rangle^2 - 2 \langle\varphi \times \mathrm{id}, \mu\rangle \langle\varphi', \mu\rangle] \\ & + \frac{1}{K^2 \sigma_K^2} \times O \left( \frac{1}{K} + \sigma_K K^{\frac{3}{2}} [1 + M_2(\mu)] + \frac{M_3(\mu)}{K} \right). \end{aligned} $$

**Step 2. Conclusion.** In similar way to Step 1, we obtain that

$$ (\mathbf{B})^K = \frac{\theta(z)m_2(z)}{K^2 \sigma_K^2} F'((\langle\varphi, \mu\rangle)) \left\langle \frac{\varphi''}{2}, \mu \right\rangle + \frac{1}{K^2 \sigma_K^2} \times O \left( \frac{1}{\sqrt{K}} + \sigma_K K M_1(\mu) \right). $$

Since, by Assumption (5), $\sigma_K K^{3/2} \ll \frac{1}{\sqrt{K}}$, the announced result follows. $\square$

## 3.3 Generators and martingales in the case of polynomials in $\mu$

### 3.3.1 For bounded test functions

In this section, we extend in Lemma 3.4 the result of Proposition 3.1 to test functions of the form

$$ P_{f,n}(\mu) := \langle f, \mu^n \rangle := \int_{\mathbb{R}} \cdots \int_{\mathbb{R}} f(x_1, \cdots, x_n) \mu(dx_1) \cdots \mu(dx_n) \quad (22) $$
**Lemma 3.7.** Let us consider the functions $f,g: \mathbb{R}^2 \to \mathbb{R}$ and $h: \mathbb{R}^3 \to \mathbb{R}$ respectively defined by $f(u,v) := u^4 v^2$, $g(u,v) := u^3 v^3$ and $h(u,v,w) := u^2 v^2 w^2$. For all $K \in \mathbb{N}^*$, the processes $(M_{t \wedge \hat{\tau}^K}^{K,P_{id^{6,1}}})_{t \ge 0}$, $(M_{t \wedge \hat{\tau}^K}^{K,P_{f,2}})_{t \ge 0}$, $(M_{t \wedge \hat{\tau}^K}^{K,P_{g,2}})_{t \ge 0}$ and $(M_{t \wedge \hat{\tau}^K}^{K,P_{h,3}})_{t \ge 0}$ satisfy for all $t \ge 0$

$$
\begin{align*}
\bullet \ M_{t \wedge \hat{\tau}^K}^{K, P_{id^{6,1}}} = M_6 (\mu_{t \wedge \hat{\tau}^K}^K) - M_6 (\mu_0^K) &+ \frac{3}{K^2 \sigma^2} \int_0^{t \wedge \hat{\tau}^K} \left\{ b (z_s^K, z_s^K) [4M_6 (\mu_s^K) - 10M_4 (\mu_s^K) M_2 (\mu_s^K)] \right. \\
& \qquad \left. - 5\theta (z_s^K) m_2 (z_s^K) M_4 (\mu_s^K) \right\} ds \\
& + \frac{1}{K^2 \sigma^2} \int_0^{t \wedge \hat{\tau}^K} O \left( \frac{M_3 (\mu_s^K)}{\sqrt{K}} + \frac{1 + M_4 (\mu_s^K)}{K} + \sigma \sqrt{K} M_5 (\mu_s^K) + \frac{M_6 (\mu_s^K)}{K^2} + \sigma K^{\frac{3}{2}} M_7 (\mu_s^K) \right) ds,
\end{align*}
$$

$$
\bullet \ M_{t \wedge \hat{\tau}^K}^{K, P_{f,2}} = M_4 (\mu_{t \wedge \hat{\tau}^K}^K) M_2 (\mu_{t \wedge \hat{\tau}^K}^K) - M_4 (\mu_0^K) M_2 (\mu_0^K) + \frac{1}{K^2 \sigma^2} \int_0^{t \wedge \hat{\tau}^K} \left\{ 2b (z_s^K, z_s^K) [6M_4 (\mu_s^K) M_2 (\mu_s^K) \right. \\
\qquad \left. - M_6 (\mu_s^K) + 4\tilde{M}_3^2 (\mu_s^K) - 6M_2^3 (\mu_s^K)] - \theta (z_s^K) m_2 (z_s^K) [6M_2^2 (\mu_s^K) + M_4 (\mu_s^K)] \right\} ds \\
\qquad + \frac{1}{K^2 \sigma^2} \int_0^{t \wedge \hat{\tau}^K} O \left( \frac{M_6 (\mu_s^K)}{K^3} + \sigma K^{\frac{3}{2}} M_7 (\mu_s^K) \right) ds,
$$

$$
\bullet \ M_{t \wedge \hat{\tau}^K}^{K, P_{g,2}} = \widetilde{M}_3^2 (\mu_{t \wedge \hat{\tau}^K}^K) - \widetilde{M}_3^2 (\mu_0^K) + \frac{1}{K^2 \sigma^2} \int_0^{t \wedge \hat{\tau}^K} \left\{ b (z_s^K, z_s^K) [-2M_6 (\mu_s^K) + 14\widetilde{M}_3^2 (\mu_s^K) \right. \\
\qquad \left. + 12M_4 (\mu_s^K) M_2 (\mu_s^K) - 18M_2^3 (\mu_s^K)] \right\} ds \\
\qquad + \frac{1}{K^2 \sigma^2} \int_0^{t \wedge \hat{\tau}^K} O \left( \frac{M_6 (\mu_s^K)}{K^3} + \sigma K^{\frac{3}{2}} M_7 (\mu_s^K) \right) ds,
$$

$$
\bullet \ M_{t \wedge \hat{\tau}^K}^{K, P_{h,3}} = M_2^3 (\mu_{t \wedge \hat{\tau}^K}^K) - M_2^3 (\mu_0^K) + \frac{1}{K^2 \sigma^2} \int_0^{t \wedge \hat{\tau}^K} \left\{ b (z_s^K, z_s^K) [-6M_4 (\mu_s^K) M_2 (\mu_s^K) \right. \\
\qquad \left. + 12M_2^3 (\mu_s^K)] - 3\theta (z_s^K) m_2 (z_s^K) M_4 (\mu_s^K) \right\} ds \\
\qquad + \frac{1}{K^2 \sigma^2} \int_0^{t \wedge \hat{\tau}^K} O \left( \frac{M_6 (\mu_s^K)}{K^3} + \sigma K^{\frac{3}{2}} M_7 (\mu_s^K) \right) ds.
$$

*Proof.* The approximation proposed is obtained in a similar way to the proof of Lemma 3.6 $\square$

**3.4 Proof of Lemma 3.4**

From Proposition 3.1, for the choice of test functions $\Phi(z, \mu) := P_{f,n}(\mu)$ given by (21) and noting that for all $x, y \in \mathbb{R}$,

$$
\left\langle f, \left( \tau_{-\frac{y-x}{K}} \# \left[ \mu - \frac{\delta_x}{K} + \frac{\delta_y}{K} \right] \right)^n \right\rangle = \left\langle f \circ \tau_{-\frac{y-x}{K}}, \left[ \mu - \frac{\delta_x}{K} + \frac{\delta_y}{K} \right]^n \right\rangle,
$$

we obtain that $\mathcal{L}^K P_{f,n}(z, \mu) = \frac{K}{K\sigma^2} ((\mathbf{A})^K + (\mathbf{B})^K)$ where

$$
(\mathbf{A})^K := \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) b(\sigma\sqrt{K}x + z, \sigma\sqrt{K}y + z) (\mathbf{C})^K_{x,y},
$$

$$
(\mathbf{B})^K := \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \theta(\sigma\sqrt{K}x + z) m(z + \sigma\sqrt{K}x, h) (\mathbf{C})^K_{x,x+h/\sqrt{K}} \, dh
$$
with $n \in \mathbb{N}^*$, $\mu \in \mathcal{M}_{1,K}^c(\mathbb{R})$, $f \in \mathcal{C}_b^3(\mathbb{R}^n, \mathbb{R})$ and where $\mu^n$ is the $n$-fold product measure of $\mu$. In Lemmas 3.6 and 3.7 we extend this result to the case of specific unbounded test functions.

For all $n \in \mathbb{N}^*$, we denote by $\mathbf{1} \in \mathbb{R}^n$, the vector whose coordinates are all 1 and by $\Delta$ the Laplacian operator on $\mathbb{R}^n$. Let us introduce, for all $n \in \mathbb{N}^*$, $\lambda > 0$ and $f \in \mathcal{C}_b^2(\mathbb{R}^n, \mathbb{R})$, the operator $B_\lambda^{(n)}$ defined by

$$B_{\lambda}^{(n)} f(x) := \frac{1}{2} \Delta f(x) - 2\lambda (\nabla f(x) \cdot \mathbf{1})(x \cdot \mathbf{1}), \quad x \in \mathbb{R}^n. \qquad (23)$$

Let us consider for all $n \in \mathbb{N}^*$ for all $i, j \in \{1, \dots, n\}$,

* $\Phi_{i,j} : \mathcal{C}_b^2(\mathbb{R}^n, \mathbb{R}) \to \mathcal{C}_b^2(\mathbb{R}^{n-1}, \mathbb{R})$, with $i \neq j$, is the function obtained from $f$ by inserting the variable $x_i$ between $x_{j-1}$ and $x_j$ when $i < j$ and by inserting the variable $x_{i-1}$ between $x_{j-1}$ and $x_j$ when $i > j$:

$$\begin{aligned} \Phi_{i,j} f(x_1, \dots, x_{n-1}) &= f(x_1, \dots, x_{j-1}, x_i, x_j, x_{j+1}, \dots, x_{n-1}) & i < j \\ \Phi_{i,j} f(x_1, \dots, x_{n-1}) &= f(x_1, \dots, x_{j-1}, x_{i-1}, x_j, x_{j+1}, \dots, x_{n-1}) & i > j. \end{aligned} \qquad (24)$$

