The stepping stone model in a random environment and the effect of local heterogneities on isolation by distance patterns

We study a one-dimensional spatial population model where the population sizes at each site are chosen according to a translation invariant and ergodic distribution and are uniformly bounded away from 0 and infinity. We suppose that the frequencies of a particular genetic type in the colonies evolve according to a system of interacting diffusions, following the stepping stone model of Kimura. We show that, over large spatial and temporal scales, this model behaves like the solution to a stochastic heat equation with Wright-Fisher noise with constant coefficients. These coefficients are the effective diffusion rate of genes within the population and the effective local population density. We find that, in our model, the local heterogeneity leads to a slower effective diffusion rate and a larger effective population density than in a uniform population. Our proof relies on duality techniques, an invariance principle for reversible random walks in a random environment and a convergence result for a system of coalescing random walks in a random environment.


Introduction
Most species occupy a spatially extended habitat, where each individual produces some quantity of offspring which disperse around them. If this dispersion is limited compared to the whole range of the population, spatial patterns of genetic diversity can build up over many generations, and individuals living far apart tend to be more genetically different than individuals living close to one another.
If sufficiently many individuals from different geographic locations are sampled and genotyped, one can estimate the parameters governing the dispersal and reproduction of individuals within the studied population [Rou97,BEKV13,RCB16]. In particular, one is interested in the diffusion coefficient of genes inside the population, corresponding to the mean squared distance between parents and their offspring, and in the effective population density, which is the mean density of reproducing individuals per generation [BDE02]. This estimation relies mainly on the comparison of the observed geographical decrease of genetic relatedness to theoretical models of isolation by distance [Wri43,Mal48,KW64].
To derive analytical formulae, these models often assume that the population density is uniform in space and constant in time [Saw76,Saw77]. Natural populations, however, are hardly likely to satisfy this assumption, either because of external conditions (e.g. climate variations or fluctuating resources) or because of demographic fluctuations. These heterogeneities are almost certain to affect the estimates of the demographic parameters, either increasing or decreasing the effective rate of diffusion and the effective population density.
In this paper, we investigate the effect of local heterogeneities in the population density on the large scale genetic composition of the population and on the corresponding demographic parameters. We assume that the population density varies in space according to a translation invariant distribution and that it is constant in time. In our model, we find that these heterogeneities lead to a slower diffusion of genes and a larger effective population density than in a population with uniform density. According to this, significant local heterogeneities could lead to an underestimation of the migration levels and an overestimation of the population densities in natural populations.
This result relies on a study of the stepping stone model introduced by Kimura in 1953 [Kim53]. In this model, the population is divided into subpopulations, or demes, sitting at the vertices of a graph (e.g. Z or Z 2 ), and in each generation, new individuals are chosen at random to form the new generation in each deme, some of which are issued from parents within the same deme while others descend from individuals in neighbouring demes. If the number of individuals in each deme is large enough and if the migration probabilities are small enough, the evolution of the frequency of a given genetic type can be described by a system of interacting diffusions solving dp t (x) = y∼x m xy (p t (y) − p t (x))dt + 1 where p t (x) is the proportion of individuals carrying the type in deme x at time t, m xy is the probability that an individual in deme x is issued from one living in deme y in the previous generation, N (x) is number of individuals in deme x and (B x t , t ≥ 0, x ∈ Z) is a family of independent Brownian motions [Eth11,Shi88]. Kimura and Weiss [KW64], followed by Sawyer [Saw76] showed how the correlations in the genetic composition of demes decrease with the distance between them. In particular, in a onedimensional space, they showed that the correlation coefficient between the genetic composition of two demes separated by a distance r decreases like e −λr , where λ > 0 depends on the parameters of the model. This model can also be extended to a (one-dimensional) continuous geographical space in the following way [Shi88, DEF + 00]. For n ≥ 1, set and assume m xy = 1 2 m if |x − y| = 1, (1 − m) if x = y and 0 otherwise and that N (x) = √ n γ for all x ∈ Z for some γ > 0. Then, as n → ∞, the sequence of processes (p n (t, ·), t ≥ 0) converges in distribution to (p t , t ≥ 0), weak solution of where W is space-time white noise and σ 2 = m. A related convergence result was proved for the long range voter model in [MT95], and for the spatial Λ-Fleming-Viot process in [EVY14].
To study the impact of local heterogeneities in the population size, we consider a stepping stone model where the population sizes are drawn at random from a translation invariant and ergodic distribution and remain fixed in time. We then suppose that, with probability 1 − m, individuals in deme x descend from an individual in the same deme, and with probability m, they descend from an individual chosen uniformly at random from the individuals living in demes {x − 1, x, x + 1}. The probability of having an ancestor in a neighbouring deme is thus proportional to the population size in this deme. More precisely, an individual in deme x has its parent in deme y with probability We show that, if the population sizes are uniformly bounded away from 0 and infinity, this stepping stone model in a random environment can be rescaled as in (2) and converges to a similar limit as in the uniform setting, i.e.
(3), as n → ∞. The two parameters σ 2 and γ of the limiting equation are expressed as functions of the distribution of the population sizes. In particular, we find that, with this particular choice of migration probabilities and under some technical assumptions , where N denotes the average deme size and 2 3 m would be the expected diffusion coefficient if the population sizes were uniform. As a result, local demographic heterogeneities appear to reduce the effective diffusion of genes within the population and to increase the effective population size. This is due to the fact that, in our model, the ancestors of current individuals are more likely to have lived in more crowded demes, from which the diffusion is slower and in which the perceived population size is larger.
This bias should in particular be taken into account when estimating levels of gene flow and effective population sizes from large scale genetic patterns. One should be careful, however, that these biases are a direct consequence of the choice made in (4), and that different migration patterns might lead to different results.
The proof of the main result relies on a duality relation between the stepping stone model (1) and a system of coalescing random walks which describes the genealogy of a random sample of individuals in the population [Eth11,Shi88]. In our model, this dual takes the form of a system of coalesing random walks in a random environment, whose jump and coalescence rates are both affected by the environment (the random walkers -or ancestral lineages -are more likely to jump to more crowded demes and coalesce more quickly if they meet in less crowded demes).
It turns out that these random walks admit a reversible measure on Z. This allows us to prove an invariance principle for the trajectory of each of these random walks [Koz85,DMFGW89,Lam14,Der15]. To characterize the coalescence time of a pair of lineages, we introduce an auxiliary process which records the sequence of demes where the two lineages meet, and we show that this process also admits a reversible measure (which happens to be the square of the previous one, see Proposition 3.3). We thus show that the rescaled dual process converges to a system of Brownian motions which coalesce at a rate proportional to their local time together. In turn, this system is known to be dual to a weak solution to equation (3), which allows us to uniquely characterise the possible limits of the rescaled stepping stone model. We then conclude the proof of the main result by proving the tightness of this sequence.
Related models have been studied before, for example [BCDG13] introduce a model in which the demes can be either occupied or empty at any time, following a realisation of the contact process. A quenched invariance principle is then shown for the random walk corresponding to a single ancestral lineage in this setting. Birkner et al. [BCD16] extended this to more general demographic dynamics with local population regulation. In his thesis, Steiber [Ste17] showed that the dual of the stepping stone model run on a realisation of the contact process converges (under a suitable rescaling) to the Brownian web, i.e. a system of Brownian motions which coalesce instantly upon meeting. Here, however, we explore the regime of so-called delayed coalescence, where lineages must spend a positive amount of time together before they can coalesce (see Theorem 4). Another related work is [GdHK17], where a large population is divided in colonies labelled by the hierarchical group of order N , and migration and reproduction can take place within each hierarchical block, and where the rates of these events for each block is random.
The paper is laid out as follows. In the first section, we define the stepping stone model in a random environment and state our main result, namely the convergence of the sequence of rescaled stepping stone models to a weak solution to equation (3). In Section 2, we define the dual of the stepping stone model in a random environment and we state the related convergence results, first for individual lineages and then for the whole dual process. These results concerning the dual are then proved in Section 3. In Section 4, we prove our main result, using estimates on the heat kernel associated to the random environment. These estimates are proved in Appendix A. Appendix B is then devoted to the proof of a local central limit theorem for the random walk in a random environment which appears in the dual process.
1 The stepping stone model in a random environment

