(or ‘potential parents’), as they are situated at the locations from which the
parental alleles are sampled. We choose to call them offspring in order to
emphasize that the dual process plays the same role as ternary branching
Brownian motion in the deterministic setting, and, indeed, much of the proof
of Theorem 2.10 carries over with minimal changes to the SLFVS setting.

Just as for the deterministic setting, the duality relation that we exploit is
between the SLFVS and the historical process of branching and coalescing
lineages,

$\Xi^{n}(t):=(\mathcal{P}^{n}_{s})_{0\leq s\leq t}.$

We write $\mathbb{P}_{x}$ for the law of $\Xi^{n}$ when $\mathcal{P}^{n}_{0}$
is the single point $x$, and $\mathbb{E}_{x}$ for the corresponding
expectation.

Just as for the branching process, we can use Ulam-Harris labels to define
lines of descent from the root individual at $x$ through $\mathcal{P}^{n}$.
More precisely, for
$\boldsymbol{i}=(i_{1},i_{2},\ldots)\in\\{1,2,3\\}^{\mathbb{N}}$, we write
$(\xi^{n}_{\boldsymbol{i}}(\cdot))_{0\leq s\leq t}\subseteq\Xi^{n}(t)$ for the
$\mathbb{R}^{\mathbbm{d}}$-valued path which jumps to the location of the
(unique) offspring when the individual in $\mathcal{P}^{n}_{s}$ at its
location is affected by a neutral event, and to the location of the $i_{k}$th
offspring, the $k$th time that it is affected by a selective event. From the
perspective of the SLFVS, $\xi_{\boldsymbol{i}}^{n}$ is an ancestral lineage,
and we shall use the terminology lineage below.

Let $p:\mathbb{R}^{\mathbbm{d}}\to[0,1]$. The voting procedure on $\Xi^{n}(t)$
is a natural modification of the one that we defined for the ternary branching
Brownian motion:

  1. 1.

Each leaf of $\Xi^{n}(t)$ independently votes $1$ with probability
$p(\xi_{i}(t))$, and $0$ otherwise;

  2. 2.

at each neutral event in $\overleftarrow{\Pi}^{n}$, all marked individuals
adopt the vote of the offspring;

  3. 3.

at each selective event in $\overleftarrow{\Pi}^{n}$, all marked individuals
adopt the majority vote of the three offspring, unless precisely one vote is
$1$, in which case they all vote $1$ with probability
$\frac{2\gamma_{n}}{3+3\gamma_{n}}$, otherwise they vote $0$.

This defines an iterative voting procedure, which runs inwards from the
‘leaves’ of $\Xi^{n}(t)$ to the ancestral individual $\emptyset$ situated at
the point $x$.

######  Definition 3.3.

With the voting procedure described above, we define
$\mathbb{V}_{p}(\Xi^{n}(t))$ to be the vote associated to the root
$\emptyset$.

We should like to have an analogue of the stochastic representation of the
solution to $(AC_{\varepsilon})$ of Proposition 2.4 for the SLFVS, but recall
from Remark 1.12 that the SLFVS is only defined up to a Lebesgue null set, and
so we cannot expect such a representation at every point of
$\mathbb{R}^{\mathbbm{d}}$. However, a weak version of the representation is
valid. The following result is easily proved using the approach introduced to
prove Proposition 1.7 in [EVY20] (the corresponding result in the case of
genic selection).

######  Theorem 3.4.

The SLFVS driven by $\Pi^{n}$,
$(w^{n}_{t}(x),x\in\mathbb{R}^{\mathbbm{d}})_{t\geq 0}$, is dual to the
process $(\Xi^{n}(t))_{t\geq 0}$ of Definition 3.1 in the sense that for every
$\psi\in C(\mathbb{R}^{\mathbbm{d}})\cap L^{1}(\mathbb{R}^{\mathbbm{d}})$, we
have

$\mathbb{E}_{p}\left[\int_{\Omega}\psi(x)w^{n}_{t}(x)\,dx\right]=\int_{\Omega}\psi(x)\mathbb{E}_{x}\left[\mathbb{V}_{p}\left(\Xi^{n}(t)\right)\right]\,dx=\int_{\Omega}\psi(x)\mathbb{P}_{x}\left[\mathbb{V}_{p}\big{(}\Xi^{n}(t)\big{)}=1\right]\,dx.$
(56)

######  Remark 3.5.

Note that the expectations on the left and right of Equation (56) are taken
with respect to different measures. The subscripts on the expectations are the
initial values for the processes on each side.

The duality reduces the proof of Theorem 1.17 to the following analogue of
Theorem 2.10.

######  Theorem 3.6.

Define $\rho_{*}=({\mathbbm{d}}-1)/\nu$ and let
$\widehat{p}(x)=1_{B(0,\rho_{*})}(x)$.

  1. 1.

Under the weak noise/selection ratio regime, for any $k\in\mathbb{N}$, there
exist $n_{*}(k)<\infty$, and $d_{*}(k)\in(0,\infty)$, such that for all $n\geq
n_{*}$ and all $t>0$,

$\displaystyle\text{for all }x\in\mathbb{R}^{\mathbbm{d}}\text{ with
}\|x\|\geq\rho_{*}+d_{*}\varepsilon_{n}|\log\varepsilon_{n}|,\text{ we have
}\mathbb{P}_{x}\left[\mathbb{V}_{\widehat{p}}(\Xi^{n}(t))=1\right]\leq\varepsilon_{n}^{k}.$
$\displaystyle\text{for all }x\in\mathbb{R}^{\mathbbm{d}}\text{ with
}\|x\|\leq\rho_{*}-d_{*}\varepsilon_{n}|\log\varepsilon_{n}|,\text{ we have
}\mathbb{P}_{x}\left[\mathbb{V}_{\widehat{p}}(\Xi^{n}(t))=1\right]\geq
1-\varepsilon_{n}^{k}.$

  2. 2.

Under the strong noise/selection ratio regime, there is a constant
$\sigma^{2}>0$ such that for every fixed $t>0$ and $\varepsilon>0$ there is a
Brownian motion, $(W_{s})_{s\geq 0}$, and $n_{*}\in\mathbb{N}$, such that for
all $n\geq n_{*}$

$\Big{|}\mathbb{P}_{x}\big{[}\mathbb{V}_{\widehat{p}}(\Xi^{n}(t))=1\big{]}-\mathbb{P}_{x}\big{[}\|W(\sigma^{2}t)\|\leq\rho_{*}\big{]}\Big{|}\leq\varepsilon.$
(57)

#### Motion of a single lineage

Our first task is to investigate the motion of a single lineage. It is a pure
jump process. By spatial homogeneity, to establish the transition rates, it
suffices to calculate the rate at which a lineage currently at the origin will
jump to the point $z\in\mathbb{R}^{\mathbbm{d}}$. This is given by

$\displaystyle m_{n}(dz)$
$\displaystyle=nu_{n}\chi_{n}n^{\mathbbm{d}\beta}\int_{0}^{\mathcal{R}_{n}}\frac{V_{r}(0,z)}{V_{r}}d\mu^{n}(r)dz,$
(58)

where $\chi_{n}=1/\widehat{u}_{n}$ in the strong noise regime and $1$
otherwise, $V_{r}$ is the volume of $B(0,r)$, and $V_{r}(0,z)$ is the volume
of $B(0,r)\cap B(z,r)$. To understand the expression (58), first observe that
in order for the lineage to jump from $0$ to $z$, it must be affected by an
event that covers both $0$ and $z$. The possible centres for an event of
radius $r$ covering both $0$ and $z$ is the region given by the intersection
of $B(0,r)$ and $B(z,r)$, which has volume $V_{r}(0,z)$, and so events fall in
this region with rate $n\chi_{n}V_{r}(0,z)$. If an event occurs with centre
$x$ in the region, then the lineage jumps with probability $u_{n}$, and if it
does so, then it jumps to a point chosen uniformly from $B(x,r)$; the chance
of that point being $z$ is $dz/V_{r}$.

######  Remark 3.7.

We can now explain why our choice of scalings in the weak and strong regimes
are appropriate for investigating the effect of increasing genetic drift.
Notice that $nu_{n}\chi_{n}=un^{2\beta}$ in the weak scaling and $n^{2\beta}$
in the strong scaling, so setting $u=1$ in (13) the transition probabilities
for a lineage will coincide under our two regimes. If the coefficients
$\boldsymbol{s}_{n}$ are the same, then so too will be the rate at which a
lineage is affected by a selective event. From the perspective of lineages,
the difference between the scalings is the probability of coalescence, driven
by $u_{n}$. The parameter $u_{n}$ determines the strength of the genetic drift
(it can be thought of as proportional to the inverse of the population
density). In particular, in two dimensions, Theorem 1.17 says that increasing
the strength of the noise can indeed break down the structure of the solution
that results from the selection term in the Allen-Cahn equation.

Integrating out over $z$ in (58) and ‘undoing’ the scaling of $\mu^{n}(dr)$,
we find that the total jump rate of the lineage is

$\displaystyle\int_{\mathbb{R}^{\mathbbm{d}}}m_{n}(dz)$
$\displaystyle=nu_{n}\chi_{n}V_{1}\int_{0}^{\mathcal{R}}r^{d}\mu(dr),$

where, as noted above, the prefactor is (up to a factor of $u$ coming from
(13)) $n^{2\beta}$. Since each jump is of size $\Theta(n^{-\beta})$, we
recognise the diffusive scaling. We can identify the diffusion constant for
the limiting Brownian motion, by noting that

$\displaystyle\sigma^{2}=\int_{\mathbb{R}^{\mathbbm{d}}}\|z\|^{2}m_{n}(dz)=\frac{u_{n}n\chi_{n}}{n^{2\beta}2{\mathbbm{d}}}\int_{0}^{\mathcal{R}}\int_{\mathbb{R}^{\mathbbm{d}}}\|z\|^{2}\frac{V_{r}(0,z)}{V_{r}(0)}dz\mu(dr).$
(59)

The following coupling is Lemma 3.8 in [EFP17].

######  Lemma 3.8.

Let $(\xi^{n}(t))$ be a pure jump process with transition rates given by
$m_{n}$ and suppose that $\sigma^{2}$ is defined by (59). For fixed $t>0$
there is a coupling of a Brownian motion $W$ and $\xi^{n}$ under which

$\mathbb{P}_{x}[|\xi^{n}(t)-W(\sigma^{2}t)|\geq
n^{-\beta/6}]=\mathcal{O}(n^{-\beta}(t\vee 1))$

(where $\xi^{n}$ and $W$ both start from the point $x$).

We need to be able to couple the lineage at the time when it is first affected
by a selective event (and so branches) with a Brownian motion. This is where
we first use Assumption 1.16, $(\log n)^{1/2}\varepsilon_{n}\to\infty$, which
guarantees that $\boldsymbol{s}_{n}=o\big{(}\log n/n^{2\beta}\big{)}$. We
abuse our Ulam-Harris based notation $\xi_{\boldsymbol{i}}$ and write simply
$\xi_{1}(\tau)$, $\xi_{2}(\tau)$, and $\xi_{3}(\tau)$, for the three possible
positions of lineages at the first branching time of $\Xi^{n}$ (corresponding
to the positions of the three offspring of the first selective event to affect
the lineage).

