# On exceptional times for generalized Fleming–Viot processes with mutations

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## Abstract

If \(\mathbf Y\) is a standard Fleming–Viot process with constant mutation rate (in the infinitely many sites model) then it is well known that for each \(t>0\) the measure \(\mathbf Y_t\) is purely atomic with infinitely many atoms. However, Schmuland proved that there is a critical value for the mutation rate under which almost surely there are exceptional times at which the stationary version of \(\mathbf Y\) is a finite sum of weighted Dirac masses. In the present work we discuss the existence of such exceptional times for the generalized Fleming–Viot processes. In the case of Beta-Fleming–Viot processes with index \(\alpha \in \,]1,2[\) we show that—irrespectively of the mutation rate and \(\alpha \)—the number of atoms is almost surely always infinite. The proof combines a Pitman–Yor type representation with a disintegration formula, Lamperti’s transformation for self-similar processes and covering results for Poisson point processes.

### Keywords

Fleming–Viot processes Mutations Exceptional times Excursion theory Jump-type SDE Self-similarity### Mathematics Subject Classification (2000)

Primary 60J80 Secondary 60G18## 1 Main result

*Fleming–Viot diffusion processes*were first introduced by Fleming and Viot [20] and have become a cornerstone of mathematical population genetics in the last decades. It is a model which describes the evolution (forward in time) of the genetic composition of a large population. Each individual is characterized by a

*genetic type*which is a point in a type-space \(E\). The Fleming–Viot process is a Markov process \((\mathbf Y _t)_{t\ge 0}\) on

*infinitely-many-alleles model*where each mutation creates a new type never seen before. Without loss of generality let the type space be \(E=[0,1]\). Then the following choice of \(A\) gives an example of an infinite site model with mutations:

- (i)
If there is no mutation, then, for all \(t>0\) fixed, the number of types is almost surely finite.

- (ii)
If the mutation parameter \(\theta \) is strictly positive, then, for all \(t>0\) fixed, the number of types is infinite almost surely.

A beautiful complement to (i) and (ii) was found by Schmuland for exceptional times that are not fixed in advance:

**Theorem 1.1**

Schmuland’s proof of the dichotomy is based on analytic arguments involving the capacity of finite dimensional subspaces of the infinite dimensional state-space. In Sect. 6 we reprove Schmuland’s theorem with a simple proof via excursion theory, that yields the result for arbitrary initial conditions.

*-Fleming–Viot processes*, a class of stochastic processes which naturally extends the class of standard Fleming–Viot processes. These processes are completely characterized by a finite measure \(\Lambda \) on \(]0,1]\) and a generator \(A\). Similarly to the standard Fleming–Viot process, these processes can be defined through their infinitesimal generator

If we chose \(\alpha \in \, ]1,2[\) and \(\mathbf Y _0\) uniform on \([0,1]\), then we find the same properties (i) and (ii) for the one-dimensional marginals \(\mathbf Y _t\) unchanged with respect to the classical case (1.1), (1.2). In fact, for a general \(\Lambda \)-Fleming–Viot process, (i) is equivalent to the requirement that the associated \(\Lambda \)-coalescent comes down from infinity (see for instance [2]). Here is our main result: contrary to Schmuland’s result, \((\alpha ,\theta )\)-Fleming–Viot processes with \(\alpha \in \, ]1,2[\) and \(\theta >0\) never have exceptional times:

**Theorem 1.2**

One way one can get a first rough understanding of why this should be true is by using a heuristic based on the duality between \(\Lambda \)-Fleming–Viot processes and \(\Lambda \)-coalescents. If \(\Lambda \)-Fleming–Viot processes describe how the composition of a population evolve forward in time, \(\Lambda \)-coalescents describe how the ancestral lineages of individuals sampled in the population merge as one goes back in time. The fact that \(\Lambda \)-coalescents describe the genealogies of \(\Lambda \)-Fleming–Viot processes can be seen from Donelly and Kurtz [14] so-called lookdown construction of Fleming–Viot processes and was also established through a functional duality relation by Bertoin and Le Gall in [5].