* $K_{i,j} : \mathcal{C}_b^2(\mathbb{R}^n, \mathbb{R}) \to \mathcal{C}_b^2(\mathbb{R}^{n+1}, \mathbb{R})$ is defined as

$$K_{i,j} f(x_1, \dots, x_n, x_{n+1}) := \partial_{ij}^2 f(x_1, \dots, x_n) x_{n+1}^2. \qquad (25)$$

We recall from [18, Definition 2.8] that the extended generator (in the sense of DYNKIN, see (27) below) $\mathcal{L}_{\text{FVc}}^{\lambda}$ of the centered FLEMING-VIOT process with resampling rate $\lambda$ is defined for any $n \in \mathbb{N}^*$, for any test functions $f \in \mathcal{C}_b^2(\mathbb{R}^n, \mathbb{R})$ by

$$\mathcal{L}_{\text{FVc}}^{\lambda} P_{f,n}(\mu) = \langle B_{\lambda}^{(n)} f, \mu^n \rangle + \lambda \sum_{\substack{i,j=1 \\ i \neq j}}^{n} \left[ \langle \Phi_{i,j} f, \mu^{n-1} \rangle - \langle f, \mu^n \rangle \right] + \lambda \sum_{i,j=1}^{n} \langle K_{i,j} f, \mu^{n+1} \rangle. \quad (26)$$

From [18, Definition 2.8] and denoting

$$\tilde{\Omega} := \left\{ X \in \mathcal{C}^0([0, +\infty), \mathcal{M}_{1}^{c,2}(\mathbb{R})) \mid \forall T > 0, \sup_{0 \le t \le T} M_2(X_t) < \infty \right\},$$

we recall that a probability measure $P_\mu$ on $\tilde{\Omega}$ is said to solve the centered FLEMING-VIOT martingale problem for polynomials with initial condition $\mu \in \mathcal{M}_{1,K}^{c,2}(\mathbb{R})$, if the canonical process $(X_t)_{t \ge 0}$ on $\tilde{\Omega}$ satisfies $P_\mu(X_0 = \mu) = 1$ and, for all $n \in \mathbb{N}^*$ and $f \in \mathcal{C}_b^2(\mathbb{R}^n, \mathbb{R})$,

$$\widehat{M}_t^{P_{f,n}} := P_{f,n}(X_t) - P_{f,n}(X_0) - \int_0^t \mathcal{L}_{\text{FVc}}^{\lambda} P_{f,n}(X_s) \mathrm{d}s \qquad (27)$$

is a $P_\mu$-martingale.

**Lemma 3.4.** The infinitesimal generator $\mathcal{L}^K$ of the $\mathbb{R} \times \mathcal{M}_{1,K}^c(\mathbb{R})$-valued MARKOV process $(z^K, \mu^K)$ given by Proposition 3.1, satisfies for all $n \in \mathbb{N}^*$ and $f \in \mathcal{C}_b^3(\mathbb{R}^n, \mathbb{R})$, the following relations:

$$\mathcal{L}^K P_{f,n}(z, \mu) = \frac{\theta(z) m_2(z)}{K^2 \sigma_K^2} \mathcal{L}_{\text{FVc}}^{\lambda(z)} P_{f,n}(\mu) + \frac{1}{K^2 \sigma_K^2} \times O \left( \frac{1}{\sqrt{K}} + \sigma_K K^{\frac{3}{2}} M_2(\mu) + \frac{M_3(\mu)}{K} \right)$$
where $(\mathbf{C})_{x,y}^K := \langle f \circ \tau_{-\frac{y-x}{K}}, [\mu - \frac{\delta_x}{K} + \frac{\delta_y}{K}]^n \rangle - \langle f, \mu^n \rangle$. Note that, from TAYLOR'S formula, we obtain that

$$ (\mathbf{C})_{x,y}^K = \sum_{k=1}^{n} \binom{n}{k} \left\langle f, \mu^{n-k} \left[ -\frac{\delta_x}{K} + \frac{\delta_y}{K} \right]^k \right\rangle - \frac{y-x}{K} \sum_{i=1}^{n} \left\langle \partial_i f, \left[ \mu - \frac{\delta_x}{K} + \frac{\delta_y}{K} \right]^n \right\rangle \\ + \frac{(y-x)^2}{2K^2} \sum_{i=1}^{n} \sum_{j=1}^{n} \left\langle \partial_{ij}^2 f, \left[ \mu - \frac{\delta_x}{K} + \frac{\delta_y}{K} \right]^n \right\rangle + O \left( \frac{|y-x|^3}{K^3} \right). $$

By abuse of notation, we do not indicate the orders of the products of measure $\mu^{n-k} [-\frac{\delta_x}{K} + \frac{\delta_y}{K}]^k$.

**Step 1. Decomposition and study of $(\mathbf{A})^K$.** Note that $(\mathbf{A})^K = \sum_{i=1}^5 (\mathbf{A})_i^K$ where

$$ (\mathbf{A})_1^K := b(z, z) \sum_{k=1}^{n} \binom{n}{k} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) \left\langle f, \mu^{n-k} \left[ -\frac{\delta_x}{K} + \frac{\delta_y}{K} \right]^k \right\rangle, $$

$$ (\mathbf{A})_2^K := -\frac{b(z, z)}{K} \sum_{i=1}^{n} \sum_{k=0}^{n} \binom{n}{k} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) (y-x) \left\langle \partial_i f, \mu^{n-k} \left[ -\frac{\delta_x}{K} + \frac{\delta_y}{K} \right]^k \right\rangle, $$

$$ (\mathbf{A})_3^K := \frac{b(z, z)}{2K^2} \sum_{i,j=1}^{n} \sum_{k=0}^{n} \binom{n}{k} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) (y-x)^2 \left\langle \partial_{ij}^2 f, \mu^{n-k} \left[ -\frac{\delta_x}{K} + \frac{\delta_y}{K} \right]^k \right\rangle, $$

$$ (\mathbf{A})_4^K := O \left( \frac{M_3(\mu)}{K^3} \right), $$

$$ (\mathbf{A})_5^K := \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) \left[ b \left( \sigma\sqrt{K}x + z, \sigma\sqrt{K}y + z \right) - b(z, z) \right] (\mathbf{C})_{x,y}^K. $$

We first study precisely the $(\mathbf{A})_2^K$ term by explaining the case $k=0, k=1$ and $k \ge 2$. We will just give the result for the other $(\mathbf{A})_i^K$, $i \in \{1,3,5\}$: the approach remains the same. By the centered assumption of $\mu$, note that for all $i \in \{1, \dots, n\}$, $\int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) (y-x) \langle \partial_i f, \mu^n \rangle = 0$. Then, using again the centered assumption of $\mu$, we obtain that

$$ \begin{aligned} & \binom{n}{1} \sum_{i=1}^{n} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) (y-x) \left\langle \partial_i f, \mu^{n-1} \left[ -\frac{\delta_x}{K} + \frac{\delta_y}{K} \right] \right\rangle \\ &= \frac{1}{K} \sum_{i=1}^{n} \sum_{j=1}^{n} \int_{\mathbb{R}} \cdots \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) \\ & \quad \times \left[ x \partial_i f (x_1, \dots, x_{j-1}, x, x_{j+1} \dots, x_n) - x \partial_i f (x_1, \dots, x_{j-1}, y, x_{j+1} \dots, x_n) \right. \\ & \quad \left. - y \partial_i f (x_1, \dots, x_{j-1}, x, x_{j+1} \dots, x_n) + y \partial_i f (x_1, \dots, x_{j-1}, y, x_{j+1} \dots, x_n) \right] \prod_{\substack{l=1 \\ l \neq j}}^{n} \mu(dx_l) \\ &= \frac{2}{K} \sum_{i=1}^{n} \sum_{j=1}^{n} \int_{\mathbb{R}} \cdots \int_{\mathbb{R}} \mu(dx) x \partial_i f (x_1, \dots, x_{j-1}, x, x_{j+1}, \dots, x_n) \prod_{\substack{l=1 \\ l \neq j}}^{n} \mu(dx_l) \\ &= \frac{2}{K} \langle (\nabla f \cdot \mathbf{1})(\bullet \cdot \mathbf{1}), \mu^n \rangle. \end{aligned} $$
where $\lambda(z) := \frac{b(z,z)}{\theta(z)m_2(z)}$. Moreover, for all $K, n \in \mathbb{N}^*$ and test functions $f \in \mathcal{C}_b^3(\mathbb{R}^n, \mathbb{R})$, the process $(M_t^{K,P_{f,n}})_{t \ge 0}$ defined by

$$M_t^{K,P_{f,n}} := P_{f,n}(\mu_t^K) - P_{f,n}(\mu_0^K) - \int_0^t \mathcal{L}^K P_{f,n}(z_s^K, \mu_s^K) ds$$

is a square integrale martingale started at 0.

The proof of this result is given in Section 3.4.

### 3.3.2 For some unbounded test functions

The following results are particular extensions of Lemma 3.4 when the test functions are no longer bounded but the processes are stopped at time $\tilde{\tau}^K$. Then, we give in Lemmas 3.6 and 3.7 DooB's semi-martingale decomposition of moments of order 2 and those whose degree is 6.

**Proposition 3.5.** For all $n \in \mathbb{N}^*$, for all $f \in \mathcal{C}^3(\mathbb{R}^n, \mathbb{R})$, the process $(M_{t \wedge \tilde{\tau}^K}^{K,P_{f,n}})_{t \ge 0}$ defined by

$$M_{t \wedge \tilde{\tau}^K}^{K,P_{f,n}} := P_{f,n}(\mu_{t \wedge \tilde{\tau}^K}^K) - P_{f,n}(\mu_0^K) - \frac{K}{K \sigma_K^2} \int_0^{t \wedge \tilde{\tau}^K} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) \\ \times b(\sigma_K \sqrt{K}x + z_s^K, \sigma_K \sqrt{K}x + z_s^K) \left[ P_{f \circ \tau_{-\frac{y-x}{K},n}} \left( \mu - \frac{\delta_x}{K} + \frac{\delta_y}{K} \right) - P_{f,n}(\mu) \right] ds \\ + \frac{K}{K \sigma_K^2} \int_0^{t \wedge \tilde{\tau}^K} \int_{\mathbb{R}} \mu(dx) \theta(\sigma_K \sqrt{K}x + z_s^K) \int_{\mathbb{R}} m(\sigma_K \sqrt{K}x + z_s^K, h) dh \\ \times \left[ P_{f \circ \tau_{-\frac{h}{K^2},n}} \left( \mu - \frac{\delta_x}{K} + \frac{\delta_{x+\frac{h}{\sqrt{K}}}}{K} \right) - P_{f,n}(\mu) \right] ds$$

is a bounded martingale.

