# The Lamperti transformation in the infinite-dimensional setting and the
genealogies of self-similar Markov processes

Arno Siri-Jégousse  IIMAS, Universidad Nacional Autónoma de México, México
Alejandro H. Wences  CIMI-LAAS, Université de Toulouse, CNRS, Toulouse, France

###### Abstract

We propose a change in focus from the prevalent paradigm based on the
branching property as a tool to analyze the structure of population models, to
one based on the self-similarity property, which we also introduce for the
first time in the setting of measure-valued processes. By extending the well-
known Lamperti transformation for self-similar Markov processes to the Banach-
valued case we are able to generalize celebrated results in population
genetics that describe the frequency-process of measure-valued stable
branching processes in terms of the subfamily of Beta-Fleming-Viot processes.
In our work we describe the frequency process of populations whose total size
evolves as any positive self-similar Markov process in terms of general
$\Lambda$-Fleming-Viot processes. Our results demonstrate the potential power
of the self-similar perspective for the study of population models in which
the reproduction dynamics of the individuals depend on the total population
size, allowing for more complex and realistic models.

Keywords:  self-similarity, Lamperti transformation, measure-valued processes,
coalescent processes.

MSC 2020:  92D25, 60G18, 60J68.

##  1 Introduction and Main Results

In the same spirit as in [5, 32], the work of [8] describes the genealogy of a
population that evolves as an stable measure-valued branching process, say
$\left(\mu_{t}\right)_{t\geq 0}$, indexed by a parameter $\beta\in(0,2]$.
Formally, the latter are Markov processes on the state space
$\mathtt{M}([0,1])$ of finite positive measures on the type space $[0,1]$, and
with generator $\mathcal{F}$ having one of the two forms:

  1. 1)

$\mathcal{F}F(\mu)=\int_{\mathcal{T}}\mu(da)\int_{0}^{\infty}h^{-1-\beta}dh\bigg{\\{}F\left(\mu+h\delta_{a}\right)-F(\mu)-hF^{\prime}(\mu;a)\bigg{\\}},$
for $\beta\in(0,2)$, or

  2. 2)

$\mathcal{F}F(\mu)=\int_{\mathcal{T}}\mu(da)F^{\prime\prime}(\mu;a,a),$ which
heuristically corresponds to the case $\beta=2$.

In the latter, we have removed the drift terms and the constants for
simplicity. The genealogy of such models is described by means of
characterizing the associated frequency process
$\left(\frac{\mu_{t}}{\left\lVert\mu_{t}\right\rVert}\right)_{t\geq 0}$, where
$\left\lVert\mu\right\rVert\coloneqq\mu([0,1])$. The authors show that, after
the time change $c_{{\beta-1}}(t)=\inf\\{s\geq
0\colon\int_{0}^{s}\left\lVert\mu_{u}\right\rVert^{1-\beta}du>t\\},$ the
frequency process becomes Markov in itself and is a member of the Beta
subfamily of $\Lambda$-Fleming-Viot (FV) processes [4]. The genealogy of the
underlying population is then understood via the well-know duality relation
between $\Lambda$-FV processes and $\Lambda$-coalescents [4].

The family of $\Lambda$-FV (resp. $\Lambda$-coalescents) generalize the
standard FV process (resp. the Kingman coalescent) to capture the population
genetics dynamics of a wider range of neutral populations, including
populations with highly skewed offspring distributions (but also many more
models, including models with selection). Still, [8] proved that only the Beta
subfamily can be obtained from a branching process by using their method based
on path-wise random time changes. For the rest of the super-processes
(measure-valued branching processes), the associated frequency process is not
Markov in itself under any random time change that is written in terms of the
total population size (see their Lemma 3.5 for further details). This apparent
lack of a branching process counterpart for the rest of the Lambda family has
motivated research seeking variants of the main results in [8] to obtain
different $\Lambda$-FV (resp. $\Lambda$-coalescents) for the frequency process
(resp. genealogy) of branching processes, but using different transformations.
An example is the culling procedure in [10] (see also [20]) who work with the
two-dimensional counterparts of FV processes, and who approximate populations
with constant size and in distribution, losing the path-wise quality. Another
example is the work of [25] who study the genealogy of multi-type branching
processes locally in time by computing the corresponding coalescence rates,
which notably depend on the size of the populations of each of the finitely-
many types. In our case it has motivated a change of perspective by now
focusing on the self-similarity property instead, enabling us to use robust
path-wise tools such as the Lamperti transformation in the infinite-type
setting. This technique also allows us to incorporate dependencies in the
reproduction dynamics of the individuals, mainly through the total population
size, and also to accommodate in a simple way the effect of this size on the
coalescence rates and the genealogy.