The coalescent which corresponds to the classical Fleming–Viot process is the celebrated Kingman’s coalescent. Kingman’s coalescent comes down from infinity at speed \(2/t\), i.e. if one initially samples infinitely many individuals in the populations, then the number of active lineages at time \(t\) in the past is \(N_t\) and \(N_t \sim 2/t\) almost surely when \(t\rightarrow 0\). It is known (see [6] or more recently [28]) that the process \((N_t, t\ge 0)\) has the same law as the process of the number of atoms of the Fleming–Viot process. For a Beta-coalescent (that is a \(\Lambda \)-coalescent where the measure \(\Lambda \) is the density of a Beta\((\alpha ,2-\alpha )\) variable) with parameter \(\alpha \in (1,2)\) we have \(N_t \sim c_\alpha t^{-1/(\alpha -1)}\) almost surely as \(t\rightarrow 0\) (see [3, Theorem 4]). Therefore Kingman’s coalescents comes down from infinity much quicker than Beta-coalescents. Since the speed at which the generalized Fleming–Viot processes looses types roughly corresponds to the speed at which the dual coalescent comes down from infinity, it is possible that \((\alpha ,\theta )\)-Fleming–Viot processes do not lose types fast enough, and hence there are no exceptional times at which the number of types is finite.

## 2 Auxiliary constructions

To prove Theorem 1.2 we construct two auxiliary objects: a particular measure-valued branching process and a corresponding Pitman–Yor type representation. Those will be used in Sect. 5 to relate the question of exceptional times to covering results for point processes. In this section we give the definitions and state their relations to the Beta-Fleming–Viot processes with mutations. All appearing stochastic processes and random variables will be defined on a common stochastic basis \((\Omega , \mathcal {G}, \mathcal {G}_{t}, {\mathbb P} )\) that is rich enough to carry all Poisson point processes (PPP in short) that appear in the sequel.

### 2.1 Measure-valued branching processes with immigration

*measure-valued branching process with interactive immigration*(MBI in short). For a textbook treatment of this subject we refer to Li [31]. Following Dawson and Li [12], we are not going to introduce the MBIs via their infinitesimal generators but as strong solutions of a system of stochastic differential equations instead. On \((\Omega , \mathcal {G}, \mathcal {{G}_{t}},{\mathbb P} )\), let us consider a Poisson point process \(\mathcal {N}=(r_i,x_i,y_i)_{i\in I}\) on \((0,\infty )\times (0,\infty )\times (0,\infty )\) adapted to \( \mathcal {{G}_{t}}\) and with intensity measure

*total mass*of the CSBP starting at time zero at the mass \(X_0=v\). We are going to consider all initial masses \(v\in [0,1]\) simultaneously, constructing a process \((\mathbf {X}_t)_{t\ge 0}\) taking values in the space \({\mathcal M} ^{F}_{[0,1]}\) of finite measures on \([0,1]\), endowed with the narrow topology, i.e. the trace of the weak-\(*\) topology of \((C[0,1])^*\). Assume that at time \(t=0\), \(\mathbf {X}_0\) is a finite measure on \([0,1]\) with cumulative distribution function \((F(v), v\in [0,1])\), and denote

**(G)**\(g : {\mathbb R}_+ \mapsto {\mathbb R}_+\) is monotone non-decreasing, continuous and locally Lipschitz continuous away from zero.

**Definition 2.1**

it is càdlàg \({\mathbb P} \)-a.s.,

for all \(v\in [0,1]\), setting \(X_t(v):=\mathbf X _t([0,v])\), \((X_t(v))_{v\in [0,1],t\ge 0}\) satisfies \({\mathbb P} \)-a.s. (2.6).

Here is a well-posedness result for (2.6):

**Theorem 2.2**

Let \(F\) and \(I\) be as above. For any immigration mechanism \(g\) satisfying Assumption **(G)**, there is a strong solution \((\mathbf X _t)_{t\ge 0}\) to (2.6) and pathwise uniqueness holds until \(T_0:= \inf \{t \ge 0 : \mathbf X _t([0,1]) =0\}\).