Definition of the model
We define a model for the evolution of a population living in discrete colonies or demes located on the one dimensional integer lattice Z. Moreover, we wish to draw the deme sizes (i.e. the effective number of individuals living in each colony) at random from some translation invariant, ergodic distribution.
For x ∈ Z, let T x be the translation operator acting on functions from Z to R defined by Thus let N = {N (x), x ∈ Z} be an R Z -valued random variable such that iii) (uniform ellipticity) there exists K > 0 such that, almost surely, for all x ∈ Z, In words, we assume that the distribution of the population sizes is invariant by translation, that any translation invariant statistic is deterministic and that they are uniformly bounded away from 0 and infinity, almost surely. Such a random variable can be defined using the usual formalism of random walks in random environments in the following way. Let (Ω, B, µ) be a probability space on which is defined a family of measurable maps (T x ) x∈Z , T x : Ω → Ω such that a) T x • T y = T x+y for all x, y ∈ Z and T 0 = Id Ω , b) T x is measure-preserving for all x ∈ Z, i.e.
c) the family (T x ) x∈Z is ergodic with respect to the measure µ, i.e.
Now let N : Ω → R + be a random variable on Ω such that there exists K > 0 with For x ∈ Z, we set Then {N (x), x ∈ Z} satisfies assumptions (i)-(iii) above. The parameter ω ∈ Ω determines the environment in which the population evolves, and the whole process will be defined on a larger (unspecified) probability space. All our results will be quenched, i.e. they will hold conditionally on ω, for µ-almost every ω in Ω.
Conditionally on the deme sizes {N (x), x ∈ Z}, we define the stepping stone model in a random environment as follows. Let F(Z, [0, 1]) denote the space of functions from Z to [0, 1].
Definition 1.1 (The stepping stone model in a random environment). Let (Ω, B, µ), (T x ) x∈Z and N be as above. Fix p 0 ∈ F(Z, [0, 1]), λ > 0 and m > 0. The stepping stone model in a random environment is defined as the F(Z, for µ-almost every ω, where (B x , x ∈ Z) is a family of independent standard Brownian motions (which is also independent from N ).
In other words, we choose the population size of deme x to be λN (x) and the probability that an individual in deme x at time t has a parent in deme This corresponds to a model where, with probability m, the parent of an individual in deme x is drawn uniformly at random from the three populations {x − 1, x, x + 1}, and with probability 1 − m, it is drawn uniformly from deme x.
Remark. The fact that the process (p t (ω, ·), t ≥ 0) is uniquely defined results from [Shi88] (Section 2) where existence and uniqueness of the stepping stone model is proved in a fixed but arbitrary environment satisfying (U.E.). Furthermore we note that for each x ∈ Z, t → p t (ω, x) is almost surely continuous.
We denote the quenched distribution of (p t (ω, ·), t ≥ 0) with initial condition p 0 by P ω p 0 (i.e. the distribution of the process conditionally on the environment ω). Expectation with respect to the quenched distribution will be denoted by E ω p 0 .