A measure-valued Markov process $\left(\mu_{t}\right)_{t\geq 0}$, and more
generally a Banach-valued Markov process $\left(X_{t}\right)_{t\geq 0}$, is
said to satisfy the _self-similarity (SS)_ property with index
$\alpha\in\mathbb{R}$ (denoted $\alpha$-SS) if

(1) $\forall
a>0,\quad\\{X_{t},\mathbbm{P}^{X}_{x}\\}=\\{aX_{a^{-\alpha}t},\mathbbm{P}^{X}_{a^{-1}x}\\}.$

On the other hand, a two-coordinate process $\left(Z_{t}\right)_{t\geq
0}=\left(\rho_{t},\xi_{t}\right)_{t\geq 0}$ on $S\times\mathbb{R}$, with $S$
being the unit sphere of a Banach space, and with lifetime $\zeta^{Z}$, is
said to be a _Markov additive process (MAP)_ if for any $y\in S$,
$z\in\mathbb{R}$, $s,t\geq 0$, and for any positive measurable function $f$ on
$S\times\mathbb{R}$, one has

(2)
$\mathbbm{E}^{Z}_{\theta,z}\left[f(\rho_{t+s},\xi_{t+s}-\xi_{t}),t+s<\zeta^{Z}|\mathcal{F}_{t}\right]=\mathbbm{E}^{Z}_{\rho_{t},0}\left[f(\rho_{s},\xi_{s}),s<\zeta^{Z}\right]\mathbbm{1}_{t<\zeta^{Z}}.$

We obtain the following (also see Theorem 4.2 below for further details).

######  Theorem 1.1.

There exists a standard measure-valued $\alpha$-SS Markov process
$\left(\mu_{t}\right)_{t\geq 0}$ with generator of the form

$\displaystyle\mathcal{F}F(\mu)=$
$\displaystyle\frac{1}{\left\lVert\mu\right\rVert^{\alpha}}\left(\mathcal{G}^{(D)}_{\kappa}F(\mu)+\mathcal{G}^{(B)}_{\sigma}F(\mu)+\mathcal{G}^{(J)}_{\Lambda}F(\mu)\right)$

where, for $\sigma\geq 0,\kappa\in\mathbb{R}$, and
$\Lambda\in\mathtt{M}((0,1))$, the operators are defined by

$\displaystyle\mathcal{G}^{(D)}_{\kappa}F(\mu)$
$\displaystyle=\int_{\mathcal{T}}\mu(da)\kappa F^{\prime}(\mu;a),$
$\displaystyle\mathcal{G}^{(B)}_{\sigma}F(\mu)$
$\displaystyle=\left\lVert\mu\right\rVert\int_{\mathcal{T}}\mu(da)\frac{\sigma^{2}}{2}F^{\prime\prime}(\mu;a,a),\text{
and }$ $\displaystyle\mathcal{G}^{(J)}_{\Lambda}F(\mu)$
$\displaystyle=\int_{\mathcal{T}}\frac{\mu(da)}{\left\lVert\mu\right\rVert}\int_{{(0,1)}}\frac{\Lambda(d\zeta)}{\zeta^{2}}\bigg{\\{}F\left(\mu+\left\lVert\mu\right\rVert\frac{\zeta}{1-\zeta}\delta_{a}\right)$
$\displaystyle\quad\quad-F(\mu)-\left\lVert\mu\right\rVert(\left\lvert\log(1-\zeta)\right\rvert\mathbbm{1}_{\zeta<1/2})F^{\prime}(\mu;a)\bigg{\\}}.$

The total size of the population
$\left(\left\lVert\mu_{t}\right\rVert\right)_{t\geq 0}$ is a general positive
$\alpha$-SS Markov process with non-negative jumps. Furthermore, letting

$c_{\alpha}(t)=\inf\\{s\geq
0\colon\int_{0}^{s}\left\lVert\mu_{u}\right\rVert^{-\alpha}du>t\\},$

the processes
$\left(\frac{\mu_{c(t)}}{\left\lVert\mu_{c(t)}\right\rVert},\log(\left\lVert\mu_{c(t)}\right\rVert)\right)_{t\geq
0}$ is a MAP. Its first coordinate is a
$(\Lambda+\frac{\sigma^{2}}{2}\delta_{0})$-Fleming-Viot process; whereas the
second coordinate is a spectrally-positive Lévy process.

Through an appropriate choice of the parameters $\sigma,\kappa,\Lambda$, the
above theorem recovers the cases when the stability index $\beta\in(1,2)$ of
[8], which correspond to measure-valued branching processes that also enjoy
the $(\beta-1)$-SS property, as seen through an adaptation of the result in
[28] from $\mathbb{R}$-valued to measure-valued processes.