The proof of Theorem 2.2 relies on ideas from recent articles on pathwise uniqueness for jump-type SDEs such as Fu and Li [22] or Dawson and Li [12]. Our equation (2.6) is more delicate since all coordinate processes depend on the total-mass \(X_t(1)\). The uniqueness statement is first deduced for the total-mass \((X_t(1))_{t\ge 0}\) and then for the other coordinates interpreting the total-mass as random environment. To construct a (weak) solution we use a (pathwise) Pitman–Yor type representation as explained in the next section.

### 2.2 A Pitman–Yor type representation for interactive MBIs

*Kuznetsov measure*, see [30, Section 4] or [31, Chapter 8]: For all \(t\ge 0\), let \(K_t(dx)\) be the unique \(\sigma \)-finite measure on \({\mathbb R}_+\) such that

*excursion measure*of the CSBP (2.3). By (2.8), it is easy to check that for any \(s>0\)

**Theorem 2.3**

**(G)**. Then, for all \(v\in [0,1]\), there is a unique càdlàg process \((Z_t(v), t\ge 0)\) on \((\Omega , \mathcal {G}, \mathcal {G}_{t}, {\mathbb P} )\) satisfying \({\mathbb P} \)-a.s.

*Remark 2.4*

The recent monograph [31] by Zenghu Li contains a full theory of this kind of Pitman–Yor type representations for measure-valued branching processes, see in particular Chapter 10. We present a different approach below which shows directly how the different Poisson point processes in (2.6) and in (2.13) are related to each other. The most important feature of our construction is that it relates the excursion construction and the SDE construction on a pathwise level.

Observe that an immediate and interesting corollary of Theorem 2.3 is the following:

**Corollary 2.5**

Let \(g\) be an immigration mechanism satisfying assumption **(G)** and let \((\mathbf {X}_t)_{t\ge 0}\) be a solution to (2.6). Then almost surely, \(\mathbf {X}_t\) is purely atomic for all \(t\ge 0\).

In the proof of our Theorem 1.2 we make use of the fact that the Pitman–Yor type representation is well suited for comparison arguments. If \(g\) can be bounded from above or below by a constant, then the righthand side of (2.6) can be compared to an explicit PPP for which general theory can be applied.

### 2.3 From MBI to Beta-Fleming–Viot processes with mutations

Let us first recall an important characterization started in [6] and completed in [12] which relates Fleming–Viot processes, defined as measure-valued Markov processes by the generator (1.3), and strong solutions to stochastic equations.

**Theorem 2.6**

Existence and uniqueness of solutions for this equation was proved in Theorem 4.4 of [12] while the characterization of the generator of the measure-valued process \({\mathbf Y} \) is the content of their Theorem 4.9.

We next extend a classical relation between Fleming–Viot processes and measure-valued branching processes which is typically known as disintegration formula. Without mutations, for the standard Fleming–Viot process this goes back to Konno and Shiga [26] and it was shown in Birkner et al. [10] that the relation extends to the generalized \(\Lambda \)-Fleming–Viot processes without immigration if and only if \(\Lambda \) is a Beta-measure. Our extension relates \((\alpha ,\theta )\)-Fleming–Viot processes to (2.6) with immigration mechanism \(g(x)=\alpha (\alpha -1)\Gamma (\alpha )\theta x^{2-\alpha }\) and for \(\theta =0\) gives an SDE formulation of the main result of [10].

**Theorem 2.7**

The proof of the theorem is different from the known result for \(\theta =0\). To prove that \(X_{S^{-1}(t)}(1)>0\) for all \(t\ge 0\), Lamperti’s representation for CSBPs was crucially used in [10]. This idea breaks down in our generalized setting since the total-mass process \(X_t(1)\) is not a CSBP. Our proof uses instead the fact that for all \(\theta \ge 0\) the total-mass process is self-similar and an interesting cancellation effect of Lamperti’s transformation for self-similar Markov processes and the time-change \(S\).

