2.1. Moran models in deterministic pure-jump environments

Consider a population of size N with two types, 0 and 1, subject to mutation and selection influenced by a deterministic environment. The latter is modeled by an at most countable collection $\zeta\,{:\!=}\, (t_k, p_k)_{k \in I}$ of points in $(\!-\!\infty,\infty) \times (0,1)$ satisfying for any $s,t\in{\mathbb R}$ with $s\leq t$ that

(2.1) \begin{equation} \sum_{t_k \in[s, t]} p_k \lt \infty.\end{equation}

We refer to $p_k$ as the peak of the environment at time $t_k$ . The individuals in the population undergo the following dynamic. Each individual independently mutates at rate $\theta_N\geq 0$ with probability $\nu_{0}\in[0,1]$ (resp. $\nu_1\,{:\!=}\, 1-\nu_0$ ) to type 0 (resp. 1). Reproduction occurs independently from mutation. Individuals of type 1 reproduce at rate 1, whereas individuals of type 0 reproduce at rate $1+\sigma_N$ , $\sigma_N\geq0$ . (The subscript N in the parameters $\sigma_N$ and $\theta_N$ emphasizes their dependence on N. In Theorem 2.2 we will require that they are asymptotically proportional to $1/N$ .) Thus, we refer to type 0 (resp. type 1) as the fit (resp. unfit) type. In addition, at time $t_k$ each type-0 individual independently reproduces with probability $p_k$ . At reproduction times, (a) each individual produces at most one offspring, which inherits the parent’s type, and (b) if n individuals are born, n individuals are randomly sampled without replacement from the population present before the reproduction event (including the parents) to die, keeping the size of the population constant.

Graphical representation. In the absence of environmental factors (i.e. $\zeta=\emptyset$ ), it is classical to describe the evolution of the population by means of the graphical representation as an interacting particle system (IPS). This decouples the randomness of the model due to the initial type configuration from the randomness due to mutations and reproductions. We now extend the graphical representation to incorporate the effect of the environment (see Section 3.1 for a more detailed description).

In the graphical representation, individuals are represented by horizontal lines at levels $i\in [N]$ (see Figure 1). Time runs forward from left to right. Potential reproduction events are depicted by arrows, with the (potential) parent at the tail and the offspring at the tip. We distinguish between neutral and selective arrows. Neutral arrows have a filled arrowhead; they occur at rate $1/N$ per pair of lines. Selective arrows have an open arrowhead; they occur in two independent ways: first, they occur at rate $\sigma_N/N$ per pair of lines, and second, at any time $t_k$ , $k\in I$ , a random number $n_k\sim\textrm{Bin}({N},{p_k})$ of lines shoot selective arrows to $n_k$ individuals in the population. Furthermore, beneficial (deleterious) mutations, depicted as circles (crosses), occur at rate $\theta_N \nu_0$ (at rate $\theta_N \nu_1$ ) per line.

Figure 1. Left: a realization of the Moran IPS; time runs forward from left to right; the environment has peaks at times $t_0$ and $t_1$ . Right: the ASG that arises from the second and third lines (from bottom to top) in the left picture, with the potential ancestors drawn in black; time runs backward from right to left; backward time $\beta\in[0,T]$ corresponds to forward time $t=s+T-\beta$ .

Note that for any $s<t$ , the number of non-environmental graphical elements present in [s, t] is almost surely finite. Moreover, thanks to Assumption (2.1), we have

(2.2) \begin{align}{\mathbb E}\left[\sum_{t_k\in[s,t]}n_k\right]=N\sum_{t_k\in[s,t]}p_k<\infty.\end{align}

Hence, $\sum_{t_k\in[s,t]}n_k<\infty$ almost surely, i.e. the number of arrows in [s, t] due to peaks of the environment is almost surely finite.

Once the graphical elements in $[s,t]\times [N]$ are drawn, we specify the initial conditions by assigning types to the N lines at time s and propagate them forward in time according to the following rules: the type of a line right after a circle (resp. cross) is 0 (resp. 1). In particular, if the type before the circle (resp. cross) is 0 (resp. 1), the mutation is silent. Type 0 propagates through neutral arrows and selective arrows; type 1 propagates only through neutral arrows.

Reading off ancestries in the Moran model. The ancestral selection graph (ASG) was introduced by Krone and Neuhauser in [Reference Krone and Neuhauser29] (see also [Reference Neuhauser and Krone35]) to study the genealogical relations in the diffusion limit of the Moran model with mutation and selection. In what follows, we briefly explain how to adapt this construction to the Moran model in deterministic environment.

Consider a realization of the IPS associated to the Moran model in the environment $\zeta\,{:\!=}\, (t_k, p_k)_{k \in I}$ in the time interval $[s,s+T]$ . Fix an untyped sample of n individuals at time $s+T$ and trace backward in time (from right to left in Figure 1) the lines of their potential ancestors (i.e. the lines that are ancestral to the sample for some type-configuration at time s), ignoring the effect of mutations; the backward time $\beta\in[0,T]$ corresponds to the forward time $t=s+T-\beta$ . We do this as follows. When a neutral arrow joins two individuals in the current set of potential ancestors, the two lines coalesce into a single one at the tail of the arrow. When a neutral arrow hits a potential ancestor from outside the current set of potential ancestors, the hit line is replaced by the line at the tail of the arrow. When a selective arrow hits the current set of potential ancestors, the individual that is hit has two possible parents, the incoming branch at the tail and the continuing branch at the tip. The true parent depends on the type of the incoming branch, but for the moment we work without types. These unresolved reproduction events can be of two types: a branching event if the selective arrow emanates from an individual outside the current set of potential ancestors, and a collision event if the selective arrow links two current potential ancestors. Note that at the peak times, multiple lines in the ASG can be hit by selective arrows, and therefore, multiple branching and collision events can occur simultaneously. Mutations are superposed on the lines of the ASG.

The object arising under this procedure up to time $\beta=T$ is called the Moran-ASG in $[s,s+T]$ under the environment $\zeta$ . It contains all the lines that are potentially ancestral (ignoring mutation events) to the lines sampled at time $t=s+T$ ; see Figure 1. Note that, since the number of events occurring in $[s,s+T]$ is almost surely finite (see (2.2)), the Moran-ASG in $[s,s+T]$ is well-defined.

Given an assignment of types to the lines present in the ASG at time $t=s$ , we can extract the true genealogy and determine the types of the sampled individuals at time $t=s+T$ . To this end, we propagate types forward in time along the lines of the ASG, taking into account mutations and reproductions, with the rule that if a line is hit by a selective arrow, the incoming line is the ancestor if and only if it is of type 0; see Figure 2. This rule is called the pecking order. Proceeding in this way, the types in $[s,s+T]$ are determined along with the true genealogy.

Figure 2. The descendant line (D) splits into the continuing line (C) and the incoming line (I). The incoming line is ancestral if and only if it is of type 0. The true ancestral line is drawn in bold.

Evolution of the type composition. Consider the set ${\mathbb D}^\star$ of non-decreasing functions $\omega\in{\mathbb D}$ satisfying the following:

1. (i) for all $s<t\in{\mathbb R}$ , $\Delta \omega(t)\,{:\!=}\, \omega(t)-\omega(t-\!)\in[0,1)$ and $\sum_{u\in[s,t]}\Delta \omega(u)<\infty$ ;

2. (ii) $\omega$ is pure-jump, i.e. for all $s<t$ , $\omega(t)=\omega(s)+\sum_{u\in(s,t]}\Delta \omega(u)$ .

Note that the set of environments $\zeta\,{:\!=}\, (t_k, p_k)_{k \in I}$ satisfying (2.1) can be identified with the set of functions $\omega\in {\mathbb D}^\star$ with $\omega(0)=0$ . Indeed, for any $\omega\in{\mathbb D}^\star$ , the collection of points $\{(t,\Delta \omega(t))\,{:}\, \Delta\omega(t)>0)\}$ is countable and satisfies (2.1) (the same collection is obtained if we add a constant to $\omega$ ; this is why we set $\omega(0)=0$ ). Conversely, for any $\zeta\,{:\!=}\, (t_k, p_k)_{k \in I}$ , the function $\omega\,{:}\,{\mathbb R}\to{\mathbb R}$ defined via

\begin{align*}\omega(t)\,{:\!=}\,\sum_{t_k\in(0,t]}p_k\quad\textrm{for $t\geq 0$, and}\quad {\omega(t)\,{:\!=}\,-\sum_{t_k\in(t,0]}p_k}\quad\textrm{for $t<0$},\end{align*}

belongs to ${\mathbb D}^\star$ . For this reason, we often abuse notation and refer to the elements of ${\mathbb D}^\star$ as environments. In addition, an environment $\omega\in{\mathbb D}^\star$ is said to be simple if $\omega$ has only a finite number of jumps in any compact time interval. We denote by $\textbf{0}$ the environment corresponding to $\zeta=\emptyset$ and refer to it as the null environment.

We denote by $Z_N^\omega(t)$ the number of fit individuals at time t in a Moran population of size N subject to the environment $\omega\in{\mathbb D}^\star$ . We refer to $Z_N^\omega\,{:\!=}\,(Z_N^\omega(t))_{t\geq 0}$ as the quenched fit-counting process. In particular, $Z_N^\textbf{0}$ is just the continuous-time Markov chain on $[N]_0$ with infinitesimal generator

\begin{align*}\mathcal{A}_N^0 f(n)&\,{:\!=}\, \left[(1+\sigma_N)\frac{n(N-n)}{N}+\theta_N\nu_0(N-n)\right](f(n+1)-f(n))\\[3pt]&\qquad\qquad\qquad\qquad\qquad\qquad\quad + \left[\frac{n(N-n)}{N}+\theta_N\nu_1 n\right](f(n-1)-f(n)).\end{align*}

Note that if $\omega$ has a jump at time t, the number n(t) of individuals placing offspring is a binomial random variable with parameters $Z_N^\omega(t-\!)$ and $\Delta\omega(t)$ . Since the n(t) individuals that will be replaced are chosen uniformly at random, the additional number of fit individuals after the reproduction event is a hypergeometric random variable with parameters N, $N-Z_N^\omega(t-\!)$ , and n(t). Therefore, the dynamic of $Z_N^\omega$ is as follows. Recall that in any finite time interval, the number of environmental reproductions is almost surely finite. Thus, we can define $(S_i)_{i\in{\mathbb N}}$ as the increasing sequence of times at which environmental reproductions take place. We set $S_0\,{:\!=}\, 0$ . By construction, $(S_i)_{i\in{\mathbb N}}$ is Markovian and its transition probabilities are given by

\begin{align*}\mathbb{P}(S_{i+1} \gt t \mid S_i = s ) = \prod_{u \in (s, t]} (1- \Delta \omega(u))^N,\quad i\in{\mathbb N}_0, 0\leq s\leq t.\end{align*}

If $Z_N^\omega(0)=n\in [N]_0$ , then $Z_N^\omega$ evolves in $[0,S_1)$ as $Z_N^\textbf{0}$ started at n. For $i\in{\mathbb N}$ , if $Z_N^\omega(S_{i}-\!)=k$ , then $Z_N^\omega(S_{i})=k+H(N,N-k,\tilde B_i(k))$ , where the random variables $H(N,N-k,b)\sim \textrm{Hyp}(N,N-k,b)$ , $b\in[k]_0$ , and $\tilde B_i(k)$ are independent, and $\tilde B_i(k)$ is a binomial random variable with parameters k and $\Delta \omega(S_i)$ conditioned to be positive. Then, $Z_N^\omega$ evolves in $[S_i,S_{i+1})$ as $Z_N^\textbf{0}$ started at $Z_N^\omega(S_{i})$ .

Let us fix $T>0$ . We end this section with our first main result, which provides the continuity in [0, T] of the fit-counting process with respect to the environment. Note that the restriction of the environment to [0, T] can be identified with an element of

(2.3) \begin{align} {\mathbb D}_T^\star\,{:\!=}\, \left\{\omega\in{\mathbb D}_{0,T}\,{:}\, \omega(0)=0,\, \Delta \omega(t)\in[0,1)\textrm{ for all $t\in[0,T]$, $\omega$ is}\right. \nonumber\\[3pt] \left.\text{non-decreasing and pure-jump}\right\}.\end{align}

Moreover, we equip ${\mathbb D}_T^\star$ with the metric $d_T^\star$ defined in Appendix A.1, Equation (A.3).

Theorem 2.1. (Continuity.) Let $\omega\in{\mathbb D}_T^\star$ and $\{\omega_k\}_{k\in{\mathbb N}}\subset{\mathbb D}_T^\star$ be such that $d_T^\star(\omega_k,\omega)\to 0$ as $k\to\infty$ . If $Z_N^{\omega_k}(0)=Z_N^\omega(0)$ for all $k\in{\mathbb N}$ , then

Theorem 2.1 is proved in Section 3.1.

2.2. Moran models in subordinator-driven environments

In contrast to Section 2.1, we consider here a random environment given by a Poisson point process $(t_i, p_i)_{i \in I}$ on $(\!-\!\infty,\infty) \times (0,1)$ with intensity measure ${\textrm{d}} t \times \mu$ , where ${\textrm{d}} t$ stands for the Lebesgue measure and $\mu$ is a measure on (0, 1) satisfying

(2.4) \begin{equation}\int_{(0,1)} x \mu({\textrm{d}} x) \lt \infty.\end{equation}

The latter implies that $(t_i, p_i)_{i \in I}$ almost surely satisfies Assumption (2.1). In particular, setting $J(t) \,{:\!=}\, \sum_{t_i\in(0,t]} p_i$ for $t\geq 0$ and $J(t)\,{:\!=}\, -\sum_{t_i\in(t,0]}p_i$ for $t<0$ , we have $J\in{\mathbb D}^\star$ almost surely. Moreover, by the Lévy–Itô decomposition, $(J(t))_{t\in{\mathbb R}}$ is a pure-jump subordinator with Lévy measure $\mu$ . If the measure $\mu$ is finite, then J is a compound Poisson process, and thus the environment J is almost surely simple.

We will see in Section 3.1 that, using the graphical representation, one can simultaneously construct Moran models for any $\omega\in{\mathbb D}^\star$ . Now, consider an independent pure-jump subordinator $(J(t))_{t\geq 0}$ , with Lévy measure $\mu$ on (0, 1) satisfying (2.4). Thanks to Theorem 2.1 the process $Z_N^J\,{:\!=}\, (Z_N^J(t))_{t\geq 0}$ is well-defined. We refer to $Z_N^J$ as the annealed fit-counting process. By definition, we have

\begin{align*}{\mathbb P}(Z_N^J \in \cdot)=\int {\mathbb P}(Z_N^\omega \in \cdot) {\mathbb P}(J \in {\textrm{d}} \omega).\end{align*}

In other words, ${\mathbb P}(Z_N^\omega \in \cdot)$ is the law of $Z_N^J$ conditionally on a realization $\omega$ of the environment (i.e. ${\mathbb P}(Z_N^\omega \in \cdot) = {\mathbb P}(Z_N^J \in \cdot \mid J=\omega)$ ) and is classically referred to as the quenched measure, while ${\mathbb P}(Z_N^J \in \cdot)$ integrates the effect of the random environment and is classically referred to as the annealed measure.

The process $Z_N^J$ is a continuous-time Markov chain on $[N]_0$ with generator

\begin{align*}\mathcal{A}_N f(n)\,{:\!=}\, \mathcal{A}_N^0 f(n) + \int_{(0,1)}\left({\mathbb E}\left[f\!\left(n+\mathcal{H}(N,N-n,B_n(u))\right)\right]-f(n)\right){\mu({\textrm{d}} u)},\quad n\in [N_0], \end{align*}

where $B_n(u)\sim \textrm{Bin}(n,u)$ , and for any $i\in[n]_0$ , $\mathcal{H}(N,N-n,i)\sim\textrm{Hyp}(N,N-n,i)$ are independent.