*Proof.* Thanks to the stopping time $\tilde{\tau}^K$, $(M_{t \wedge \tilde{\tau}^K}^{K,P_{f,n}})_{t \ge 0}$ is bounded. Hence the martingality of this process follows from Lemma 3.4. $\square$

(a) Moment of order 2. The following result will be useful in Section 4.

**Lemma 3.6.** For all $K \in \mathbb{N}^*$, the process $(M_{t \wedge \tilde{\tau}^K}^{K,P_{id}^{2,1}})_{t \ge 0}$ defined in Proposition 3.5 satisfies for all $t \ge 0$

$$M_{t \wedge \tilde{\tau}^K}^{K,P_{id}^{2,1}} = M_2(\mu_{t \wedge \tilde{\tau}^K}^K) - M_2(\mu_0^K) \\ + \frac{1}{K^2 \sigma_K^2} \int_0^{t \wedge \tilde{\tau}^K} \left\{ 2b(z_s^K, z_s^K) M_2(\mu_s^K) - \theta(z_s^K) m_2(z_s^K) \right\} ds \\ + \frac{1}{K^2 \sigma_K^2} \int_0^{t \wedge \tilde{\tau}^K} O\left(\frac{1}{K} + \sigma_K \sqrt{K} M_1(\mu_s^K) + \sigma_K K^{\frac{3}{2}} M_3(\mu_s^K)\right) ds$$
As $\nabla f$ is bounded, we deduce that for all $k \ge 2$, for all $i \in \{1, \dots, n\}$,

$$ \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) (y-x) \left\langle \partial_i f, \mu^{n-k} \left[ -\frac{\delta_x}{K} + \frac{\delta_y}{K} \right]^k \right\rangle = O \left( \frac{M_1(\mu)}{K^2} \right). $$

Therefore, we deduce that

$$ (\mathbf{A})_2^K = - \frac{2b(z, z)}{K^2} \langle (\nabla f \cdot \mathbf{1})(\bullet \cdot \mathbf{1}), \mu^n \rangle + O \left( \frac{M_1(\mu)}{K^3} \right). $$

In similar way and using Assumptions (A) we have that

$$ (\mathbf{A})_1^K = \frac{b(z, z)}{K^2} \sum_{i=1}^{n} \sum_{\substack{j=1 \\ j \neq i}}^{n} \left[ \langle \Phi_{i,j} f, \mu^{n-1} \rangle - \langle f, \mu^n \rangle \right] + O \left( \frac{1}{K^3} \right) $$

$$ (\mathbf{A})_3^K = \frac{b(z, z)}{K^2} \sum_{i=1}^{n} \sum_{j=1}^{n} \langle K_{i,j} f, \mu^{n+1} \rangle + O \left( \frac{M_2(\mu)}{K^3} \right) $$

$$ (\mathbf{A})_5^K = O \left( \frac{\sigma \sqrt{K}}{K} [M_1(\mu) + M_2(\mu)] \right). $$

**Step 2. Decomposition and study of (B)**$^K$. For all $\alpha, h \in \mathbb{R}$, we denote $\theta_m(\alpha, h) := \theta(\alpha) m(\alpha, h)$. Note that $(\mathbf{B})^K = \sum_{i=1}^5 (\mathbf{B})_i^K$ where

$$ (\mathbf{B})_1^K := \sum_{k=1}^{n} \binom{n}{k} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \theta_m(z, h) \left\langle f, \mu^{n-k} \left[ -\frac{\delta_x}{K} + \frac{\delta_{x+\frac{h}{\sqrt{K}}}}{K} \right]^k \right\rangle \mathrm{d}h, $$

$$ (\mathbf{B})_2^K := -\frac{1}{K^{\frac{3}{2}}} \sum_{i=1}^{n} \sum_{k=0}^{n} \binom{n}{k} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \theta_m(z, h) h \left\langle \partial_i f, \mu^{n-k} \left[ -\frac{\delta_x}{K} + \frac{\delta_{x+\frac{h}{\sqrt{K}}}}{K} \right]^k \right\rangle \mathrm{d}h, $$

$$ (\mathbf{B})_3^K := \frac{1}{2K^3} \sum_{i,j=1}^{n} \sum_{k=0}^{n} \binom{n}{k} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \theta_m(z, h) h^2 \left\langle \partial_{ij}^2 f, \mu^{n-k} \left[ -\frac{\delta_x}{K} + \frac{\delta_{x+\frac{h}{\sqrt{K}}}}{K} \right]^k \right\rangle \mathrm{d}h, $$

$$ (\mathbf{B})_4^K := O \left( \frac{1}{K^{\frac{9}{2}}} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \theta_m(z, h) |h|^3 \mathrm{d}h \right), $$

$$ (\mathbf{B})_5^K := \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \left[ \theta_m \left( \sigma \sqrt{K} x + z, h \right) - \theta_m(z, h) \right] (\mathbf{C})_{x,x+h/\sqrt{K}}^K \mathrm{d}h. $$

Note that, from TAYLOR's formula and Assumptions (A), we have that

$$ \begin{aligned} & \binom{n}{1} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \theta_m(z, h) \left\langle f, \mu^{n-1} \left[ -\frac{\delta_x}{K} + \frac{\delta_{x+\frac{h}{\sqrt{K}}}}{K} \right] \right\rangle \mathrm{d}h \\ & = \frac{1}{K} \sum_{i=1}^{n} \int_{\mathbb{R}} \cdots \int_{\mathbb{R}} \mu(dx_i) \int_{\mathbb{R}} \theta_m(z, h) \left[ \frac{h}{\sqrt{K}} \partial_i f(x_1, \dots, x_i, \dots, x_n) \right. \\ & \qquad \left. + \frac{h^2}{2K} \partial_{ii}^2 f(x_1, \dots, x_i, \dots, x_n) + O \left( \frac{|h|^3}{K^{\frac{3}{2}}} \right) \right] \mu(dx_1) \cdots \mu(dx_{i-1}) \mu(dx_{i+1}) \cdots \mu(dx_n) \end{aligned} $$
is a square integrable martingale with quadratic variation:

$$ \langle M^{K, P_{\text{id}^2,1}} \rangle_{t \wedge \tilde{\tau}^K} = \frac{2}{K^2 \sigma_K^2} \int_0^{t \wedge \tilde{\tau}^K} b(z_s^K, z_s^K) \{M_4(\mu_s^K) - M_2^2(\mu_s^K)\} \mathrm{d}s \\ + \frac{1}{K^2 \sigma_K^2} \int_0^{t \wedge \tilde{\tau}^K} O\left(\frac{\sigma_K \sqrt{K}}{K^2} + \frac{M_2(\mu_s^K)}{K} + \frac{\sigma_K \sqrt{K} M_3(\mu_s^K)}{K} + \sigma_K \sqrt{K} M_5(\mu_s^K)\right) \mathrm{d}s. \quad (28) $$

*Proof. Step 1. Approximation of the DOOB decomposition.* Note that for all $x, y \in \mathbb{R}$,

$$ P_{\text{id}^2 \circ \tau_{-\frac{y-x}{K},1}} \left( \mu - \frac{\delta_x}{K} + \frac{\delta_y}{K} \right) - P_{\text{id}^2,1}(\mu) = \frac{1}{K} (y^2 - x^2) - \frac{1}{K^2} (y-x)^2. $$

From TAYLOR'S formula, it follows that

$$ \frac{K}{K \sigma_K^2} \int_0^t \mathrm{d}s \int_{\mathbb{R}} \mu(\mathrm{d}x) \int_{\mathbb{R}} \mu(\mathrm{d}y) b \left( \sigma_K \sqrt{K} x + z_s^K, \sigma_K \sqrt{K} y + z_s^K \right) \\ \times \left[ P_{\text{id}^2 \circ \tau_{-\frac{y-x}{K},1}} \left( \mu - \frac{\delta_x}{K} + \frac{\delta_y}{K} \right) - P_{\text{id}^2,1}(\mu) \right] \\ = \frac{1}{K^2 \sigma_K^2} \int_0^t \left[ -2b(z_s^K, z_s^K) M_2(\mu_s^K) + O \left( \sigma_K K^{\frac{3}{2}} M_3(\mu_s^K) \right) \right] \mathrm{d}s $$

and

$$ \frac{K}{K \sigma_K^2} \int_0^t \mathrm{d}s \int_{\mathbb{R}} \mu(\mathrm{d}x) \int_{\mathbb{R}} m \left( \sigma_K \sqrt{K} x + z_s^K, h \right) \theta \left( \sigma_K \sqrt{K} x + z_s^K \right) \\ \times \left[ P_{\text{id}^2 \circ \tau_{-\frac{h}{K^{3/2}},1}} \left( \mu - \frac{\delta_x}{K} + \frac{\delta_{x+\frac{h}{\sqrt{K}}}}{K} \right) - P_{\text{id}^2,1}(\mu) \right] \mathrm{d}h \\ = \frac{1}{K^2 \sigma_K^2} \int_0^t \left[ \theta(z_s^K) m_2(z_s^K) + O \left( \frac{1}{K} + \sigma_K \sqrt{K} M_1(\mu_s^K) \right) \right] \mathrm{d}s. $$

We deduce the first announced result.