**Corollary 2.8**

As \(t \rightarrow \infty \) the probability-valued process \(\left( {\mathbf {X}_{S^{-1}(t)}(dv) \over X_{S^{-1}(t)}(1)}\right) _{t\ge 0}\) converges weakly to the unique invariant measure of \((\mathbf {Y}_t, t\ge 0)\).

This corollary is a direct consequence of the result, due to Donnelly and Kurtz [13, 14], that the \((\alpha , \theta )\)-Fleming–Viot process (as well as its lookdown particle system) is strongly ergodic and of Theorem 2.7. For the sake of self-containdeness, a sketch of the proof is given in Sect. 7 which specialize and explicits the arguments of Donelly and Kurtz to our case.

## 3 Proof of Theorems 2.2 and 2.3

### 3.1 The Pitman–Yor type representation with predictable random immigration

#### 3.1.1 Definition of \({\mathcal N} \)

- (1)
\((r_j^i)_{j\in J^i}\) is the family of jump times of \(r\mapsto w^i_r\);

- (2)
for each \(r_j^i\) we set

**Proposition 3.1**

\({\mathcal N} \) is a PPP with intensity \(\nu (dr , dx , dy) = dr\otimes c_\alpha \, x^{-1-\alpha }\, dx\otimes dy\).

*Proof*

- (1)
Conditionally on \(w^k_t\) and \(s_k < t\) the process \(w^k_{\cdot +t}\) has law \(P_{w^k_t}\) (this follows for instance from (2.7)).

- (2)Let \((w_t, t\ge 0)\) be a CSBP started from \(w_0\) with law \({\mathbb P} _{w_0}\). Let \({\mathcal M} =(r_i,x_i, y_i)\) be a point process which is defined from \(w\) and a sequence of i.i.d. uniform variables on \([0,1]\) as \({\mathcal N} ^k\) is constructed from \(w^k\) and \((U_{ij})_{i,j\in {\mathbb N} }\). Then for any positive function \(f=f(r,x,y)\)$$\begin{aligned}&{\mathbb E} \left[ \int \limits _{[0,T] \times {\mathbb R}_+\times {\mathbb R}_+} f(r,x,y) {\mathcal M} (dr,dx,dy) \right] \\&\quad ={\mathbb E} _{w_0} \left[ \int \limits _{[0,T] \times {\mathbb R}_+\times {\mathbb R}_+} f(r,x,y) \mathbf {1}_{y\le w_{r-}} \nu (dr,dx,dy) \right] . \end{aligned}$$

Proposition 3.1 tells us how to construct a Poisson noise \({\mathcal N} \) from the \((s_i,u_i,a_i,w^i)\). Let us now show that \(Z\) solves (2.6) with this particular noise.

**Proposition 3.2**

*Proof*

**Lemma 3.3**

*Proof*

**Lemma 3.4**

- (1)
\(\lim _{n\rightarrow \infty }\int \limits _{{\mathcal E} } (z_{\frac{1}{n}})^2 \, {\small 1}\!\!1_{{(z_{\frac{1}{n}}\le 1)}} \, {\mathbb Q} (dz)=0.\)

- (2)
\(\lim _{n\rightarrow \infty }\int \limits _{{\mathcal E} } z_{\frac{1}{n}}\, {\small 1}\!\!1_{{(z_{\frac{1}{n}}\ge 1)}} \, {\mathbb Q} (dz)=0.\)

*Proof*

**Lemma 3.5**

*Proof*

The proof of Proposition 3.2 is complete.

### 3.2 Proof of Theorem 2.3

### 3.3 Proof of Theorem 2.2

It remains to prove pathwise uniqueness. Let \((\mathbf X ^i_t)_{t\ge 0}\), \(i=1,2\), be two solutions to (2.6) driven by the same Poisson noise \({\mathcal N} \) and let us set \(X^i_t(v):=\mathbf X ^i_t([0,v])\), \(v\in [0,1]\). Let us first consider the case \(v=1\): then \((X^i_t(1), t\ge 0)\), \(i=1,2\), solves a particular case of the equation considered by Dawson and Li [12, (2.1)]; therefore, by [12, Theorem 2.5], \({\mathbb P} (X^1_t(1)=X^2_t(1), \, \forall \, t\ge 0)=1\).