Main result -rescaling limit of the stepping stone model in a random environment
Individuals are related when they share at least one common ancestor some time in the past. If two individuals are sampled at a distance √ n of each other, they need to look back at least n generations in the past to expect to have ancestors living in the same deme. Over time scales of the order of n generations, individuals sampled from the same deme will have ancestors living in the same deme around √ n generations. Each time they do, they have a probability 1/N of having a common genealogical ancestor in the previous generation. Hence if N is of the order of √ n, a positive proportion of individuals living in these demes should be related after a time of the order of n generations.
For this reason, we shall study the behaviour of (p t (ω, ·), t ≥ 0) on spatial scales of the order of √ n and at times of the order of n with λ = √ n as n tends to infinity, for a fixed environment ω. When the deme sizes are uniform, it is well known that the stepping stone model rescaled in this way converges in distribution to a weak solution of the stochastic heat equation with Wright-Fisher noise (3) [MT95,Shi88]. Our main result states that on these spatial and temporal scales, the process forgets the details of the environment and evolves as an effective population with uniform demographic parameters.
For n ≥ 1, let p 0 n ∈ F(Z, [0, 1]) and set λ n = √ n. Let (p n t (ω, ·), t ≥ 0) be the solution of (5) with initial condition p 0 n and λ = λ n , and set, for n ≥ 1, It is convenient to view (p n t (·), t ≥ 0) as a process taking values in the space Ξ of Radon measures on R through the identification Let C ∞ c (R) be the space of smooth and compactly supported real-valued functions on R. For p ∈ Ξ and φ ∈ C ∞ c (R), set In this way, for any compactly supported function φ : R → R, The space Ξ is endowed with the topology of vague convergence (a sequence of measures defines a metric for the vague topology on Ξ. Also let C([0, T ], Ξ) denote the space of continuous functions from [0, T ] to (Ξ, d), endowed with the uniform topology. The main result of this paper is the following.
Theorem 1 (Convergence to the stochastic heat equation with Wright-Fisher noise). Fix T > 0 and suppose that (Ω, B, µ), (T x ) x∈Z , N satisfy (a-b-c) and (U.E.). Assume that p n 0 converges vaguely to p 0 ∈ Ξ and that p 0 is absolutely continuous with respect to the Lebesgue measure. Further assume that p n 0 satisfies the following uniform Hölder estimate: Then, µ(dω)-almost surely, as n → ∞, the sequence of Ξ-valued processes (p n t (ω, ·), t ≥ 0) converges in distribution in C([0, T ], Ξ) to a Ξ-valued process (p t ) t≥0 such that p t is absolutely continuous with respect to the Lebesgue measure for every t ≥ 0 almost surely and such that, for is a continuous local martingale with quadratic variation where this term is well defined since p s is absolutely continuous and where σ 2 and γ are given by Theorem 1 states that, over large spatial and temporal scales, the stepping stone model in a random environment behaves as if it were in a homogeneous environment with effective parameters σ 2 and γ. It is worth noting that, by Jensen's inequality and the Cauchy-Schwartz inequality, Hence σ 2 ≤ 2 3 m, which would be the expected variance if N were constant. The local variations in the population density are thus seen to reduce the effective dispersion of genes in the population. Similarly, if n → E N N 2 3 N = n is a non decreasing function of n (which is the case for example if N , T 1 N and T −1 N are independent) one can show that, To see this, write This shows that the effective population density in a heterogeneous population is larger than the mean population density. We shall see below that this is due to the fact that individuals are more likely to descend from more crowded regions, where the coalescence rate is lower and the perceived population density is larger. Theorem 1 is proved in Section 4 where we show that the sequence (p n t (ω, ·), t ≥ 0) is tight in C([0, T ], Ξ) µ(dω)-almost surely and we identify the limit through a duality relation.