The dynamic of the graphical representation is as follows. For each $i,j\in [N]$ with $i\neq j$ , selective (resp. neutral) arrows from level i to level j appear at rate $\sigma_N/N$ (resp. $1/N$ ). For each $i\in [N]$ , open circles (resp. crosses) appear at level i at rate $\theta_N\nu_0$ (resp. $\theta_N\nu_1$ ). For each $k \in [N]$ , every group of k lines is subject to simultaneous potential reproductions at rate

\begin{align*}\sigma_{N,k}\,{:\!=}\, \int_{(0,1)}y^k (1-y)^{N-k}\mu({\textrm{d}} y),\end{align*}

resulting in the appearance of k selective arrows from the lines of this group (of potential parents) to the lines of a group of size k (the potential descendants) that is chosen uniformly at random among subsets of size k of the N individuals. The k selective arrows are drawn uniformly at random from the k potential parents to the k potential descendants. Recall that only type 0 propagates through selective arrows, while both types propagate through neutral arrows. The appearance of a selective arrow is therefore silent when the potential parent at its tail is of type 1.

2.3. The Wright–Fisher diffusion in random environment

In this section we are interested in the Wright–Fisher diffusion in random environment described in the introduction as the solution to the SDE (1.3). In Section 3 we will see that, indeed, for any $x_0\in[0,1]$ , this SDE has a pathwise unique strong solution starting at $x_0$ (see Proposition 3.3).

Consider a pure-jump subordinator $J=(J(s))_{s\geq 0}$ with Lévy measure satisfying (2.4) and an independent standard Brownian motion $B=(B(s))_{s\geq 0}$ . For any $T>0$ , the solution to (1.3) in [0, T] is a measurable function of $(B(s),J(s))_{s\in[0,T]}$ , which we denote by F(B, J). A regular version of the conditional law ${\mathbb P}(F(B,J) \in \cdot \mid J=\omega)$ of F(B, J) given J is classically referred to as the quenched probability measure. It is defined for almost every realization $\omega$ of J. ${\mathbb P}(F(B,J) \in \cdot)$ integrates the effect of the random environment and is classically referred to as the annealed measure. As before, the quenched and annealed measures are related via

\begin{align*}{\mathbb P}(F(B,J) \in \cdot)=\int {\mathbb P}(F(B,J) \in \cdot \mid J=\omega)\, {\mathbb P}(J \in {\textrm{d}} \omega).\end{align*}

We write X and $X^\omega$ for the solutions to (1.3) under the annealed and quenched measures, respectively. For $\omega$ simple, the process $X^\omega$ starting at $x_0$ can be alternatively defined as follows. Denote by $t_1<\cdots<t_k$ the consecutive jump times of $\omega$ in [0, T], and set $t_0\,{:\!=}\, 0$ and $X^\omega(0)\,{:\!=}\, x_0$ . In the intervals $[t_i,t_{i+1})$ , $X^\omega$ evolves as the solution to (1.1) starting at $X^\omega(t_i)$ . Moreover, if $X^\omega(t_i-\!)=x$ , then $X^\omega(t_i)\,{:\!=}\, x+x(1-x)\Delta\omega(t_i)$ ; see Figure 3 for an illustration.

Figure 3. An illustration of a path of the process $X^\omega$ in the interval [0, T]. The grey vertical lines represent the peaks of the environment $\omega$ ; $t_1$ , $t_2$ , $t_3$ , and $t_4$ are the jump times of $\omega$ .

The next result states the convergence of the type frequency process in the Moran model to the Wright–Fisher diffusion in random environment, as population size grows to $\infty$ and time is suitably accelerated.

The proof of Theorem 2.2 is given in Section 3.2. The reason for using the environment $J_N$ or $\omega_N$ is to compensate for the fact that time is sped up by a factor of N. In this way, $X_N$ and X share the same environment.

2.4. The ancestral selection graph in random/deterministic environment

The aim of this section is to associate an ASG to the Wright–Fisher diffusion in random/deterministic environment. In contrast to the Moran model setting described in Section 2.1, it is not straightforward to set up a graphical representation for the forward process. To circumvent this problem, we proceed as follows. We first consider the graphical representation of a Moran model with parameters $\sigma/N$ , $\theta/N$ , $\nu_0$ , $\nu_1$ , and environment $\omega_N(\!\cdot\!)=\omega(\!\cdot\!/N)$ , and we speed up time by N. Next, we sample n individuals at time T and we construct the ASG as in Section 2.1.

Now, replace $\omega$ by a pure-jump subordinator J with Lévy measure $\mu$ supported in (0,1). Note that the Moran-ASG in [0, T] evolves according to the time reversal of J. The latter is the subordinator $\bar{J}^T\,{:\!=}\,(\bar{J}^T(\beta))_{\beta\in[0,T]}$ with $\bar{J}^T(\beta)\,{:\!=}\, J(T)-J((T-\beta)-\!)$ , which has the same law as J (its law does not depend on T). A simple asymptotic analysis of the rates and probabilities for the possible events leads to the following definition.

Let $\omega\,{:}\,{\mathbb R}\to{\mathbb R}$ be a fixed environment. The quenched ancestral selection graph with parameters $\sigma,\theta,\nu_0,\nu_1$ , and environment $\omega$ of a sample of size n at time T is a branching–coalescing particle system $\mathcal{G}_T^\omega\,{:\!=}\,(\mathcal{G}_T^\omega(\beta))_{\beta\geq 0}$ starting at $\beta=0-$ with n lines and evolving as the annealed ASG but with (iii) replaced by the following:

1. (iii^′) If at time $\beta$ we have $\Delta \omega(T-\beta)>0$ , then any line splits into two lines, an incoming line and a continuing line, with probability $\Delta \omega(T-\beta)$ , independently from the other lines.

See Figure 4 for an illustration of the type frequency process $X^\omega$ and the killed ASG $\mathcal{G}_T^\omega$ . The branching–coalescing system $\mathcal{G}_T^\omega$ is clearly well-defined for $\omega$ simple. The justification of the previous definition for general environments is more involved and will be given in Section 4.1.

Figure 4. Illustration of a path of the process $X^\omega$ (grey path) and the killed ASG $\mathcal{G}_T^\omega$ (black lines) embedded in the same picture. Forward time t runs from left to right; backward time $\beta\,{:\!=}\, T-t$ runs from right to left. The environment $\omega$ jumps at forward times $t_0$ , $t_1$ , and $t_2$ .

2.5. Type frequency via the killed ASG

The aim of this section is to relate the type-0 frequency process X to the ASG. To this end, assume that the proportion of fit individuals at time 0 is equal to $x\in[0,1]$ . Conditionally on X(T), the probability of independently sampling n unfit individuals at time T equals $(1-X(T))^n$ . Now, consider the annealed ASG associated to the n sampled individuals in [0, T], and randomly assign types independently to each line in the ASG at time $\beta=T$ according to the initial distribution $(x,1-x)$ . In the absence of mutations, the n sampled individuals are unfit if and only if all the lines in the ASG at time $\beta=T$ are assigned the unfit type (because at any selective event a fit individual can only be replaced by another fit individual). Therefore, if R(T) denotes the number of lines present in the ASG at time $\beta=T$ , then conditionally on R(T), the probability that the n sampled individuals are unfit is $(1-x)^{R(T)}$ . We would then expect to have

\begin{align*}{\mathbb E}[(1-X(T))^n\big| X(0)=x]={\mathbb E}[(1-x)^{R(T)}\big| R(0)=n].\end{align*}

Mutations determine the types of some of the lines in the ASG even before we assign types to the lines at time $\beta=T$ . Hence, we can prune away from the ASG all the sub-ASGs arising from a mutation event. If in the pruned ASG there is a line ending in a beneficial mutation, we can infer that at least one of the sampled individuals is fit. If all the lines end up in a deleterious mutation, we can infer directly that all the sampled individuals are unfit. In the remaining case, the sampled individuals are all unfit if and only if all the lines present at time $\beta=T$ in the pruned ASG are assigned the unfit type. We use this idea in Section 4.2 to construct, for a given sample of the population at time $t=T$ , branching–coalescing systems $\bar{\mathcal{G}}\,{:\!=}\, (\bar{\mathcal{G}}(\beta))_{\beta\geq 0}$ and $\bar{\mathcal{G}}_T^\omega\,{:\!=}\, (\bar{\mathcal{G}}_T^\omega(\beta))_{\beta\geq 0}$ in the annealed and quenched settings, respectively. Both processes have a cemetery state $\dagger$ . The main feature of $\bar{\mathcal{G}}$ (resp. $\bar{\mathcal{G}}^\omega_T$ ) is that for any $\beta\geq0$ , the individuals in the sample are all unfit if and only if $\bar{\mathcal{G}}\neq \dagger$ (resp. $\bar{\mathcal{G}}^\omega_T\neq \dagger$ ) and all the lines present at time $\beta$ in $\bar{\mathcal{G}}$ (resp. $\bar{\mathcal{G}}^\omega_T$ ) are unfit. We refer to $\bar{\mathcal{G}}$ and $\bar{\mathcal{G}}^\omega_T$ as the annealed and quenched killed ASGs (k-ASGs), respectively.

Moment dualities. We will now establish duality relations between the process X and the line-counting process of the k-ASG. For each $\beta\geq 0$ , we denote by $R(\beta)$ the number of lines present in the annealed k-ASG at time $\beta$ , with the convention that $R(\beta)=\dagger$ if ${\bar{\mathcal{G}}}(\beta)=\dagger$ . The process $R\,{:\!=}\, (R(\beta))_{\beta\geq 0}$ , called the annealed line-counting process of the k-ASG, is a continuous-time Markov chain with values on ${\mathbb N}_0^\dagger\,{:\!=}\, \mathbb{N}_0\cup\{\dagger\}$ and infinitesimal generator matrix $Q^\mu_\dagger\,{:\!=}\, \left(q^\mu_\dagger(i,j)\right)_{i,j\in{\mathbb N}_0^\dagger}$ defined via

(2.6) \begin{equation} q^\mu_\dagger(i,j)\,{:\!=}\, \left\{\begin{array}{l@{\quad}l} i(i-1)+i \theta\nu_1 & \text{if $j=i-1$},\\ \\[-8pt] \binom{i}{k}(\sigma_{i,k}+\sigma 1_{\{k=1\}}) & \text{if $j=i+k,\, i\geq k\geq 1$},\\ \\[-8pt] i\theta\nu_0& \textrm{if $j=\dagger$},\\ \\[-8pt] -i(i-1+\theta+\sigma)-\int_{(0,1)}(1-(1-y)^i)\mu({\textrm{d}} y)& \textrm{if $j=i\in{\mathbb N}_0$}, \end{array}\right.\end{equation}

where the coefficients $\sigma_{m,k}$ are defined in Equation (2.5). All other entries are zero.

Similarly, for $T \in \mathbb{R}$ and a fixed environment $\omega\in{\mathbb D}^\star$ , we denote by $R_T^\omega \,{:\!=}\, (R_T^\omega(\beta))_{ \beta \geq 0}$ the line-counting process associated to the quenched k-ASG ${\bar{\mathcal{G}}_T^\omega}$ . The process $R_T^\omega$ , called the quenched line-counting process of the k-ASG, is a continuous-time (inhomogeneous) Markov process with values in ${\mathbb N}_0^\dagger$ . It jumps from $i\in{\mathbb N}$ to $j\in{\mathbb N}_0^\dagger \setminus \{i\}$ at rate $q^0_\dagger(i,j)$ , where $q^0_\dagger$ is the matrix defined in (2.6) with $\mu=0$ . In addition, at each time $\beta \geq 0$ with $\Delta\omega(T-\beta)>0$ , conditionally on $\{R_T^\omega(\beta-\!)=i\}$ , $i\in{\mathbb N}$ , we have $R_T^\omega(\beta) \sim i + \textrm{Bin}({i},{\Delta \omega(T-\beta)})$ . If $\theta \gt0$ and $\nu_0 \in (0,1)$ , the states 0 and $\dagger$ are absorbing for R and $R_T^\omega$ .

Let J be a pure-jump subordinator with Lévy measure $\mu$ supported in (0, 1). We write here $X^J$ instead of X to stress the dependency of the (strong) solution to (1.3) on the environment J. Similarly, we write $X^\omega$ for its quenched version (as introduced in Section 2.3). Since, in the annealed case, backward and forward environments have the same law, we can construct the line-counting process of the k-ASG as the strong solution to an SDE involving J and four other independent Poisson processes encoding the non-environmental events. We denote it by $(R^J(\beta))_{\beta\geq 0}$ . The next result establishes a formal relation between $X^J$ and $R^J$ : a reinforced moment duality, which allows us to derive moment dualities in the annealed and quenched settings (see Figure 4 to visualize forward and backward processes in the same picture).

Theorem 2.3. (Reinforced, annealed, and quenched moment dualities.) For all $x\in[0,1]$ , $n\in\mathbb{N}$ , and $T\geq 0$ , and any function $f\in\mathcal{C}^2([0,\infty))$ with compact support,

(2.7) \begin{equation}\mathbb{E}[(1-X^J(T))^n f(J(T))\mid X^J(0)=x]=\mathbb{E}[(1-x)^{R^J(T)} f(J(T))\mid R^J(0)=n], \end{equation}

with the convention $(1-x)^\dagger=0$ for all $x\in[0,1]$ . In particular, if $f=1$ we recover the moment duality (we will often drop the superscript J when using this relation, unless we want to emphasize the dependency on J):

(2.8) \begin{equation}\mathbb{E}[(1-X^J(T))^n \mid X^J(0)=x]=\mathbb{E}[(1-x)^{R^J(T)} \mid R^J(0)=n]. \end{equation}

For almost every (with respect to the law of J) environment $\omega\in{\mathbb D}^\star$ ,

(2.9) \begin{equation}\mathbb{E} \left [ (1-X^\omega(T))^n\mid X^\omega(0)=x \right ]= \mathbb{E} \left [ (1-x)^{R_T^\omega(T-\!)}\mid R_T^\omega(0-\!)=n \right ]. \end{equation}

We prove (2.7) and (2.8) in Section 5.1. The proof of the quenched duality (2.9) is given in Section 6.1. Moreover, Theorem 7.1 extends (2.9) to any simple environment.

Asymptotic type composition. Assume now that $\theta \gt0$ and $\nu_0, \nu_1 \in (0,1)$ . In particular, the processes X and $X^\omega$ are not absorbed in $\{0,1\}$ . We will describe the asymptotic behavior of these processes using Theorem 2.3. The quenched case is particularly delicate, because for a given environment $\omega$ , $X^\omega(t)$ strongly depends on the environment in the recent past, and only weakly on the environment in the distant past (see Figure 3). Hence, unless $\omega$ is constant after some fixed time $t_0$ (i.e. $\omega$ has no jumps after $t_0$ ), $X^\omega(t)$ will not converge as $t\to\infty$ (see Remark 2.6 for the case of periodic environments). In contrast, for a given environment $\omega$ in $(\!-\!\infty,0]$ , we will see that $X^\omega(0)$ , conditionally on $X^\omega(\!-\tau)=x$ , converges in distribution as $\tau\to\infty$ , and we will characterize its law; the setting is illustrated in Figure 5 (compare with Figure 3). To this end, for $n\in{\mathbb N}_0$ define

(2.10) \begin{align}\pi_n &\,{:\!=}\, \mathbb{P}(\exists \beta\geq 0\,{:}\, R(\beta)=0 \mid R(0)=n),\nonumber\\[3pt]\Pi_n (\omega)&\,{:\!=}\, \mathbb{P}(\exists \beta\geq 0 \,{:}\, R_0^\omega(\beta)=0 \mid R_{0}^\omega(0-\!)=n), \end{align}

and set $\pi_\dagger\,{:\!=}\, 0$ and $\Pi_\dagger(\omega)\,{:\!=}\, 0$ . Clearly, $\pi_0=1$ and $\Pi_0(\omega)=1$ .

Figure 5. An illustration of two paths of $X^\omega$ : the black (resp. grey) path is defined in the interval $[\!-(\tau+h),0]$ (resp. $[\!-\tau,0]$ ) starting at $X^\omega(\!-(\tau+h))=x$ (resp. $X^\omega(\!-\tau)=x$ ). The peaks of the environment $\omega$ are depicted as grey vertical lines.

The setting of Theorem 2.4(1) is illustrated in Figure 3, and its proof is given in Section 5.1; the setting of Theorem 2.4(2) is illustrated in Figure 5, and its proof is given in Section 6.1. Moreover, Theorem 7.2 extends Theorem 2.4(2) to any simple environment. A refinement of Theorem 2.4(2) is given in Theorem 7.3 for simple environments under additional conditions.