######  Corollary 3.9.

Let $\tau$ be the first branching of $\Xi^{n}$. Then there is a Brownian
motion $W$ and a coupling of $\Xi^{n}$ and $W$ under which the following
holds. The branching time $\tau$ and $W$ are independent,
$\tau\sim\mathtt{Exp(}\eta(1+\gamma_{n})\varepsilon_{n}^{-2})$ with
$\eta=uV_{1}\int_{0}^{\mathcal{R}}r^{\mathbbm{d}}\mu(dr)$, and for $i=1,2,3$,

$\mathbb{P}_{x}\left[|\xi_{i}^{n}(\tau)-W(\sigma^{2}\tau)|\geq
n^{-\beta/6}\right]=\mathcal{O}(n^{-\beta}).$

where $\xi^{n}$ and $W$ both have the same starting point $x$.

###### Sketch of proof.

By Poisson thinning, we can express the position of the lineage at time $t$ as
the sum of the jumps due to neutral events and those due to selective events.
Lemma 3.8 allows us to couple the part corresponding to neutral events with a
Brownian motion at time $\sigma^{2}(1-\boldsymbol{s}_{n})t$. Since
$\boldsymbol{s}_{n}=o\big{(}\log n/n^{2\beta}\big{)}$, Chebyshev’s inequality
gives

$\mathbb{P}\left[\|W(\sigma^{2}t)-W(\sigma^{2}(1-\boldsymbol{s}_{n})t)\|\geq
n^{-\beta/6}\right]=o\left(\frac{\log n}{n^{2\beta}}n^{\beta/3}(t\vee
1)\right).$

The proof is completed by an application of the triangle inequality, using
Poisson thinning to partition over the value of $\tau$, and using that
$|\xi_{i}^{n}(\tau)-\xi_{1}^{n}(\tau-)|\leq
2{\mathcal{R}}_{n}=2n^{-\beta}{\mathcal{R}}$. ∎

###  3.2 Proof of Theorem 3.6, weak regime

We outline the main steps of the proof of Theorem 3.6 in the weak scaling
regime, that is one in which selection overwhelms noise.

There are two ways in which the dual to the SLFVS differs from our ternary
branching Brownian motion. The first is that lineages can coalesce during
reproduction events; the second is that lineages follow a continuous time and
space random walk which only converges to Brownian motion in the scaling
limit.

Following [EFP17], the first step in the proof of the limiting result in the
weak scaling regime is to show that with high probability $\Xi^{n}(t)$ can be
coupled to a branching jump process.

######  Definition 3.10 (Branching Jump Process).

For a given $n\in\mathbb{N}$ and starting point
$x\in\mathbb{R}^{\mathbbm{d}}$, $(\Psi^{n}(t),t\geq 0)$ is the historical
process of the branching random walk described as follows.

  1. 1.

Each individual has an independent lifetime, which is exponentially
distributed with parameter $\eta(1+\gamma_{n})\varepsilon_{n}^{-2}$.

  2. 2.

During its lifetime, each individual, independently, evolves according to a
pure jump process with jump rates given by
$(1-(1+\gamma_{n})\boldsymbol{s}_{n})m_{n}$.

  3. 3.

At the end of its lifetime an individual branches into three offspring. The
locations of these offspring are determined as follows. First choose
$r\in(0,\mathcal{R}_{n}]$ according to

$\frac{r^{\mathbbm{d}}\mu^{n}(dr)}{\int_{0}^{\mathcal{R}_{n}}\widetilde{r}^{\mathbbm{d}}\mu^{n}(d\widetilde{r})}.$

If the individual is at point $z$ then the location of each offspring is
sampled independently and uniformly from $B(z,r)$.

The process $\Psi^{n}(t)$ differs from $\Xi^{n}(t)$ only in that we have
suppressed the coalescence events. Since, in this weak noise regime,
coalescence events are extremely rare, we can couple the two processes in such
a way that they coincide with high probability. The following is Lemma 3.12 of
[EFP17].

######  Lemma 3.11.

Let $T^{*}\in(0,\infty)$, $k\in\mathbb{N}$ and $z\in\mathbb{R}^{\mathbbm{d}}$.
There exists $n_{*}\in\mathbb{N}$ such that, for all $n\geq n_{*}$, there is a
coupling of $\Xi^{n}$ and $\Psi^{n}$, both started with one particle at $z$
such that, with probability at least $1-\varepsilon_{n}^{k}$ we have:

$\Xi^{n}(T^{*})=\Psi^{n}(T^{*}).$

###### Sketch of proof.

The key idea is to modify the dual process of Definition 3.1 in such a way
that individuals are ‘preemptively’ marked. More precisely, at time zero each
individual is marked with probability $u_{n}$. When an individual is in the
region covered by a reproduction event, it will be affected only if it is
marked. After the event, the marks of individuals within the region covered
are removed and new marks are assigned (including to the offspring, if any)
independently with probability $u_{n}$. Once marked, individuals remain marked
until they are covered by an event. Unless both are marked, any two lineages
evolve independently. The probability that two individuals are marked at time
zero is $u_{n}^{2}$. We must control the probability that for a pair of ‘root
to leaf rays’ in $\Xi^{n}(T^{*})$ a reproduction event occurs during
$[0,T^{*}]$ after which both are marked. If they were not both marked at time
zero, then in order for both to be marked, one of them must be affected by an
event, after which the probability that they are both marked is $u_{n}^{2}$.
Since order $nT^{*}$ events affect a lineage over $[0,T^{*}]$, for any pair of
root to leaf rays, the probability that there is a reproduction event after
which both are marked is
${\mathcal{O}}(nu_{n}^{2})={\mathcal{O}}(n^{4\beta-1})$. (This is why we
restrict to $\beta<1/4$ in the weak noise/selection regime.)

We then use Assumption 1.16 for the second time. This time, it allows one to
control the total number of such root to leaf rays in $\Xi^{n}$ (by the number
in a regular ternary tree of height $b\log n$ for a suitable constant $b$). A
union bound over pairs of such rays shows that the probability that there is
any time in $[0,T^{*}]$ when at least two rays are marked (which is required
for a coalescence event to take place) is $\mathcal{O}(n^{-\alpha})$ for some
$\alpha>0$. ∎

Since we have already checked that the motion of a lineage in $\Xi^{n}$, and
therefore in $\Psi^{n}$, is close to a Brownian motion at the first branch
time, we can already see why Theorem 3.6 should hold in the weak regime. Of
course, there is still some work to do; as $\varepsilon_{n}\to 0$ the number
of branches in $\Psi^{n}$ is very large, and it is not obvious that the
convergence to Brownian motion along a single lineage will translate into
sufficiently rapid convergence on the whole tree. The proof follows the same
pattern as the deterministic case.

#### Generation of interface for the SLFVS

The first step is to show that, analogously to Proposition 2.13, the SLFVS
generates an interface in a time window that is of order
$\varepsilon_{n}^{2}|\log(\varepsilon_{n})|$. Evidently it suffices to work
with $\Psi^{n}$.

######  Proposition 3.12.

Let $k\in\mathbb{N}$. Then there exists $n_{*}(k),a_{*}(k),b_{*}(k)>0$ such
that, for all $n\geq n_{*}$, if we set:

$\delta_{*}(k,n):=a(k)\varepsilon_{n}^{2}|\log(\varepsilon_{n})|\>\>\text{and}\>\>\delta_{*}^{\prime}(k,n):=(a(k)+\eta^{-1}(k+1))\varepsilon_{n}^{2}|\log(\varepsilon_{n})|,$
(60)

then, for $t\in[\delta,\delta^{\prime}]$, we have that,

$\displaystyle\text{for any }x\text{ such that
}\|x\|\geq\rho_{*}+d_{*}\varepsilon_{n}|\log\varepsilon_{n}|,\text{ we have
}\mathbb{P}_{x}^{\varepsilon}[\mathbb{V}_{\widehat{p}}^{\gamma}(\Psi^{n}(t))=1]\leq\varepsilon_{n}^{k}.$
$\displaystyle\text{for any }x\text{ such that
}\|x\|\leq\rho_{*}-d_{*}\varepsilon_{n}|\log\varepsilon_{n}|,\text{ we have
}\mathbb{P}_{x}^{\varepsilon}[\mathbb{V}_{\widehat{p}}^{\gamma}(\Psi^{n}(t))=1]\geq
1-\varepsilon_{n}^{k}.$

Our proof in the deterministic setting required that we could find a large
ternary tree sitting within $\mathcal{T}(\boldsymbol{W})$. Here we also need
the converse to prove an analogue of (2.5).

######  Lemma 3.13 ([EFP17], Lemma 3.16).

Let $k\in\mathbb{N}$ and let $A(k)$ be as in Lemma 2.7. There exists
$a_{*}(k),B_{*}(k)\in(0,\infty)$ and $n_{*}(k)<\infty$ such that for all
$n\geq n_{*}$ and $\delta_{*},\delta_{*}^{\prime}$ as defined in (60),

$\displaystyle\mathbb{P}\left[\mathcal{T}(\Psi^{n}(\delta_{*}))\supseteq\mathcal{T}_{A(k)|\log(\varepsilon_{n})|}^{reg}\right]$
$\displaystyle\geq 1-\varepsilon_{n}^{k},$ (61)
$\displaystyle\mathbb{P}\left[\mathcal{T}(\Psi^{n}(\delta_{*}^{\prime}))\subseteq\mathcal{T}_{B(k)|\log(\varepsilon_{n})|}^{reg}\right]$
$\displaystyle\geq 1-\varepsilon_{n}^{k}.$ (62)

###### Sketch of Proof of Proposition 3.12.

The proof is based on the proof of Proposition 2.13, where the new ingredient
is that now we use Lemma 3.8 to control the distance between the jump process
with transitions given by $m_{n}$ (which governs the lineage motion) and a
Brownian motion, over the time interval $[\delta_{*},\delta_{*}^{\prime}]$.
Since $\varepsilon_{n}^{-2}=o(\log n)$, for any constant $d$ we can arrange
that for large enough $n$, $d\varepsilon_{n}|\log(\varepsilon_{n})|\geq
2n^{-\beta/6}$. Combining with our previous estimates for the Brownian motion
(Proposition 2.12), for any root to leaf ray in
$\Psi^{n}(\delta_{*}^{\prime})$, we can use this to control the probability
that the leaf is more than
$\frac{1}{2}d_{*}\varepsilon_{n}|\log(\varepsilon_{n})|$ from its starting
point at any time in $[\delta_{*},\delta_{*}^{\prime}]$. We can extend this to
the whole of $\Psi^{n}$ using Equation (62) and a union bound.

The proof of the first inequality of Proposition 3.12 now mirrors that of
Proposition 2.13. A symmetric computation gives the second inequality. ∎

#### Final steps for the weak noise/selection ratio regime of Theorem 3.6

To conclude the proof of the weak noise/selection ratio regime of Theorem 3.6.
We need the following modification of Lemma 2.15. For simplicity, we write
$\|x\|_{\rho_{*}}:=\|x\|-\rho_{*}$.