Coalescing random walks in a random environment
It is well known that the stepping stone model of Definition 1.1 admits a moment dual in the form of a system of coalescing random walks [Shi88]. These random walks describe the positions of the ancestors of a random sample of individuals in the population. Each pair of random walks (also called ancestral lineages) coalesces at the first time in the past when the two sampled individuals have a common ancestor. The lineages are affected by the heterogeneity of the environment in two ways: they are more likely to jump to more crowded demes, and they coalesce more quickly in less crowded demes.
In this section, we first define the system of coalescing random walks in a random environment which is dual to the stepping stone model of Definition 1.1. We then state several results on this dual which will be used to identify the limit in the proof of Theorem 1. First we state the convergence of the rescaled random walk to Brownian motion (Theorem 2) and a local central limit theorem for this random walk (Theorem 3). Then we state the convergence of the whole dual process to a system of coalescing Brownian motions (Theorem 4).

The dual of the stepping stone model in a random environment
Definition 2.1 (The dual of the stepping stone model in a random environment). Fix m > 0, independently of the others, iii) each pair of lineages sitting in the same colony x ∈ Z coalesces at rate independently of other pairs.
For t ≥ 0, we denote the number of lineages in A ω t by N t .
.,x k } denote the expectation with respect to this probability. When k = 1, P ω {x} = P ω x is the quenched distribution of a random walk in a random environment with transition rates given by (9). In the following, we show that this random walk satisfies a quenched invariance principle (Theorem 2 below).
When k = 2, we can give a more precise construction of (A ω t ) t≥0 . Fix {x 1 , x 2 } ∈ Z 2 and let ξ 1 t t≥0 and ξ 2 t t≥0 be two independent random walks on Z with transition rates given by (9) and started from x 1 and x 2 , respectively. Let E be an independent exponential random variable with parameter 1 and define, for t ≥ 0, Now define the coalescence time of the two lineages as Setting we obtain a version of the process in Definition 2.1 started from two lineages. This construction can be extended to more than two lineages [Lia09], but we will not need it in such generality here.
The next proposition states the duality relation between the stepping stone model and (A ω t ) t≥0 . Proposition 2.2 (Duality, [Shi88]). For any p 0 : Z → [0, 1] and for any k ≥ 1, {x 1 where (p t (ω, ·), t ≥ 0) is given by Definition 1.1 and (A ω t ) t≥0 by Definition 2.1. We use this proposition to characterise the limiting behaviour of (p n t ) t≥0 via the study of the scaling limit of (A ω t ) t≥0 . We show below that a rescaled version of (A ω t ) t≥0 converges in distribution to a system of independent Brownian motions which coalesce at a rate proportional to the local time at 0 of their difference (Theorem 4 below).