Remark 2.5. Simpson’s index is a popular tool for describing population diversity. It represents the probability that two individuals chosen uniformly at random from the population have the same type. In our case it is given by ${\text{Sim}}(t) \,{:\!=}\, X(t)^2 + (1-X(t))^2$ . If the types represent different species, it gives a measure of bio-diversity. If the types represent two alleles of a gene for a given species, it measures homozygosity. As a consequence of Theorem 2.4, one can express the moments of ${\text{Sim}}(\infty)$ in terms of the coefficients $(\pi_n)_{n\geq 0}$ . In particular, we have

\begin{align*} \mathbb{E}[{\text{Sim}}(\infty)]=\mathbb{E}\!\left[X(\infty)^2 + (1-X(\infty))^2\right] = 1 - 2 \pi_1 + 2\pi_2. \end{align*}

Mixed environments. We present now an application illustrating the advantage of studying both quenched and annealed settings. We consider a population evolving from the distant past in a (stationary) random environment and analyze the effect of a recent perturbation of the environment on the type composition at present. To this end, we assume we only know the distribution of the environment before the perturbation.

In the absence of perturbations, the environment is given by a pure-jump subordinator J in $(\!-\!\infty,0]$ with Lévy measure $\mu$ satisfying (2.4). The perturbation occurs in $(\!-\tau_\star^{},0]$ (for some $\tau_\star^{} \gt 0$ ) and is given by a deterministic environment $\omega$ . Let $X^{J\otimes_{\tau_\star^{}}^{} \omega}$ be the solution to (1.3) under the environment $J\otimes_{\tau_\star^{}}^{} \omega$ , which coincides with J and $\omega$ in $(\!-\!\infty,-{\tau_\star^{}}]$ and $(\!-{\tau_\star^{}},0]$ , respectively; see Figure 6 for an illustration. Recall that $ (R_0^\omega(\beta))_{\beta \in [0,{\tau_\star^{}})}$ is the line-counting process associated to the quenched k-ASG (see Section 2.5). We are interested in the distribution of $X^{J\otimes_{\tau_\star^{}}^{} \omega}(0)$ . The next result provides the moments of this random variable.

Figure 6. An illustration of two paths (thin and thick) of $X^{J\otimes_{\tau_\star^{}}^{}\omega}$ in $[\!-\tau,0]$ starting at x. Both paths are subject to the same deterministic environment $\omega$ in $[\!-{\tau_\star^{}},0]$ ; the vertical lines in $(\!-{\tau_\star^{}},0]$ represent the peaks of $\omega$ . The environment in $[\!-\tau,-\tau_\star^{})$ is random and driven by J; the peaks of the realization of J giving rise to the thick (resp. thin) path in $[\!-\tau,-{\tau_\star^{}})$ are depicted as solid (resp. dotted) vertical lines.

Proposition 2.1. Assume that $\theta>0$ and $\nu_0,\nu_1\in(0,1)$ . For any $\tau_\star^{}>0$ , $n\in{\mathbb N}$ , $x\in[0,1]$ , and almost every (with respect to the law of J) $\omega\in{\mathbb D}^\star$ , we have

(2.15) \begin{equation}\lim_{\tau\to\infty}{\mathbb E}\left[(1-X^{J\otimes_{\tau_\star^{}}^{} \omega}(0))^n\mid X^{J\otimes_{\tau_\star^{}}^{} \omega}(\!-\tau)=x\right]={\mathbb E}\left[\pi_{R_0^{\omega}({\tau_\star^{}}-\!)}\mid R_0^\omega(0-\!)=n\right].\end{equation}

Proposition 2.1 is proved in Section 6.1; a refinement of this result is given for simple environments under the additional condition $\sigma=0$ in Proposition 7.2.

2.6. Ancestral type via the pruned lookdown ASG

In this section we are interested in the type distribution at present of the individuals that will be successful in the long run. This distribution may differ substantially from the type composition at present and may show a bias towards the fit type.

Consider a sample of n individuals at some time T in the future and trace their ancestral lines using the ASG. We will see in Section 5.2 that the number of lines in the ASG is positive recurrent (see Lemma 5.2). Hence, the ASG has bottlenecks, and if T is sufficiently large, the n individuals share a common ancestor at time 0. Assigning types to the lines in the ASG at time 0 and propagating the types along using the pecking order, we determine the types in the sample as well as the true genealogy. In particular, we obtain the type of the common ancestor of the sample. What it means for T to be sufficiently large depends on n and on the realization of the ASG, but this dependency vanishes as $T\to \infty$ . Because we are interested in the type of the individual that is successful in the long run, we can work under this limit consideration. In what follows we formalize this idea.

Consider a realization $\mathcal{G}_{[0,T]}^{}\,{:\!=}\, (\mathcal{G}(\beta))_{\beta\in[0,T]}$ of the annealed ASG in [0, T] started with one line, representing an individual sampled at forward time T. If t denotes forward time, we set $\beta=T-t$ to denote the backward time (see Figure 4). For $\beta\in[0,T]$ , let $V_\beta$ be the set of lines present at time $\beta$ in $\mathcal{G}_{[0,T]}^{}$ . Consider a function $c\,{:}\,V_T\to\{0,1\}$ representing an assignment of types to the lines in $V_T$ . Given $\mathcal{G}_{[0,T]}^{}$ and c, we propagate types (forward in time) along the lines of $\mathcal{G}_{[0,T]}$ , keeping track, at any time $\beta\in[0,T]$ , of the true ancestor in $V_T$ of each line in $V_\beta$ . We denote by $a_c(\mathcal{G}_{[0,T]}^{})$ the type of the ancestor in $V_T$ of the single line in $V_0$ . Assume that, under ${\mathbb P}_x$ , c assigns independently to each line type 0 with probability x and type 1 with probability $1-x$ . The annealed ancestral type distribution at time T is

\begin{align*}h_T(x)\,{:\!=}\, {\mathbb P}_x(a_c(\mathcal{G}_{[0,T]})=0),\quad x\in[0,1].\end{align*}

In the quenched setting we proceed in the same way, but using $\mathcal{G}_{[0,T]}^\omega\,{:\!=}\, (\mathcal{G}_T^\omega(\beta))_{\beta\in[0,T]}$ , the quenched ASG in [0, T] in the environment $\omega$ of an individual sampled at time T, instead of $\mathcal{G}_{[0,T]}^{}$ . The quenched ancestral type distribution at time T is

\begin{align*}h_T^\omega(x)\,{:\!=}\, {\mathbb P}_x\left(a_c(\mathcal{G}_{[0,T]}^\omega)=0\right),\quad x\in[0,1],\end{align*}

where, under ${\mathbb P}_x$ , c assigns independently to each line present in $\mathcal{G}_{[0,T]}^\omega$ at time $\beta=T$ type 0 with probability x and type 1 with probability $1-x$ .

In the absence of mutations, the ancestor of the sampled individual is fit if and only if there is at least one fit line in the ASG having type 0 at time $\beta=T$ . In the presence of mutations, determining the type of the ancestor is more involved. In [Reference Lenz, Kluth, Baake and Wakolbinger31] the ancestral type distribution was obtained for the null environment using the line-counting process of a pruned version of the ASG, called the pruned lookdown ASG (pLD-ASG). In Section 4.3 we generalize this construction to incorporate the effect of the environment. The main feature of the pLD-ASG is that the type of the ancestor at time $t=0$ of the sampled individual at time $t=T$ is 0 if and only if there is at least one line in the pLD-ASG at time $\beta=T$ that has type 0 (see Lemma 4.3). Hence, $h_T(x)$ and $h_T^\omega(x)$ can be represented via the corresponding line-counting processes (which can easily be inferred from the description of the pLD-ASG given in Section 5.2).

The line-counting process of the annealed pLD-ASG, denoted by $L\,{:\!=}\, (L(\beta))_{\beta \geq 0}$ , is a continuous-time Markov chain with values on $\mathbb{N}$ and generator matrix $Q^\mu\,{:\!=}\, (q^\mu(i,j))_{i,j\in{\mathbb N}}$ given by

(2.16) \begin{equation}q^\mu(i,j)\,{:\!=}\, \left\{\begin{array}{l@{\quad}l} i(i-1)+(i-1) \theta\nu_1 + \theta\nu_0 & \text{if $j=i-1$},\\ \\[-7pt] i(\sigma + \sigma_{i,1}) &\text{if $j=i+1$},\\ \\[-7pt] \binom{i}{k}\sigma_{i,k} &\text{if $j=i+k,\ i\geq k\geq 2$},\\ \\[-7pt] \theta\nu_0 &\textrm{if $1 \leq j \lt i-1$},\\ \\[-7pt] -(i-1)(i+\theta)-i\sigma-\int_{(0,1)}(1-(1-y)^i)\mu({\textrm{d}} y) &\textrm{if $j=i$},\\ \end{array}\right.\end{equation}

where $\sigma_{m,k}$ is defined in (2.5); all other entries are 0.

The pLD-ASG associated to $\omega\in{\mathbb D}^\star$ is well-defined and almost surely contains finitely many lines at any time; we show this in Section 5.2. The corresponding line-counting process $(L_T^\omega(\beta))_{ \beta \geq 0}$ started at time T is a continuous-time (inhomogeneous) Markov process with values in $\mathbb{N}$ . It jumps from $i\in{\mathbb N}$ to $j\in{\mathbb N} \setminus \{i\}$ at rate $q^0(i,j)$ , where $q^0$ is the matrix defined in (2.16) with $\mu=0$ , and in addition, at each time $\beta \geq 0$ such that $\Delta\omega(T-\beta)>0$ , conditionally on $\{L_T^\omega(\beta-\!)=i\}$ , $i\in{\mathbb N}$ , we have $L_T^\omega(\beta) \sim i+\textrm{Bin}({i},{\Delta \omega(T-\beta)})$ .

We now state the main result of this section, describing the asymptotic behavior of $h_T(x)$ and $h_T^\omega(x)$ .

The proof of Theorem 2.5(1) is given in Section 5.2; Theorem 2.5(2) is proved in Section 6.2. Theorem 7.4 extends Theorem 2.5(2) to the case $\theta\nu_0=0$ for simple environments under additional conditions. A refinement of Theorem 2.5(2) is given in Theorem 7.5 for simple environments under additional conditions.

In the case $\theta=0$ , Theorem 2.5 yields the following result about the boundary behavior of X.

Corollary 2.1. (Accessibility of the boundaries.) If $\theta=0$ , then for any $T>0$ and $x\in[0,1]$ ,

\begin{align*}h_T(x)=\mathbb{E} [X(T) \mid X(0) = x].\end{align*}

Moreover, conditionally on $\{X(0)=x\}$ , X(T) converges almost surely as $T\to\infty$ to a Bernoulli random variable with parameter h(x). In particular, the absorbing states 0 and 1 are both accessible from any $x\in(0,1)$ .

Remark 2.7. Corollary 2.1 is not a direct consequence of [Reference González Casanova, Spanò and Wilke-Berenguer20, Theorem 3.2], whose statement does not cover SDEs with a diffusion term (the term $\sqrt{2X(t)(1-X(t))}{\textrm{d}} B(t)$ ). We close this section with an application of our results to the comparison of the (isolated) effects of the environment and of (genic) selection. To this end, we fix a non-zero measure $\mu$ on (0, 1) satisfying (2.4) and we consider two models, both without mutations. The first model has selection parameter

(2.21) \begin{equation}\sigma=\sigma_\mu\,{:\!=}\, \int_{(0,1)}y\mu({\textrm{d}} y),\end{equation}

and no environment (i.e. in (1.3) we take $S(t)\,{:\!=}\, \sigma_\mu t$ ). The second one has selection parameter $\sigma=0$ and an environment given by a subordinator with Lévy measure $\mu$ (i.e. in (1.3) we take $S(t)\,{:\!=}\, J(t)$ ). We will use the superscript ‘sel’ (resp. ‘env’) to refer to the first (resp. second) model.

For $n\in{\mathbb N}$ , set $\rho_n^{{\text{env}}}\,{:\!=}\, {\mathbb P}(L^{{\text{env}}}(\infty)=n)$ and $\rho_n^{{\text{sel}}}\,{:\!=}\, {\mathbb P}(L^{{\text{sel}}}(\infty)=n)$ . Consider the probability generating functions

\begin{align*}p^{{\text{env}}}(z)\,{:\!=}\, \sum_{n=1}^\infty \rho_n^{{\text{env}}}z^{n}\quad\textrm{and}\quad p^{{\text{sel}}}(z)\,{:\!=}\, \sum_{n=1}^\infty \rho_n^{{\text{sel}}}z^{n},\quad z\in[0,1].\end{align*}

Note that $p^{{\text{env}}}(z)=1-h^{{\text{env}}}(1-z)$ and $p^{{\text{sel}}}(z)=1-h^{{\text{sel}}}(1-z)$ .

Proposition 2.2. For any non-zero measure $\mu$ on (0,1) satisfying (2.4) we have

\begin{align*}\rho_1^{{\text{env}}}>\rho_1^{{\text{sel}}}=\frac{\sigma_\mu}{e^{\sigma_\mu}-1}\quad{and}\quad p^{{\text{env}}}(z)\leq \frac{\rho_1^{{\text{env}}}}{\rho_1^{{\text{sel}}}}\, p^{{\text{sel}}}(z)=\rho_1^{{\text{env}}}\,\left(\frac{e^{\sigma_\mu z}-1}{\sigma_\mu}\right),\quad z\in[0,1].\end{align*}

In particular, there is $x_c\in(0,1)$ such that, for $x\in[x_c,1)$ ,

\begin{equation*}h^{{\text{env}}}\!(x)\,{=}\,{\mathbb P}\!\left(\lim_{t\to\infty}\!X^{{\text{env}}}(t)\,{=}\,1 \!\mid\! X^{{\text{env}}}(0)\,{=}\,x\!\right)\!<{\mathbb P}\!\left(\lim_{t\to\infty}\!X^{{\text{sel}}}(t)\,{=}\,1 \!\mid\! X^{{\text{sel}}}(0)\,{=}\,x\!\right)\,{=}\,h^{{\text{sel}}}(x).\end{equation*}

Remark 2.8. As a consequence of Proposition 2.2 one recovers the classical result of Kimura [Reference Kimura28],

\begin{align*}h^{{\text{sel}}}(x)=\frac{1-e^{-\sigma_\mu x}}{1-e^{-\sigma_\mu}},\quad x\in[0,1].\end{align*}

3.1. Results related to Section 2.1

Graphical representation. We start by making more precise the description of the graphical representation of the Moran model as an IPS. This will allow us to decouple the randomness of the model coming from the initial type configuration, the randomness coming from mutations and reproductions, and the randomness coming from the environment. Non-environmental events are as usual encoded via a family of independent Poisson processes $\Lambda\,{:\!=}\, \{\lambda_{i}^{0},\lambda_{i}^{1},\{\lambda_{i,j}^{\vartriangle},\lambda_{i,j}^{\blacktriangle}\}_{j\in{[N] \setminus \{i\}}} \}_{i\in[N]}$ , where (a) for each $i,j\in [N]$ with $i\neq j$ , $(\lambda_{i,j}^{\vartriangle}(t))_{t\in{\mathbb R}}$ and $(\lambda_{i,j}^{\blacktriangle}(t))_{t\in{\mathbb R}}$ are Poisson processes with rates $\sigma_N/N$ and $1/N$ , respectively, and (b) for each $i\in [N]$ , $(\lambda_{i}^{0}(t))_{t\in{\mathbb R}}$ and $(\lambda_{i}^{1}(t))_{t\in{\mathbb R}}$ are Poisson processes with rates $\theta_N\nu_0$ and $\theta_N\nu_1$ , respectively. We call $\Lambda$ the basic background. The environment introduces a new independent source of randomness into the model, which we describe via the collection

\begin{align*}\Sigma\,{:\!=}\, \{(U_i(t))_{i\in[N], t\in{\mathbb R}}, (\tau_A^{}(t,\cdot))_{A\subset[N], t\in{\mathbb R}}\},\end{align*}

where (c) $(U_i(t))_{i\in[N], t\in{\mathbb R}}$ is an $[N] \times {\mathbb R}$ -indexed family of independent and identically distributed (i.i.d.) random variables with $U_i(t)$ being uniformly distributed on [0, 1], and (d) $(\tau_A(t,\cdot))_{A\subset[N], t\in{\mathbb R}}$ is a family of independent random variables with $\tau_A(t,\cdot)$ being uniformly distributed on the set of injections from A to [N]. We call $\Sigma $ the environmental background. We assume that the basic and environmental backgrounds are independent, and we call $(\Lambda,\Sigma)$ the background.