######  Lemma 3.14.

Let $l\in\mathbb{N}$ with $l\geq 4$, $K_{1}>0$. There exists $n_{*}$ such that
for all $n\geq n_{*}$, $x\in\mathbb{R}^{\mathbbm{d}}$,
$s\in[\sigma^{2}\varepsilon_{n}^{l+3},\sigma^{2}(l+1)\eta^{-1}\varepsilon_{n}^{2}|\log(\varepsilon_{n})|]$
and $t\in[s,\infty)$,

$\displaystyle\mathbb{E}_{x}\left[g\left(\mathbb{P}_{\|W_{s}\|_{\rho_{*}}-\nu(t-s)+K_{1}\varepsilon_{n}|\log(\varepsilon_{n})|+3n^{-\beta/6}}[\mathbb{V}^{\gamma}(\mathbf{B}(t-s)=1]+\varepsilon_{n}^{l}\right)\right]$
$\displaystyle\leq\frac{3+5\gamma_{n}}{4(1+\gamma_{n})}\varepsilon_{n}^{l}+\mathbb{E}_{\|x\|_{\rho_{*}}}\left[g\left(\mathbb{P}_{B_{s}-\nu
t+K_{1}\varepsilon_{n}|\log(\varepsilon_{n})|}[\mathbb{V}^{\gamma}(\mathbf{B}(t-s))=1]\right)\right]+1_{s\leq\varepsilon_{n}^{3}}\varepsilon_{n}^{l}.$
(63)
$\displaystyle\mathbb{E}_{x}\left[g\left(\mathbb{P}_{\|W_{s}\|_{\rho_{*}}-\nu(t-s)-K_{1}\varepsilon_{n}|\log(\varepsilon_{n})|+3n^{-\beta/6}}[\mathbb{V}^{\gamma}(\mathbf{B}(t-s)=0]+\varepsilon_{n}^{l}\right)\right]$
$\displaystyle\leq\frac{3+5\gamma_{n}}{4(1+\gamma_{n})}\varepsilon_{n}^{l}+\mathbb{E}_{\|x\|_{\rho_{*}}}\left[g\left(\mathbb{P}_{B_{s}-\nu
t-K_{1}\varepsilon_{n}|\log(\varepsilon_{n})|}[\mathbb{V}^{\gamma}(\mathbf{B}(t-s))=0]\right)\right]+1_{s\leq\varepsilon_{n}^{3}}\varepsilon_{n}^{l}.$
(64)

###### Proof.

The proof is identical to that of Lemma 2.15 except that we also approximate
the jump process of a lineage by Brownian motion. Since
$n^{-\beta/6}=o(s\varepsilon_{n}|\log(\varepsilon_{n})|)$, the additional term
$3n^{-\beta/6}$ is negligible for large $n$. For the case in which
$\|W_{s}\|_{\rho_{*}}$ is close to $0$, we use Lemma 2.11, where the distance
is now to a sphere of fixed radius $\rho_{*}$ (but the statement of the result
does not change). ∎

The equivalent of Proposition 2.14 for $\Psi^{n}$ is:

######  Proposition 3.15.

Let $l\in\mathbb{N}$ with $l\geq 4$. Let $a_{*}(l)$ and
$\delta_{*}(l,\varepsilon)$ given by Proposition 3.12. There exists $K_{1}(l)$
and $n_{*}(l,K_{1})>0$ such that, for all $n\geq n_{*}$,
$t\in[\delta_{*}(l,n),\infty)$ we have:

$\displaystyle\sup_{x\in\mathbb{R}^{\mathbbm{d}}}\left(\mathbb{P}_{x}^{\varepsilon}[\mathbb{V}_{\widehat{p}}^{\gamma}(\Psi^{n}(t))=1]-\mathbb{P}^{\varepsilon_{n}}_{\|x\|_{\rho_{*}}-\nu
t+K_{1}\varepsilon_{n}|\log(\varepsilon_{n})|}[\mathbb{V}^{\gamma}(\mathbf{B}(t))=1]\right)\leq\varepsilon_{n}^{l}.$
$\displaystyle\sup_{x\in\mathbb{R}^{\mathbbm{d}}}\left(\mathbb{P}_{x}^{\varepsilon}[\mathbb{V}_{\widehat{p}}^{\gamma}(\Psi^{n}(t))=0]-\mathbb{P}^{\varepsilon_{n}}_{\|x\|_{\rho_{*}}-\nu
t-K_{1}\varepsilon_{n}|\log(\varepsilon_{n})|}[\mathbb{V}^{\gamma}(\mathbf{B}(t))=0]\right)\leq\varepsilon_{n}^{l}.$

###### Proof.

The proof is essentially identical to that of Proposition 2.14, except that we
use Lemma 3.14 in place of Lemma 2.15, and Proposition 3.12 in place of
Proposition 2.13. ∎

###### Proof of Theorem 3.6, weak noise regime.

As in the deterministic setting, it suffices to prove the result for
sufficiently large $k\in\mathbb{N}$. By Lemma 3.11, it suffices to work with
$\Psi^{n}$, and the result then follows from Proposition 3.15, in the same
vein as the proof of Theorem 2.10. ∎

###  3.3 Proof of Theorem 3.6, strong regime

We now turn to the strong noise/selection ratio regime.

The total rate at which a lineage is affected by a selective event, and so a
new particle is created in the dual, is proportional to:

$\frac{n}{\widehat{u}_{n}}u_{n}\boldsymbol{s}_{n}=\boldsymbol{s}_{n}n^{2\beta}.$

The first case of (15) corresponds to not seeing any creation of particles in
the dual process in the limit, and we include it only for completeness.

The second condition in (15) is more interesting, and more complex. In this
case, the parameter $u_{n}$ is sufficiently large relative to the rate of
creation of lineages that even though, asymptotically, new lineages may be
created infinitely fast in the dual, they are effectively instantly annulled
by coalescence. This is, for example, the case if $u_{n}=(\log n)^{-1/2}$ and
$\boldsymbol{s}_{n}=(\log n)^{1/3}/n^{2\beta}$.

We remark that this cannot be achieved in $\mathbbm{d}\geq 3$; the distance
between two lineages in the dual is itself a homogeneous jump process with
bounded jump size, which will be transient in $\mathbbm{d}\geq 3$, so that
there is always a positive probability of the two lineages ‘escaping’ from one
another and never coalescing. In $\mathbbm{d}=2$, even if two lineages
initially move apart, there will, with probability one, be a later time at
which they are close enough together to be covered by the same event, and
therefore have a chance to coalesce. Indeed, in the strong noise/selection
ratio regime, most lineages will coalesce with their ‘siblings’ very soon
after being created, with the result that, in a way that we shall make precise
below, the ‘effective’ rate of creation of new lineages in the dual is of
order

$\frac{n}{\widehat{u}_{n}}\frac{\boldsymbol{s}_{n}u_{n}}{u_{n}\log
n}=\frac{\boldsymbol{s}_{n}n^{2\beta}}{u_{n}\log n}.$

Here is a rigorous statement of the key result that we must prove.

######  Lemma 3.16.

Let $\big{(}\mathcal{P}^{n}(s)\big{)}_{s\geq
0}:=\big{(}\xi_{1}^{n}(s),...,\xi_{N(s)}^{n}(s)\big{)}_{s\geq 0}$ be the dual
process to the SLFVS introduced in Definition 3.1. Let
$x\in\mathbb{R}^{\mathbbm{d}}$ and $t>0$ be fixed. In the strong
noise/selection ratio regime there is a Brownian motion $(W_{s})_{s\geq 0}$
and a coupling of $\mathcal{P}^{n}$ and $W$ such that, for any
$\varepsilon>0$, there is $n_{*}$ such that for all $n\geq n_{*}$;

$\mathbb{P}_{x}\left[\big{\\{}N(t)=1\big{\\}}\cap\big{\\{}\|\xi_{1}^{n}(t)-W(\sigma^{2}t)\|<n^{-\beta/6}\big{\\}}\right]\geq
1-\varepsilon.$ (65)

where $\mathcal{P}^{n}$ starts from a single particle at $x$, which is also
the starting point of $W$.

Armed with Lemma 3.16, the strong noise/ selection regime of Theorem 3.6
follows easily.

###### Proof of Theorem 3.6, strong noise/selection regime.

From Lemma 3.16, there is a Brownian motion $W$ such that, for $n$
sufficiently large,

$\displaystyle\left|\mathbb{P}_{x}\left[\mathbb{V}_{\widehat{p}}(\Xi^{n}(t))=1\right]-\mathbb{P}_{x}[\|W(\sigma^{2}t)\|\leq\rho_{*}]\right|\
$
$\displaystyle\leq\left|\mathbb{P}_{x}\big{[}\|\xi_{1}^{n}(t)-W(\sigma^{2}t)\|\leq
n^{-\beta/6},\|\xi_{1}^{n}(t)\|\leq\rho_{*}\big{]}-\mathbb{P}_{x}[\|W(\sigma^{2}t)\|\leq\rho_{*}]\right|+\varepsilon$

Note that

$\displaystyle\mathbb{P}_{x}[\|W(\sigma^{2}t)\|\leq\rho_{*}]-\mathbb{P}_{x}\Big{[}|\|W(\sigma^{2}t)\|-\rho_{*}|\leq
n^{-\beta/6}\Big{]}$
$\displaystyle\leq\mathbb{P}_{x}\big{[}|\xi_{1}^{n}(t)-W(\sigma^{2}t)|\leq
n^{-\beta/6},\|\xi^{n}_{1}(t)\|\leq\rho_{*}\big{]}$
$\displaystyle\leq\mathbb{P}_{x}\big{[}\|W(\sigma^{2}t)\|\leq\rho_{*}\big{]}+\mathbb{P}\Big{[}|\|W(\sigma^{2}t)\|-\rho_{*}|\leq
n^{-\beta/6}\Big{]},$ (66)

To conclude, observe that

$\mathbb{P}_{x}[|\|W(\sigma^{2}t)\|-\rho_{*}|\leq n^{-\beta/6}]\leq
C({\mathbbm{d}},\rho_{*},x)n^{-\beta/6}\leq\varepsilon,$ (67)

where the last inequality is valid for sufficiently large values of $n_{*}$.
Using (67) in (66) gives the result. ∎

#### Proof of Lemma 3.16

Lemma 3.16 is a consequence of Lemma 3.8 and the following proposition.

######  Proposition 3.17.

Let $t>0,\varepsilon>0$. Then in the strong noise/selection scaling ratio
regime there is $n_{*}\in\mathbb{N}$ such that for all $n\geq n_{*}$

$\mathbb{P}_{x}\big{[}N(t)>1\big{]}\leq\varepsilon,$ (68)

where $N(s)$ is the number of particles alive at time $s$ in the dual process
$(\mathcal{P}_{s}^{n})_{0\leq s\leq t}$ of Lemma 3.1.