The central limit theorem for reversible random walks in a random environment
Here, we state the results on the motion of a single lineage in the dual process. For ω ∈ Ω, let (ξ t ) t≥0 be a random walk on Z with transition rates given by (9), i.e., conditionally on the environment ω, it jumps from x ∈ Z to y ∈ {x − 1, x + 1} at rate m N (ω, y) N 3 (ω, x) .
The most notable property of this random walk is that it admits a reversible measure on Z, given by (Note that we have normalised π so that π(ω)µ(dω) is a probability measure on Ω.) Together with (U.E.), this implies the central limit theorem for the random walk [Lam14, with σ 2 as in (8). In Section 3, we prove that this extends to a quenched invariance principle for the random walk (ξ t , t ≥ 0). For n ≥ 1, set For T > 0, let D ([0, T ], R) denote the space of càdlàg real-valued functions endowed with the usual Skorokhod topology.
The corresponding central limit theorem was proved for the random walk among random conductances in [Lam14] (see also [Lam12]). Although our case doesn't take the form of a random walk among random conductances, the proof in [Lam12, Chapter 4] only requires the reversibility of the random walk, and can be applied by replacing the conductivity with N (ω)T 1 N (ω) and the reversible measure with N (ω)N 3 (ω). We give the details in Section 3.
In [Der15], Derrien proves a local central limit theorem for the random walk among random conductances which also extends to our setting. We prove a slightly stronger version of his result, namely Theorem 3 below. For ω ∈ Ω, t ≥ 0 and x, y ∈ Z, set and for t > 0, x ∈ R, define Theorem 3 (Local central limit theorem for reversible random walks). For all ε > 0, T > 0 and R > 0 and for µ-almost every environment ω, In Section 4, we shall use several estimates on the kernel g ω to show the tightness of the sequence (p n t (·), t ≥ 0). These estimates are proved in Appendix A and are then used to prove Theorem 3 in Appendix B.

Delayed coalescence for random walks in a random environment
Now that we know how each lineage behaves over large scales, we state the corresponding result for the dual process (A ω t ) t≥0 . We limit ourselves to the dual started from two lineages, as this is enough to identify the limit of (p n t (·), t ≥ 0) in the proof of Theorem 1. Let us start with the definition of the limiting process. It will be defined as a system of independent Brownian motions which coalesce at a rate proportional to their local time together. For σ 2 > 0 and γ > 0, let X 1 t t≥0 and X 2 t t≥0 be two independent Brownian motions with variance σ 2 , started from x 1 and x 2 , respectively. Let E be an independent exponential random variable with parameter γ, and let t → L 0 t (X 1 − X 2 ) denote the local time at 0 of X 1 − X 2 . Set Definition 2.3 (Brownian flow with delayed coalescence, [Lia09]). The process (D t ) t≥0 defined by is called the Brownian flow with delayed coalescence with parameters (σ 2 , γ).
The name Brownian flow with delayed coalescence is used in [Lia09] for a more general process started from an arbitrary number of lineages (which also coalesce at a rate proportional to their local time together). Here we only use the process started from two lineages.
Remark. The process (D t , t ≥ 0) takes values in the disjoint union of R and R 2 , and we let D([0, T ], R∪R 2 ) denote the space of càdlàg functions on [0, T ] taking values in this space, endowed with the usual Skorokhod topology.
The following result is proved in [Lia09] (see in particular Theorem 6.2 and Proposition 7.2).
Proposition 2.4. There exists a unique Feller Markov process (p t ) t≥0 taking values in the subspace p ∈ Ξ : 0 ≤ p, φ ≤ R φ(x)dx, ∀φ ≥ 0 such that, for any k ≥ 1, for any φ 1 , . . . , φ k in C ∞ c (R), where D t = {X 1 t , . . . , X Nt t } is the Brownian flow with delayed coalescence. In addition, for any twice continuously differentiable function f : R → R and any φ ∈ C ∞ c (R), is a local martingale with respect to the natural filtration of (p t ) t≥0 .

It follows that the process defined in Proposition 2.4 is a weak solution to the stochastic heat equation with Wright-Fisher noise.
We now state our result on the scaling limit of the dual of the stepping stone model in a random environment. For n ≥ 1, take λ n = √ n, x n 1 , x n 2 ∈ Z and let (A ω,λn t , t ≥ 0) be the process of Definition 2.1 with λ = λ n started from {x n 1 , x n 2 }. For n ≥ 1, t ≥ 0, set (note that N t ∈ {1, 2}).
Theorem 4 (Convergence to the Brownian flow with delayed coalescence). Assume that Then for µ-almost every environment ω, the process (A n t ) t≥0 of (14) converges in distribution in D [0, T ], R ∪ R 2 to the Brownian flow with delayed coalescence (D t ) t≥0 of Definition 2.3 with σ 2 and γ given by (8).
We prove Theorem 4 in Subsection 3.3. We already know from Theorem 2 that each lineage converges in distribution to Brownian motion with variance σ 2 . It thus remains to show that the process L(t) defined in (10) becomes asymptotically proportional to L 0 t (X 1 − X 2 ). This is done by applying the ergodic theorem to an auxiliary process which is defined in Subsection 3.2.
Durrett and Restrepo [DR08] already showed that, in the stepping stone model with uniform population sizes and in the long range voter model, the time spent together by two lineages before they coalesce is asymptotically exponential (see also [Mar71]). Theorem 4 extends this result to the stepping stone model in a random environment.
Remark. The above result also holds if the process (A ω t ) t≥0 is started from more than two lineages. The rescaled process (A n t ) t≥0 then converges to the Brownian flow with delayed coalescence started from the corresponding number of lineages, as defined in [Lia09].