Recall that in the graphical representation individuals are represented by horizontal lines at levels $i\in [N]$ (see Figure 1). The random appearance of selective and neutral arrows, circles, and crosses is prescribed by the background as follows. At the arrival times of $\lambda_{i,j}^{\vartriangle}$ (resp. $\lambda_{i,j}^{\blacktriangle}$ ), we draw selective (resp. neutral) arrows from level i to level j. At the arrival times of $\lambda_{i}^{0}$ (resp. $\lambda_{i}^{1})$ , we draw an open circle (resp. a cross) at level i. Given an environment $\zeta\,{:\!=}\, (t_k, p_k)_{k \in I}$ satisfying (2.1), we define, for each $k\in I$ ,

\begin{align*}I_{\zeta}(k)\,{:\!=}\, \{i\in[N]\,{:}\,U_i(t_k)\leq p_k\}\quad\textrm{and}\quad n_{\zeta}(k)\,{:\!=}\, |I_{\zeta}(k)|,\end{align*}

and we draw, at time $t_k$ , for each $i\in I_{\zeta}(k)$ a selective arrow from level i to level $\tau_{I_{\zeta}(k)}^{}(t,i)$ .

Continuity with respect to the environment. Now we embark on the proof of Theorem 2.1, which states the continuity of the type composition in a Moran model with respect to the environment. The paths of the fit-counting process are considered as elements of ${\mathbb D}_{0,T}$ , which is endowed with the $J_1$ -Skorokhod topology, i.e. the topology induced by the Billingsley metric $d_T^0$ defined in (A.2). Recall also that the restriction of an environment to [0, T] is described by means of a function in ${\mathbb D}_T^\star$ (see (2.3)), which is endowed with the topology induced by the metric $d_T^\star$ defined in (A.3).

Let us denote by $\mu_N(\omega)$ the law of $(Z_N^\omega(t))_{t\in[0,T]}$ (recall that $Z_N^\omega(t)$ is the number of fit individuals at time t in a Moran population of size N subject to environment $\omega$ ). Theorem 2.1 states the continuity of the mapping $\omega\mapsto \mu_N(\omega)$ , where the set of probability measures on ${\mathbb D}_{0,T}$ is equipped with the topology of weak convergence of measures. We will use the fact that the topology of weak convergence of probability measures on a complete and separable metric space (E, d) is induced by the metric $\varrho_E$ defined in (A.6).

First, we get rid of the small jumps of the environment. To this end, we introduce the following notation. For $\delta>0$ and $\omega\in{\mathbb D}_T^\star$ , we define $\omega^\delta, \omega_\delta\in {\mathbb D}_T^\star$ via

(3.1) \begin{align}\omega^\delta(t)\,{:\!=}\, \sum_{u\in[0,t]:\Delta \omega(u)\geq \delta} \Delta \omega(u)\quad\textrm{and}\quad \omega_\delta(t)\,{:\!=}\, \sum_{u\in[0,t]:\Delta \omega(u)\lt \delta} \Delta \omega(u).\end{align}

Clearly, $\omega^\delta$ is simple and $\omega=\omega^\delta+\omega_\delta$ . Moreover, $\omega_\delta \to 0$ pointwise as $\delta\to 0$ , and hence for any $t\in[0,T]$ ,

\begin{align*}d_t^\star(\omega,\omega^\delta)\leq \sum_{u\in[0,T]}\lvert \Delta \omega(u)-\Delta \omega^\delta(u)\rvert= \omega_\delta(T) \xrightarrow[\delta\to 0]{}0.\end{align*}

In addition, for $\omega\in{\mathbb D}_T^\star$ , $n\in{\mathbb N}$ , and $\vec{r}\,{:\!=}\, (r_i)_{i\in[n]}\in[0,T]^n$ , we denote by $\mu_N^{\vec{r}}(\omega)$ the law of $(Z_N^{\omega}(r_i))_{i\in[n]}$ , where $[0,N]^n$ is equipped with the distance $d_1$ defined via

(3.2) \begin{equation}d_1( (x_i)_{i\in[n]}, (y_i)_{i\in[n]}) \,{:\!=}\, \sum_{i\in[n]}|x_i - y_i|.\end{equation}

Proposition 3.1. Let $\omega\in{\mathbb D}_T^\star$ . Assume that for any $\delta>0$ we have $Z_N^{\omega^\delta}(0)=Z_N^{\omega}(0)$ ; then

(3.3) \begin{equation}\varrho_{[0,T]^n}^{}(\mu_N^{\vec{r}}\left(\omega^\delta\right),\mu_N^{\vec{r}}(\omega))\leq nN\,\omega_\delta(r_*)e^{\sigma_N r_*+ \omega(r_*)}, \ \forall \vec{r}\in[0,T]^n, n\in{\mathbb N},\end{equation}

where $r_*\,{:\!=}\, \max_{i\in[n]}r_i$ . Moreover,

(3.4) \begin{equation}\varrho_{{\mathbb D}_{0,T}}^{}(\mu_N\left(\omega^\delta\right),\mu_N(\omega))\leq N\omega_\delta(T)e^{(1+\sigma_N) T+\omega(T)}.\end{equation}

In particular,

\begin{align*}(Z_N^{\omega^\delta}(t))_{t\in[0,T]}\xrightarrow[\delta\to 0]{(d)}(Z_N^{\omega}( t))_{t\in[0,T]}.\end{align*}

Proof. For $\delta>0$ , we couple in [0, T] a Moran model with parameters $(\sigma_N,\theta_N,\nu_0,\nu_1)$ and environment $\omega$ to a Moran model with parameters $(\sigma_N,\theta_N,\nu_0,\nu_1)$ and environment $\omega^\delta$ (both of size N) by using the same initial type configuration, the same basic background, and the same environmental background. For any $t\in[0,T]$ and $a,b\in\{0,1\}$ , we denote by $Y_N^{a,b}(t)$ the number of individuals that at time t have type a under the environment $\omega$ and type b under the environment $\omega^\delta$ . Clearly, we have

\begin{align*}\left\lvert Z_N^{\omega^{\delta}}(t)-Z_N^{\omega}(t)\right\rvert=\left\lvert Y_N^{1,0}(t) - Y_N^{0,1}(t) \right\rvert\leq Y_N^{1,0}(t) + Y_N^{0,1}(t)\,{:\!=}\, Y_N^{\neq}(t).\end{align*}

Note that $Y_N^{\neq}(t)$ is the number of individuals that have different types at time t under $\omega$ and $\omega^\delta$ . In particular, we have $Y_N^{\neq}(t) \leq N$ almost surely. Let us assume that at time t a graphical element arises in the basic background, i.e. t is an arrival time of one of the Poisson processes in the family $\Lambda$ . If the graphical element is a mutation, then $Y_N^{\neq}(t)\leq Y_N^{\neq}(t-\!)$ . If the graphical element is a neutral arrow, we have

\begin{align*}{\mathbb E}\left[Y_N^{\neq}(t)\mid Y_N^{\neq}(t-\!)\right] &=Y_N^{\neq}(t-\!)+\frac{1}{N}Y_N^{\neq}(t-\!)(N-Y_N^{\neq}(t-\!))-\frac{1}{N}(N-Y_N^{\neq}(t-\!))Y_N^{\neq}(t-\!)\\[3pt] &=Y_N^{\neq}(t-\!).\end{align*}

If the graphical element is a selective arrow, then $Y_N^{\neq}(t)$ can increase by 1 only if the individual at the tail of the arrow has a different type at time t under $\omega$ and $\omega^\delta$ . Thus

\begin{align*}{\mathbb E}\left[Y_N^{\neq}(t)\mid Y_N^{\neq}(t-\!)\right]\leq\left(1+\frac1N\right)Y_N^{\neq}(t-\!).\end{align*}

Now, let $0\leq s<t\leq T$ and assume that there are neither jumps of $\omega^\delta$ nor selective events in (s, t). In particular, in (s, t) only the population driven by $\omega$ is affected by the environment. Moreover, since neutral reproductions and mutations do not increase the expected value of $Y_N^{\neq}$ , we obtain

\begin{align*}{\mathbb E}\left[Y_N^{\neq}(t-\!)\mid Y_N^{\neq}(s)\right]\leq Y_N^{\neq}(s) +N\sum_{u\in(s,t)}\Delta \omega(u).\end{align*}

In addition, if at time t the environment $\omega^\delta$ jumps (there are only finitely many of these jumps), then

\begin{align*}{\mathbb E}\left[Y_N^{\neq}(t)\mid Y_N^{\neq}(t-\!)\right]\leq Y_N^{\neq}(t-\!)(1+\Delta \omega (t)).\end{align*}

Let $0\leq t_1<\cdots<t_m\leq T$ be the jump times of $\omega^\delta$ . The previous discussion yields

(3.5) \begin{equation} {\mathbb E}\left[Y_N^{\neq}(t_{i+1})\mid Y_N^{\neq}(t_i)\right]\leq {\mathbb E}\left[\left(1+\frac1N\right)^{K_i} \right]\left(Y_N^{\neq}(t_i)+N\epsilon_i(\delta)\right)(1+\Delta \omega(t_{i+1})), \end{equation}

where $\epsilon_i(\delta)\,{:\!=}\, \sum_{u\in(t_i,t_{i+1})}\Delta \omega(u)$ and $K_i$ is the number of selective events in $(t_i,t_{i+1})$ . Note that $K_i$ has a Poisson distribution with parameter $N\sigma_N(t_{i+1}-t_i)$ . Hence,

\begin{align*}{\mathbb E}\left[Y_N^{\neq}({t_{i+1}})\mid Y_N^{\neq}(t_i)\right]\leq e^{\sigma_N(t_{i+1}-t_i)}\left(Y_N^{\neq}(t_i)+N\epsilon_i(\delta)\right)(1+\Delta \omega(t_{i+1})).\end{align*}

Iterating this formula and using that $Y_N^{\neq}(0)=0$ yields

(3.6) \begin{equation} {\mathbb E}\left[Y_N^{\neq}(t)\right]\leq e^{\sigma_N t} N\omega_\delta(t)\prod_{t_i\leq t}(1+\Delta \omega(t_i))\leq N\,\omega_\delta(t)\,e^{\sigma_N t+\sum_{u\in[0,t]}\Delta \omega(u)}.\end{equation}

Recall the definition of the space $\textrm{BL}(E)$ in Appendix A.2. We equip $[0,N]^n$ with the distance $d_1$ defined in (3.2). For any $n \geq 1$ and $F\in \textrm{BL}([0,N]^n)$ , we have

\begin{align*} \left\lvert \int F {\textrm{d}}\mu_N^{\vec{r}}\left(\omega^\delta\right)- \int F {\textrm{d}}\mu_N^{\vec{r}}(\omega)\right\rvert = \left\lvert \mathbb{E} \left [ F((Z_N^{\omega^\delta}(r_j))_{j\in[n]}) \right ] - \mathbb{E} \left [ F((Z_N^\omega(r_j))_{j\in[n]}) \right ] \right\rvert. \end{align*}

Hence, if $\lVert F\rVert_{\textrm{BL}}\leq 1$ (see (A.5) for the definition of $\lVert \cdot\rVert_{\textrm{BL}}$ ) and we couple $Z_N^{\omega^{\delta}}(t)$ and $Z_N^\omega(t)$ as before, we get that

\begin{align*} &\left\lvert \mathbb{E} \left[ F((Z_N^{\omega^\delta}(r_j))_{j\in[n]}) \right] - \mathbb{E} \left[ F((Z_N^{\omega}(r_j))_{j\in[n]}) \right] \right\rvert \\ &\quad \leq \left\lvert \mathbb{E} \left [ d_1((Z_N^{\omega^\delta}(r_j))_{j\in[n]}, (Z_N^\omega(r_j))_{j\in[n]}) \right ] \right\rvert= \mathbb{E} \left [ \sum_{j\in[n]}|Y_N^{\neq}(r_j)| \right ]\\ &\quad \leq \sum_{j\in[n]} N\omega_\delta(r_j)e^{\sigma_N r_j+\sum_{u\in[0,r_j]}\Delta \omega(u)},\end{align*}

where the last bound comes from (3.6) applied at $r_j$ , $j\in[n]$ . Taking the supremum over all $F\in \textrm{BL}([0,N]^n)$ with $\lVert F\rVert_{\textrm{BL}}\leq 1$ and using the definition of the distance $\varrho_{[0,N]^n}$ in (A.6), we get (3.3). Now, define $Y_N^*(t)\,{:\!=}\, \sup_{u\in[0,t]}Y_N^{\neq}(u)$ . If at time t a neutral event occurs, then

\begin{align*}{\mathbb E}[Y_N^*(t)\mid Y_N^*(t-\!)]\leq\left(1+\frac1N\right)Y_N^*(t-\!).\end{align*}

Other events can be treated as before, leading to (3.5) with $K_i$ being this time the number of selective and neutral events in $(t_i,t_{i+1})$ . Hence, Equation (3.4) follows similarly to (3.3). The convergence of $Z_N^{\omega^\delta}$ towards $Z_N^\omega$ is a direct consequence of (3.4).

Proposition 3.2. Let $\omega\in{\mathbb D}_T^\star$ and $\{\omega_k\}_{k\in{\mathbb N}}\subset {\mathbb D}_T^\star$ be such that $d_T^\star(\omega_k,\omega)\to 0$ as $k\to\infty$ . If $\omega$ is simple and, for any $k\in{\mathbb N}$ , $Z_N^\omega(0)=Z_N^{\omega_k}(0)$ , then

(3.7) \begin{equation}(Z_N^{\omega_k}(t))_{t\in[0,T]}\xrightarrow[k\to \infty]{(d)}(Z_N^\omega(t))_{t\in[0,T]}.\end{equation}

Proof. The proof consists of two parts. In the first part, we construct a time deformation $\lambda_k\in\mathcal{C}_T^\uparrow$ with suitable properties. In the second part, we compare $Z_N^{\omega_k}\circ\lambda_k$ and $Z_N^\omega$ under an appropriate coupling of the underlying Moran models.

Part 1: We assume, without loss of generality, that $d_T^\star(\omega_k,\omega)>0$ for all $k\in{\mathbb N}$ . Set $\epsilon_k\,{:\!=}\, 2d_T^\star(\omega_k,\omega)$ , so that $d_T^\star(\omega_k,\omega)\lt \epsilon_k$ . By definition of the metric $d_T^\star$ in (A.3), there is $\varphi_k\in \mathcal{C}_T^\uparrow$ such that

\begin{align*}\lVert \varphi_{k}\rVert_T^0\leq \epsilon_k\quad\textrm{and}\quad\sum_{u\in[0,T]}|\Delta \omega(u)-\Delta (\omega_k\circ\varphi_{k})(u)|\leq \epsilon_k,\end{align*}

where $\lVert \cdot \rVert_T^0$ is defined in (A.1). Denote by $r_1<\cdots<r_n$ the consecutive jump times of $\omega$ in [0, T]. We assume without loss of generality that $0<r_1\leq r_n<T$ . The case where $\omega$ jumps at T can be reduced to the previous case, by extending $\omega_k$ , $k\in{\mathbb N}$ , and $\omega$ to $[0,T+\varepsilon]$ as constants in $[T,T+\varepsilon]$ . Set $\gamma_k\,{:\!=}\, T\,\sqrt{e^{\epsilon_k}-1}$ . In the remainder of the proof we assume that k is sufficiently large, so that $\gamma_k\leq \min_{i\in[n]_0}(r_{i+1}-r_{i})/3$ , where $r_0\,{:\!=}\, 0$ and $r_{n+1}\,{:\!=}\, T$ . This condition ensures that the intervals $I_i^k\,{:\!=}\, [r_i-\gamma_k,r_i+\gamma_k]$ , $i\in[n]$ , are disjoint and contained in [0, T]. Now, define $\lambda_k\,{:}\,[0,T]\to[0,T]$ via the following:

1. (i) For $u\notin \cup_{i=1}^nI_i^k$ : $\lambda_k(u)\,{:\!=}\, u$ .