We focus on the case $u_{n}\log n\rightarrow\infty$ and
$\boldsymbol{s}_{n}n^{2\beta}/(u_{n}\log n)\rightarrow 0$ (note that $\liminf
u_{n}\log n$ diverging implies that $\lim u_{n}\log n\to\infty$, and so we
will use the latter condition in what follows). The proof is heavily inspired
by [Eth+17]. In that work, it was shown that, in the case of genic selection
(in which exactly two offspring are produced in the dual at a selective
event), if $\beta=1/2$, $u_{n}\equiv u$ and $\boldsymbol{s}_{n}=\log(n)/n$,
there is an equilibrium of branching and coalescence in the dual, so that in
the limit as $n\to\infty$ we see a branching Brownian motion (with an
‘effective’ branching rate). In our notation, this condition on
$\boldsymbol{s}_{n}$ corresponds to $\boldsymbol{s}_{n}n^{2\beta}/(u_{n}\log
n)$ being ${\mathcal{O}}(1)$. In the case $u_{n}\equiv u$, if a pair of
lineages is covered by a reproduction event, then there is a probability of
order one that they will coalesce. As a result, even for large $n$, they
coalesce having been hit by a finite number of events. In our setting, if one
lineage is affected by an event, the probability that the second lineage is
also affected is $u_{n}$ and, as a consequence, a pair of lineages must come
together of the order of $1/u_{n}$ times before they coalesce. The factor
$\log n$ corresponds to the number of times that two lineages will come close
enough together to be covered by the same event, before they escape to a
separation of order one; once they do that, we can expect to wait a long time
before they next come close together.

###### Sketch of proof of Proposition 3.17..

Most of the work was done in [Eth+17], and indeed we don’t require the precise
estimates obtained there. First suppose that two lineages,
$\xi^{n,1},\xi^{n,2}$, are created in a selective event occurring at time $0$.
Let $\eta^{n}=\xi^{n,1}-\xi^{n,2}$. The key step is to show that with high
probability the two lineages will coalesce before time $1/(\log n)^{c}$ where
$c$ can be chosen to be at least $3$. (We shall use a union bound on the
complementary event to estimate the probability that all three lineages
created in a selection event coalesce on this timescale.)

Much of the work arises from the fact that $\xi^{n,1}$ and $\xi^{n,2}$ only
evolve independently when their separation is more than $2{\mathcal{R}}_{n}$.

Following [Eth+17], we consider three possible scenarios:

  1. 1.

The separation $\|\eta^{n}\|$ exceeds $(\log n)^{-c}$ at some time before
$(\log n)^{-c}$. This occurs with probability
$\mathcal{O}\left(\frac{1}{u_{n}\log n}\right)$, and we say that $\eta^{n}$
diverges.

  2. 2.

The quantity $\|\eta^{n}\|$ does not exceed $(\log n)^{-c}$, and neither do
the two lineages coalesce before time $(\log n)^{-c}$. This happens with
probability $\mathcal{O}\left(\frac{1}{(\log n)^{c-3/2}u_{n}}\right)$, and we
say that $\eta^{n}$ overshoots.

  3. 3.

The lineages coalesce within time $(\log n)^{-c}$.

The idea, which can be traced to Lemma 4.2 of [EV12], is to characterise the
behaviour of $\eta^{n}$ in terms of ‘inner’ and ‘outer’ excursions, defined
through sequences of stopping times. Set $\tau_{0}^{\mathrm{out}}=0$ and
define

$\displaystyle\tau_{i}^{\mathrm{in}}=\inf\\{s>\tau_{i}^{\mathrm{out}}:|\eta_{s}^{n}|\geq
5\mathcal{R}_{n}\\},$ (69)
$\displaystyle\tau_{i+1}^{\mathrm{out}}=\inf\\{s>\tau_{i}^{\mathrm{in}}:|\eta_{s}^{n}|\leq
4\mathcal{R}_{n}\\}.$ (70)

The interval $[\tau_{i}^{\mathrm{out}},\tau_{i}^{\mathrm{in}})$ (and the path
of $\eta^{n}$ during it) is the $i$th inner excursion, and similarly
$[\tau_{i-1}^{\mathrm{in}},\tau_{i}^{\mathrm{out}})$ (and the corresponding
path) is the $i$th outer excursion.

We use $\mathbb{P}_{[r_{1},r_{2}]}$ to denote results that hold for any
$|\eta_{0}^{n}|\in[r_{1},r_{2}]$ and we set

$L_{n}=(\log n)^{-c},$

for a fixed $c\geq 3$. We then define

$\displaystyle\tau^{\mathrm{coal}}=\inf\\{s>0:|\eta_{s}^{n}|=0\\},$
$\displaystyle\tau^{\mathrm{div}}=\inf\\{s>0:|\eta_{s}^{n}|\geq L_{n}\\},$
$\displaystyle\tau^{\mathrm{type}}=\tau^{\mathrm{coal}}\wedge\tau^{\mathrm{div}}\wedge
L_{n}.$

We say $\eta^{n}$ coalesces if $\tau^{type}=\tau^{coal}$, diverges if
$\tau^{type}=\tau^{div}$, and overshoots otherwise. We are also going to need
the stopping times:

$\displaystyle\tau_{r}=\inf\\{s>0:|\eta_{s}^{n}|\leq r\\},$ (71)
$\displaystyle\tau^{r}=\inf\\{s>0:|\eta_{s}^{n}|\geq r\\}.$ (72)

Lemma 4.7 in [Eth+17] yields that, as $n\rightarrow\infty$,

$\mathbb{P}_{[5\mathcal{R}_{n},7\mathcal{R}_{n}]}[\tau^{L_{n}}<\tau_{4\mathcal{R}_{n}}]=\mathcal{O}\left(\frac{1}{\log
n}\right).$ (73)

The proof is unchanged in our setting; during outer excursions, $\eta^{n}$
behaves like the difference between two independent jump processes and it
takes $\mathcal{O}(\log n)$ such excursions to achieve separation $L_{n}$.

In order to control $\mathbb{P}[\eta^{n}\text{ diverges}]$, we control the
number of inner excursions before we see a coalescence. We first note there
exists $M>0$ such that

$\mathbb{P}_{[0,5\mathcal{R}_{n}]}[\tau_{0}\leq\tau^{5\mathcal{R}_{n}}]\geq
Mu_{n}$ (74)

Indeed, from a separation of $5{\mathcal{R}}_{n}$, there is a strictly
positive probability that within $3$ jumps $\|\eta^{n}\|\leq{\mathcal{R}}_{n}$
and then a strictly positive probability that the next event will cover both
lineages, in which they coalesce with probability $u_{n}$. This allows us to
bound above the number of inner excursions before coalescence by a geometric
random variable with success probability $Mu_{n}$, and hence the probability
of divergence has order at most $1/(u_{n}\log n)$. Again the details follow
exactly as in [Eth+17].

The proof that the probability that $\eta^{n}$ overshoots is
${\mathcal{O}}\left(\frac{1}{(\log n)^{c-3/2}}\right)$ is a minor modification
of that of [Eth+17], Lemma 4.4. The idea is that if $\|\eta^{n}_{t}\|=x$, with
$x\in(0,5{\mathcal{R}}_{n})$, then with strictly positive probability
$\theta$, $\eta^{n}$ will either coalesce or exit $B(0,5{\mathcal{R}}_{n})$
within three jumps. The number of jumps that $\eta^{n}$ makes before either
exiting $B(0,5{\mathcal{R}}_{n})$ or coalescence is therefore stochastically
bounded by three times a geometric random variable with success probability
$\theta$. Since $\eta^{n}$ jumps at least as fast as $\xi^{n,1}$, which jumps
at rate $m_{n}$ given by (58), this allows us to estimate

$\mathbb{P}_{(0,5{\mathcal{R}}_{n})}\left[\tau_{0}\wedge\tau^{5{\mathcal{R}}_{n}}>n^{-\beta}\right]={\mathcal{O}}\left(n^{-\beta/3}+(1-\theta)^{n^{2\beta/3}}\right),$

where the first term is an estimate of the probability that the sum of
$n^{2\beta/3}$ independent exponential random variables with parameter $m_{n}$
exceeds $n^{-\beta}$ and the second is the probability that a Geometric random
variable with parameter $\theta$ exceeds $n^{2\beta/3}$.

Now let $i^{*}$ denote the index of the excursion during which
$\tau^{\mathrm{type}}$ occurs. From the argument that we used to control the
probability of divergence, we can bound the number of inner excursions before
coalescence above by a geometric random variable with success probability
$Mu_{n}$. Let $n$ be large enough that

$\frac{(\log n)^{1/2}}{u_{n}}\left(n^{-\beta}+(\log n)^{-c-2}\right)\leq(\log
n)^{-c}.$

Note that this is possible since $u_{n}\log n\to\infty$ as $n\to\infty$, by
assumption. If $\eta^{n}$ overshoots, and $i^{*}<(\log n)^{1/2}/u_{n}$, then
at least one inner excursion must have lasted at least $n^{-\beta}$ or at
least one outer excursion must have lasted at least $(\log n)^{-c-1}$. Again
using that when $\|\eta^{n}\|>2{\mathcal{R}}_{n}$, $\eta^{n}$ behaves as the
difference of two independent walks (and Skorohod embedding), Lemma 4.7 of
[Eth+17] shows that

$\mathbb{P}_{[5{\mathcal{R}}_{n},7{\mathcal{R}}_{n}]}\left[\tau^{L_{n}}\wedge\tau_{4{\mathcal{R}}_{n}}>(\log
n)^{-c-2}\right]=\mathcal{O}\left(\frac{1}{(\log n)^{c}}\right).$

Combining the above,

$\mathbb{P}\left[\eta^{n}\mbox{ overshoots}\right]\leq\frac{(\log
n)^{1/2}}{u_{n}}\Bigg{\\{}\mathbb{P}_{(0,5{\mathcal{R}}_{n})}\left[\tau_{0}\wedge\tau^{5{\mathcal{R}}_{n}}>n^{-\beta}\right]\\\
+\mathbb{P}_{[5{\mathcal{R}}_{n},7{\mathcal{R}}_{n}]}\left[\tau^{L_{n}}\wedge\tau_{4{\mathcal{R}}_{n}}>(\log
n)^{-c-2}\right]\Bigg{\\}}+\mathbb{P}\left[i^{*}>\frac{(\log
n)^{1/2}}{u_{n}}\right]\\\ \leq\frac{(\log
n)^{1/2}}{u_{n}}\left(n^{-\beta}+(\log n)^{-c}\right)+(1-Mu_{n})^{(\log
n)^{1/2}/u_{n}}=\mathcal{O}\Big{(}\frac{1}{(\log n)^{c-3/2}u_{n}}\Big{)}.$

To conclude the proof of Proposition 3.17, let us write $\xi$ for a root to
leaf ray connecting $\xi_{0}=x$ to $\xi^{n}_{1}(t)$ (the first individual in
${\cal P}^{n}(t)$). Let $S_{m}$ denote the time of the $m$-th selective event
to affect the lineage, creating offspring $\xi^{n,1},\xi^{n,2},\xi^{n,3}$.
Then, by the Markov property,

$\displaystyle\mathbb{P}$ $\displaystyle\left[\text{all individuals created at
time }S_{m}\text{ have not coalesced by time }S_{m}+\frac{1}{(\log
n)^{c}}\right]$ $\displaystyle\leq
3\mathbb{P}[\eta^{n}=(\xi^{n,1}-\xi^{n,2})\text{ diverges or overshoots}]$
$\displaystyle=\mathcal{O}\left(\frac{1}{u_{n}\log
n}\right)+\mathcal{O}\left(\frac{1}{u_{n}(\log
n)^{c-3/2}}\right)=\mathcal{O}\left(\frac{1}{u_{n}\log n}\right).$

Since selective events affect $\xi$ according to a Poisson process of rate
proportional to
$\boldsymbol{s}_{n}u_{n}n/\widehat{u}_{n}=\boldsymbol{s}_{n}n^{2\beta}$,

$\displaystyle\mathbb{P}[N_{t}^{n}\not=1]$
$\displaystyle\leq\mathbb{P}\Big{[}\text{a particle created in }[0,t-(\log
n)^{-c}]\text{ did not coalesce}\Big{]}$
$\displaystyle+\mathbb{P}\Big{[}\text{a selective event occurred in }[t-(\log
n)^{-c},t]\Big{]}$
$\displaystyle=\mathcal{O}\left(\frac{\boldsymbol{s}_{n}n^{2\beta}t}{u_{n}\log
n}\right)+\mathcal{O}\left(\frac{\boldsymbol{s}_{n}n^{2\beta}}{(\log
n)^{-c}}\right)\leq\mathcal{O}\left(\frac{\boldsymbol{s}_{n}n^{2\beta}(t\vee
1)}{u_{n}\log n}\right),$

where our conditions on $\boldsymbol{s}_{n}$ guarantee that all the terms on
the right hand side tend to $0$ as $n\to\infty$, and the proof is complete. ∎

##  Appendix A Proof of Lemma 2.7

For completeness, in the section we reproduce the proof of Lemma 2.7 from
[Goo18].