Large scale behaviour of the dual of the stepping stone model in a random environment
The aim of this section is to prove Theorem 4. We start in Subsection 3.1 by showing the quenched functional central limit theorem for the random walk (ξ t ) t≥0 (Theorem 2). In Subsection 3.2, we introduce an auxiliary process called the environment viewed by the two random walks, and we show that it is ergodic and give its stationary distribution. Theorem 4 is then proved in Subsection 3.3.

Quenched functional central limit theorem for the nearest neighbour reversible random walk
Here we prove Theorem 2 using a martingale approximation of the random walk (ξ n t ) t≥0 introduced in [Lam12]. We start by defining a function F (ω, ·) : Z → R such that F (ω, ξ t ) is a martingale. Let F : Ω × Z → R be a real-valued measurable function such that, for all k, h ∈ Z and ω ∈ Ω, and Such a function is given by By the pointwise ergodic theorem, We then decompose ξ n t = 1 √ n ξ nt as Since, from (15) and (16), the process M ω n is a martingale with respect to the natural filtration of (ξ n t ) t≥0 .
Lemma 3.1. For µ-almost every ω and for any T > 0, where the expectation is taken with respect to the distribution of ξ n started at ξ n 0 in the environment ω (recall that ξ n 0 → x 0 ). Before proving Lemma 3.1, let us show how it implies Theorem 2. The only point left to be shown is that M ω n converges in distribution to Brownian motion with the right variance.
Proof of Theorem 2. We begin by noting that the quadratic variation of M ω n is with, after a simple calculation, Let us define the so-called process of the environment viewed from the random walk as We wish to use the ergodic theorem to show that M ω n t converges to a constant times t. To do this, we show that the Ω-valued Markov process (Z t ) t≥0 with initial distribution π(ω)µ(dω) is stationary and ergodic (recall that π was defined in (11)). Let L denote the infinitesimal generator of (Z t ) t≥0 . It is symmetric in L 2 (π), i.e.
Hence any function f in L 2 (π) such that Lf = 0 must be invariant by translation. By the ergodicity of (T x ) x∈Z (assumption c), such an f is constant. As a result (Z t ) t≥0 is a stationary and ergodic Markov process with reversible measure π(ω)µ(dω). As a consequence, for any t ≥ 0, This, together with the observation that imply that (M ω n (t)) t≥0 converges in distribution in D ([0, T ], R) to Brownian motion with variance hπdµ for µ almost every ω [Reb80, Proposition 1]. To conclude the proof of Theorem 2, we note that Let us now prove Lemma 3.1.
Proof of Lemma 3.1. This proof follows closely that of Proposition 4.2.2 in [Lam12]. From the convergence (18), we see that for any ε > 0, there exists M ε = M ε (ω) such that, for all x ∈ Z, Taking ε < c, we have Combining (22) and (23), we obtain Then for all x ∈ Z, Replacing x with ξ nt , dividing by √ n and squaring both sides, we obtain By Doob's inequality, for some C > 0 using (20) and (U.E.) (the bound on M ω n (0) comes from the fact that |F (ω, x)| ≤ C |x| and that ξ n 0 converges). As a result, the expectation of the first term in (25) can be made arbitrarily small, uniformly in t ∈ [0, T ], by choosing ε small enough, while the second term vanishes as n → ∞ for each ε > 0. Hence In passing, we can prove the following, which will be useful later in the proof of Theorem 1.