2. (ii) For $u\in [r_i-\gamma_k,r_i]\,:$ $\lambda_k(u)\,{:\!=}\, \varphi_k(r_i)+m_i(u-r_i)$ , where $m_i\,{:\!=}\, (\varphi_k(r_i)-r_i+\gamma_k)/\gamma_k.$

3. (iii) For $u\in (r_i,r_i+\gamma_k]\,:$ $\lambda_k(u)\,{:\!=}\, \varphi_k(r_i)+\bar{m}_i(u-r_i)$ , where $\bar{m}_i\,{:\!=}\, (r_i+\gamma_k-\varphi_k(r_i))/\gamma_k.$

For k sufficiently large, so that $\epsilon_k \lt \log 2$ , we can infer from $\lVert \varphi_{k}\rVert_T^0\leq \epsilon_k$ and from $\gamma_k=T\,\sqrt{e^{\epsilon_k}-1}$ that $m_i$ and $\bar{m}_i$ are positive. It is then straightforward to check that $\lambda_k\in\mathcal{C}_T^\uparrow$ , $\lambda_k(I_i^k)=I_i^k$ , $i\in[n]$ , and that

\begin{align*}\sum_{u\in[0,T]}|\Delta \omega(u)-\Delta \bar{\omega}_k(u)|\leq \epsilon_k,\end{align*}

where $\bar{\omega}_k\,{:\!=}\, \omega_k\circ\lambda_k$ . Moreover, since $\lVert \varphi_{k}\rVert_T^0\leq \epsilon_k$ , we infer that $\varphi_k(r_i)\in[e^{-\epsilon_k}r_i,e^{\epsilon_k}r_i]$ . It follows that, for k sufficiently large,

\begin{align*}1-2\sqrt{e^{\epsilon_k}-1} \leq m_i \leq 1+2\sqrt{e^{\epsilon_k}-1},\quad i\in[n],\end{align*}

and the same holds for $\bar{m}_i$ . Note that we can write $\lambda_k(t) = \int_0^t p_k(u) {\textrm{d}} u$ , with $p_k\,{:}\,[0,T] \mapsto \mathbb{R}$ taking only the values $(m_i)_{i\in[n]}$ , $(\bar{m}_i)_{i\in[n]}$ , and 1. In particular, we have $|p_k(u)-1|\leq 2\sqrt{e^{\epsilon_k}-1}$ for all $u\in[0,T]$ . Thus, for any $s,t\in[0,T]$ with $s\neq t$ , the slope $(\lambda_k(s)-\lambda_k(t))/(s-t)$ belongs to $[1-2\sqrt{e^{\epsilon_k}-1}, 1+2\sqrt{e^{\epsilon_k}-1}]$ . Therefore, for k sufficiently large, we have

\begin{align*}\frac{\lambda_k(s)-\lambda_k(t)}{s-t}, \,\frac{s-t}{\lambda_k(s)-\lambda_k(t)}\leq 1+3\sqrt{e^{\epsilon_k}-1},\quad i\in[n].\end{align*}

Hence, using that $\log(1+x)\leq x$ for $x>-1$ , we obtain for k sufficiently large

(3.8) \begin{equation}\lVert \lambda_k\rVert_T^0\leq 3\sqrt{e^{\epsilon_k}-1}.\end{equation}

Part 2: For $\delta>0$ , we couple in [0, T] a Moran model with parameters $(\sigma_N,\theta_N,\nu_0,\nu_1)$ and environment $\omega$ to a Moran model with parameters $(\sigma_N,\theta_N,\nu_0,\nu_1)$ and environment $\omega_k$ (both of size N) by using (1) the same initial type configuration and (2) the same basic background, and (3) by using in the second population the environmental background of the first one, time-changed by $\lambda_k^{-1}$ . Under this coupling and by construction of the function $\lambda_k$ , the Moran model associated to $\omega$ and the Moran model associated to $\omega_k$ (time-changed by $\lambda_k$ ) experience the same basic events out of the time intervals $I_i^k$ . Moreover, at the times $r_i$ , the success of simultaneous environmental reproductions is decided according to the same uniform random variables.

For $t\in[0,T]$ , let $Y_N^{\neq}(t)$ be the number of individuals that have different types at time t for $\omega$ and at time $\lambda_k(t)$ for $\omega_k$ , and set $Y_N^*(t)\,{:\!=}\, \sup_{u\in[0,t]}Y_N^{\neq}(u)$ .

Consider the event $E_{k}\,{:\!=}\, {\textrm{there are no basic events in}} $ ${ \cup _{i \in [n]}}I_i^k$ , and note that

(3.9) \begin{equation}{\mathbb P}(E_{k}^c)\leq n\left(1-e^{-2N(1+\sigma_N+\theta_N)\gamma_k}\right).\end{equation}

Moreover, on the event $E_{k}$ , only the population driven by $\omega_k$ can change in $(r_i,r_i+\gamma_k]$ , and this can only be due to environmental events. Hence,

(3.10) \begin{equation} {\mathbb E}[Y_N^*(r_i+\gamma_k)1_{E_k}]\leq {\mathbb E}[Y_N^*(r_i)1_{E_{k}}] +N\sum_{u\in(r_i,r_i+\gamma_k]}\Delta\bar{\omega}_k(u).\end{equation}

A similar argument yields

(3.11) \begin{equation} {\mathbb E}[Y_N^*(r_{i+1}-\!)\,1_{E_{k}}]\leq {\mathbb E}[Y_N^*(r_{i+1}-\gamma_k)1_{E_{k}}]+ N\sum_{u\in(r_{i+1}-\gamma_k,r_{i+1})}\Delta\bar{\omega}_k(u).\end{equation}

Since in the interval $J_{i}^{k}\,{:\!=}\,(r_i+\gamma_k,r_{i+1}-\gamma_k]$ there are no simultaneous jumps of the two environments, we can proceed as in the proof of Proposition 3.1 to obtain

(3.12) \begin{equation} {\mathbb E}\left[Y_N^*(r_{i+1}-\gamma_k)1_{E_{k}}\right]\leq e^{(1+\sigma_N)(r_{i+1}-r_{i})}\left({\mathbb E}[Y_N^*(r_i+\gamma_k)1_{E_{k}}]+N\sum_{u\in J_{i}^k}\Delta\bar{\omega}_k(u)\right).\end{equation}

Moreover, at time $r_{i+1}$ , there are two possible contributions to take into account: (i) the contribution of selective arrows arising simultaneously in both environments, and (ii) the contribution of selective arrows arising only on the environment with the biggest jump. This leads to

(3.13) \begin{align} &{\mathbb E}\left[Y_N^*(r_{i+1})\,1_{E_{k}}\right]\leq {\mathbb E}[Y_N^*(r_{i+1}-\!)\,1_{E_{k}}](1+\Delta \omega(r_{i+1})\wedge \Delta \bar{\omega}_k(r_{i+1}))\nonumber\\[3pt]&\quad + N|\Delta \omega(r_{i+1})-\Delta \bar{\omega}_k(r_{i+1})|.\end{align}

Using (3.10), (3.11), (3.12), and (3.13), we obtain

\begin{align*}{\mathbb E}[Y_N^*(r_{i+1})\,1_{E_k}]\leq e^{(1+\sigma_N)(r_{i+1}-r_{i})}&\left[{\mathbb E}[Y_N^*(r_{i})\,1_{E_k}]\vphantom{+N\sum_{u\in(r_i,r_{i+1}]}|\Delta \omega(u)-\Delta \bar{\omega}_k(u)|}\right.\\[3pt]&\left.+N\sum_{u\in(r_i,r_{i+1}]}|\Delta \omega(u)-\Delta \bar{\omega}_k(u)|\right](1+\Delta \omega(r_{i+1})).\end{align*}

Iterating this inequality, using that $Y_N^*(0)=0$ , and adding the contribution of the interval $(r_n+\gamma_k,T]$ , we obtain

(3.14) \begin{equation}{\mathbb E}\left[Y_N^*(T)\,1_{E_{k}}\right]\leq N\epsilon_k e^{(1+\sigma_N)T+\sum_{u\in(0,T]}\Delta \omega(u)}.\end{equation}

Using (3.9), (3.14), (3.8), and the definition of $d_T^0$ in (A.2), we obtain for k large enough

\begin{align*} {\mathbb E}\left[d_T^0(Z_N^\omega,Z_N^{\omega_k})\right] &\leq 2nN\left(1-e^{-2N(1+\sigma_N+\theta_N)\gamma_k}\right)\\[3pt]&\quad + {3} \sqrt{e^{\epsilon_k}-1}\vee\left(N\epsilon_k\, e^{(1+\sigma_N)T+\sum_{u\in(0,T]}\Delta \omega(u)} \right).\end{align*}

The result follows from letting $k\to\infty$ and using that $\gamma_k\to 0$ and $\epsilon_k\to 0$ as $k\to\infty$ .

Proof of Theorem 2.1 (continuity).If $\omega$ has a finite number of jumps, the result follows directly from Proposition 3.2. In the general case, note that for any $\delta>0$ ,

(3.15) \begin{align} {\varrho_{{\mathbb D}_{0,T}^{}}^{}}({\mu_N(\omega_k)},{\mu_N(\omega)})&\leq {\varrho_{{\mathbb D}_{0,T}^{}}^{}}\left({\mu_N(\omega_k)},{\mu_N\left(\omega_{k}^{\delta}\right)}\right)\nonumber\\[3pt]&\quad + {\varrho_{{\mathbb D}_{0,T}^{}}^{}}\left({\mu_N\left(\omega_{k}^{\delta}\right)},{\mu_N\left(\omega^\delta\right)}\right)+{\varrho_{{\mathbb D}_{0,T}^{}}^{}}({\mu_N(\omega^{\delta})},{\mu_N(\omega)}), \end{align}

where $\omega^{\delta}$ is as in (3.1) and, similarly, $\omega_{k}^{\delta}(t)\,{:\!=}\, \sum_{u\in[0,t]: \Delta\omega_k(u)\geq \delta}\Delta \omega_k(u)$ . Recall the definition of $d_T^\star$ in (A.3). We claim that, for any $\delta\in A_\omega\,{:\!=}\, \{d>0\,{:}\,\Delta \omega(u)\neq d \textrm{ for any u $\in$ [0, T]}\}$ , we have

(Claim 1) \begin{equation}d_T^\star(\omega_{k}^{\delta},\omega^\delta)\xrightarrow[k\to\infty]{}0.\end{equation}

Assume that Claim 1 is true and let $\delta\in A_\omega$ . Note that for any $\lambda\in\mathcal{C}^\uparrow_T$ , we have

\begin{align*} &\omega_{k,\delta}(T) \,{:\!=}\, \sum_{u\in[0,T]: \Delta\omega_k(u)\lt \delta}\Delta \omega_k(u)=\omega_k(T)-\omega_{k}^{\delta}(T)=\omega_k(\lambda(T))-\omega_{k}^{\delta}(\lambda(T))\\ &\quad \leq |\omega_k(\lambda(T))-\omega(T)|+|\omega(T)-\omega^\delta(T)|+|\omega^\delta(T)-\omega_{k}^{\delta}(\lambda(T))|\\ &\quad \leq d_T^0(\omega,\omega_k)+ \omega_\delta(T)+d_T^0(\omega_{k}^{\delta},\omega^\delta)\leq d_T^\star(\omega,\omega_k)+ \omega_\delta(T)+d_T^\star(\omega_{k}^{\delta},\omega^\delta), \end{align*}

where we used the definition of $d_T^0$ in (A.2) and then Lemma A.1. Combining this with Claim 1 and Proposition 3.1, we obtain

\begin{equation*} \limsup_{k\to\infty}{\varrho_{{\mathbb D}_{0,T}^{}}^{}}({\mu_N(\omega_k)},{\mu_N\left(\omega_{k}^{\delta}\right)})\leq N\omega_\delta(T) e^{(1+\sigma_N)T+\omega(T)}.\end{equation*}

Proposition 3.2 and Claim 1 in turn yield $\limsup_{k\to\infty}^{}{\varrho_{{\mathbb D}_{0,T}^{}}^{}}\left({\mu_N\left(\omega_{k}^{\delta}\right)},{\mu_N(\omega^{\delta})}\right)=0$ . Hence, letting $k\to\infty$ in (3.15) and using Proposition 3.1, we obtain

\begin{align*}\limsup_{k\to\infty}{\varrho_{{\mathbb D}_{0,T}^{}}^{}}({\mu_N(\omega_k)},{\mu_N(\omega)})\leq 2N\omega_\delta(T) e^{(1+\sigma_N)T+\omega(T)}.\end{align*}

The previous inequality holds for any $\delta\in A_\omega$ . It is plain to see that $\inf A_\omega=0$ . Hence, letting $\delta\to 0$ with $\delta\in A_\omega$ in the previous inequality yields the result.

It remains to prove Claim 1. Let $\delta\in A_\omega$ . Since $d_T^\star(\omega_k,\omega)\to 0$ as $k\to\infty$ , we see from the definition of $d_T^\star$ in (A.3) that there exists $(\lambda_k)_{k\in{\mathbb N}}$ with $\lambda_k\in\mathcal{C}_T^\uparrow$ such that

\begin{align*}\lVert \lambda_k\rVert_T^0\xrightarrow[k\to\infty]{}0\quad\textrm{ and }\quad \epsilon_k\,{:\!=}\, \sum_{u\in[0,T]}|\Delta(\omega_k\circ\lambda_k)(u)-\Delta \omega(u)| \xrightarrow[k\to\infty]{}0.\end{align*}

Set $\bar{\omega}_k\,{:\!=}\, \omega_k\circ\lambda_k$ . Clearly, $\Delta\bar{\omega}_k(u)\leq \epsilon_k+\Delta \omega(u)$ and $\Delta\omega (u)\leq \epsilon_k+\Delta \bar{\omega}_k(u)$ , $u\in[0,T].$ Therefore,

\begin{align*}\omega_{k}^{\delta} (\lambda_k(t)) -\omega^\delta(t)=\sum_{\substack{u\in[0,t]\\ \Delta\bar{\omega}_k(u)\geq \delta}}\!\!\!\!\Delta \bar{\omega}_k(u)-\sum_{\substack{u\in[0,t]\\ \Delta\omega(u)\geq \delta}}\!\!\!\!\Delta \omega(u)\leq \sum_{\substack{u\in[0,t]\\\Delta\omega(u)\geq \delta-\epsilon_k^{}}}\!\!\!\!\!\!\!\!\!\!\Delta \bar{\omega}_k(u)-\sum_{\substack{u\in[0,t]\\ \Delta\omega(u)\geq \delta}}\!\!\!\!\!\Delta \omega(u)\\ =\sum_{\substack{u\in[0,t]\\ \Delta\omega(u)\geq \delta-\epsilon_k}}\!\!\!\!\!\!\!(\Delta \bar{\omega}_k(u)-\Delta \omega(u))+\sum_{\substack{u\in[0,t]\\ \Delta\omega(u)\in[\delta-\epsilon_k,\delta)}}\!\!\!\!\!\!\!\!\Delta \omega(u)\leq d_T^\star(\omega_k,\omega)+\sum_{\substack{u\in[0,T]\\ \Delta\omega(u)\in[\delta-\epsilon_k,\delta)}}\!\!\! \!\!\!\!\!\!\!\Delta \omega(u).\end{align*}

Similarly, we obtain

\begin{align*}& \omega^\delta (t)-\omega_{k}^{\delta}(\lambda_k(t))=\sum_{\substack{u\in[0,t]\\ \Delta\omega(u)\geq \delta}}\!\!\!\!\Delta \omega(u)-\sum_{\substack{u\in[0,t]\\ \Delta\bar{\omega}_k(u)\geq \delta}}\!\!\!\!\Delta \bar{\omega}_k(u)\\[3pt] &\quad \leq \sum_{\substack{u\in[0,t]\\ \Delta\omega(u)\in[\delta,\delta+\epsilon_k)}}\!\!\!\!\!\!\!\!\!\!\Delta \omega(u)+\sum_{\substack{u\in[0,t]\\ \Delta\bar{\omega}_k(u)\geq \delta}}\!\!\!\!(\Delta \omega(u)-\Delta \bar{\omega}_k(u))\leq \sum_{\substack{u\in[0,T]\\ \Delta\omega(u)\in[\delta,\delta+\epsilon_k)}}\!\!\!\!\!\!\!\!\Delta \omega(u)+d_T^\star(\omega_k,\omega).\end{align*}

Thus, using the definition of $d_T^0$ in (A.2), we deduce that

\begin{align*}d_T^0(\omega_k^\delta,\omega^\delta)\leq d_T^\star(\omega_k,\omega)+\sum_{\substack{u\in[0,T]\\ \Delta\omega(u)\in(\delta-\epsilon_k,\delta+\epsilon_k)}}\!\!\!\!\!\!\!\!\!\!\Delta \omega(u).\end{align*}

Since $\delta\in A_\omega$ , letting $k\to\infty$ in the previous inequality yields $\lim_{k\to\infty }d_T^0(\omega_k^\delta,\omega^\delta)=0$ . Since $\omega^\delta$ has a finite number of jumps, Claim 1 follows using Lemma A.1.