We begin with the first statement. We shall carry out two phases of iteration
of $g$. First, we will show that it takes
${\mathcal{O}\left(|\log\varepsilon|\right)}$ iterations to obtain

$g^{(n)}\big{(}\tfrac{1+\gamma_{\varepsilon}}{2}+\varepsilon\big{)}\geq\tfrac{1}{2}+\sqrt{\tfrac{1+\gamma_{\varepsilon}^{2}}{8}}.$
(75)

Note that $\tfrac{1+\gamma}{2}<\tfrac{1}{2}+\sqrt{\tfrac{1+\gamma^{2}}{8}}<1$
for $\gamma\in(0,1)$. After establishing this, we note that
${\mathcal{O}\left(k|\log\varepsilon|\right)}$ iterations are required to
obtain

$g^{(n)}\left(\tfrac{1}{2}+\sqrt{\tfrac{1+\gamma_{\varepsilon}^{2}}{8}}\right)\geq
1-\varepsilon^{k},$ (76)

and then, since $g$ is monotone, combining the two phases will complete the
proof of the first statement.

For the first phase, note that if
$\delta\in(0,\sqrt{(1+\gamma_{\varepsilon}^{2})/8}-\gamma_{\varepsilon}/2)$
then a straightforward calculation gives

$\begin{split}g(\tfrac{1+\gamma_{\varepsilon}}{2}+\delta)=&\tfrac{1+\gamma_{\varepsilon}}{2}+\tfrac{\delta}{2(1+\gamma_{\varepsilon})}(3+2\gamma_{\varepsilon}-(\gamma_{\varepsilon}+2\delta)^{2})\\\
\geq&\tfrac{1+\gamma_{\varepsilon}}{2}+\tfrac{5-\gamma_{\varepsilon}}{4}\delta.\end{split}$

Thus if
$g^{(n)}(\frac{1+\gamma_{\varepsilon}}{2}+\varepsilon)-\frac{1}{2}<\sqrt{\tfrac{1+\gamma_{\varepsilon}^{2}}{8}}$,
setting
$\delta=g^{(n)}(\frac{1+\gamma_{\varepsilon}}{2}+\varepsilon)-\frac{1+\gamma_{\varepsilon}}{2}$,
we have

$\displaystyle
g^{(n+1)}(\tfrac{1+\gamma_{\varepsilon}}{2}+\varepsilon)-\tfrac{1+\gamma_{\varepsilon}}{2}\geq\tfrac{5-\gamma_{\varepsilon}}{4}\left(g^{(n)}(\tfrac{1+\gamma_{\varepsilon}}{2}+\varepsilon)-\tfrac{1+\gamma_{\varepsilon}}{2}\right)\geq(\tfrac{5-\gamma_{\varepsilon}}{4})^{n}\varepsilon.$

It follows immediately that ${\mathcal{O}\left(|\log\varepsilon|\right)}$
iterations are required to achieve (75).

For the second phase, as $g$ is monotone increasing on $[0,1]$, it is easy to
see that

$\displaystyle 1-g(1-\delta)=$
$\displaystyle\tfrac{\delta}{1+\gamma_{\varepsilon}}\left((3-\gamma_{\varepsilon})\delta+2\gamma_{\varepsilon}-2\delta^{2}\right)$
$\displaystyle\leq$
$\displaystyle\tfrac{\delta}{1+\gamma_{\varepsilon}}\left((3-\gamma_{\varepsilon})\left(\tfrac{1}{2}-\sqrt{\tfrac{1+\gamma_{\varepsilon}^{2}}{8}}\right)+2\gamma_{\varepsilon}\right)$
$\displaystyle:=$ $\displaystyle a_{\gamma_{\varepsilon}+}\delta,$

where the inequality holds for
$0\leq\delta\leq\tfrac{1}{2}-\sqrt{\tfrac{1+\gamma_{\varepsilon}^{2}}{8}}$.
Another simple calculation shows that $0<a_{\gamma_{\varepsilon}+}<1$, which
means that the sequence created by iterating $1-g(1-\delta)$ starting with
$0\leq\delta\leq\tfrac{1}{2}-\sqrt{\tfrac{1+\gamma_{\varepsilon}^{2}}{8}}$
remains in the interval
$\left[0,\tfrac{1}{2}-\sqrt{\tfrac{1+\gamma_{\varepsilon}^{2}}{8}}\right]$.
Thus,

$1-g^{(n+1)}\left(\tfrac{1}{2}+\sqrt{\tfrac{1+\gamma_{\varepsilon}^{2}}{8}}\right)\leq
a_{\gamma_{\varepsilon}+}\left(1-g^{(n)}\left(\tfrac{1}{2}+\sqrt{\tfrac{1+\gamma_{\varepsilon}^{2}}{8}}\right)\right)\leq
a_{\gamma_{\varepsilon}+}^{n+1}\left(\tfrac{1}{2}-\sqrt{\tfrac{1+\gamma_{\varepsilon}^{2}}{8}}\right).$

Noting again that
$0<\tfrac{1}{2}-\sqrt{\tfrac{1+\gamma_{\varepsilon}^{2}}{8}}<\tfrac{1}{2}$, it
follows easily that the number of iterations required to obtain (76) is
${\mathcal{O}\left(k|\log\varepsilon|\right)}$.

We now turn to the second statement. The proof of this statement is very
similar to the first phase, but it does not simply follow by symmetry, so we
include it here. Again, we split the proof into two phases. For the first
phase, if
$\delta\in(0,\sqrt{(1+\gamma_{\varepsilon}^{2})/8}+\gamma_{\varepsilon}/2)$
then almost the same calculation as above shows that

$\begin{split}g(\tfrac{1+\gamma_{\varepsilon}}{2}-\delta)=&\tfrac{1+\gamma_{\varepsilon}}{2}-\tfrac{\delta}{2(1+\gamma_{\varepsilon})}(3+2\gamma_{\varepsilon}-(\gamma_{\varepsilon}-2\delta)^{2})\\\
\leq&\tfrac{1+\gamma_{\varepsilon}}{2}-\tfrac{5-\gamma_{\varepsilon}}{4}\delta.\end{split}$

Thus if
$\frac{1}{2}-g^{(n)}(\frac{1+\gamma_{\varepsilon}}{2}-\varepsilon)<\sqrt{\tfrac{1+\gamma_{\varepsilon}^{2}}{8}}$,
we have

$\displaystyle\tfrac{1+\gamma_{\varepsilon}}{2}-g^{(n+1)}(\tfrac{1+\gamma_{\varepsilon}}{2}-\varepsilon)\geq\tfrac{5-\gamma_{\varepsilon}}{4}\left(\tfrac{1+\gamma_{\varepsilon}}{2}-g^{(n)}(\tfrac{1+\gamma_{\varepsilon}}{2}-\varepsilon)\right)\geq(\tfrac{5-\gamma_{\varepsilon}}{4})^{n}\varepsilon.$

It follows immediately that ${\mathcal{O}\left(|\log\varepsilon|\right)}$
iterations are required to achieve

$g^{(n)}(\tfrac{1+\gamma_{\varepsilon}}{2}-\varepsilon)\leq\tfrac{1}{2}-\sqrt{\tfrac{1+\gamma_{\varepsilon}^{2}}{8}}.$
(77)

This time for the second phase observe that for
$0\leq\delta\leq\tfrac{1}{2}-\sqrt{\tfrac{1+\gamma_{\varepsilon}^{2}}{8}}$, we
have

$\displaystyle
g(\delta)\leq\frac{3+\gamma_{\varepsilon}}{1+\gamma_{\varepsilon}}\delta^{2}\leq\frac{3+\gamma_{\varepsilon}}{1+\gamma_{\varepsilon}}\left(\tfrac{1}{2}-\sqrt{\tfrac{1+\gamma_{\varepsilon}^{2}}{8}}\right)\delta:=a_{\gamma_{\varepsilon}-}\delta.$

Another simple calculation shows that $0<a_{\gamma-}<1$, which means that
$g^{(n)}(\delta)\in[0,\tfrac{1}{2}-\sqrt{\tfrac{1+\gamma_{\varepsilon}^{2}}{8}}]$
if $0\leq\delta\leq\tfrac{1}{2}-\sqrt{\tfrac{1+\gamma_{\varepsilon}^{2}}{8}}$.
Thus,

$g^{(n+1)}\left(\tfrac{1}{2}-\sqrt{\tfrac{1+\gamma_{\varepsilon}^{2}}{8}}\right)\leq
a_{\gamma_{\varepsilon}-}g^{(n)}\left(\tfrac{1}{2}-\sqrt{\tfrac{1+\gamma_{\varepsilon}^{2}}{8}}\right)\leq
a_{\gamma_{\varepsilon}-}^{n+1}\left(\tfrac{1}{2}-\sqrt{\tfrac{1+\gamma_{\varepsilon}^{2}}{8}}\right).$

Noting again that
$0<\tfrac{1}{2}-\sqrt{\tfrac{1+\gamma_{\varepsilon}^{2}}{8}}<\tfrac{1}{2}$, it
follows easily that it takes ${\mathcal{O}\left(k|\log\varepsilon|\right)}$
iterations to achieve

$g^{(n)}\left(\tfrac{1}{2}-\sqrt{\tfrac{1+\gamma_{\varepsilon}^{2}}{8}}\right)\leq\varepsilon^{k}.$
(78)

Combining (77) and (78) gives the second statement, which concludes the proof.