*Step 2. Quadratic variation.* In similar way to the proof of Proposition 3.1, we have

$$ \langle M^{K, P_{\text{id}^2,1}} \rangle_t = \frac{K}{K \sigma_K^2} \int_0^t \mathrm{d}s \int_{\mathbb{R}} \mu_s^K(\mathrm{d}x) \int_{\mathbb{R}} \mu_s^K(\mathrm{d}y) b \left( \sigma_K \sqrt{K} x + z_s^K, \sigma_K \sqrt{K} y + z_s^K \right) \\ \times \left[ \frac{1}{K} (y^2 - x^2) - \frac{1}{K^2} (y-x)^2 \right]^2 \\ + \frac{K}{K \sigma_K^2} \int_0^t \mathrm{d}s \int_{\mathbb{R}} \mu_s^K(\mathrm{d}x) \int_{\mathbb{R}} m \left( \sigma_K \sqrt{K} x + z_s^K, h \right) \theta \left( \sigma_K \sqrt{K} x + z_s^K \right) \\ \times \left[ \frac{h^2}{K^2} - \frac{h^2}{K^3} + \frac{2xh}{K^{\frac{3}{2}}} \right]^2 \mathrm{d}h. $$

The announced result follows from TAYLOR'S formula and straightforward computations. □

(b) Moments of degree 6. Let us denote for all $\ell \in \mathbb{N}$, $\tilde{M}_\ell(\mu) := \langle \mathrm{id}^\ell, \mu \rangle$.
$$= \frac{\theta(z)m_2(z)}{K^2} \left\langle \frac{\Delta f}{2}, \mu^n \right\rangle + O\left(\frac{1}{K^{\frac{5}{2}}}\right).$$

In similar way to Step 1, we obtain that

$$(\mathbf{B})_1^K = \frac{\theta(z)m_2(z)}{K^2} \left\langle \frac{\Delta f}{2}, \mu^n \right\rangle + O\left(\frac{1}{K^{\frac{5}{2}}}\right)$$

$$(\mathbf{B})_2^K = (\mathbf{B})_3^K = O\left(\frac{1}{K^3}\right), \quad (\mathbf{B})_4^K = O\left(\frac{1}{K^{\frac{9}{2}}}\right) \quad \text{and} \quad (\mathbf{B})_5^K = O\left(\frac{\sigma\sqrt{K}}{K^{\frac{3}{2}}}M_1(\mu)\right).$$

and using Assumptions (5) the announced result follows.

**Step 3. Martingality.** By classical arguments, since $M^{K,P_{f,n}}$ is bounded, we obtain that $M^{K,P_{f,n}}$ is a square integrable martingale started at 0.

# 4 Moments estimates

Lemma 3.6 and 3.7 lead us to look for moment estimates. In Lemma 3.6 we have given the DooB semi-martingale decomposition of the second-order moment and note that the latter involves the third-order moment in the error term. The presence of the higher order moment generates a difficulty to obtain a fine control of the second order moment. This difficulty is overcome by introducing the stopping time $\tilde{\tau}^K$, given by (14). In Section 4.1, we give estimates of the moment of order 6 and some corollaries useful for Sections 5 and 6. In Section 4.2 we establish estimates in expectation and probability of the second order moment up to the stopping time $\tilde{\tau}^K$ and we also control the martingale bracket of the DooB decomposition given in Lemma 3.6. In Sections 4.3 and 4.4, we prove that the stopping time $\tau^K := \tilde{\tau}^K \wedge \hat{\tau}^K$, given by (12), converges in probability to $+\infty$ when $K \to +\infty$ using coupling arguments between $M_2(\mu^K)$ and a biased random walk on $\mathbb{N}^*$, reflected in 1, and large deviations estimates for this random walk.

## 4.1 Estimates of the moment of order 6

**Lemma 4.1.** There exists $K_0 \in \mathbb{N}^*$ large enough and a constant $C > 0$ such that for all $K \ge K_0$ and $t \ge 0$,

$$\mathbb{E} \left( \left[ \frac{3}{4} M_6 (\mu_t^K) + 3M_4 (\mu_t^K) M_2 (\mu_t^K) + \frac{3}{2} M_2^3 (\mu_t^K) \right] \mathbf{1}_{t \le \tilde{\tau}^K} \right) \le \left( \frac{37}{4} C_6^* - C \right) \exp \left( - \frac{3b}{4K^2 \sigma^2 t} \right) + C$$

*Proof. Step 1. Pathwise inequality of* $\frac{3}{4}M_6(\mu) + 3M_4(\mu)M_2(\mu) + \frac{3}{2}M_2^3(\mu)$. Note that for all $t \ge 0$,

$$M_7(\mu_{t \wedge \tilde{\tau}^K}^K) \le \mathrm{Diam}(\mathrm{Supp} \, \mu_{t \wedge \tilde{\tau}^K}^K) M_6(\mu_{t \wedge \tilde{\tau}^K}^K) \le \frac{M_6(\mu_{t \wedge \tilde{\tau}^K}^K)}{\sigma K^{\frac{3+\varepsilon}{2}}} \quad (28)$$
**Lemma 3.7.** Let us consider the functions $f,g: \mathbb{R}^2 \to \mathbb{R}$ and $h: \mathbb{R}^3 \to \mathbb{R}$ respectively defined by $f(u,v) := u^4 v^2$, $g(u,v) := u^3 v^3$ and $h(u,v,w) := u^2 v^2 w^2$. For all $K \in \mathbb{N}^*$, the processes $(M_{t \wedge \hat{\tau}^K}^{K,P_{id}^{6,1}})_{t \ge 0}$, $(M_{t \wedge \hat{\tau}^K}^{K,P_{f,2}})_{t \ge 0}$, $(M_{t \wedge \hat{\tau}^K}^{K,P_{g,2}})_{t \ge 0}$ and $(M_{t \wedge \hat{\tau}^K}^{K,P_{h,3}})_{t \ge 0}$ satisfy for all $t \ge 0$

$$
\begin{aligned}
\bullet \quad & M_{t \wedge \hat{\tau}^K}^{K,P_{id}^{6,1}} = M_6 (\mu_{t \wedge \hat{\tau}^K}^K) - M_6 (\mu_0^K) + \frac{3}{K^2 \sigma_K^2} \int_0^{t \wedge \hat{\tau}^K} \left\{ b (z_s^K, z_s^K) [4M_6 (\mu_s^K) \right. \\
& \qquad \left. -10M_4 (\mu_s^K) M_2 (\mu_s^K)] - 5\theta (z_s^K) m_2 (z_s^K) M_4 (\mu_s^K) \right\} \mathrm{d}s \\
& + \frac{1}{K^2 \sigma_K^2} \int_0^{t \wedge \hat{\tau}^K} O \left( \frac{M_3 (\mu_s^K)}{\sqrt{K}} + \frac{1+M_4 (\mu_s^K)}{K} + \sigma_K \sqrt{K} M_5 (\mu_s^K), \right. \\
& \qquad \left. + \frac{M_6 (\mu_s^K)}{K^2} + \sigma_K K^{\frac{3}{2}} M_7 (\mu_s^K) \right) \mathrm{d}s
\end{aligned}
$$

$$
\begin{aligned}
\bullet \quad & M_{t \wedge \hat{\tau}^K}^{K,P_{f,2}} = M_4 (\mu_{t \wedge \hat{\tau}^K}^K) M_2 (\mu_{t \wedge \hat{\tau}^K}^K) - M_4 (\mu_0^K) M_2 (\mu_0^K) \\
& \qquad + \frac{1}{K^2 \sigma_K^2} \int_0^{t \wedge \hat{\tau}^K} \left\{ 2b (z_s^K, z_s^K) [6M_4 (\mu_s^K) M_2 (\mu_s^K) - M_6 (\mu_s^K) + 4\tilde{M}_3^2 (\mu_s^K) \right. \\
& \qquad \left. -6M_2^3 (\mu_s^K)] - \theta (z_s^K) m_2 (z_s^K) [6M_2^2 (\mu_s^K) + M_4 (\mu_s^K)] \right\} \mathrm{d}s \\
& \qquad + \frac{1}{K^2 \sigma_K^2} \int_0^{t \wedge \hat{\tau}^K} O \left( \frac{M_6 (\mu_s^K)}{K^3} + \sigma_K K^{\frac{3}{2}} M_7 (\mu_s^K) \right) \mathrm{d}s,
\end{aligned}
$$

$$
\begin{aligned}
\bullet \quad & M_{t \wedge \hat{\tau}^K}^{K,P_{g,2}} = \tilde{M}_3^2 (\mu_{t \wedge \hat{\tau}^K}^K) - \tilde{M}_3^2 (\mu_0^K) + \frac{1}{K^2 \sigma_K^2} \int_0^{t \wedge \hat{\tau}^K} \left\{ b (z_s^K, z_s^K) [-2M_6 (\mu_s^K) \right. \\
& \qquad \left. + 14\tilde{M}_3^2 (\mu_s^K) + 12M_4 (\mu_s^K) M_2 (\mu_s^K) - 18M_2^3 (\mu_s^K)] \right\} \mathrm{d}s \\
& \qquad + \frac{1}{K^2 \sigma_K^2} \int_0^{t \wedge \hat{\tau}^K} O \left( \frac{M_6 (\mu_s^K)}{K^3} + \sigma_K K^{\frac{3}{2}} M_7 (\mu_s^K) \right) \mathrm{d}s,
\end{aligned}
$$

$$
\begin{aligned}
\bullet \quad & M_{t \wedge \hat{\tau}^K}^{K,P_{h,3}} = M_2^3 (\mu_{t \wedge \hat{\tau}^K}^K) - M_2^3 (\mu_0^K) + \frac{1}{K^2 \sigma_K^2} \int_0^{t \wedge \hat{\tau}^K} \left\{ b (z_s^K, z_s^K) [12M_2^3 (\mu_s^K) \right. \\
& \qquad \left. - 6M_4 (\mu_s^K) M_2 (\mu_s^K)] - 3\theta (z_s^K) m_2 (z_s^K) M_4 (\mu_s^K) \right\} \mathrm{d}s \\
& \qquad + \frac{1}{K^2 \sigma_K^2} \int_0^{t \wedge \hat{\tau}^K} O \left( \frac{M_6 (\mu_s^K)}{K^3} + \sigma_K K^{\frac{3}{2}} M_7 (\mu_s^K) \right) \mathrm{d}s.
\end{aligned}
$$