3.2. Results related to Section 2.3: the Wright–Fisher process as a large-population limit

We start this section by proving that the SDE (1.3) is well-posed.

Proof. We prove the existence and pathwise uniqueness of strong solutions to (1.3) via [Reference Li and Pu32, Theorems 3.2 and 5.1]. To this end, we first extend (1.3) to an SDE on ${\mathbb R}$ and write it in the form of [Reference Li and Pu32, Equation 2.1]. Note that by Lévy–Itô decomposition, the pure-jump subordinator J can be expressed as $J(t)=\int_{(0,1)} x\, N(t,{\textrm{d}} x),$ where $N({\textrm{d}} s,{\textrm{d}} x)$ is a Poisson random measure with intensity measure $\mu$ . We define the functions $a,b\,{:}\,{\mathbb R}\to{\mathbb R}$ and $g\,{:}\,{\mathbb R}\times(0,1)\to{\mathbb R}$ via

\begin{align*}a(x)\,{:\!=}\, \sqrt{2x(1-x)},\quad b(x)\,{:\!=}\,\sigma x(1-x)+\theta\nu_0(1-x)-\theta\nu_1 x, \quad g(x,u)\,{:\!=}\, x(1-x)u,\end{align*}

for $x\in[0,1],$ $u\in(0,1)$ ; $a(x)\,{:\!=}\, 0$ , $g(x,u)\,{:\!=}\, 0$ for $x\notin[0,1]$ ; $b(x)\,{:\!=}\, \theta\nu_0$ for $x\lt 0$ and $b(x)\,{:\!=}\, -\theta\nu_1$ for $x>1$ . Thus, Equation (1.3) can be extended to the following SDE on ${\mathbb R}$ :

(3.16) \begin{equation} X(t)=X(0)+\int_0^t a(X(s)){\textrm{d}} B(s)+\int_0^t\int_{(0,1)}g(X(s-\!),u)N({\textrm{d}} s,{\textrm{d}} u)+\int_0^t b(X(s)){\textrm{d}} s.\end{equation}

Thus, any solution $(X(t))_{t\geq 0}$ of (3.16) such that $X(t)\in[0,1]$ for any $t\geq 0$ is a solution to (1.3), and vice versa. Note that the functions a, b, g are continuous. Moreover, $b=b_1- b_2$ , where

\begin{align*}b_1(x)\,{:\!=}\, \theta\nu_0+\sigma (x\wedge 1)_+\quad\text{and}\quad b_2(x)\,{:\!=}\, \theta(x\wedge 1)_+ +\sigma (x\wedge 1)_+^2. \end{align*}

In addition, $b_2$ is non-decreasing. Thus, in order to apply [Reference Li and Pu32, Theorems 3.2 and 5.1], we only need to verify the sufficient conditions (3.a), (3.b), and (5.a) therein. Condition (3.a) in our case amounts to proving that $x\mapsto b_1(x)+\int_{(0,1)}g(x,u)\mu({\textrm{d}} u)$ is Lipschitz continuous. In fact, a straightforward calculation shows that

\begin{align*}|b_1(x)-b_1(y)|+\int_{(0,1)}|g(x,u)-g(y,u)|\mu({\textrm{d}} u)\leq \left(\sigma+\int_{(0,1)}u\mu({\textrm{d}} u)\right)|x-y|,\quad x,y\in{\mathbb R},\end{align*}

and hence (3.a) follows. Condition (3.b) amounts to proving that $x\mapsto a(x)$ is $1/2$ -Hölder, which is already shown in the proof of [Reference González Casanova and Spanò19, Lemma 3.6]. Therefore, [Reference Li and Pu32, Theorem 3.2] yields the pathwise uniqueness for (3.16). Condition (5.a) follows from the fact that the functions a, b, $x\mapsto \int_{(0,1)} g(x,u)^2\mu({\textrm{d}} u)$ , and $x\mapsto \int_{(0,1)} g(x,u)\mu({\textrm{d}} u)$ are bounded on ${\mathbb R}$ . Hence, [Reference Li and Pu32, Theorem 5.1] ensures the existence of a strong solution to (3.16). It remains to show that any solution to (3.16) with $X(0)\in[0,1]$ is such that $X(t)\in[0,1]$ for any $t\in[0,1]$ . Sufficient conditions implying such a result are provided in [Reference Fu and Li17, Proposition 2.1]. The conditions on the diffusion and drift coefficients are satisfied; namely, a is 0 outside [0, 1], and b(x) is positive for $x\leq 0$ and negative for $x\geq 1$ . However, the condition on the jump coefficient, $x+g(x,u)\in[0,1]$ for every $x\in{\mathbb R}$ , is not fulfilled. Nevertheless, the proof of [Reference Fu and Li17, Proposition 2.1] works without modifications under the alternative condition $x+g(x,u)\in[0,1]$ for $x\in[0,1]$ and $g(x,u)=0$ for $x\notin[0,1]$ , which is in turn satisfied. This ends the proof.

Lemma 3.1. The solution to the SDE (1.3) is a Feller process with generator $\mathcal{A}$ satisfying, for all $f\in \mathcal{C}^2([0,1],{\mathbb R})$ ,

\begin{align*}\mathcal{A} f(x)=x(1-x)f''(x)+ (\sigma x(1-x)&+\theta\nu_0(1-x)-\theta\nu_1 x) f{'}(x)\\[3pt]&\quad +\int_{(0,1)}\left(f\left(x+ x(1-x)u\right)-f(x)\right)\mu({\textrm{d}} u).\end{align*}

Moreover, $\mathcal{C}^\infty([0,1],{\mathbb R})$ is an operator core for $\mathcal{A}$ .

Now we will prove the first part of the main result of Section 2.3, i.e. the annealed convergence of a sequence of Moran models towards the solution to the SDE (1.3).

Proof of Theorem 2.2(1) (annealed convergence.)Let $\mathcal{A}_N^*$ and $\mathcal{A}$ be the infinitesimal generators of the processes $(X_N(t))_{t\geq 0}$ and $(X(t))_{t\geq 0}$ , respectively. Note that $(X_N(t))_{t\geq 0}$ has state space

(3.17) \begin{equation} E_N\,{:\!=}\, \{k/N\,{:}\,k\in[N]_0\}.\end{equation}

We will prove that, for all $f\in\mathcal{C}^\infty([0,1],{\mathbb R})$ ,

(Claim 2) \begin{equation}\sup\limits_{x\in E_N}|\mathcal{A}_N^* f(x)-\mathcal{A} f(x)|\xrightarrow[N\to\infty]{} 0.\end{equation}

Provided Claim 2 is true, since X is Feller and $\mathcal{C}^\infty([0,1],{\mathbb R})$ is an operator core for $\mathcal{A}$ (see Lemma 3.1), the result follows from applying [Reference Kallenberg24, Theorem 17.25]. Thus, it remains to prove Claim 2. To this end, we write $\mathcal{A}$ as $\mathcal{A}^{1}+\mathcal{A}^{2}+\mathcal{A}^3+\mathcal{A}^4$ , where

\begin{align*}\mathcal{A}^1f(x) &\,{:\!=}\, x(1-x)f{''}(x),\\[3pt]\mathcal{A}^2f(x) &\,{:\!=}\, (\sigma x(1-x)+\theta\nu_0(1-x)-\theta\nu_1 x) f{'}(x),\\[3pt]\mathcal{A}^3f(x) &\,{:\!=}\, \int_{(0,\varepsilon_N)}\left(f\left(x+ x(1-x)u\right)-f(x)\right)\mu({\textrm{d}} u),\\[3pt]\mathcal{A}^4f(x) &\,{:\!=}\, \int_{(\varepsilon_N,1)}\left(f\left(x+ x(1-x)u\right)-f(x)\right)\mu({\textrm{d}} u).\end{align*}

We also write $\mathcal{A}_N^*=\mathcal{A}_N^{1}+\mathcal{A}_N^{2}+\mathcal{A}_N^3+\mathcal{A}_N^4$ , where

\begin{align*}\mathcal{A}_N^1f(x) &\,{:\!=}\, N^2x(1-x)\left[\Delta_{\frac1N}f(x)+\Delta_{-\frac1N}f(x)\right],\\[3pt]\mathcal{A}_N^2f(x) &\,{:\!=}\, N^2(\sigma_N x(1-x)+\theta_N\nu_0(1-x))\left[\Delta_{\frac1N}f(x)\right]+N^2\theta_N\nu_1x\left[\Delta_{-\frac1N}f(x)\right],\\[3pt]\mathcal{A}_N^3f(x) &\,{:\!=}\, \int_{(0,\varepsilon_N^{})}{\mathbb E}\left[f\left(x+\xi_N(x,u)\right)\right]-f(x)\mu({\textrm{d}} u),\\[3pt]\mathcal{A}_N^4f(x) &\,{:\!=}\, \int_{(\varepsilon_N^{},1)}{\mathbb E}\left[f\left(x+\xi_N(x,u)\right)\right]-f(x)\mu({\textrm{d}} u),\end{align*}

where $\Delta_hf(x)\,{:\!=}\, f(x+h)-f(x)$ , and $\xi_N(x,u)\,{:\!=}\, \mathcal{H}(N,N(1-x),B_{Nx}(u))/N$ , with $\mathcal{H}(N,N(1-x),k)\sim\textrm{Hyp}({N},{N(1-x)},{k})$ , and $B_{Nx}(u)\sim\textrm{Bin}({Nx},{u})$ being independent; $\varepsilon_N>0$ will be chosen later in an appropriate way.

Let $f\in\mathcal{C}^\infty([0,1],{\mathbb R})$ and note that

(3.18) \begin{equation} \sup_{x\in E_N} |\mathcal{A}_N^* f(x)-\mathcal{A} f(x)|\leq \sum_{i=1}^4 \sup_{x\in E_N} |\mathcal{A}_N^i f(x)-\mathcal{A}^i f(x)|.\end{equation}

Taylor expansions of order three around x for $f(x+1/N)$ and $f(x-1/N)$ yield

(3.19) \begin{equation} \sup\limits_{x\in E_N}|\mathcal{A}_N^1 f(x)-\mathcal{A}^1 f(x)|\leq \frac{\lVert f'''\rVert_\infty}{3N}.\end{equation}

Similarly, the triangle inequality and appropriate Taylor expansions of order two yield

(3.20) \begin{equation} \sup\limits_{x\in E_N}|\mathcal{A}_N^2 f(x)-\mathcal{A}^2 f(x)|\leq {\frac{(N\sigma_N+N\theta_N) \lVert f''\rVert_\infty}{2N}}+ (|\sigma-N\sigma_N|+|\theta-N\theta_N|)\lVert f'\rVert_{\infty}.\end{equation}

Since $N\sigma_N\to\sigma$ and $N\theta_N\to\theta$ , the right-hand side in (3.20) converges to 0 as $N\to\infty$ . In addition, since ${\mathbb E}[\xi_N(x,u)]=x(1-x)u$ , we have

\begin{align*}|\mathcal{A}^3_Nf(x)|\leq \lVert f'\rVert_\infty\int_{(0,\varepsilon_N)} u\mu({\textrm{d}} u),\quad x\in[0,1],\end{align*}

and hence,

(3.21) \begin{equation} \sup\limits_{x\in E_N}|\mathcal{A}_N^3 f(x)-\mathcal{A}^3 f(x)|\leq 2 ||f'||_\infty\int_{(0,\varepsilon_N^{})} u\mu({\textrm{d}} u).\end{equation}

For the last term, note first that

\begin{align*} \left|{\mathbb E}\left[f\!\left(x+\xi_N(x,u)\right)\right.\right. \left.\left.-f(x+x(1-x)u)\right]\right| \leq \lVert f'\rVert_{\infty}{\mathbb E}\left[\left|\xi_N(x,u)-x(1-x)u\right|\right]\\ \leq \lVert f'\rVert_{\infty}\sqrt{{\mathbb E}\left[\left(\xi_N(x,u)-x(1-x)u\right)^2\right]}\leq\lVert f'\rVert_{\infty} \sqrt{\frac{u}{N}},\end{align*}

where in the last inequality we used that

(3.22) \begin{equation} {\mathbb E}\left[\left(\xi_N(x,u)-x(1-x)u\right)^2\right]=\frac{x(1-x)^2u(1-u)}{N}+\frac{Nx^2(1-x)u^2}{N^2(N-1)}\leq \frac{u}{N},\end{equation}

which is obtained from standard properties of the hypergeometric and binomial distributions. Hence,

(3.23) \begin{equation} \sup\limits_{x\in E_N}|\mathcal{A}_N^4 f(x)-\mathcal{A}^4 f(x)|\leq ||f'||_\infty\int_{(\varepsilon_N^{},1)} \sqrt{\frac{u}{N}}\,\mu(du)\leq \frac{||f'||_\infty}{\sqrt{N\varepsilon_N}}\int_{(\varepsilon_N^{},1)} u\,\mu({\textrm{d}} u) .\end{equation}

Now, choose $\varepsilon_N\,{:\!=}\, 1/\sqrt{N}$ . Since $\int_{(0,1)} u\mu({\textrm{d}} u)<\infty$ , Claim 2 follows from plugging (3.19), (3.20), (3.21), and (3.23) into (3.18) and letting $N\to\infty$ .

Before embarking on the proof of the second part of Theorem 2.2, we prove the following estimates for the Moran model with null environment.