##  Appendix B A geometric computation

######  Proposition B.1.

Let
$\Omega=\\{(x_{1},x^{\prime}),x_{1}\in\mathbb{R},x^{\prime}\in\phi(x_{1})\subseteq\mathbb{R}^{{\mathbbm{d}}-1}$
with $\phi(x_{1})=H-h(x_{1})$. Let $z>0$,
$a=-z+\sqrt{\mathbbm{r}^{2}-(H+h(z))^{2}}$. Let $x\in\partial\Omega$,
$v_{1}(x)$ be the vector pointing from $x$ to $(a,0,..,0)$ and $v_{2}(x)$ the
(inward) normal vector of $\partial\Omega$ at $x$. Then,

$\langle
v_{1}(x),v_{2}(x)\rangle=H+h(-x_{1})+h^{\prime}(-x_{1})(-z-x_{1}-\sqrt{\mathbbm{r}^{2}-(H+h(z))^{2}}).$
(79)

###### Proof.

This is mostly a matter of writing the vectors and then the dot product
explicitly. Without loss of generality, by rotational symmetry, we can assume
$x=(x_{1},x_{2},0,\ldots,0)$. In fact, since $x$ is in the boundary of
$\partial\Omega$, we can assume that $x=(x_{1},H+h(-x_{1}),0,\ldots,0)$.
Therefore we can write $v_{1}(x)$ as,

$v_{1}(x)=(a-x_{1},-(H+h(-x_{1})),0,\ldots,0).$

On the other hand, since the boundary of the domain has a contour given by
$h(-x_{1})$, one can see $v_{2}(x)$ is given by,

$v_{2}(x)=(h^{\prime}(-x_{1}),-1,0,\ldots,0).$

Hence we have that,

$\displaystyle\langle v_{1}(x),v_{2}(x)\rangle$
$\displaystyle=h^{\prime}(-x_{1})(a-x_{1})+(H-h(-x_{1}))$
$\displaystyle=H+h(-x_{1})+h^{\prime}(-x_{1})(-z-x_{1}-\sqrt{\mathbbm{r}^{2}-(H+h(z))^{2}}).$

∎

##  Appendix C A SLFVS with reflecting boundary condition

Since our main focus in the deterministic setting was on the interaction
between selection and the shape of the domain, for completeness, we should
like analogous results in the stochastic setting. As is evident from the
results in the deterministic setting, such results should depend on local
effects around a spherical shell, and therefore follow from our work in
Section 3. However, the SLFVS has only previously been studied on the whole of
Euclidean space, or on a torus. In this section we therefore suggest a way in
which the SLFVS can be extended to a process with reflecting boundary
conditions, at least for a class of domains that includes $\Omega$ of Figure
1. The idea is simple: when an event overlaps the boundary of the domain, we
sample parents in such a way that the transition densities of the motion of
ancestral lineages will be what we obtained by reflecting the motion each time
it intersects the domain. To be more precise, to obtain a reflected motion we
will use the method of images, a classical way of obtaining solutions of
equations with reflecting boundary conditions, see e.g. [Smy88]. In the
interests of simplifying the notation, we present the details in
$\mathbbm{d}=2$, but the ideas are easily adapted to $\mathbbm{d}\geq 3$.

We reserve the symbol $\Omega$ for the domain of Figure 1, and use ${\cal D}$
to refer to the class of domains for which we now define the SLFVS with
reflecting boundary conditions.

#### Geometric definitions.

We will need two notions: reflected points at the boundary, and a
classification of balls depending on the way in which they intersect the
boundary of ${\cal D}$. Both will rest on a decomposition of the boundary of
${\cal D}$ into lines, and we only consider domains for which this is possible
(which of course includes $\Omega$). In higher dimensions we would replace
‘line’ by ‘plane’ and below we shall use the term ‘affine hyperplane’ for a
doubly infinite straight line and ‘line’ to mean a segment of such a line.

We define a line (in dimension $2)$ as a closed convex subset of an affine
hyperplane. We suppose that $\partial{\cal D}$ can be decomposed as a finite
union of lines. That is, there exist lines $(L_{i})_{i=1}^{N}$ such that

$\partial{\cal D}=\bigcup_{i=1}^{N}L_{i}.$ (80)

We suppose that the lines are maximal, in the sense that

$|L_{j}\cap L_{i}|\leq 1,\text{ and if }L_{j}\cap L_{i}\not=\emptyset\text{
then }H_{L_{i}}\not=H_{L_{j}}\>\>\>\forall j\not=i,$ (81)

where $H_{L_{i}}$ is the affine hyperplane that contains $L_{i}$. Note that
$\partial\Omega$ admits a unique decomposition into lines. Indeed, in that
case, it is elementary to write down explicit expressions; for example, one
line and its corresponding hyperplane are given by

$L=\\{(x_{1},R_{0})\in\Omega|x_{1}\leq 0\\}\subseteq
H_{L}=\\{(x_{1},R_{0})|x_{1}\in\mathbb{R}\\}.$

From now on we write $\mathcal{L}_{\cal D}$ for the set of lines such that
(80) and (81) hold for a domain $\mathcal{D}$.

Let $L\in\mathcal{L}_{\cal D}$, then let $H_{L}$ be the affine hyperplane such
that $L\subseteq H_{L}$, and write $\widehat{n}_{H_{L}}$ for the corresponding
normal vector. The orthogonal projection of $z$ into $H_{L}$ is given by
$z-\langle z,\widehat{n}_{H_{L}}\rangle\widehat{n}_{H_{L}}$. If this element
lies in $L$, then we define the reflection of $z$ with respect to $L$ as its
reflection with respect to $H_{L}$; that is the unique (other) point with the
same orthogonal projection and distance to $H_{L}$ as $z$. We also need to
define reflection with respect to a corner of our domain. A formal definition
is below, and we provide an illustration in Figure 8.

(i) (ii)

Figure 8: (i) Example of how we reflect a point with respect to a line. (ii)
Example of how we reflect a point with respect to a corner.

######  Definition C.1 (Reflected points).

For $L\in\mathcal{L}_{\cal D}$ and $z\in{\cal D}$ we define $RP(z,L)$, the
reflected point of $z$ with respect to $L$ as:

$RP(z,L):=\\{w\in{\cal D}^{c}|z-\langle
z,\widehat{n}_{H_{L}}\rangle\widehat{n}_{H_{L}}=w-\langle
w,\widehat{n}_{H_{L}}\rangle\widehat{n}_{H_{L}}\in L\text{ and
}d(z,H_{L})=d(w,H_{L})\\}.$

For $L_{1},L_{2}\in\mathcal{P}_{\Omega}$ with $L_{1}\cap L_{2}\not=\emptyset$
we define the set of reflected points with respect to the corner,
$RC(z,L_{1},L_{2})$, as:

$\displaystyle RC(z,L_{1},L_{2})=\\{$ $\displaystyle w\in{\cal
D}^{c}|(w\not\in RP(z,L_{1})\cup RP(z,L_{2}))\wedge(\exists\widehat{z}\in
RP(z,L_{2});$
$\displaystyle\widehat{z}-\langle\widehat{z},\widehat{n}_{H_{L_{1}}}\rangle\widehat{n}_{H_{L_{1}}}=w-\langle
w,\widehat{n}_{H_{L_{1}}}\rangle\in\Omega^{c})\wedge(d(\widehat{z},H_{L_{1}})=d(w,H_{L_{1}}))\\}.$

We also define the ‘completed’ versions:

$\overline{RP(z,L)}=RP(z,L)\cup\\{z\\},$
$\overline{RC(z,L_{1},L_{2})}=RC(z,L_{1},L_{2})\cup RC(z,L_{2},L_{1})\cup
RP(z,L_{1})\cup RP(z,L_{2})\cup\\{z\\}.$

We note that $|RP(z,L)|\leq 1$, as for any hyperplane there are two points
with the same orthogonal projection and distance to it. Also we remark that
for $L_{1},L_{2}\in\mathcal{L}_{\cal D}$ with $L_{1}\cap L_{2}\not=\emptyset$,
and $w\in\mathbb{R}$, setting

$\overline{RP}^{-1}(w,L_{1})=\\{z\in\Omega|z\in\overline{RP}(w,L_{1})\\},$
$\overline{RC}^{-1}(w,L_{1},L_{2})=\\{z\in\Omega|z\in\overline{RC}(w,L_{1},L_{2})\\},$

for any $z\in\Omega$ we have that

$\overline{RP}^{-1}(z,L_{1})=\overline{RC}^{-1}(z,L_{1},L_{2})=\\{z\\}.$ (82)

We now define a classification of balls depending on their intersection with
$\mathcal{D}$. For any $A\subset\mathbb{R}^{\mathbbm{d}}$ we set

$A\cap\mathcal{L}_{\cal D}:=\\{L\in\mathcal{L}_{\cal D}|A\cap
L\not=\emptyset\text{ or }A\cap H_{L}\cap{\cal D}^{c}\not=\emptyset\\}.$

######  Definition C.2.

[Classification of balls in $\mathbb{R}^{2}$] Let $x\in\mathbb{R}^{2}$ and
$r>0$.

  1. 1.

We say $B(x,r)$ is inside ${\cal D}$ if $B(x,r)\subset{\cal D}$.

  2. 2.

We say it is outside ${\cal D}$ if $B(x,r)\subset{\cal D}^{c}$.

  3. 3.

We say $B(x,r)$ intersects a side of ${\cal D}$ if
$|B(x,r)\cap\mathcal{L}_{\cal D}|=1$ and $B(x,r)$ is not outside of ${\cal
D}$.

  4. 4.

We say $B(x,r)$ intersects a corner if $|B(x,r)\cap\mathcal{L}_{\cal
D}|=|\\{L_{1},L_{2}\\}|=2$ with $L_{1}\cap L_{2}\not=\emptyset$ and $B(x,r)$
is not outside of ${\cal D}$.

  5. 5.

We say $B(x,r)$ intersects a hallway if $|B(x,r)\cap\mathcal{L}_{\cal
D}|=|\\{L_{1},L_{2}\\}|=2$ with $L_{1}\cap L_{2}=\emptyset$ and $B(x,r)$ is
not outside of ${\cal D}$.

Note that Definition C.2 provides a complete classifications of balls in
$\mathbb{R}^{2}$, of radius at most $r$, given that
$|B(x,r)\cap\mathcal{L}_{\cal D}|\leq 2$ for all $x\in\mathbb{R}^{2}$. With
this we can define the concept of reflected sampling in $B(x,r)$.

######  Definition C.3 (Reflected sampling).

Let $x\in\mathbb{R}^{2}$ and $r>0$ such that $B(x,r)$ is not outside ${\cal
D}$, does not intersect a hallway and
$|B(x,r)\cap\mathcal{L}_{\mathcal{D}}|\leq 2$. We define the reflected
sampling in $B(x,r)$ as follows:

  1. 1.

If $B(x,r)$ is inside ${\cal D}$ then it is just the uniform sampling in
$B(x,r)$.

  2. 2.

If $B(x,r)$ intersects a side of ${\cal D}$, that is
$B(x,r)\cap\mathcal{L}_{\cal D}=\\{P\\}$, then we pick a point $z$ uniformly
at random in $B(x,r)$, take the only point in $\overline{RP}^{-1}(z,P)$.

  3. 3.

If $B(x,r)$ intersects a corner of ${\cal D}$, that is
$B(x,r)\cap\mathcal{L}_{\cal D}=\\{L_{1},L_{2}\\}$, then we pick a point $z$
uniformly at random in $B(x,r)$, then pick a point uniformly at random in
$\overline{RC}^{-1}(z,L_{1},L_{2})$.

Reflected sampling can be thought of as sampling from the complete ball and
then, if the ball intersects the boundary, folding it along the boundary in
such a way that the content is all inside ${\cal D}$. This is illustrated for
the domain $\Omega$ in Figure 9.