*Proof.* The approximation proposed is obtained in a similar way to the proof of Lemma 3.6 $\square$

**3.4 Proof of Lemma 3.4**

From Proposition 3.1, for the choice of test functions $\Phi(z, \mu) := P_{f,n}(\mu)$ given by (22) and noting that for all $x, y \in \mathbb{R}$,

$$
\left\langle f, \left( \tau_{-\frac{y-x}{K}} \sharp \left[ \mu - \frac{\delta_x}{K} + \frac{\delta_y}{K} \right] \right)^n \right\rangle = \left\langle f \circ \tau_{-\frac{y-x}{K}}, \left[ \mu - \frac{\delta_x}{K} + \frac{\delta_y}{K} \right]^n \right\rangle,
$$
and from CAUCHY-SCHWARZ'S inequality $\tilde{M}_3^2(\mu) \le M_4(\mu)M_2(\mu)$. Then Lemma 3.7, (28) and Assumptions (A) imply that there exists a constant $C > 0$ such that for all $t \ge 0$,

$$
\begin{align*}
& \frac{3}{4} M_6 (\mu_{t \wedge \hat{\tau}^K}^K) + 3M_4 (\mu_{t \wedge \hat{\tau}^K}^K) M_2 (\mu_{t \wedge \hat{\tau}^K}^K) + \frac{3}{2} M_2^3 (\mu_{t \wedge \hat{\tau}^K}^K) \\
& \le \frac{3}{4} M_6 (\mu_{t \wedge \hat{\tau}^K}^K) + 7M_4 (\mu_{t \wedge \hat{\tau}^K}^K) M_2 (\mu_{t \wedge \hat{\tau}^K}^K) - 4\tilde{M}_3^2 (\mu_{t \wedge \hat{\tau}^K}^K) + \frac{3}{2} M_2^3 (\mu_{t \wedge \hat{\tau}^K}^K) \\
& \le \frac{3}{4} M_6 (\mu_0^K) + 7M_4 (\mu_0^K) M_2 (\mu_0^K) - 4\tilde{M}_3^2 (\mu_0^K) + \frac{3}{2} M_2^3 (\mu_0^K) \\
& \quad - \frac{1}{K^2 \sigma^2} \int_0^{t \wedge \hat{\tau}^K} \left\{ \left( b - \frac{C}{K^{\frac{\epsilon}{2}}} \right) \left[ 3M_6 (\mu_s^K) + \frac{9}{2} M_4 (\mu_s^K) M_2 (\mu_s^K) + 6M_2^3 (\mu_s^K) \right] \right. \\
& \qquad \left. - \bar{\theta} \bar{m}_2 \left[ \frac{91}{4} M_4 (\mu_s^K) + 42M_2^2 (\mu_s^K) \right] \right\} ds + \mathrm{Mart}_{t \wedge \hat{\tau}^K}
\end{align*}
$$

where $(\mathrm{Mart}_t)_{t \ge 0}$ is a martingale. Note that, choosing $K_0$ large enough, for all $K \ge K_0$, $b - CK^{-\epsilon/2} > b/2$. From the inequalities $M_2^2(\mu) \le M_4(\mu)$ and $M_4(\mu) \le M_6^{2/3}(\mu) \le \alpha M_6(\mu) + 1/\alpha^2$ with $\alpha := 2b/259\bar{\theta}\bar{m}_2$, we deduce that

$$
\frac{3b}{2} M_6(\mu) - \frac{259 \bar{\theta} \bar{m}_2}{4} M_4(\mu) \geq b M_6(\mu) - \frac{259^2 \bar{\theta}^2 \bar{m}_2^2}{4b}
$$

Thus, for all $t \ge 0$,

$$
\begin{align*}
& \frac{3}{4} M_6 (\mu_{t \wedge \hat{\tau}^K}^K) + 3M_4 (\mu_{t \wedge \hat{\tau}^K}^K) M_2 (\mu_{t \wedge \hat{\tau}^K}^K) + \frac{3}{2} M_2^3 (\mu_{t \wedge \hat{\tau}^K}^K) \\
& \le \frac{3}{4} M_6 (\mu_0^K) + 7M_4 (\mu_0^K) M_2 (\mu_0^K) + \frac{3}{2} M_2^3 (\mu_0^K) + \mathrm{Mart}_{t \wedge \hat{\tau}^K} \\
& \quad - \frac{b}{2K^2 \sigma^2} \int_0^{t \wedge \hat{\tau}^K} \left[ 2M_6 (\mu_s^K) + \frac{9}{2} M_4 (\mu_s^K) M_2 (\mu_s^K) + 6M_2^3 (\mu_s^K) - 2 \left( \frac{259 \bar{\theta} \bar{m}_2}{2b} \right)^2 \right] ds \\
& \le \frac{3}{4} M_6 (\mu_0^K) + 7M_4 (\mu_0^K) M_2 (\mu_0^K) + \frac{3}{2} M_2^3 (\mu_0^K) + \mathrm{Mart}_{t \wedge \hat{\tau}^K} \\
& \quad - \frac{3b}{4K^2 \sigma^2} \int_0^{t \wedge \hat{\tau}^K} \left[ \frac{3}{4} M_6 (\mu_s^K) + 3M_4 (\mu_s^K) M_2 (\mu_s^K) + \frac{3}{2} M_2^3 (\mu_s^K) - \frac{4}{3} \left( \frac{259 \bar{\theta} \bar{m}_2}{2b} \right)^2 \right] ds.
\end{align*}
$$

**Step 2. Conclusion.** Let us consider the stochastic process $(N(\mu_t^K))_{t \ge 0}$ defined by

$$
N(\mu_t^K) := \begin{cases} \tilde{N}_t & \text{if } t \le \check{\tau}^K \\ (\tilde{N}_{\check{\tau}^K} - C) \exp\left(-\frac{3b}{4K^2\sigma^2}(t - \check{\tau}^K)\right) + C & \text{if } t > \check{\tau}^K \end{cases}
$$

where $\tilde{N}_s := \frac{3}{4}M_6(\mu_s^K) + 3M_4(\mu_s^K)M_2(\mu_s^K) + \frac{3}{2}M_2^3(\mu_s^K)$, $s \ge 0$ and $C := \frac{4}{3}\left(\frac{259\bar{\theta}\bar{m}_2}{2b}\right)^2$. From Step 1, for all $t \ge 0$,

$$
N(\mu_t^K) \le \mathbb{E}(\tilde{N}_0 + 4M_4(\mu_0^K)M_2(\mu_0^K)) - \frac{3b}{4K^2\sigma^2} \int_0^t [N(\mu_s^K) - C] ds + \mathrm{Mart}_{t \wedge \hat{\tau}^K}.
$$
we obtain that $\mathcal{L}^K P_{f,n}(z, \mu) = \frac{K}{K\sigma_K^2} ((\mathbf{A})^K + (\mathbf{B})^K)$ where

$$ (\mathbf{A})^K := \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) b\left(\sigma_K \sqrt{K}x + z, \sigma_K \sqrt{K}y + z\right) (\mathbf{C})_{x,y}^K, $$

$$ (\mathbf{B})^K := \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \theta\left(\sigma_K \sqrt{K}x + z\right) m\left(z + \sigma_K \sqrt{K}x, h\right) (\mathbf{C})_{x,x+h/\sqrt{K}}^K \, dh $$

where $(\mathbf{C})_{x,y}^K := \langle f \circ \tau_{-\frac{y-x}{K}}, [\mu - \frac{\delta_x}{K} + \frac{\delta_y}{K}]^n \rangle - \langle f, \mu^n \rangle$. Note that, from TAYLOR'S formula, we obtain that

$$ (\mathbf{C})_{x,y}^K = \sum_{k=1}^{n} \binom{n}{k} \left\langle f, \mu^{n-k} \left[ -\frac{\delta_x}{K} + \frac{\delta_y}{K} \right]^k \right\rangle - \frac{y-x}{K} \sum_{i=1}^{n} \left\langle \partial_i f, \left[ \mu - \frac{\delta_x}{K} + \frac{\delta_y}{K} \right]^n \right\rangle \\ + \frac{(y-x)^2}{2K^2} \sum_{i=1}^{n} \sum_{j=1}^{n} \left\langle \partial_{ij}^2 f, \left[ \mu - \frac{\delta_x}{K} + \frac{\delta_y}{K} \right]^n \right\rangle + O\left(\frac{|y-x|^3}{K^3}\right). $$

By abuse of notation, we do not indicate the orders of the products of measure $\mu^{n-k} [-\frac{\delta_x}{K} + \frac{\delta_y}{K}]^k$.