The environment viewed from the two random walks
We now introduce an auxiliary process which records the environment viewed by the two independent random walks when they meet. Let (ξ 1 t ) t≥0 and (ξ 2 t ) t≥0 be two independent random walks on Z with transition rates given by (9) and started from x 1 and x 2 respectively. Define, for t ≥ 0, the time spent together by the two random walks up to time t, Also let L −1 0 (·) be the right-continuous inverse of L 0 , i.e.
As in (21), let (Z(t)) t≥0 be the environment viewed from the first random walk, i.e.
The environment viewed from the two random walks is then defined as The main result of this subsection is the following.
Proposition 3.3. The process (Y (t)) t≥0 , started from the initial distribution is a stationary and ergodic Markov process.
Proof. Let τ 0 denote the first time at which the two random walks ξ 1 and ξ 2 meet, i.e.
Let F 0 (Z) be the space of real-valued functions on Z which are zero on all but a finite number of points. For f ∈ F 0 (Z), define be the rate at which ξ i jumps from x to x + z. The process (ξ 1 L −1 0 (t) , t ≥ 0) then jumps from x to y at rate where δ y (x) = 1 if x = y and 0 otherwise. Setting q(ω, y) = q(ω, 0, y), the generator of (Y (t)) t≥0 takes the form Proposition 3.3 will then be proved if we show that for µ almost every ω ∈ Ω and every y ∈ Z, π(ω) 2 q(ω, y) = π(T y ω) 2 q(T y ω, −y).
We now prove (29). To avoid writing infinite sums, we first restrict ourselves to random walks on a bounded region [−A, A]. For A ≥ 1, define Let (ξ A,1 t ) t≥0 and (ξ A,2 t ) t≥0 be two independent random walks on [−A, A] which jump from x to x + z at rate j A (ω, x, z) (i.e. they behave as ξ i but the jumps leading outside of [−A, A] are suppressed). Let τ A 0 be the first time at which ξ A,1 and ξ A,2 meet and define, for f ∈ F 0 (Z), Then, since π(ω, x)j A (ω, x, z) = π(ω, x + z)j A (ω, x + z, −z), for any f, g ∈ F 0 (Z), As a result, But, by the Markov property and the definition of E ω , for any x = y ∈ [−A, A] and for any f ∈ F 0 (Z), As a result, the only non-zero terms in the sums appearing in (30) are those for which x = y.
We thus obtain that, for any f, g ∈ F 0 (Z), Taking f = δ y and g = δ x , we obtain We now wish to let A → ∞ in order to obtain (29). Clearly, for A large enough, j A (ω, x, z) = j(ω, x, z). Let T A be such that Then, since T A → ∞ almost surely as A → ∞, ξ A,i τ A 0 converges in distribution to ξ i τ 0 . Hence E A,ω δ y (x, x + z) → E ω δ y (x, x + z) and (29) is proved.

Convergence to the Brownian flow with delayed coalescence
We now set out to prove Theorem 4. Recall the construction of (A ω,λ ) t≥0 started from two lineages in Subsection 2.1. In particular, recall that (ξ 1 t ) t≥0 and (ξ 2 t ) t≥0 are two independent random walks on Z with transition rates given by (9) and and that the two lineages coalesce at time T c such that where E is an independent exponential random variable with parameter 1. For n ≥ 1, we have set By Theorem 2, ( 1 √ n ξ i nt ) t≥0 converges µ-almost surely in distribution in D ([0, T ], R) to Brownian motion with variance σ 2 given by (8). Since ξ 1 and ξ 2 are independent, by the Skorokhod representation theorem, for µ-almost every ω, there exists a probability space on which two sequences of processes (ξ 1,n t , ξ 2,n t ) t≥0 and two Brownian motions (X 1 t , X 2 t ) t≥0 are defined and such that ii) ξ 1,n and ξ 2,n are independent for all n ≥ 1,  Then (L n (t)) t≥0 is distributed like (L(nt)) t≥0 (with λ = √ n).
Proof. For t ≥ 0 and n ≥ 1, let is distributed like (Y (t), t ≥ 0). We can then write L n (t) as ds.
Since ( 1 √ n ξ i,n nt , t ≥ 0) converges to (X i t ) t≥0 almost surely in D ([0, T ], R), ( 1 √ n L 0 n (nt), t ≥ 0) converges almost surely to (L 0 t (X 1 − X 2 ), t ≥ 0) in the uniform topology. Furthermore, by Proposition 3.3 and by the pointwise ergodic theorem, for each n ≥ 1, Fix ε > 0 and T > 0. For each n ≥ 1, there exists a (random) t n 0 > 0 such that Furthermore, there exists n 0 ≥ 1 such that, for any n ≥ n 0 , As a result, for n ≥ n 0 , Since the (t n 0 ) n≥1 are identically distributed, 1 √ n t n 0 converges to 0 in probability as n → ∞, and the result follows.
Let us now prove Theorem 4.
Proof of Theorem 4. Define, on the same probability space as the ξ i,n an independent exponential random variable E with parameter 1, and set T n c = inf{t ≥ 0 : L n (t) > E}. Then is distributed as the process A n of Theorem 4. Lemma 3.4 implies As a result, (A n 4 Convergence to the stochastic heat equation with Wright-Fisher