Let us now consider \(0\le v<1\); in this case the equation satisfied by \((X^i_t(v), t\ge 0)\) depends on \((X^i_t(1), t\ge 0)\) and therefore the uniqueness result by Dawson and Li does not apply directly. Instead, we consider the difference \(D_t:=X^1_t(v)-X^2_t(v)\) so that the drift terms cancel since \(X^1(1)=X^2(1)\). Hence, \((D_t, t\ge 0)\) can be treated as if \(g\) were identically equal to 0. The same proof as in [12] shows that \({\mathbb P} (X^1_t(v)=X^2_t(v), \, \forall \, t\ge 0)=1\). Finally, since a.s. the two finite measures \(\mathbf X ^1_t\) and \(\mathbf X ^2_t\) are equal on each interval \(]a,b]\), \(a,b\in {\mathbb Q} \cap [0,1]\), they coincide. Therefore, pathwise uniqueness holds for (2.6).

Finally, in order to obtain existence of a strong solution, we apply the classical Yamada-Watanabe argument, for instance in the general form proved by Kurtz [25, Theorem 3.14].

## 4 Proof of Theorem 2.7

**Lemma 4.1**

The right-hand side of (2.15) is well-defined for all \(v\in [0,1]\) and \(t\ge 0\).

*Proof*

We can now show how to construct on a pathwise level the Beta-Fleming–Viot processes with mutations the measure-valued branching process.

*Proof of Theorem 2.7*

Suppose \(\mathcal {N}\) is the PPP with compensator measure \(\nu \) that drives the strong solution of (2.6) with atoms