Step 1. Decomposition and study of $(\mathbf{A})^K$. Note that $(\mathbf{A})^K = \sum_{i=1}^5 (\mathbf{A})_i^K$ where

$$ (\mathbf{A})_1^K := b(z, z) \sum_{k=1}^{n} \binom{n}{k} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) \left\langle f, \mu^{n-k} \left[ -\frac{\delta_x}{K} + \frac{\delta_y}{K} \right]^k \right\rangle, $$

$$ (\mathbf{A})_2^K := -\frac{b(z, z)}{K} \sum_{i=1}^{n} \sum_{k=0}^{n} \binom{n}{k} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) (y-x) \left\langle \partial_i f, \mu^{n-k} \left[ -\frac{\delta_x}{K} + \frac{\delta_y}{K} \right]^k \right\rangle, $$

$$ (\mathbf{A})_3^K := \frac{b(z, z)}{2K^2} \sum_{i,j=1}^{n} \sum_{k=0}^{n} \binom{n}{k} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) (y-x)^2 \left\langle \partial_{ij}^2 f, \mu^{n-k} \left[ -\frac{\delta_x}{K} + \frac{\delta_y}{K} \right]^k \right\rangle, $$

$$ (\mathbf{A})_4^K := O\left(\frac{M_3(\mu)}{K^3}\right), $$

$$ (\mathbf{A})_5^K := \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) \left[ b\left(\sigma_K \sqrt{K}x + z, \sigma_K \sqrt{K}y + z\right) - b(z, z) \right] (\mathbf{C})_{x,y}^K. $$

We first study precisely the $(\mathbf{A})_2^K$ term by explaining the case $k=0, k=1$ and $k \ge 2$. We will just give the result for the other $(\mathbf{A})_i^K$, $i \in \{1, 3, 5\}$: the approach remains the same. By the centered assumption of $\mu$, note that for all $i \in \{1, \dots, n\}$,

$$ \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) (y-x) \langle \partial_i f, \mu^n \rangle = 0. $$
From (7) then (4), note that

$$M_6(\mu_0^K) = \langle \mathrm{id}^6, \mu_0^K \rangle = \frac{1}{K^3 \sigma^6} \langle (\mathrm{id} - x_0)^6, \nu_0^K \rangle \le C_6^*.$$

Hence for all $t \ge 0$

$$\mathbb{E}(N(\mu_t^K)) \le \frac{37}{4} C_6^* - \frac{3b}{4K^2\sigma^2} \int_0^t [\mathbb{E}(N(\mu_s^K)) - C] \, ds$$

and the announced result follows.

<figure><img src="image_3.png" /></figure>

The following lemma will be useful in the proof of Theorem 5.1.

**Lemma 4.2.** For all $t \ge 0$,

$$\sup_{K \in \mathbb{N}^*} \mathbb{E} \left( \int_0^{t \wedge \hat{\tau}^K} M_6(\mu_s^K) \, ds \right) < \infty.$$

*Proof.* Let $t \ge 0$ be fixed. Note that

$$\int_0^t M_6(\mu_s^K) \mathbf{1}_{s \wedge \hat{\tau}^K} ds \le \frac{4}{3} \int_0^t \left( \frac{3}{4} M_6(\mu_s^K) + 3M_4(\mu_s^K) M_2(\mu_s^K) + \frac{3}{2} M_2^3(\mu_s^K) \right) \mathbf{1}_{s \le \hat{\tau}^K} ds.$$

By FUBINI'S theorem and Lemma 4.1, there exists a constant $C > 0$ such that

$$\mathbb{E} \left( \int_0^t M_6 (\mu_s^K) \mathbf{1}_{s \wedge \hat{\tau}^K} ds \right) \le \frac{4}{3} \left( \left( \frac{37}{4} C_6^* - C \right) \int_0^t \exp \left( -\frac{3b}{4K^2\sigma^2} s \right) ds + Ct \right) \le \frac{37C_6^* + 4C}{3} t$$

which ends the proof.

<figure><img src="image_4.png" /></figure>

The last lemma has the following consequence, used in Sections 5 and 6.

**Corollary 4.3.** For all $T > 0$, the family

$$\left( \int_0^{t \wedge \hat{\tau}^K} M_5 (\mu_s^K) \, ds \right)_{t \in [0, T], K \in \mathbb{N}^*}$$

is uniformly integrable.

*Proof.* Note that for all $K \in \mathbb{N}^*$, for all $t \in [0, T]$, for all $A \in (0, +\infty)$,

$$\mathbf{1}_{\int_0^t M_5(\mu_s^K) \mathbf{1}_{s \le \hat{\tau}^K} ds \ge A} \int_0^t M_5(\mu_s^K) \mathbf{1}_{s \le \hat{\tau}^K} ds \le \frac{1}{\sqrt[5]{A}} \left( \int_0^t M_5(\mu_s^K) \mathbf{1}_{s \le \hat{\tau}^K} ds \right)^{6/5},$$

we deduce from HÖLDER' inequality that, for all $t \in [0, T]$, for all $A \in (0, +\infty)$,

$$\sup_{K \in \mathbb{N}^*} \mathbb{E} \left( \mathbf{1}_{\int_0^t M_5(\mu_s^K) \mathbf{1}_{s \le \hat{\tau}^K} ds \ge A} \int_0^t M_5(\mu_s^K) \mathbf{1}_{s \le \hat{\tau}^K} ds \right) \le \frac{T}{\sqrt[5]{A}} \sup_{K \in \mathbb{N}^*} \mathbb{E} \left( \int_0^t M_5^{6/5}(\mu_s^K) \mathbf{1}_{s \le \hat{\tau}^K} ds \right) \\ \le \frac{T}{\sqrt[5]{A}} \sup_{K \in \mathbb{N}^*} \mathbb{E} \left( \int_0^t M_6(\mu_s^K) \mathbf{1}_{s \le \hat{\tau}^K} ds \right).$$

From Lemma 4.2, we deduce that the right hand side of the previous inequality goes to 0 when $A \to +\infty$ and so the announced result.

<figure><img src="image_5.png" /></figure>
Then, using again the centered assumption of $\mu$, we obtain that

$$ 
\begin{aligned}
& \binom{n}{1} \sum_{i=1}^{n} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) (y-x) \left\langle \partial_i f, \mu^{n-1} \left[ -\frac{\delta_x}{K} + \frac{\delta_y}{K} \right] \right\rangle \\
&= \frac{1}{K} \sum_{i=1}^{n} \sum_{j=1}^{n} \int_{\mathbb{R}} \cdots \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) \\
& \quad \times \left[ x \partial_i f(x_1, \cdots, x_{j-1}, x, x_{j+1} \cdots, x_n) - x \partial_i f(x_1, \cdots, x_{j-1}, y, x_{j+1} \cdots, x_n) \right. \\
& \quad \left. - y \partial_i f(x_1, \cdots, x_{j-1}, x, x_{j+1} \cdots, x_n) + y \partial_i f(x_1, \cdots, x_{j-1}, y, x_{j+1} \cdots, x_n) \right] \prod_{\substack{l=1 \\ l \neq j}}^{n} \mu(dx_l) \\
&= \frac{2}{K} \sum_{i=1}^{n} \sum_{j=1}^{n} \int_{\mathbb{R}} \cdots \int_{\mathbb{R}} \mu(dx) x \partial_i f(x_1, \cdots, x_{j-1}, x, x_{j+1}, \cdots, x_n) \prod_{\substack{l=1 \\ l \neq j}}^{n} \mu(dx_l) \\
&= \frac{2}{K} \left\langle (\nabla f \cdot \mathbf{1})(\bullet \cdot \mathbf{1}), \mu^n \right\rangle.
\end{aligned}
$$

As $\nabla f$ is bounded, we deduce that for all $k \ge 2$, for all $i \in \{1, \cdots, n\}$,

$$ \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \mu(dy) (y-x) \left\langle \partial_i f, \mu^{n-k} \left[ -\frac{\delta_x}{K} + \frac{\delta_y}{K} \right]^k \right\rangle = O \left( \frac{M_1(\mu)}{K^2} \right). $$

Therefore, we deduce that

$$ (\mathbf{A})_2^K = -\frac{2b(z, z)}{K^2} \left\langle (\nabla f \cdot \mathbf{1})(\bullet \cdot \mathbf{1}), \mu^n \right\rangle + O\left(\frac{M_1(\mu)}{K^3}\right). $$

In similar way and using Assumptions (A) we have that

$$ (\mathbf{A})_1^K = \frac{b(z, z)}{K^2} \sum_{i=1}^{n} \sum_{\substack{j=1 \\ j \neq i}}^{n} \left[ \langle \Phi_{i,j} f, \mu^{n-1} \rangle - \langle f, \mu^n \rangle \right] + O\left(\frac{1}{K^3}\right), $$

$$ (\mathbf{A})_3^K = \frac{b(z, z)}{K^2} \sum_{i=1}^{n} \sum_{j=1}^{n} \langle K_{i,j} f, \mu^{n+1} \rangle + O\left(\frac{M_2(\mu)}{K^3}\right), $$

$$ (\mathbf{A})_5^K = O\left(\frac{\sigma_K \sqrt{K}}{K} [M_1(\mu) + M_2(\mu)]\right). $$

**Step 2.** Decomposition and study of $(\mathbf{B})^K$. For all $\alpha, h \in \mathbb{R}$, we denote $\theta_m(\alpha, h) :=$
## 4.2 Some inequalities on the moment of order 2

**Lemma 4.4.** There exists $K_0 \in \mathbb{N}^*$ large enough such that for all $K \ge K_0$, the process $M_2(\mu^K)$ satisfies for all $t \ge 0$ the following inequalities:

$$ M_2(\mu_0^K) - \frac{5\bar{b}}{2K^2\sigma^2} \int_0^{t\wedge\bar{\tau}^K} \left[ M_2(\mu_s^K) - \frac{2\bar{\theta}\bar{m}_2}{5\bar{b}} + \frac{2\bar{\theta}\bar{m}_2}{5\bar{b}K} \right] ds + M_{t\wedge\bar{\tau}^K}^{K,P_{\text{id}2,1}} \\ \le M_2(\mu_{t\wedge\bar{\tau}^K}^K) \le M_2(\mu_0^K) - \frac{\bar{b}}{K^2\sigma^2} \int_0^{t\wedge\bar{\tau}^K} \left[ M_2(\mu_s^K) - \frac{\bar{\theta}\bar{m}_2}{\bar{b}} \right] ds + M_{t\wedge\bar{\tau}^K}^{K,P_{\text{id}2,1}} \tag{29} $$

where $M_{t\wedge\bar{\tau}^K}^{K,P_{\text{id}2,1}}$ is defined in Proposition 3.5. Moreover, for $t \ge 0$,

$$ \mathbb{E} \left( M_2 \left( \mu_t^K \right) 1_{t \le \bar{\tau}^K} \right) \le \mathbb{E} \left( M_2 \left( \mu_0^K \right) - \frac{\bar{\theta} \bar{m}_2}{\bar{b}} \right) \exp \left( - \frac{\bar{b}}{K^2 \sigma^2 t} \right) + \frac{\bar{\theta} \bar{m}_2}{\bar{b}}. $$