Lemma 3.2. For any $x\in E_N$ (see (3.17) for its definition) and $t\geq 0$ , we have

\begin{align*}{\mathbb E}\left[\left(X_N^{{\textbf{0}}}(t)-x\right)^2\mid X_N^{{\textbf{0}}}(0)=x\right]\leq \left(\frac12 + N(\sigma_N+3\theta_N)\right)t,\end{align*}

and

\begin{align*}-N\theta_N\nu_1 t\leq {\mathbb E}\left[X_N^{\textbf{0}}(t)-x\mid X_N^{{\textbf{0}}}(0)=x\right]\leq N(\sigma_N+\theta_N\nu_0) t.\end{align*}

Proof. Fix $x\in E_N$ and consider the functions $f_x,g_x\,{:}\,E_N\to[0,1]$ defined via $f_x(z)\,{:\!=}\,$ $(z-x)^2$ and $g_x(z)\,{:\!=}\, z-x$ , $z\in E_N$ . The process $X_N^{\textbf{0}}$ is a Markov chain with generator $\mathcal{A}_N^{\star,0}\,{:\!=}\, \mathcal{A}_N^1+\mathcal{A}_N^2$ , where $\mathcal{A}_N^1$ and $\mathcal{A}_N^2$ are defined in the proof of Theorem 2.2. Moreover, for every $z\in E_N$ , we have

\begin{align*} \mathcal{A}_N^{\star,0} f_x(z) & =2z(1-z)\\[3pt] &\quad +N\left[(\sigma_N z+\theta_N \nu_0)(1-z)\left(2(z-x)+\frac1N\right)+\theta_N\nu_1 z\left(2(x-z)+\frac1N\right)\right]\\[3pt] &\quad \leq \frac12+N\left[3\left(\frac{\sigma_N}{4}+\theta_N \nu_0\right)+3\theta_N\nu_1\right]\leq \frac12 + N(\sigma_N+3\theta_N), \end{align*}

and

\begin{align*} \mathcal{A}_N^{\star,0} g_x(z) =N\left[(\sigma_N z+\theta_N \nu_0)(1-z)-\theta_N\nu_1 z\right]\in[\!-N\theta_N\nu_1,N(\sigma_N+\theta_N\nu_0)]. \end{align*}

Hence, Dynkin’s formula applied to $X_N^0$ with $f_x$ leads to

\begin{align*} {\mathbb E}\left[\left(X_N^{\textbf{0}}(t)-x\right)^2\mid X_N^{{\textbf{0}}}(0)=x\right] &=\int_0^t {\mathbb E}\left[\mathcal{A}_N^{\star,0} f_x(X_N^{\textbf{0}}(s))\mid X_N^{{\textbf{0}}}(0)=x\right]{\textrm{d}} s\\ &\leq \left(\frac12 + N(\sigma_N+3\theta_N)\right)t. \end{align*}

Similarly, applying Dynkin’s formula to $X_N^0$ with $g_x$ , we obtain

\begin{align*} {\mathbb E}\left[X_N^{\textbf{0}}(t)-x\mid X_N^{{\textbf{0}}}(0)=x\right] &=\int_0^t {\mathbb E}\left[\mathcal{A}_N^{\star,0} g_x(X_N^{\textbf{0}}(s))\mid X_N^{{\textbf{0}}}(0)=x\right]{\textrm{d}} s\\[3pt]&\in[\!-N\theta_N\nu_1 t,N(\sigma_N+\theta_N\nu_0)t], \end{align*}

which ends the proof.

Since, for all $t\geq 0$ and $N\geq 1$ , $X_N^\omega(t)\in E_N \subset [0,1]$ (see (3.17) for the definition of $E_N$ ), Condition (A1) is satisfied. Now, we claim that there are constants $c,C>0$ , independent of N, such that

(Claim 3) \begin{equation}{\mathbb E}\left[(X_N^\omega(t)-X_N^\omega(s))^2\mid \mathcal{F}_s^N\right]\leq c\sum_{u\in[s,t]}\Delta \omega(u)+ C (t-s),\quad\textrm{for all $0\leq s\leq t$}.\end{equation}

If Claim 3 is true, then Condition (A2) is satisfied with $\alpha=2$ and $F_N(t)=F(t)=c\sum_{u\in[0,t]}\Delta \omega(u)+Ct$ , and the result follows from [Reference Bansaye, Kurtz and Simatos5, Theorem 1]. The rest of the proof is devoted to proving Claim 3.

For $x\in E_N$ and $t\geq 0$ , we set $\psi_x(\omega,t)\,{:\!=}\, {\mathbb E}_x[(X_N^\omega(t)-x)^2].$ For $s\geq 0$ , we set $\omega_s(\!\cdot\!)\,{:\!=}\,$ $\omega(s+\cdot\!)$ . From the definition of $X_N^\omega$ , it follows that for any $0\leq s\lt t$ ,

(3.24) \begin{equation} {\mathbb E}\left[(X_N^\omega(t)-X_N^\omega(s))^2\Big| \mathcal{F}_s^N\right]=\psi_{X_N^\omega(s)}(\omega_s,t-s).\end{equation}

Let $0\leq s\lt t$ . We split the proof of Claim 3 into three cases.

Case 2: $\omega$ has n jumps in (s, t]. Let $t_1,\ldots,t_n\in(s,t]$ be the jump times of $\omega$ in (s, t] in increasing order. We set $t_0\,{:\!=}\, s$ and $t_{n+1}=t$ . For any $i\in[n+1]$ and any $r\in(t_{i-1},t_{i})$ , $\omega$ has no jumps in $(t_{i-1},r]$ . In particular, $(t_{i-1},r]$ falls into Case 1. Using Claim 3 in $(t_{i-1},r]$ and letting $r\to t_i$ , we obtain

\begin{align*}{\mathbb E}\left[(X_N^\omega(t_i -\!)-X_N^\omega(t_{i-1}))^2\mid \mathcal{F}_{t_{i-1}}^N\right]\leq C_1(t_i-t_{i-1}).\end{align*}

Moreover,

\begin{align*}{\mathbb E}\left[(X_N^\omega(t_i)-X_N^\omega(t_{i}-\!))^2\mid \mathcal{F}_{t_{i}-}^N\right]\leq {\mathbb E}\left[\left(\frac{B_{N}(\Delta \omega(t_i))}{N}\right)^2\right]\leq \Delta \omega(t_i),\end{align*}

where $B_{N}(\Delta \omega(t_i)) \sim \textrm{Bin}({N},{\Delta \omega(t_i)})$ . Using the two previous inequalities and the tower property of the conditional expectation, we get

(3.25) \begin{equation} {\mathbb E}\left[(X_N^\omega(t_i)-X_N^\omega(t_{i-1}))^2\mid \mathcal{F}_{s}^N\right]\leq 2C_1(t_i-t_{i-1})+2\Delta\omega(t_i).\end{equation}

Now, note that

\begin{align*} (X_N^\omega(t)-X_N^\omega(s))^2&=\left(\sum_{i=0}^n( X_N^\omega(t_{i+1})-X_N^\omega(t_i))\right)^2 \\&=\sum_{i=0}^n( X_N^\omega(t_{i+1})-X_N^\omega(t_i))^2+2\sum_{i=0}^n( X_N^\omega(t_{i+1})-X_N^\omega(t_i))(X_N^\omega(t_i)-X_N^\omega(s)).\end{align*}

Using Equation (3.25), we see that

\begin{align*}{\mathbb E}\left[\sum_{i=0}^n(X_N^\omega(t_{i+1})-X_N^\omega(t_{i}))^2\mid \mathcal{F}_{s}^N\right]\leq 2C_1(t-s)+2\sum_{i=1}^n\Delta\omega(t_i).\end{align*}

Moreover, we have

\begin{align*} {\mathbb E}\left[(X_N^\omega(t_{i+1})-X_N^\omega(t_{i}))(X_N^\omega(t_{i})-X_N^\omega(s))\mid \mathcal{F}_{t_i}^N\right]=\varphi_{X_N^\omega(s),X_N^\omega(t_i)}(\omega_{t_i},t_{i+1}-t_{i}),\end{align*}

where for $x,y\in E_N$ and $t\geq 0$ we set $\varphi_{x,y}(\omega,t)\,{:\!=}\, (y-x){{\mathbb E}[X_N^\omega(t)-y \mid X_N^\omega(0)=y]}$ . Since, for any $r\in(t_i,t_{i+1})$ , $\omega_{t_i}$ has no jumps in $(0,r-t_i]$ , we can use Lemma 3.2 to infer that for any $x,y\in E_N$ ,

\begin{align*}\varphi_{x,y}(\omega_{t_i}, r-t_i)\leq N((\sigma_N+\theta_N\nu_0)\vee\theta_N\nu_1)(r-t_i).\end{align*}

Note that $(y-x){\mathbb E}_y[X_N^{\omega_{t_i}}(t_{i+1}-t_i)-X_N^{\omega_{t_i}}((t_{i+1}-t_i)-\!)]\leq \Delta \omega(t_{i+1})$ . Hence, letting $r\to t_{i+1}$ , we get

\begin{align*}\varphi_{x,y}(\omega_{t_i}, t_{i+1}-t_i)\leq N((\sigma_N+\theta_N\nu_0)\vee\theta_N\nu_1)(t_{i+1}-t_i)+\Delta\omega(t_{i+1}),\end{align*}

for any $x,y\in E_N$ , hence in particular for $x=X_N^\omega(s)$ and $y=X_N^\omega(t_i)$ . Altogether, we obtain

\begin{align*}{\mathbb E}\left[(X_N^\omega(t)-X_N^\omega(s))^2\mid \mathcal{F}_{s}^N\right]\leq C_2(t-s)+3\sum_{i=1}^n\Delta\omega(t_i),\end{align*}

where $C_2\,{:\!=}\, 2C_1+\sup_{N\in{\mathbb N}}N((\sigma_N+\theta_N\nu_0)\vee\theta_N\nu_1)$ . Hence, Claim 3 holds for any $C\geq C_2$ and $c\geq 3$ .

Case 3: $\omega$ has infinitely many jumps in (s, t]. For any $\delta$ , we consider $\omega^\delta$ as in (3.1) and couple the processes $X_N^\omega$ and $X_N^{\omega^\delta}$ as in the proof of Proposition 3.1. Note that $\omega^\delta$ has only a finite number of jumps in any compact interval; thus $\omega^\delta$ falls into Case 2. Moreover, we have

\begin{align*} \psi_x(\omega,t) &\leq 2{\mathbb E}_x[(X_N^\omega(t)-X_N^{\omega^\delta}(t))^2]+2{\mathbb E}_x\left[(X_N^{\omega^\delta}(t)-x)^2\right]\\[3pt] & \leq 2{\mathbb E}_x[|X_N^\omega(t)-X_N^{\omega^\delta}(t)|]+2{\mathbb E}_x\left[(X_N^{\omega^\delta}(t)-x)^2\right]\\[3pt] & \leq 2e^{N\sigma_N t+\omega(t)} \sum_{\substack{u\in[0,t]\\ \Delta \omega(u)<\delta}}\!\!\Delta \omega(u)+2{\mathbb E}_x\left[(X_N^{\omega^\delta}(t)-x)^2\right],\end{align*}

where in the last inequality we used Proposition 3.1. Now, using Claim 3 for $X_N^{\omega^\delta}$ and the previous inequality, we obtain

\begin{align*}{\mathbb E}\left[(X_N^\omega(t)-X_N^\omega(s))^2\mid \mathcal{F}_{s}^N\right]\leq e^{N\sigma_N (t-s)+\omega(t-s)}\!\!\!\!\sum_{\substack{u\in[s,t]\\ \Delta \omega(u)<\delta}}\!\!\!\!\!\Delta \omega(u)+2C_2(t-s)+6\!\!\sum_{u\in[s,t]}\Delta\omega(u).\end{align*}

We let $\delta\to 0$ and conclude that Claim 3 holds for any $C\geq 2C_2$ and $c\geq 6$ .

Now we proceed to prove the quenched convergence of the sequence of Moran models to the Wright–Fisher diffusion, under the assumption that the environment is simple.

Proof of Theorem 2.2(2) (quenched convergence).Let $B\,{:\!=}\, (B(t))_{t\geq 0}$ be a standard Brownian motion. We denote by $X^{\textbf{0}}$ the unique strong solution to (1.3) associated to B and the null environment. Theorem 2.2(1) implies in particular that $X_N^{\textbf{0}}$ converges to $X^{\textbf{0}}$ as $N\to\infty$ .

Now, assume that $\omega\neq{\textbf{0}}$ is simple. We denote by $T_\omega$ the (discrete but possibly infinite) set of jump times of $\omega$ in $(0,\infty)$ . Moreover, for $0<i\lt |T_\omega|+1$ , we denote by $t_i\,{:\!=}\, t_i(\omega)\in T_\omega$ the time of the ith jump of $\omega$ . We set $t_0 \,{:\!=}\, 0$ and $t_{|T_\omega|+1} \,{:\!=}\, \infty$ . We need to prove that

\begin{align*}(X_N^\omega(t))_{t\geq 0}\xrightarrow[N\to\infty]{(d)} (X^\omega(t))_{t\geq 0}.\end{align*}

Recall that the process $X^\omega$ starting at $x_0$ can be constructed as follows:

1. (i) (i) $X^\omega(0)=x_0$ .

2. (ii) For $i\in{\mathbb N}$ with $i\leq |T_\omega|+1$ , the restriction of $X^{\omega}$ to the interval $(t_{i-1},t_i)$ is given by a version of $X^{\textbf{0}}$ started at $X^\omega(t_{i-1})$ .

3. (iii) For $0<i\lt |T_\omega|+1$ , conditionally on $X^\omega(t_i-\!)$ ,

   \begin{align*}X^\omega(t_i)=X^\omega(t_i-\!)+X^\omega(t_i-\!)(1-X^\omega(t_i-\!))\Delta\omega(t_i).\end{align*}

Since the sequence $(X_N^\omega)_{N\in{\mathbb N}}$ is tight (see Proposition 3.4), it is enough to prove the convergence at the level of the finite-dimensional distributions. More precisely, we will prove by induction on $i\in{\mathbb N}$ with $i\leq |T_\omega|+1$ that for any finite set $I\subset[0,t_i)$ , we have

\begin{align*}((X_N^\omega(t))_{t\in I},X_N^\omega(t_i-\!))\xrightarrow[N\to\infty]{(d)} ((X^\omega(t))_{t\in I},X^\omega(t_i-\!)).\end{align*}

For $i=|T_\omega|+1<\infty$ we remove the components $X_N^\omega(t_i-\!)$ and $X^\omega(t_i-\!)$ since they do not make sense. Since $X_N^\omega(t_1-\!)=X_N^{\textbf{0}}(t_1)$ and $X^\omega(t_1-\!)=X^{\textbf{0}}(t_1)$ almost surely, the result for $i=1$ follows from Theorem 2.2(1). Now, assume that the result is true for some $i<|T_\omega|+1$ and let $I\subset(0,t_{i+1})$ . Without loss of generality we assume that $I=\{s_1,\ldots,s_k, t_i,r_1,\ldots, r_m\}$ , with $s_1<\cdots<s_k<t_i<r_1<\cdots<r_m$ . We also assume that $i<|T_\omega|$ ; the other case, i.e. $i=|T_\omega|<\infty$ , follows by an analogous argument.