Figure 9: Reflected sampling in $\Omega$. Each of the balls intersects
$\Omega$. For the ball contained completely within $\Omega$, the green point
is uniformly sampled in the usual way. For balls intersecting the boundary,
points in purple are sampled uniformly from the intersection of the ball with
$\Omega^{c}$, corresponding to a uniformly sampled point falling outside the
domain. The points in green correspond to the cases in which reflected
sampling leads to a unique point within the domain. The points in yellow
correspond to a case in which reflection gives two possible outcomes for the
reflected sampling and we need to choose one of them uniformly at random. The
purple lines join sampled points to the corresponding possible outcomes of
reflected sampling.

#### The SLFVS on $\Omega$.

The SLFVS on $\Omega$ can now be defined in the obvious way. It is driven by
the same Poisson Point Process $\Pi$ of events as in Definition 1.13, but now
for a point $(t,x,r)\in\Pi$, if $B(x,r)\cap\Omega=\emptyset$ then nothing
happens, and when $B(x,r)$ intersects the boundary $\partial\Omega$, parental
locations are chosen according to reflected sampling, and the allele
frequencies are, of course, only updated in $\Omega$. This construction will
work for any domain ${\cal D}$ that fulfils the following conditions:

  1. 1.

There exists a set of lines $\mathcal{L}_{\cal D}$ such that ${\cal D}$
fulfils (80) and (81).

  2. 2.

There is $\mathcal{R}_{\cal D}>0$ such that, for all $x\in\mathbb{R}^{2}$, we
have $|B(x,\mathcal{R}_{\cal D})\cap\mathcal{L}_{\cal D}|\leq 2$ and
$B(x,\mathcal{R}_{\mathcal{D}})$ does not intersect a hallway.

  3. 3.

For any $r<\mathcal{R}_{\cal D}$ and $x\in\mathbb{R}^{2}$, if $B(x,r)$
intersects a corner given by $L_{1},L_{2}$ then
$\overline{RC}^{-1}(w,L_{1},L_{2})\not=\emptyset$ for all $w\in B(x,r)$. If
$B(x,r)$ intersects a side given by the line $L$, then
$\overline{RP}^{-1}(w,P)\not=\emptyset$ for all $w\in B(x,r)$.

We can then construct the SLFVS on ${\cal D}$, provided that events are
assumed to have radius bounded by $\mathcal{R}_{{\cal D}}$. These conditions
are sufficient to ensure that the reflected sampling that we use to choose a
parental location will be well defined. The third condition prevents there
being no pre-image through the reflection of the parental location; the second
removes the possibility of an event intersecting two walls, for which we have
not defined the corresponding reflected points. We could, of course, extend
our definitions further, but we are primarily interested in the domain
$\Omega$, and one can check that $\Omega$ fulfils these conditions with
$\mathcal{R}_{\Omega}\leq r_{0}\wedge R_{0}-r_{0}$.

#### Blocking in the stochastic setting

Using the approach adopted in Theorem 1.17 we can prove a stochastic analogue
of Theorem 1.6. While we omit the details, the key tool is a branching and
coalescing dual for the SLFVS on $\Omega$. This mirrors the dual for the
process on the whole Euclidean space, introduced in Definition 3.1, except
that uniform sampling of offspring locations is replaced by reflected
sampling.

The key ideas in the proof of Theorem 1.17 can then be adapted in the obvious
way to this setting. Since we deliberately constructed the reflected sampling
in such a way that transition probabilities of the jump process followed by a
lineage could be obtained from those on Euclidean space via the method of
images, it should be no surprise that Lemma 3.8 can be translated to this
setting on replacing Brownian motion by reflected Brownian motion. The
technical details will appear in the doctoral thesis of the third author.

## References

  * [AHM64] M. Alfaro, H. Hilhorst and H. Matano  “The singular limit of the Allen–Cahn equation and the FitzHugh–Nagumo system”  In _J. Differ. Equ._ 245.2, 1964, pp. 505–565
  * [Alf+18] M. Alfaro, D. Antonopoulou, G. Karali and H. Matano  “Generation of fine transition layers and their dynamics for the stochastic Allen-Cahn equation”, 2018  URL: arXiv:1812.03804v3
  * [BBC16] H. Berestycki, J. Bouhours and G. Chapuisat  “Front blocking and propagation in cylinders with varying cross section”  In _Cacl. Var. Partial Dif._ 55.3, 2016, pp. 44  DOI: 10.1007/s00526-016-0962-2
  * [BH91] R.. Bass and P. Hsu  “Some potential theory for reflecting Brownian motion in Holder and Lipschitz domains.”  In _Ann. Probab._ 19.2, 1991, pp. 486–508
  * [Che92] X. Chen  “Generation and propagation of interfaces for reaction-diffusion equations”  In _J. Diff. Eq._ 96, 1992, pp. 116–141
  * [DFL86] A. De Masi, P.. Ferrari and J.. Lebowitz  “Reaction-diffusion equations for interacting particle system”  In _J. Stat. Phys._ 44, 1986, pp. 589–644  DOI: 10.1007/BF01011311
  * [EFP17] A.. Etheridge, N. Freeman and S. Penington  “Branching Brownian motion, mean curvature flow and the motion of hybrid zones”  In _Electronic Journal of Probability_ 22 The Institute of Mathematical Statisticsthe Bernoulli Society, 2017
  * [Eth+17] A.M. Etheridge, N. Freeman, S. Penington and D. Straulino  “Branching Brownian motion and selection in the spatial $\Lambda$-Fleming–Viot Process”  In _Ann. Appl. Probab._ 27.5, 2017, pp. 2605–2645  DOI: 10.1007/s00208-008-0258-8
  * [EV12] A.. Etheridge and A. Véber  “The spatial $\Lambda$-Fleming–Viot process on a large torus: Genealogies in the presence of recombination.”  In _Ann. Appl. Probab._ 22.6, 2012, pp. 2165–2209  DOI: 10.1214/12-AAP842
  * [EVY20] A.. Etheridge, A. Véber and F. Yu  “Rescaling limits of the spatial Lambda-Fleming-Viot process with selection”  In _Electron. J. Probab._ 25, 2020, pp. 1–89  DOI: 10.1214/20-EJP523
  * [Fun95] T. Funaki  “The scaling limit for a stochastic PDE and the separation of phases”  In _Probab. Th. Rel. Fields_ 102, 1995, pp. 221–288  DOI: 10.1007/BF01213390
  * [Fun99] T. Funaki  “Singular limit for stochastic reaction-diffusion equation and generation of random interfaces”  In _Acta Mathematica Sinica_ 15, 1999, pp. 407–438  DOI: https://doi.org/10.1007/BF02650735
  * [Goo18] M. Gooding  “Long term behaviour of spatial population models with heterozygous or asymmetric homozygous selection”, 2018  URL: https://ora.ox.ac.uk/objects/uuid:ef35b918-35ad-4f90-98ea-325fc692f1eb
  * [HRW12] M. Hairer, M.. Ryser and H. Weber  “On the triviality of the 2D Allen-Cahn equation”  In _Electron. J. Probab._ 17.39, 2012, pp. 14
  * [Lee18] K. Lee  “Generation of interfaces for multi-dimensional stochastic Allen–Cahn equation with a noise smooth in space”  In _Stochastics_ 90.6 Taylor & Francis, 2018, pp. 836–860  DOI: 10.1080/17442508.2018.1426586
  * [McK67] H.. McKean  “Propagation of chaos for a class of non-linear parabolic equations”  In _Lecture Series in Differential Equations_ 2, 1967, pp. 177–194
  * [McK75] H.. McKean  “Application of Brownian motion to the equation of Kolmogorov-Petrovski-Piskunov”  In _Common. Pure Appl. Math._ 28, 1975, pp. 323–331
  * [MNL06] H. Matano, K. Nakamura and B. Lou  “Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit”  In _Networks and Heterogeneous Media_ 1.4, 2006, pp. 537–568
  * [Sko64] A.. Skorohod  “Branching diffusion processes.”  In _Th. Prob. Appl._ 9, 1964, pp. 492–497
  * [Smy88] W.B. Smythe  “Static and dynamic electricity”  United States: Hemisphere Publishing., 1988
  * [VW15] A. Véber and A. Wakolbinger  “The spatial Lambda-Fleming–Viot process: An event-based construction and a lookdown representation”  In _Ann. Inst. H. Poincaré Probab. Statist._ 51.2, 2015, pp. 570–598  DOI: 10.1214/13-AIHP571


## System Architecture

The LeafAI query engine was designed using a modular, micro service-based
architecture with a central Application Program Interface (API) which
orchestrates end-to-end query generation. Inter-module communication is
performed using gRPC [grpc], a robust open-source remote procedure call
framework which enables language-agnostic service integration. This allows
individual modules to be implemented (and substituted) in programming
languages and using libraries well-suited to a given task. A diagram of the
LeafAI query engine architecture is shown in Figure 1.

Figure 1: LeafAI query architecture. Inter-module communication is performed
using the gRPC framework. Individual modules are deployed as Docker [docker]
containers and communicate solely with the central API, which orchestrates
query generation and handles query generation requests.

At a high level, query generation is performed in the following steps:

  1. 1.

A query request is received by the API in the form of inclusion and exclusion
criteria as free-text strings.

  2. 2.

The input texts are tokenized and named entity recognition is performed to
determine spans of text representing conditions, procedures, and so on.

  3. 3.

Relation extraction is performed to determine relations between the entities,
such as Caused-By or Numeric-Filter.

  4. 4.

The input texts are transformed by replacing spans of ”raw” text with logical
form names. For example, ”Diagnosed with diabetes” would become ”Diagnosed
with cond(”diabetes”).” The resulting input texts are in turn transformed into
an output logical representation using a Sequence to Sequence (Seq2Seq)
architecture, in the form of a string.

  5. 5.

A logical form interpreter module implemented as a recursive descent parser
[johnstone1998generalised] reads the logical form string input and
instantiates it as an abstract syntax tree (AST) of nested in-memory logical
form objects.

  6. 6.

”Named” logical form objects (i.e., specified with quoted text, such as
”cond(”diabetes”)”) are normalized into one or more corresponding UMLS
concepts. UMLS child concepts are also added using our KB. For example,
”cond(”type 2 diabetes”)” would also include concepts for type 2 diabetes with
kidney complications (C2874072).

  7. 7.

Working recursively inside-to-outside the AST structure, each logical form
object calls a Reason() method which executes various rules depending on
context.

  8. 8.

Each reasoning rule is performed as one or more pre-defined SPARQL queries to
the KB, concept by concept.

  9. 9.

The final normalized, reasoned, logical form AST is thus a nested structure of
UMLS concepts. Each AST criterion is mapped to zero or more corresponding
entries in the semantic metadata mapping (SMM), which in turn lists meanings,
roles, and relations of a database schema in the form of UMLS concepts.

  10. 10.

The final mapped AST object is transformed into a series of database queries,
one per line of eligibility criteria text. The output SQL query can either be
executed directly on a database or returned to the API caller.

Figure 2 illustrates an example of this process. In the following subsections
we examine these steps in detail.