*Proof.* Note that for all $t \ge 0$,

$$ M_3(\mu_{t\wedge\bar{\tau}^K}^K) \le \text{Diam} (\text{Supp } \mu_{t\wedge\bar{\tau}^K}^K) M_2(\mu_{t\wedge\bar{\tau}^K}^K) \le \frac{M_2(\mu_{t\wedge\bar{\tau}^K}^K)}{\sigma K^{\frac{3+\epsilon}{2}}}. \tag{30} $$

From Lemma 3.6, (30) and Assumptions (A), there exists two constants $C, \tilde{C} > 0$ such that for all $t \ge 0$

$$ M_2(\mu_0^K) - \frac{1}{K^2\sigma^2} \int_0^{t\wedge\bar{\tau}^K} \left( \left[ 2\bar{b} + \frac{C}{K^{\frac{\epsilon}{2}}} \right] M_2(\mu_s^K) - \frac{\bar{\theta}\bar{m}_2}{K} \right) ds + M_{t\wedge\bar{\tau}^K}^{K,P_{\text{id}2,1}} \\ \le M_2(\mu_{t\wedge\bar{\tau}^K}^K) \le M_2(\mu_0^K) - \frac{1}{K^2\sigma^2} \int_0^t \left( \left[ 2\underline{b} - \frac{\tilde{C}}{K^{\frac{\epsilon}{2}}} \right] M_2(\mu_s^K) - \frac{\bar{\theta}\bar{m}_2}{K} \right) ds + M_{t\wedge\bar{\tau}^K}^{K,P_{\text{id}2,1}}. $$

Noting that for all $K \ge K_0$, $2\underline{b} - \tilde{C}K^{-\epsilon/2} > \underline{b}$ and $2\bar{b} + CK^{-\epsilon/2} < 5\bar{b}/2$, the announced first result follows. In similar way to Step 2 of the proof of Lemma 4.1, we obtain the second part of the announced result. $\square$

**Corollary 4.5.** For all $t \ge 0$, $\mathbb{E} (M_2(\mu_t^K) 1_{t \le \bar{\tau}^K}) \le \max \{C_2^*, \bar{\theta}\bar{m}_2\}$.

*Proof.* From (7) and (4), note that

$$ M_2(\mu_0^K) = \langle \text{id}^2, \mu_0^K \rangle = \frac{1}{K\sigma^2} C_2^*. $$

The announced result follows from Lemma 4.4. $\square$

The goal of the following result is to bound the martingale bracket $\langle M^{K,P_{\text{id}2,1}} \rangle_{t\wedge\bar{\tau}^K}$. To do this, we exploit the martingale approximation of the moment of order 6 in order to control the dominant term $M_4(\mu_s^K) - M_2^2(\mu_s^K)$ in (27).
$\theta(\alpha)m(\alpha, h)$. Note that $(\mathbf{B})^K = \sum_{i=1}^5 (\mathbf{B})_i^K$ where

$$ (\mathbf{B})_1^K := \sum_{k=1}^{n} \binom{n}{k} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \theta_m(z, h) \left\langle f, \mu^{n-k} \left[ -\frac{\delta_x}{K} + \frac{\delta_{x+\frac{h}{\sqrt{K}}}}{K} \right]^k \right\rangle \mathrm{d}h, $$

$$ (\mathbf{B})_2^K := -\frac{1}{K^{\frac{3}{2}}} \sum_{i=1}^{n} \sum_{k=0}^{n} \binom{n}{k} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \theta_m(z, h) h \left\langle \partial_i f, \mu^{n-k} \left[ -\frac{\delta_x}{K} + \frac{\delta_{x+\frac{h}{\sqrt{K}}}}{K} \right]^k \right\rangle \mathrm{d}h, $$

$$ (\mathbf{B})_3^K := \frac{1}{2K^3} \sum_{i,j=1}^{n} \sum_{k=0}^{n} \binom{n}{k} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \theta_m(z, h) h^2 \left\langle \partial_{ij}^2 f, \mu^{n-k} \left[ -\frac{\delta_x}{K} + \frac{\delta_{x+\frac{h}{\sqrt{K}}}}{K} \right]^k \right\rangle \mathrm{d}h, $$

$$ (\mathbf{B})_4^K := O\left(\frac{1}{K^{\frac{9}{2}}} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \theta_m(z, h) |h|^3 \mathrm{d}h\right), $$

$$ (\mathbf{B})_5^K := \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \left[ \theta_m \left( \sigma_K \sqrt{K} x + z, h \right) - \theta_m(z, h) \right] (\mathbf{C})_{x,x+h/\sqrt{K}}^K \mathrm{d}h. $$

Note that, from TAYLOR'S formula and Assumptions (A), we have that

$$ \begin{aligned} & \binom{n}{1} \int_{\mathbb{R}} \mu(dx) \int_{\mathbb{R}} \theta_m(z, h) \left\langle f, \mu^{n-1} \left[ -\frac{\delta_x}{K} + \frac{\delta_{x+\frac{h}{\sqrt{K}}}}{K} \right] \right\rangle \mathrm{d}h \\ &= \frac{1}{K} \sum_{i=1}^{n} \int_{\mathbb{R}} \cdots \int_{\mathbb{R}} \int_{\mathbb{R}} \mu(dx_i) \int_{\mathbb{R}} \theta_m(z, h) \left[ \frac{h}{\sqrt{K}} \partial_i f(x_1, \cdots, x_i, \cdots, x_n) \right. \\ & \qquad \left. + \frac{h^2}{2K} \partial_{ii}^2 f(x_1, \cdots, x_i, \cdots, x_n) + O\left(\frac{|h|^3}{K^{\frac{3}{2}}}\right) \right] \prod_{\substack{\ell=1 \\ \ell \neq i}}^{n} \mu(dx_\ell) \\ &= \frac{\theta(z)m_2(z)}{K^2} \left\langle \frac{\Delta f}{2}, \mu^n \right\rangle + O\left(\frac{1}{K^{\frac{5}{2}}}\right). \end{aligned} $$

In similar way to Step 1, we obtain that

$$ (\mathbf{B})_1^K = \frac{\theta(z)m_2(z)}{K^2} \left\langle \frac{\Delta f}{2}, \mu^n \right\rangle + O\left(\frac{1}{K^{\frac{5}{2}}}\right), \quad (\mathbf{B})_2^K = O\left(\frac{1}{K^3}\right), $$

$$ (\mathbf{B})_3^K = O\left(\frac{1}{K^3}\right), \quad (\mathbf{B})_4^K = O\left(\frac{1}{K^{\frac{9}{2}}}\right) \quad \text{and} \quad (\mathbf{B})_5^K = O\left(\frac{\sigma_K \sqrt{K}}{K^{\frac{3}{2}}} M_1(\mu)\right). $$

and using Assumptions (5) the announced result follows.

*Step 3. Martingality.* By classical arguments, since $M^{K,P_{f,n}}$ is bounded, we obtain that $M^{K,P_{f,n}}$ is a square integrable martingale started at 0.

# 4 Moments estimates

Lemmas 3.6 and 3.7 lead us to look for moment estimates. In Lemma 3.6 we have given the DOOB semi-martingale decomposition of the second-order moment and note that involves the third-order moment in the error term. The presence of the higher order moment generates a
**Lemma 4.6.** There exists a constant $C > 0$ such that for all $K \in \mathbb{N}^*$, for all $t \ge 0$ the martingale bracket $\langle M^{K, P_{\text{id}2,1}} \rangle_t$ satisfies

$$ \mathbb{E} \left( \langle M^{K, P_{\text{id}2,1}} \rangle_{t \wedge \check{\tau}^K} \right) \le \frac{C}{K^2 \sigma^2} \mathbb{E} \left( t \wedge \check{\tau}^K \right). $$

*Proof.* From Lemma 3.6 and Hölder's inequalities $M_2^2(\mu) \le M_4(\mu) \le M_6^{2/3} \le 1 + M_6(\mu)$ and $M_5(\mu) \le M_6(\mu)^{5/6} \le 1 + M_6(\mu)$, we deduce that there exists a constant $C > 0$ such that for all $t \ge 0$

$$ \langle M^{K, P_{\text{id}2,1}} \rangle_{t \wedge \check{\tau}^K} \le \frac{C}{K^2 \sigma^2} \int_0^{t \wedge \check{\tau}^K} \left(1 + M_6(\mu_s^K)\right) \left(1 + \frac{1}{K}\right) \mathrm{d}s. $$

The announced result follows from FUBINI's theorem and Lemma 4.1. $\square$

## 4.3 Convergence to $+\infty$ of the stopping time $\tau^K$ and support concentration property

The main result of this section is the following convergence result for the stopping time $\tau^K$, defined by (12), when $K \to +\infty$.

**Proposition 4.7.** Under Assumption (5), $\tau^K$ converges in probability to $+\infty$ when $K \to +\infty$.

The proof is based on the next lemma proved in Section 4.4 below. Thanks to Proposition 4.7, we deduce the support concentration property of $(\nu_{t/K\sigma^2}^K)$ given by (6). Indeed, with probability which tends to 1 when $K \to +\infty$, $\text{Diam}(\mu_t^K) \le \frac{1}{\sigma K^{\frac{3+\epsilon}{2}}}$ for all $t \in [0, T]$. Hence,

$$ \text{Diam} (\nu_{t/K\sigma^2}^K) = \sigma \sqrt{K} \text{Diam} (\mu_t^K) \le \frac{1}{K^{1+\frac{\epsilon}{2}}} $$

and (6) follows.