Tightness of the sequence
Since the sequence of processes (p n t (ω, ·), t ≥ 0) takes values in a compact subspace of Ξ almost surely (see (33) above), to prove that it is tight in C([0, T ], Ξ) it is enough to show that for any φ ∈ C ∞ c (R) and f ∈ C 2 (R), the sequence of real-valued processes (f ( p n t , φ ), t ≥ 0) is tight in C([0, T ], R) [EK86, Theorem 3.9.1]. To do so, we want to use the Aldous-Rebolledo criterion and show that both the finite variation part and the quadratic variation part of (f ( p n t , φ ), t ≥ 0) are tight. In fact, we show that this is true for another process (f ( p n t , φ ), t ≥ 0), defined below, and we show that the difference between the two processes vanishes as n → ∞. Let us first state two lemmas.
Lemma 4.2 is proved in Subsection 4.3. Recall the definition of π in (11). For n ≥ 1, δ > 0 and x ∈ 1 √ n Z, set and extend x → Π n,δ (x) to R by linear interpolation.
Proof. By the pointwise ergodic theorem, for each x ∈ R and δ > 0, Moreover, for any x, y ∈ R, by (U.E.), Tightness of (f ( p n t , φ ), t ≥ 0) By the definition of p n and (5), where j(ω, x, z) was defined in (28) and (B x , x ∈ Z) is a family if independent Brownian motions. Note that, by the reversibility of π, By Taylor's theorem, In view of (19), equation (36) equals m c 2 √ n x∈Z z∈{−1,1} π(ω, x)j(ω, x, z)p n nt (x)R Since |p n nt (x)| ≤ 1 and by assumption (U.E.), this is smaller than for some constant C > 0. In turn, the above expression is uniformly bounded as n → ∞ since φ ∈ C ∞ c (R) (also note (24)). We have thus proved the existence of a constant C 1 > 0 such that, for all n ≥ 1, We now turn to the quadratic variation part of p n t , φ . Since the Brownian motions (B x , x ∈ Z) are independent, it is p n ns (x)(1 − p n ns (x))ds.
Then, by (U.E.) and since φ ∈ C ∞ c (R) there exists a constant C 2 > 0 such that We then write F V t (f, φ) and QV t (f, φ) for the finite variation part and the quadratic variation part of f ( p n t , φ ). Then, by (37) and (38) and by the Itô formula, where the supremum is taken over a suitable compact set (depending on φ). Hence both (F V t (f, φ), t ≥ 0) and (QV t (f, φ), t ≥ 0) are tight, and by the Aldous-Rebolledo criterion, . Since for all n ≥ 1, t → f ( p n t , φ ) is continuous almost surely, so are its potential limit points, hence the sequence is tight in C([0, T ], Ξ).

Conclusion
Let us now show that the difference between p n t , φ and p n t , φ vanishes as n → ∞. First note that, by (24), for any ε > 0, The first term on the right hand side vanishes as n → ∞ for any fixed ε while the second term can be made arbitrarily small for all n ≥ 1 by choosing ε small enough. Hence To conclude, we write, for δ > 0, π(ω, y) √ n y∈B(x,δ √ n)∩Z π(ω, y) p n nt (x)φ(x/ √ n) − p n nt (y)φ(y/ √ n) .
The first term on the right hand side vanishes as n → ∞ for any δ > 0 by Lemma 4.3 (recall that φ is compactly supported Proof of Lemma 4.4. We first note that the second part of the statement follows at once from Lemma A.2 and (U.E.), noting that h ω t (x, y) = h ω t (y, x) by reversibility. For the first statement, we have, using Lemma A.2, (g ω ns (x, z) − g ω ns (y, z)) 2 ≤ C(ns) −3/4 |x − y| 1/2 (g ω ns (x, z) + g ω ns (y, z)) .

B The local central limit theorem for reversible random walks in a random environment
Armed with Lemma A.1 and Lemma A.2, we can prove the local central limit theorem for the random walk (ξ t ) t≥0 (Theorem 3). Again, we follow the method used in [Der15].
By the central limit theorem, F ω n converges pointwise to F as n → ∞, for µ almost every ω ∈ Ω. Moreover, by the functional central limit theorem (Theorem 2), F ω n (·, x, y) converges to F (·, x, y) uniformly on [0, T ] for each x, y ∈ R (using Fatou's lemma). Also, x → F ω n (t, x, y) and y → F ω n (t, x, y) are both monotone (and so are x → F (t, x, y) and y → F (t, x, y)), and F is continuous. It then follows that F ω n converges to F uniformly on compact sets of R + × R 2 , µ(dω) almost surely. Now note that |F ω n (t, x, y) − F (t, x, y)| .
By the uniform convergence of F ω n on compact sets, we thus have, for any δ > 0, Combining (49), (50), (51) and (52), we see that δ can be chosen so that B δ n and D δ n are arbitrarily small for all n ≥ 1, and then we can choose n 0 such that for all n ≥ n 0 , A δ n and C δ n are arbitrarily small. This concludes the proof of Theorem 3.