Figure 2: LeafAI query generation processes

## Named entity recognition and relation extraction

Named entity recognition (NER) refers to the segmentation and identification
of tokens within an input sentence as ”entities”, such as conditions or
procedures. We used the Leaf Clinical Trials (LCT) corpus [dobbins2022leaf] to
train two BERT-based [devlin2018bert] NER extractors, one each for LCT
general- and fine-grained-entities (see [dobbins2022leaf] for more information
on LCT entity types). Next, we perform relation extraction between named
entity pairs similarly using a BERT-based model also trained on the LCT
corpus.

## Logical form transformation

One of the core challenges of generating queries for eligibility criteria is
the problem of logical representation. Generating queries directly based on
named entities and relations alone, while practical, performs poorly in cases
of nested or particularly complex logic. An alternative to this approach is to
use a so-called intermediate representation (IR), which transforms the
original natural language by removing ”noise” unnecessary to a given task and
which more logically represents underlying semantics (see Herzig et al
[herzig2021unlocking] for an examination of IR-based SQL generation
approaches). Similar to earlier work using Description Logics, Roberts and
Demner-Fushman [roberts2016annotating] proposed a representation of questions
on EHR databases using a comparatively compact but flexible format using first
order logic expressions, for example, representing ”Is she wheezing this
morning?” as

> $\delta(\lambda x.has\\_problem(x,C0043144,status)\wedge
> time\\_within(x,\mathrm{"this\ morning"}))$

This style of representation is powerfully generalizable, but also difficult
to translate directly into SQL statements as multiple predicates (e.g.,
has_problem and time_within) may actually correspond to one or many SQL
statements, depending on context, complicating direct transformation into
queries.

We thus chose a similar intermediate representation (hereafter simply ”logical
forms”) as proposed by Roberts and Demner-Fushman but more closely resembling
a nested functional structure in programming languages such as Python or
JavaScript and more amenable to SQL generation. A criterion such as ”Diabetic
women and men over age 65” would be represented by our logical forms as

> $intersect(\\\ \mathrm{\ \ \ \ }cond("Diabetic"),\\\ \mathrm{\ \ \ \
> }union(female(),male()),\\\ \mathrm{\ \ \ \
> }age().num\\_filter(eq(op(GT),val("65")))\\\ )$

A description of our logical forms annotation schema, corpus, called the Leaf
Logical Forms (LLF) corpus, annotation process, and performance metrics can be
found in Appendix A.

After NER and relation extraction are performed, we leverage T5
[raffel2020exploring], a state-of-the-art Seq2Seq architecture we fine-tuned
for predicting logical forms on the LLF corpus. As inputs to the Seq2Seq model
we use the original eligibility criteria with named entity spans replaced by
logical form representations, since we found this improved performance
compared to training with raw inputs. Thus the above example input would be
transformed to

> ”cond(“Diabetic”) female() and male() over age() eq(op(GT), val(“65”))”

The returned logical form string is then instantiated into an abstract syntax
tree (AST) of nested in-memory logical form objects using a recursive descent
parser [johnstone1998generalised] within our API.

## Concept normalization

Normalization refers to the mapping of free-text string values (e.g.,
”diabetes mellitus”) to coded representations (e.g., UMLS, ICD-10, SNOMED, or
LOINC). We normalize ”named” logical forms to UMLS concepts using MetaMapLite
[aronson2001effective, demner2017metamap]. We consider a logical form ”named”
if it contains a free-text value surrounded by quotes. For example, cond() is
unnamed and refers to any condition or disease, while cond(”hypertension”) is
named as it refers to a specific condition.

Normalization using MetaMapLite can often result in high recall but low
precision, as MetaMapLite has no NER component and tends to return UMLS
concepts which match a given phrase syntactically but refer to abstract
concepts not of interest (e.g., a search for ”BMI” may return ”body mass
index” (C1305855), but also organic chemical ”BMI 60” (C0910133)). To improve
normalization precision, we employ two strategies. First, our NER component
filters predicted UMLS concepts to only those of specific semantic types. For
example, we limit condition concepts to only those related to semantic types
of diseases or syndrome (dsyn) and so on. Next, using term-frequencies pre-
computed across UMLS concept phrases, we compare term frequency-inverse
document frequency (tf-idf) on MetaMapLite predictions, removing UMLS concepts
whose summed matched spans have a tf-idf score lower than that of unmatched
spans in a given named entity. For example, for the string ”covid-19
infection”, MetaMapLite predicts both ”COVID-19” (C5203670) as well as several
concepts related to general infections. Our tf-idf strategy removes general
infection concepts because “infection” has a lower tf-idf score than the
summed scores for “covid” + “-” + “19”.

Laboratory values present a particular challenge, as LeafAI expects predicted
lab concepts to have directly associated LOINC codes, while MetaMapLite
typically normalizes lab test strings to UMLS concepts of semantic type
”laboratory test or finding”, but which do not have direct mappings to LOINC
codes. For example, a search for ”platelet count” returns the concept
”Platelet Count Measurement” (C0032181), but not the needed concept of
”Platelet # Bld Auto” (C0362994) with LOINC code “777-3”. Thus similar to Lee
and Uzuner with medications [lee2020normalizing], we trained a BERT model for
sequence classification to normalize lab tests. We trained this model to
identify UMLS concepts associated with LOINC codes most frequently used in
eligibility criteria [rafee2022elapro], with each CUI as a possible class.

## Reasoning using an integrated knowledge base

For reasoning and derivation of ICD-10, LOINC, and other codes for UMLS
concepts, we designed a KB accessible via SPARQL queries and stored as
Resource Description Framework (RDF) [manola2004rdf] triples. The core of our
KB is the UMLS, derived using a variation of techniques created for ontologies
in BioPortal [noy2009bioportal]. To further augment the UMLS, we mapped and
integrated the Disease Ontology [schriml2012disease], Symptom Ontology
[sayers2010database], COVID-19 Ontology [sargsyan2020covid], Potential Drug-
Drug Interactions [ayvaz2015toward], LOINC2HPO [zhang2019semantic], and the
Disease-Symptom Knowledge Base [wang2008automated], as well as ICD-9, ICD-10,
SNOMED, and other general equivalence mappings not present in the UMLS . We
then developed SPARQL queries parameterized by UMLS concepts for various
scenarios which leveraged our KB, such as contraindications to treatments,
symptoms of diseases, and so on. Using LOINC2HPO mappings further allows us to
infer phenotypes by lab test results rather than using ICD-10 or SNOMED codes
alone.

Our KB, nested logical forms, and inside-to-outside normalization methods
enable ”multi-hop” reasoning on eligibility criteria over several steps. For
example, given the non-specific criterion ”Contraindications to drugs for
conditions which affect respiratory function”, our system successfully reasons
that (among other results),

  1. 1.

Asthma causes changes to respiratory function

  2. 2.

Methylprednisolone can be used to treat asthma

  3. 3.

Mycosis (fungal infection) is a contraindication to methylprednisolone

These features allow LeafAI to reason upon fairly complex non-specific
criteria.

## Query generation using semantic metadata mapping

To enable data model-agnostic query generation, we leveraged a subset of codes
within the UMLS in what we define as a semantic metadata mapping, or SMM. An
SMM includes a listing of available databases, tables, columns, and so on
within a given database schema. Critically, these database artifacts are
”tagged” using UMLS concepts. An example of this can be seen in Figure 3,
which shows strategies by which a given criterion can be used to generate
schema-specific queries by leveraging different SMMs. In cases where the
LeafAI query engine finds more than one means of querying a concept (e.g., two
SQL tables for diagnosis codes), the queries are combined in a UNION
statement.

Figure 3: The LeafAI query engine’s SQL query generation process using two
hypothetical database schema to generate queries for platelet counts (shown in
logical form after normalization). This example illustrates the flexibility of
LeafAI’s semantic metadata mapping system (represented here in JSON format) in
adapting to virtually any data model. On the left, ”Tall Table Structure”,
platelet counts must be filtered from within a general purpose ”labs” table.
The LeafAI KB recognizes that labs may be stored as LOINC codes, and the
corresponding SMM indicates that records in this table can be filtered to
LOINC values. On the right, ”Pivoted Table Structure”, platelet counts are
stored as a specific column in a ”complete_blood_counts” table, and thus can
be directly queried without further filtering. Additional metadata, columns,
tables, types and so on needed in SMMs are omitted for brevity.

## Evaluation

It is reasonable to expect that an NLP-based system for finding patients based
on eligibility criteria would find many patients who actually enrolled in a
real clinical trial — assuming that patients enrolled in those trials met the
necessary criteria as determined by study investigators. While there are
caveats to this approach (for example, certain diagnosis codes may be missing
for some patients, etc.), we suggest that tools such as LeafAI be evaluated by
their ability to handle real-world eligibility criteria and clinical data.

In this study we compared LeafAI’s results to that of a human database
programmer with over 5 years of experience as a skilled analyst using SQL
databases and other tools related to our EHR. This study was approved by the
University of Washington’s Institutional Review Board. Our evaluation was
performed as follows:

  1. 1.

We extracted metadata on 168 clinical trials from our EHR between January 2017
and December 2021 where at least 10 patients were indicated as enrolled and
not withdrawn, and the total number of raw lines of free-text within the
eligibility criteria (besides the phrases ”Inclusion Criteria” and ”Exclusion
Criteria”) was less than or equal to 30.

  2. 2.

By manual review, we excluded 22 trials with multiple sub-groups, as it would
not be possible to know which eligibility criteria applied to which sub-group
of enrolled patients.

  3. 3.

To narrow the scope of our evaluation, we chose to evaluate only trials
studying the following 7 disease groups: Cardiology, COVID-19, Crohn’s
Disease, Multiple Sclerosis (MS), Diabetes Mellitus, Hepatitis C, and Cancer.
Using the ”condition” field for each trial within metadata from
https://clinicaltrials.gov, we filtered and grouped the remaining 146 trials
into only those studying our diseases of interest. These diseases were
selected to provide a diverse representation of potential queries that not
only span different systems or medical specialties (infection, malignancy,
cardiac, neurologic, endocrinologic) but also included conditions (COVID-19,
Crohn’s disease, diabetes) commonly studied in clinical trials.

  4. 4.

We randomly chose 1 trial from each group, with the exception of Cancer, where
given the large number of trials and variety of cancer types, we chose 2
trials. 427 patients were enrolled across the chosen 8 clinical trials.

  5. 5.

Both LeafAI and the human programmer were provided the raw text of eligibility
criteria from https://clinicaltrials.gov, (LeafAI using API calls). Each then
created queries to find patients based on each eligibility criteria, which we
executed on an OMOP database derived from our EHR containing our institution’s
entire research-eligible patient population.

  6. 6.

To ensure results returned would be limited to only data available during the
time of each trial, we replaced references to the SQL function for generating
a current timestamp (GETDATE()) with that of each trial’s end date, and
similarly replaced OMOP table references with SQL views filtering data to only
that existing prior to the end of a trial.

  7. 7.

To ensure queries would be comparable to LeafAI, the human programmer was
instructed to (1) ignore criteria which cannot be computed, such as “Willing
and able to participate in the study” or criteria subject to “the opinion of
the investigator”, as the intents and consents of patients and investigators
are unavailable in our data, (2) make a best effort to reason upon non-
specific criteria (e.g., symptoms for a condition), (3) not check whether
patients found by a human query enrolled within a trial, and (4) skip criteria
which cause an overall query to find no eligible patients. While our team did
review examples of each of these cases, we did not define formal guidelines
and the human programmer instead used their best judgment.