**Lemma 4.8.** Under Assumption (5), for all $T \ge 0$,

$$ \lim_{K \to +\infty} \mathbb{P} (\hat{\tau}^K \le \check{\tau}^K \wedge T) = \lim_{K \to +\infty} \mathbb{P} \left( \sup_{0 \le t \le T \wedge \check{\tau}^K} M_2 (\mu_t^K) \ge K^\epsilon \right) = 0. $$

*Proof of Proposition 4.7.* Note that for all $T \ge 0$,

$$ \mathbb{P} (\check{\tau}^K < T \wedge \hat{\tau}^K) \le \mathbb{P} \left( \exists t < T \wedge \hat{\tau}^K, \text{Diam} (\text{Supp } \mu_t^K) > \frac{1}{\sigma K^{\frac{3+\epsilon}{2}}} \right) $$

As for all $t \ge 0$, for all $K \in \mathbb{N}^*$, $\text{Diam}(\text{Supp } \mu_t^K)^2 \le (2 \max \{|x| \mid x \in \text{Supp } \mu_t^K\})^2 \le 4KM_2(\mu_t^K)$, we deduce that

$$ \begin{aligned} & \mathbb{P} (\exists t < T \wedge \hat{\tau}^K, \text{Diam} (\text{Supp } \mu_t^K) > \frac{1}{\sigma K^{\frac{3+\epsilon}{2}}}) \\ & \le \mathbb{P} (\exists t < T \wedge \hat{\tau}^K M_2 (\mu_t^K) > \frac{1}{4\sigma^2 K^{4+\epsilon}}). \end{aligned} $$
difficulty to obtain a fine control of the second order moment. This difficulty is overcome by introducing the stopping time $\tilde{\tau}^K$, given by (15). In Section 4.1, we give estimates of the moment of order 6 and some corollaries useful for Sections 5 and 6. In Section 4.2 we establish estimates in expectation and probability of the second order moment up to the stopping time $\tilde{\tau}^K$ and we also control the martingale bracket of the DOOB decomposition given in Lemma 3.6. In Sections 4.3 and 4.4, we prove that the stopping time $\tau^K := \tilde{\tau}^K \wedge \hat{\tau}^K$, given by (13), converges in probability to $+\infty$ when $K \to +\infty$ using coupling arguments between $M_2(\mu^K)$ and a biased random walk on $\mathbb{N}^*$, reflected in 1, and large deviations estimates for this random walk.

## 4.1 Estimates of the moment of order 6

In this section, we establish a technical lemma useful for Lemma 4.2 below. The idea is to obtain an explicit upper bound of the moments of order 6.

**Lemma 4.1.** Given any continuous function $y(t)$ satisfying

$$y(t) \le y(s) + \int_s^t F(y(r)) dr, \quad \forall s \le t \qquad (29)$$

for some LIPSCHITZ function $F: \mathbb{R} \to \mathbb{R}$, then $y(s) \le z(s)$ for all $s \ge 0$, where $z(t)$ is the unique solution of

$$z(t) = z(s) + \int_s^t F(z(r)) dr, \quad \forall s \le t \qquad (30)$$

such that $z(0) = y(0)$.

*Proof.* First, it is sufficient to prove that, for all $\varepsilon > 0$, $y(s) \le z_\varepsilon(s)$ for all $s \ge 0$, where $z_\varepsilon$ is the unique solution of

$$z_\varepsilon(t) = z_\varepsilon(s) + \int_s^t [F(z_\varepsilon(r)) + \varepsilon] dr, \quad \forall s \le t \qquad (31)$$

such that $z_\varepsilon(0) = y(0) + \varepsilon$. Indeed, it follows from (30) and (31) that, for all $t \ge 0$ and $\varepsilon > 0$

$$|z(t) - z_\varepsilon(t)| \le \varepsilon(1 + t) + C_{\text{Lip}} \int_0^t |z(s) - z_\varepsilon(s)| ds$$

where $C_{\text{Lip}}$ if the LIPSCHITZ constant of $F$. GRONWALL'S lemma then entails

$$|z(t) - z_\varepsilon(t)| \le \varepsilon(1 + t) \exp(C_{\text{Lip}}t) \underset{\varepsilon \to 0}{\longrightarrow} 0$$

so that $y(t) \le z_\varepsilon(t)$, for all $\varepsilon > 0$ indeed implies $y(t) \le z(t)$.

So, let $\varepsilon > 0$ be fixed and let us prove that $y(t) \le z_\varepsilon(t)$ for all $t \ge 0$. Define

$$\vartheta_\varepsilon := \inf \left\{ t \ge 0 \mid y(t) \ge z_\varepsilon(t) \right\}.$$

Our goal is to prove that $\vartheta_\varepsilon = +\infty$.

By continuity of $y$, $\vartheta_\varepsilon > 0$ and, if $\vartheta_\varepsilon < \infty$, $y(\vartheta_\varepsilon) = z_\varepsilon(\vartheta_\varepsilon)$. Assume by contradiction that $\vartheta_\varepsilon < \infty$. Then, using (29) and (31), for all $\delta \in (0, \vartheta_\varepsilon)$,

$$\frac{z_\varepsilon(\vartheta_\varepsilon) - y(\vartheta_\varepsilon) - (z_\varepsilon(\vartheta_\varepsilon - \delta) - y(\vartheta_\varepsilon - \delta))}{\delta} \ge \varepsilon + \frac{1}{\delta} \int_{\vartheta_\varepsilon - \delta}^{\vartheta_\varepsilon} [F(z_\varepsilon(r)) - F(y(r))] dr.$$
According to (5), we deduce that the following inequality $4\sigma^2 K^{4+\epsilon} \le 1/K^\epsilon$ holds true when $K \to +\infty$ and $\sigma \to 0$. Hence, we deduce that

$$ \mathbb{P} \left( \exists t < T \wedge \hat{\tau}^K, M_2 (\mu_t^K) > \frac{1}{4\sigma^2 K^{4+\epsilon}} \right) \le \mathbb{P} \left( \exists t < T \wedge \hat{\tau}^K, M_2 (\mu_t^K) > K^\epsilon \right) = 0. $$

Now, for all $T > 0$, $\{\check{\tau}^K \ge T \wedge \hat{\tau}^K\} = \{\check{\tau}^K \ge \hat{\tau}^K\} \cup \{\check{\tau}^K \ge T\}$, and so

$$ \begin{aligned} \mathbb{P} \left( \{\check{\tau}^K \ge T \wedge \hat{\tau}^K\} \cap \{\tau^K < T\} \right) &\le \mathbb{P} \left( \left[ \{\hat{\tau}^K < T\} \cap \{\hat{\tau}^K \le \check{\tau}^K\} \right] \cup \{\hat{\tau}^K < T \le \check{\tau}^K\} \right) \\ &\le \mathbb{P} \left( \hat{\tau}^K \le T \wedge \check{\tau}^K \right). \end{aligned} $$

Note that for all $T > 0$,

$$ \mathbb{P} (\tau^K < T) \le \mathbb{P} (\check{\tau}^K < T \wedge \hat{\tau}^K) + \mathbb{P} (\hat{\tau}^K \le T \wedge \check{\tau}^K). $$

Hence, from Lemma 4.8, we have for all $T \ge 0$, $\lim_{K \to +\infty} \mathbb{P} (\tau^K < T) = 0$ and the announced result follows. $\square$

## 4.4 Proof of Lemma 4.8

Note that if $M_2(\mu_s^K)$ becomes larger than $\overline{\theta m_2}/b$, the drift term in the right-hand side of (29) is then negative preventing $M_2(\mu_s^K)$ to become excessively large unless the martingale $M^{K,P_{id^2,1}}$ has an exceptional path with large increments on a small time interval. We then expect to have large deviations estimates and bounds on the time of exit from attractive domains for $M_2(\mu^K)$. However, we cannot consider establishing directly a large deviation principle on $M_2(\mu^K)$, in particular because of the slow-fast limit. The approach considered below is based on coupling arguments between $M_2(\mu^K)$ and a simpler process for which large deviations estimates are known.

Let us introduce $u_0 := 0$ and for all $\ell \in \mathbb{N}^*$ the real number $u_\ell := 3^\ell K^{\frac{\epsilon}{2}}$ and the interval $\mathcal{I}_\ell := [u_{\ell-1}, u_{\ell+1})$. We will look at the process $M_2(\mu^K)$ at successive exit times of $(\mathcal{I}_\ell)_{\ell \in \mathbb{N}^*}$. We set $L_0^K := 1$ when $M_2(\mu_0^K) \in [0, 2u_1+1]$ and $L_0^K := \ell_0 \in \mathbb{N}^*$ when $M_2(\mu_0^K) \in (2u_{\ell_0-1}+1, 2u_{\ell_0}+1]$. We also set $T_0^K := 0$ and by induction on $k \in \mathbb{N}^*$, if $(L_i^K)_{1 \le i \le k}$ and $(T_i^K)_{1 \le i \le k}$ are constructed,

$$ T_{k+1}^K := \inf \left\{ t \ge T_k^K \mid M_2 (\mu_t^K) \notin \mathcal{I}_{L_k^K} \right\} \quad \text{and} \quad L_{k+1}^K := \min \left\{ \ell \in \mathbb{N} \mid M_2 (\mu_{T_{k+1}^K}^K) \le 2u_\ell + 1 \right\}. $$

Note that from (4), for $K$ large enough, $L_0^K = 1$ because $M_2(\mu_0^K) \le C_2^*$. Since $(\mu_t^K)_{t \ge 0}$ is a pure jump process, note that $(M_2(\mu_t^K))_{t \ge 0}$ is also a pure jump process. The definition of $u_\ell$ is motivated by the fact that, for all $k \ge 0$, $L_{k+1}^K \le L_k^K + 1$, which follows from the next lemma.

**Lemma 4.9.** Let $\vartheta_1$ be the first jump time of $\mu^K$. Then, for $K$ large enough

$$ M_2 (\mu_{\vartheta_1}^K) \le 2M_2 (\mu_0^K) + 1. $$

In particular, for all $t > 0$, $\Delta M_2(\mu_t^K) := M_2(\mu_t^K) - M_2(\mu_{t-}^K) \le M_2(\mu_{t-}^K) + 1$.