**Step 1:**- We haveTo verify the third equality, first note that due to Lemma II.2.18 of [24] the compensation can be split from the martingale part and then can be canceled by the compensator integral since integrating-out the \(y\)-variable yields$$\begin{aligned}&\quad R_t(v) =\frac{X_{S^{-1}(t)}(v)}{X_{S^{-1}(t)}(1)} = \\&=\frac{X_0(v)}{X_0(1)}+\int \limits _0^{S^{-1}(t)}\int \limits _0^\infty \int \limits _0^\infty \Bigg [\frac{X_{r-}(v)+ x\mathbf 1_{(y\le X_{r-}(v))}}{X_{r-}(1)+ x\mathbf 1_{(y\le X_{r-}(1))}}-\frac{X_{r-}(v)}{X_{r-}(1)}\Bigg ] (\mathcal {N}-\nu )(dr,dx,dy) \\&\quad +\int \limits _0^{S^{-1}(t)}\int \limits _0^\infty \int \limits _0^\infty \Bigg [\frac{X_{r-}(v)+ x\mathbf 1_{(y\le X_{r-}(v))}}{X_{r-}(1)+ x\mathbf 1_{(y\le X_{r-}(1))}}-\frac{X_{r-}(v)}{X_{r-}(1)}-\frac{ x\mathbf 1_{(y\le X_{r-}(v))} }{X_{r-}(1)}\\&\quad \quad +\frac{ x\mathbf 1_{(y\le X_{r-}(1))} X_{r-}(v)}{X_{r-}(1)^2}\Bigg ] \nu (dr,dx,dy)\\&\quad + \alpha (\alpha -1)\Gamma (\alpha )\theta v\int \limits _0^{S^{-1}(t)}\frac{1}{X_{r}(1)}X_r(1)^{2-\alpha } \,dr- \theta \int \limits _0^{S^{-1}(t)}\frac{X_r(v)}{X_{r}(1)^2} X_r(1)^{2-\alpha }\,dr\\&=v+\int \limits _0^{S^{-1}(t)}\int \limits _0^\infty \int \limits _0^\infty \Bigg [\frac{X_{r-}(v)+ x\mathbf 1_{(y\le X_{r-}(v))}}{X_{r-}(1)+ x\mathbf 1_{(y\le X_{r-}(1))}}-\frac{X_{r-}(v)}{X_{r-}(1)}\Bigg ] \mathcal {N}(dr,dx,dy)\\&\quad + \alpha (\alpha -1)\Gamma (\alpha )\theta v\int \limits _0^{S^{-1}(t)}X_r(1)^{1-\alpha } \,dr- \theta \int \limits _0^{S^{-1}(t)}\frac{X_r(v)}{X_{r}(1)} X_r(1)^{1-\alpha }\,dr. \end{aligned}$$To replace the jumps governed by the PPP \(\mathcal {N}\) by jumps governed by \(\mathcal M\) note that by the definition of \(\mathcal M\) we find, for measurable non-negative test-functions \(h\) for which the first integral is defined, the almost sure transfer identity$$\begin{aligned}&\quad \int \limits _0^{S^{-1}(t)}\int \limits _0^\infty \int \limits _0^\infty \Bigg [-\frac{ x\mathbf 1_{(y\le X_{r-}(v))} }{X_{r-}(1)}+\frac{ x\mathbf 1_{(y\le X_{r-}(1))} X_{r-}(v)}{X_{r-}(1)^2}\Bigg ] c_\alpha x^{-1-\alpha } dr\,dx\,dy =0. \end{aligned}$$or in an equivalent but more suitable form$$\begin{aligned}&\int \limits _{0}^{S^{-1}(t)}\int \limits _{0}^{\infty }\int \limits _{0}^{\infty } h\left( S(r),\frac{x}{X_{r-}(1)+ x\mathbf 1_{(y\le X_{r-}(1))}},\frac{y}{X_{r-}(1)}\right) \mathcal {N}(dr,dx,dy)\nonumber \\&\quad =\int \limits _{0}^{t}\int \limits _{0}^{1}\int \limits _{0}^{\infty } h(s, z, u)\,\mathcal {M}(ds,d z,d u) \end{aligned}$$(4.4)Since the integrals are non-compensated we actually defined \(\mathcal {M}\) in such a way that the integrals produce exactly the same jumps. Let us now rewrite the equation found for \(R\) in such a way that (4.5) can be applied:$$\begin{aligned}&\quad \int \limits _0^{S^{-1}(t)}\int \limits _0^\infty \int \limits _0^\infty h\left( r,\frac{x}{X_{S^{-1}(r-)}(1)+ x\mathbf 1_{(y\le X_{S^{-1}(r-)}(1))}},\frac{y}{X_{S^{-1}(r-)}(1)}\right) \mathcal {N}(dr,dx,dy)\nonumber \\&=\int \limits _0^{t}\int \limits _0^1\int \limits _0^\infty h\big (S^{-1}(s), z, u\big )\,\mathcal {M}(d s,d z,d u). \end{aligned}$$(4.5)The stochastic integral driven by \(\mathcal {N}\) can now be replaced by a stochastic integral driven by \(\mathcal {M}\) via (4.5):$$\begin{aligned} R_t(v)&=v+\int \limits _0^{S^{-1}(t)}\int \limits _0^\infty \int \limits _0^\infty \Bigg [\frac{x\mathbf 1_{(y\le X_{r-}(v))}X_{r-}(1)-X_{r-}(v) x\mathbf 1_{(y\le X_{r-}(1))}}{(X_{r-}(1)+ x\mathbf 1_{(y\le X_{r-}(1))})X_{r-}(1)}\Bigg ]\\&\quad \times \mathcal {N}(dr,dx,dy) + \alpha (\alpha -1)\Gamma (\alpha )\theta \int \limits _0^{S^{-1}(t)}\\&\quad \times \Big [vX_r(1)^{1-\alpha } -\frac{X_r(v)}{X_{r}(1)} X_r(1)^{1-\alpha }\Big ]\,dr\\&=v+\int \limits _0^{S^{-1}(t)}\int \limits _0^\infty \int \limits _0^\infty \frac{x}{X_{r-}(1)+ x\mathbf 1_{(y\le X_{r-}(1))}} \\&\quad \quad \times \left[ \mathbf 1_{(y\le X_{r-}(v))}-\frac{X_{r-}(v)}{X_{r-}(1)}\mathbf 1_{(y\le X_{r-}(1))}\right] \mathcal {N}(dr,dx,dy)\\&\quad + \alpha (\alpha -1)\Gamma (\alpha )\theta \int \limits _0^{S^{-1}(t)}\Big [vX_r(1)^{1-\alpha } -\frac{X_r(v)}{X_{r}(1)} X_r(1)^{1-\alpha }\Big ]\,dr. \end{aligned}$$By monotonicity in \(v\), \(R_t(v)\le 1\) so that the \(du\)-integral in fact only runs up to \(1\) and the second indicator can be skipped:$$\begin{aligned} R_t(v)&= v+\int \limits _0^{t}\int \limits _0^1\int \limits _0^\infty z\bigg [\mathbf 1_{( uX_{S^{-1}( s)-}(1)\le {X_{S^{-1}( s)-}(v)})} - \\&\quad - R_{S^{-1}( s)-}( v)\mathbf 1_{( uX_{S^{-1}( s)-}(1)\le {X_{S^{-1}( s)-}(1)})} \bigg ]\mathcal {M}(d s,d z,d u)\\&\quad + \theta \int \limits _0^{t}\big [v-R_s(v)\big ]\,ds\\&=v+\int \limits _0^{t}\int \limits _0^1\int \limits _0^\infty z\bigg [\mathbf 1_{(u\le R_{s-}(v))}- R_{ s-}( v)\mathbf 1_{( u\le 1)} \bigg ]\mathcal {M}(d s,d z,d u) \\&\quad + \theta \int \limits _0^{t}\big [v-R_s(v)\big ]\,ds. \end{aligned}$$This is precisely the equation we wanted to derive.$$\begin{aligned} R_t(v)&= v+\int \limits _0^{t}\int \limits _0^1\int \limits _0^1 z\bigg [\mathbf 1_{(u\le R_{s-}(v))}- R_{ s-}( v) \bigg ]\mathcal {M_|}(d s,d z,d u)\\&+ \theta \int \limits _0^{t}\big [v-R_s(v)\big ]\,ds. \end{aligned}$$
- Step 2:
The proof is complete if we can show that the restriction \(\mathcal {M}_|\) of \(\mathcal M\) to \((0,\infty )\times [0,1]\times [0,1]\) is a PPP with intensity \(\mathcal M'(ds,dz,du)=ds \otimes C_\alpha z^{-1-\alpha }(1-z)^{\alpha -1}dz\otimes du\). For this sake, we choose a non-negative measurable predictable function \(W:\Omega \times (0,\infty )\times (0,1)\times (0,1)\rightarrow {\mathbb R}\) bounded in the second and third variable and compactly supported in the first, plug-in the definition of \(\mathcal M_|\) and use the compensator measure \(\nu \) of \(\mathcal {N}\) to obtain via (4.4)