Let $F\,{:}\,[0,1]^{k+1}\to{\mathbb R}$ be a Lipschitz function with $\lVert F\rVert_{\textrm{BL}}\leq 1$ (see (A.5) for the definition of $\lVert \cdot\rVert_{\textrm{BL}}$ ). Note that

\begin{align*} {\mathbb E}\left[F\left(\left(X_N^\omega\left(s_j\right)\right)_{j=1}^k,X_N^\omega(t_i)\right)\right] ={\mathbb E}\left[F(\left(X_N^\omega\left(s_j\right)\right)_{j=1}^k,X_N^\omega(t_i-\!)+\xi_N(X_N^\omega(t_i-\!),\Delta\omega(t_i)))\right],\end{align*}

where for $x\in E_N$ (see (3.17) for the definition of $E_N$ ) and $u\in(0,1)$ , we let $\xi_N(x,u)\,{:\!=}\, \mathcal{H}(N,N(1-x),B_{Nx}(u))/N$ with $\mathcal{H}(N,N(1-x),k)\sim\textrm{Hyp}({N},{N(1-x)},{k})$ , $k\in[N]_0$ , and $B_{Nx}(u)\sim\textrm{Bin}({Nx},{u})$ being independent of each other and independent of $X_N^\omega$ . Now, set

\begin{align*} D_N &\,{:\!=}\, {\mathbb E}\left[F(\left(X_N^\omega\left(s_j\right)\right)_{j=1}^k,X_N^\omega(t_i-\!)+\xi_N(X_N^\omega(t_i-\!),\Delta\omega(t_i)))\right]\\ &\quad -{\mathbb E}\left[F(\left(X_N^\omega\left(s_j\right)\right)_{j=1}^k,X_N^\omega(t_i-\!)+X_N^\omega(t_i-\!)(1-X_N^\omega(t_i-\!))\Delta\omega(t_i))\right].\end{align*}

Using that $\lVert F\rVert_{\textrm{BL}}\leq 1$ and (3.22), we see that $|D_N| \leq\sqrt{\Delta\omega(t_i)/N}\to 0$ as $N\to\infty$ . Therefore, the induction hypothesis yields

\begin{align*} &{\mathbb E} \left[F\left(\left(X_N^\omega\left(s_j\right)\right)_{j=1}^k,X_N^\omega(t_i)\right)\right]\\ &\quad=D_N+{\mathbb E}\left[F(\left(X_N^\omega\left(s_j\right)\right)_{j=1}^k,X_N^\omega(t_i-\!)+X_N^\omega(t_i-\!)(1-X_N^\omega(t_i-\!))\Delta\omega(t_i))\right]\\ &\quad\xrightarrow[N\to\infty]{}\,{\mathbb E}\left[F((X^\omega\left(s_j\right))_{j=1}^k,X^\omega(t_i-\!)+X^\omega(t_i-\!)(1-X^\omega(t_i-\!))\Delta\omega(t_i))\right]\\ &\quad={\mathbb E}\left[F\left(\left(X^\omega\left(s_j\right)\right)_{j=1}^k,X^\omega(t_i)\right)\right].\end{align*}

Therefore,

(3.26) \begin{equation}\left(\left(X_N^\omega\left(s_j\right)\right)_{j=1}^k,X_N^\omega(t_i)\right)\xrightarrow[N\to\infty]{(d)} \left(\left(X^\omega\left(s_j\right)\right)_{j=1}^k,X^\omega(t_i)\right).\end{equation}

Let $G\,{:}\,[0,1]^{k+m+2}\to{\mathbb R}$ be a Lipschitz function with $\lVert G\rVert_{\textrm{BL}}\leq 1$ . For $x\in E_N$ , define

\begin{align*}H_N(z,x)\,{:\!=}\, {\mathbb E}_x[G(z,x,(X_N^{\textbf{0}}(r_j-t_i))_{j=1}^m,X_N^{\textbf{0}}(t_{i+1}-t_i))], \ \forall z \in{\mathbb R}^k.\end{align*}

Note that

(3.27) \begin{equation} {\mathbb E}[G(\left(X_N^\omega\left(s_j\right)\right)_{j=1}^k,X_N^\omega(t_i),(X_N^\omega(r_j))_{j=1}^m,X_N^\omega(t_{i+1}-\!))]={\mathbb E}\left[H_N\left(\left(X_N^\omega\left(s_j\right)\right)_{j=1}^k,X_N^\omega(t_i)\right)\right].\end{equation}

Similarly, for $x\in[0,1]$ , define

\begin{align*}H(z,x)\,{:\!=}\, {\mathbb E}_x[G(z,x,(X^{\textbf{0}}(r_j-t_i))_{j=1}^m,X^{\textbf{0}}(t_{i+1}-t_i))],\quad z\in{\mathbb R}^k,\end{align*}

and note that

(3.28) \begin{equation} {\mathbb E}[G((X^\omega\left(s_j\right))_{j=1}^k,X^\omega(t_i),(X^\omega(r_j))_{j=1}^m,X^\omega(t_{i+1}-\!))]={\mathbb E}\left[H\left(\left(X^\omega\left(s_j\right)\right)_{j=1}^k,X^\omega(t_i)\right)\right].\end{equation}

Using (3.26) and the Skorokhod representation theorem, we can assume that the random variables $\left(\left(X_N^\omega\left(s_j\right)\right)_{j=1}^k,X_N^\omega(t_i)\right)_{N\geq1}$ and $\left(\left(X^\omega\left(s_j\right)\right)_{j=1}^k,X^\omega(t_i)\right)$ are defined on the same probability space and that the convergence holds almost surely. In particular, we can write

(3.29) \begin{equation} \left|{\mathbb E}\left[H_N\left(\left(X_N^\omega\left(s_j\right)\right)_{j=1}^k,X_N^\omega(t_i)\right)\right]-{\mathbb E}\left[H\left(\left(X^\omega\left(s_j\right)\right)_{j=1}^k,X^\omega(t_i)\right)\right]\right|\leq R_N^1+R_N^2,\end{equation}

where

\begin{align*}R_N^1 &\,{:\!=}\, \left|{\mathbb E}\left[H_N\left(\left(X_N^\omega\left(s_j\right)\right)_{j=1}^k,X_N^\omega(t_i)\right)\right]-{\mathbb E}\left[H_N((X^\omega\left(s_j\right))_{j=1}^k,X_N^\omega(t_i))\right]\right|,\\R_N^2 &\,{:\!=}\, \left|{\mathbb E}\left[H_N((X^\omega\left(s_j\right))_{j=1}^k,X_N^\omega(t_i))\right]-{\mathbb E}\left[H\left(\left(X^\omega\left(s_j\right)\right)_{j=1}^k,X^\omega(t_i)\right)\right]\right|.\end{align*}

Using that $\lVert G\rVert_{\textrm{BL}}\leq 1$ , we obtain

(3.30) \begin{equation} R_N^1\leq \sum_{j=1}^k{\mathbb E}[|X_N^\omega\left(s_j\right)-X^\omega\left(s_j\right)|]\xrightarrow[N\to\infty]{}0.\end{equation}

Moreover, since $X_N^\omega(t_i)$ converges to $X^\omega(t_i)$ almost surely, we conclude using Theorem 2.2 that, for any $z\in[0,1]^k$ , $H_N(z,X_N^\omega(t_i))$ converges to $H(z,X^\omega(t_i))$ almost surely. Therefore, using the dominated convergence theorem, we conclude that

(3.31) \begin{equation} R_N^2\xrightarrow[N\to\infty]{}0.\end{equation}

Plugging (3.30) and (3.31) into (3.29) and using (3.27) and (3.28) yields the result.

4.1. Results related to Section 2.4: the quenched ASG

The aim of this section is to prove the following result.

Proof. We will explicitly construct a branching–coalescing particle system $(\mathcal{G}_T^\omega(\beta))_{\beta\geq 0}$ with the desired properties. The main difficulty is that the environment $\omega$ may have infinitely many jumps on each compact interval. Fix $T \gt 0$ and $n \in {\mathbb N}$ (sampling size) and define

\begin{align*}\Lambda_{\textrm{mut}}\,{:\!=}\,\left\{\lambda_{i}^{0},\lambda_{i}^{1} \right\}_{i \geq 1}, \qquad \Lambda_{\textrm{sel}}\,{:\!=}\,\left\{\lambda_{i}^{\vartriangle}\right\}_{i \geq 1}, \qquad \Lambda_{\textrm{coal}}\,{:\!=}\,\left\{\lambda_{i,j}^{\blacktriangle} \right\}_{i,j \geq 1, i \neq j},\end{align*}

where $\lambda_{i}^{0}$ , $\lambda_{i}^{1}$ , $\lambda_{i}^{\vartriangle}$ , and $\lambda_{i,j}^{\blacktriangle}$ are independent Poisson processes on [0, T] with parameters $\theta \nu_0$ , $\theta \nu_1$ , $\sigma$ , and 1, respectively. For $\beta \in [0,T]$ , let $\tilde \omega (\beta) \,{:\!=}\, \omega (T) - \omega ((T-\beta)-\!)$ and $I_{\tilde \omega} \,{:\!=}\, \{ \beta \in [0,T]\,{:}\, \Delta \tilde \omega (\beta) \gt 0 \}$ ; $I_{\tilde \omega}$ is the countable set of jump times of $\tilde \omega$ . Let $\mathcal{U}_{\tilde \omega} \,{:\!=}\, \{ U_i(\beta) \}_{i \geq 1, \beta \in I_{\tilde \omega}}$ be an i.i.d. family of uniform random variables on (0, 1). Assume, without loss of generality, that the arrival times of $\lambda_i^0$ , $\lambda_i^1$ , $\lambda_i^{\vartriangle}$ , and $\lambda_{i,j}^{\blacktriangle}$ , $i,j\in{\mathbb N}$ , $i\neq j$ , are countable, distinct from each other, and distinct from the jump times of $\tilde \omega$ . Let $I_{\textrm{coal}}$ (resp. $I_{\textrm{sel}}$ ) be the set of arrival times of $\Lambda_{\textrm{coal}}$ (resp. $\Lambda_{\textrm{sel}}$ ).

We first construct a set $\mathcal{V}^\omega\subset {\mathbb N} \times [0,T]$ of virtual lines, representing the lines that would be part of the ASG if there were no coalescences. In particular, once a line enters this set, it will remain there. The set $\mathcal{V}^\omega$ is constructed on the basis of the set of potential branching times $I_{\textrm{bran}}\,{:\!=}\, I_{\tilde{\omega}}\cup I_{\textrm{sel}}$ as follows. Consider the (countable) set

\begin{align*}S_{\textrm{bran}}\,{:\!=}\,\{(\beta_1,\ldots,\beta_k)\,{:}\,k\in{\mathbb N},\, 0\leq \beta_1<\cdots<\beta_k, \beta_i\in I_{\textrm{bran}}, i\in[k]\},\end{align*}

and fix an injective function $i_\star\,{:}\,[n]\times S_{\textrm{bran}}\to{\mathbb N}\setminus[n]$ . The set $\mathcal{V}^\omega$ is determined as follows:

1. 1. For any $i\in[n]$ (i.e. in the initial sample) and $\beta\in[0,T]$ : $(i,\beta)\in\mathcal{V}^\omega$ .

2. 2. For any $(\beta_1,\ldots,\beta_k)\in S_{\textrm{bran}}$ , $j\in[n]$ , and $\beta\in[\beta_k,T]$ : $(i_\star(j,\beta_1,\ldots,\beta_k),\beta)\in\mathcal{V}^\omega$ if and only if

   • • for any $\ell\in[k]$ with $\beta_\ell\in I_{\tilde{\omega}}$ , $U_{i_\star(j,\beta_1,\ldots,\beta_{\ell-1})}(\beta_\ell)\leq \Delta \tilde \omega(\beta_\ell)$ (or $U_{j}(\beta_1) \leq \Delta \tilde \omega(\beta_1)$ if $\ell=1$ ),

   • • for any $\ell\in[k]$ such that $\beta_\ell\in I_{\textrm{sel}}$ , $\beta_\ell$ is a jump time of $\lambda_{i_\star(j,\beta_1,\ldots,\beta_{\ell-1})}^{\vartriangle}$ (or of $\lambda_{j}^{\vartriangle}$ if $\ell=1$ ),

and these are all possible virtual lines; see Figure 7.

Let $V^\omega(\beta)\,{:\!=}\, \{i\in{\mathbb N}\,{:}\, (i,\beta)\in\mathcal{V}^\omega\}$ . According to Lemma 4.1 below, $V^\omega(\beta)$ is almost surely finite for all $\beta \in [0,T]$ . Now, for $\beta \in I_{\textrm{coal}}$ , let $(a_{\beta},b_{\beta})$ be the pair (i, j) such that $\beta$ is an arrival time of $\lambda_{i,j}^{\blacktriangle}$ . Since the Poisson processes $\lambda_{i,j}^{\blacktriangle}$ , $i\neq j$ , have distinct jump times, $(a_{\beta},b_{\beta})$ is uniquely defined. Let

\begin{align*}\tilde I_{\textrm{coal}}\,{:\!=}\, \{ \beta \in I_{\textrm{coal}} \,{:}\, a_{\beta},b_{\beta} \in V^\omega(\beta) \}\quad\textrm{and}\quad\tilde I_{\textrm{bran}}\,{:\!=}\, \{ \beta \in I_{\textrm{bran}} \,{:}\, V^\omega(\beta-\!)\subsetneq V^\omega(\beta) \}.\end{align*}

Since $V^\omega(T)$ is independent of $\Lambda_{\textrm{coal}}$ and almost surely finite, it follows that $\tilde I_{\textrm{coal}}$ and $\tilde I_{\textrm{bran}}$ are almost surely finite. Let $\beta_1 \lt \cdots \lt \beta_m$ be the elements of $\tilde I_{\textrm{coal}} \cup \tilde I_{\textrm{bran}}$ (set $\beta_{0} \,{:\!=}\, 0$ and $\beta_{m+1} \,{:\!=}\, T$ for convenience). We define $V_{\text{on}}^\omega(\beta)\subset V^\omega(\beta)$ , the set of active lines at time $\beta$ , as follows (see also Figure 7). For $\beta=0$ we set $V_{\text{on}}^\omega(0)\,{:\!=}\, V^\omega(0)$ , and for $\beta\in(\beta_\ell,\beta_{\ell+1})$ we set $V_{\text{on}}^\omega(\beta)\,{:\!=}\, V_{\text{on}}^\omega(\beta_\ell)$ . For $\beta=\beta_\ell\in \tilde I_{\textrm{coal}}$ , we set $V_{\text{on}}^\omega(\beta_\ell)\,{:\!=}\, V_{\text{on}}^\omega(\beta_\ell-\!)\setminus \{ a_{\beta_\ell} \}$ if $\{a_{\beta_\ell},b_{\beta_\ell}\}\subset V_{\text{on}}^\omega(\beta_\ell-\!)$ , and $V_{\text{on}}^\omega(\beta_\ell)\,{:\!=}\, V_{\text{on}}^\omega(\beta_\ell-\!)$ otherwise. Finally, for $\beta=\beta_\ell\in \tilde I_{\textrm{bran}}$ , we set $V_{\text{on}}^\omega(\beta_\ell)\,{:\!=}\, V_{\text{on}}^\omega(\beta_\ell-\!) \cup J_\ell$ , where the set $J_\ell$ consists of the integers $i \in V^\omega(\beta_\ell) \setminus$ $V^\omega(\beta_\ell\!-\!)$ such that $i=i_\star(j,\beta_\ell)$ for some $j\in [n]\cap V_{\text{on}}^\omega(\beta_\ell-\!)$ , or $i=i_\star(j,\hat \beta_1,\ldots,\hat \beta_k,\beta_\ell)$ for some $(j,\hat \beta_{1},\ldots,\hat \beta_{k})\in i_\star^{-1}(V_{\text{on}}^\omega(\beta_\ell-\!)\setminus [n])$ with $\hat{\beta}_k<\beta_\ell$ .

Figure 7. Illustration of the construction of the quenched ASG. The environment $\omega$ has jumps at forward times $t_0$ , $t_1$ , $t_2$ ; backward times $\beta_1,\ldots,\beta_5$ belong to the set of potential branching times $\tilde I_{{\text{bran}}}$ . Virtual lines are depicted in grey or black; active lines are black. The ASG in [0, T] consists of the set of active lines together with their connections and mutation marks.

The ASG on [0, T] is then the branching–coalescing system starting with n lines at levels in [n], consisting at any time $\beta\in[0,T]$ of the lines in $V_{\text{on}}^\omega(\beta)$ , and where the following hold:

1. (i) For $\beta \in I_{\textrm{bran}}$ such that $V_{\text{on}}^\omega(\beta-\!)\subsetneq V_{\text{on}}^\omega(\beta)$ and $i\in V_{\text{on}}^\omega(\beta)\setminus V_{\text{on}}^\omega(\beta-\!)$ , either there is $(j,\hat \beta_{1},\ldots,\hat \beta_{k})\in[n]\times S_{\textrm{bran}}$ with $\hat \beta_{k}<\beta$ such that $i=i_\star(j,\hat \beta_1,\ldots,\hat \beta_k,\beta)$ , or there is $j\in[n]$ such that $i=i_\star(j,\beta)$ . In the first case, line $i_\star(j,\hat \beta_1,\ldots,\hat \beta_k)$ branches at time $\beta$ into $i_\star(j,\hat \beta_1,\ldots,\hat \beta_k)$ (continuing line) and i (incoming line). In the second case, line j branches at time $\beta$ into j (continuing line) and i (incoming line).

2. (ii) For $\beta \in I_{\textrm{coal}}$ such that $V_{\text{on}}^\omega(\beta)\subsetneq V_{\text{on}}^\omega(\beta-\!)$ and $i\in V_{\text{on}}^\omega(\beta-\!)\setminus V_{\text{on}}^\omega(\beta)$ , $i=a_\beta$ and $b_{\beta}\in V_{\text{on}}^\omega(\beta)$ . Thus, lines i and $b_\beta$ merge into $b_\beta$ at time $\beta$ .

3. (iii) At each $\beta \in [0,T]$ that is an arrival time of $\lambda_{i}^{0}$ (resp. $\lambda_{i}^{1}\big)$ for some $i \in V_{\text{on}}^\omega(\beta)$ , we mark line i with a beneficial (resp. deleterious) mutation at time $\beta$ .

Clearly the branching–coalescing particle system thus constructed satisfies the requirements (i), (ii), (iii $^{\prime}$ ), (iv), and (v) of Definition 2.1. This ends the proof.