## 5 Proof of Theorem 1.2

*immigrated types*alive is infinite at all times, therefore proving that the result in Theorem 1.2 is indeed independent of the starting configuration \(\mathbf Y _0\).

**Lemma 5.1**

*Proof*

The proof is an iterated use of Shepp’s result for the sequence of restricted Poisson point processes \(\Pi _k\) obtained by removing all the atoms \((s_i,h_i)\) with \(h_i>\frac{1}{k}\) from \(\Pi \), i.e. restricting the intensity measure to \([0,\frac{1}{k}]\). Since Shepp’s criterion (5.2) only involves the intensity measure around zero, the shadows of all point processes \(\Pi _k\) cover the half line. Consequently, if there is some \(t>0\) such that \(t\) is only covered by the shadows of finitely many points \((s_i,h_i)\in \Pi \), then \(t\) is not covered by the shadows generated by \(\Pi _{k'}\) for some \(k'\) large enough. But this is a contradiction to Shepp’s result applied to \(\Pi _{k'}\). \(\square \)

*Proof of Theorem 1.2*

## 6 A Proof of Schmuland’s Theorem

In this section we sketch how our lines of arguments can be adopted for the continuous case corresponding to \(\alpha =2\). The proofs go along the same lines (reduction to a measure-valued branching process and then to an excursion representation for which the covering result can be applied) but are much simpler due to a constant immigration structure. The crucial difference, leading to the possibility of exceptional times, occurs in the final step via Shepp’s covering results.

*Proof of Schmuland’s Theorem 1.1*

## 7 Proof of Corollary 2.8

The fact that the \((\alpha ,\theta )\)-Fleming–Viot process \((\mathbf Y_t,t\ge 0)\) converges in distribution to its unique invariant distribution and that this invariant distribution is not trivial (i.e. it charges measures with at least two atoms) is proven by Donelly and Kurtz in [14] at the end of Sect. 5.1 and [13] Sect. 4.1. Here we re-sketch their argument that relies on the so-called *lookdown* construction of \((\mathbf Y_t,t\ge 0)\) which was introduced in the same papers. Let us very briefly describe how the lookdown construction works (for more details we refer to [9, 14]).

*type*of the

*level*\(i\) at time \(t\). The types evolve by two mechanisms :

*lookdown events:*with rate \(x^{-2}\Lambda (dx)\) a proportion \(x\) of lineages are selected by i.i.d. Bernoulli trials. Call \(i_1, i_2,\ldots \) the selected levels at a given event at time \(t\). Then, \(\forall k >1, \xi _{i_k(t)} =\xi _{i_1}(t-)\), that is the levels all adopt the type of the smallest participating level. The type \(\xi _{i_k}(t-)\) which was occupying level \(i_k\) before the event is pushed up to the next available level.*mutation events:*On each level \(i\) there is an independent Poisson point process \((t^{(i)}_j, j\ge 1)\) of rate \(\theta \) of mutation events. At a mutation event \(t^{(i)}_j\) the type \(\xi _i(t^{(i)}_j-)\) is replaced by a new independent variable uniformly distributed on \([0,1]\) and the previous type is pushed up by one level (as well as all the types above him).

*exchangeable*process. Recall from Corollary 2.5 that for each \(t\ge 0\) fixed, \(\Xi _t\) is almost surely purely atomic. Alternatively this can be seen from the lookdown construction since at a fixed time \(t>0\), the level one has been looked down upon by infinitely many level above since the last mutation event on level one. We can thus write

For each \(n\ge 1, \) let us consider \(\pi ^{(n)}(t) = \pi _{|[n]}(t)\) the restriction to \(\{1,\ldots ,n\}\) of \(\pi (t)\). Then, for all \(n\ge 1\), the process \((\pi ^{(n)}_t, t\ge 0)\) is an irreducible Markov process on a finite state-space and thus converges to its unique invariant distribution. This now implies that \((\pi (t), t\ge 0)\) must also converges to its invariant distribution. By Kingman continuity Theorem (see [33, Theorem 36] or [4, Theorem 1.2]) this implies that the ordered sequence of the atom masses \((a_i(t))\) converges in distribution as \(t\rightarrow \infty \). Because \((x_i(t), i\ge 1)\) also converges in distribution this implies that \(\Xi _t\) itself converges in distribution to its invariant measure. (Alternatively, this second part of the Corollary could be deduced from Theorem 2 in [28])

Furthermore it is also clear that the invariant distribution of \((\pi ^{(n)}_t, t\ge 0)\) must charge configurations with at least two non-singleton blocks. Since \(\pi \) is an exchangeable process, so is its invariant distribution. Exchangeable partitions have only two types of blocks: singletons and blocks with positive asymptotic frequency so this proves that the invariant distribution of \(\pi \) charges partition with at least two blocks of positive asymptotic frequency.

## Notes

### Acknowledgments

The authors are very grateful for stimulating discussions with Zenghu Li and Marc Yor and to an anonymous referee whose insightful comments greatly helped us improve this manuscript. L. Döring was supported by the Fondation Science Mathématiques de Paris, L. Mytnik is partly supported by the Israel Science Foundation. J. Berestycki, L. Mytnik and L. Zambotti thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for a very pleasant stay during which part of this work was produced. L. Mytnik thanks the Laboratoire de Probabilités et Modèles Aléatoires for the opportunity to visit it and carry out part of this research there

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