Our method is based on a generalization of the Lamperti transformation [30, 1,
11] to the infinite-dimensional setting, which we now describe (see section 2
for further details). Let $E$ be a conic subset of a normed vector space
$(\mathbb{V},\left\lVert\cdot\right\rVert)$, and suppose that $(E,\mathtt{d})$
is a metric space such that the map $x\to\left\lVert x\right\rVert$ is
continuous. In the application above, $E$ is the space of positive measures
$\mathtt{M}(\mathcal{T})$ over a type space $\mathcal{T}$, and $\mathtt{d}$ is
a metrization of the weak topology on $\mathtt{M}(\mathcal{T})$. We also
assume $(E,\mathtt{d})$ to be locally-compact and second-countable, and
augment it to $\hat{E}=E\cup\\{\infty\\}$ where $\infty$ is a point at
infinity if $E$ is not compact, or just an isolated point if $E$ is compact.

We consider only standard Markov process with values in
$D([0,\infty),\hat{E})$. The latter is the space of càdlàg functions with
values in $\hat{E}$ that are absorbed at $\\{\infty,0\\}$. Also, for a general
process $X$ we write $\zeta_{\infty}^{X}$ for its absorption time to $\infty$,
$\zeta^{X}_{0}$ for its absorption time to $0\in E$, and define its lifetime
$\zeta^{X}\coloneqq\zeta_{\infty}^{X}\wedge\zeta_{0}^{X}$. The self-similar
Lamperti transformation can be expressed as follows.

######  Theorem 1.2.

Let $\left(X_{t}\right)_{t\geq 0}$ be a standard $\alpha$-SS Markov process.
Consider the additive functional $t\to\int_{0}^{t}\left\lVert
X_{u}\right\rVert^{-\alpha}du$ for $t\in[0,\zeta^{X}]$, and its generalized
inverse

$c_{\alpha}(t)\coloneqq\inf\left\\{s>0\colon\int_{0}^{s}\left\lVert
X_{u}\right\rVert^{-\alpha}du\geq t\right\\},\quad
t\in[0,\int_{0}^{\zeta^{X}}\left\lVert X_{u}\right\rVert^{-\alpha}du].$

Then the process $\left(Y_{t}\right)_{t\geq
0}=\left(\frac{X_{c_{\alpha}(t)}}{\left\lVert
X_{c_{\alpha}(t)}\right\rVert},\log(\left\lVert
X_{c_{\alpha}(t)}\right\rVert)\right)_{t\geq 0}$ is a standard MAP with
lifetime
$\zeta^{Y}=\zeta_{\infty}^{Y}\wedge\zeta_{0}^{Y}\underset{a.s.}{=}\int_{0}^{\zeta^{X}}\left\lVert
X_{u}\right\rVert^{-\alpha}du$.

Conversely, let $\left(\rho_{t},\xi_{t}\right)_{t\geq 0}$ be a standard MAP,
and let $\zeta^{\xi}=\zeta_{\infty}^{\xi}\wedge\zeta^{\xi}_{0}$ be its
lifetime, after which it is absorbed at some extra state. For any $\alpha\geq
0$, consider the inverse additive functional

$\gamma_{\alpha}(t)\coloneqq\inf\left\\{s>0\colon\int_{0}^{s}e^{\alpha\xi_{u}}du\geq
t\right\\},\quad t\in[0,\int_{0}^{\zeta^{\xi}}e^{\alpha\xi_{u}}du].$

Then the process $\left(X_{t}\right)_{t\geq
0}=\left(\rho_{\gamma_{\alpha}(t)}e^{\xi_{\gamma_{\alpha}(t)}}\right)_{t\geq
0}$ is a standard $\alpha$-SS Markov process with lifetime
$\zeta^{X}=\zeta_{\infty}^{X}\wedge\zeta_{0}^{X}\underset{a.s.}{=}\int_{0}^{\zeta_{\infty}^{\xi}\wedge\zeta^{\xi}_{0}}e^{\alpha\xi_{u}}du$.

Our results show an unexplored link between the fields of mathematical
population genetics and self-similar Markov processes in infinite dimensions,
motivating new research and opening new questions in both of these fields
separately, but also at their intersection. For instance, the populations
driven by $\mathcal{F}$ above, characterized by four parameters (the Lévy
triplet and the self-similarity index) constitute only a sub-class of the
entire family of measure-valued self-similar processes which is yet to be
fully described. Whether one will need a new model to describe their
corresponding genealogies, outside or extending the family of $\Lambda$ (and
more generally $\Xi$) coalescent processes, is still an open question. At the
same time, measure-valued processes, together with the well-established
analytic tools available in population genetics such as duality methods, may
serve as a suitable template for the development of the theory of self-similar
Markov processes and their Lamperti transformations in infinite dimensions.

The rest of the manuscript is structured as follows. Section 2 is devoted to
the development of our main methodological result, the Lamperti transformation
in the infinite-dimensional setting (Theorem 1.2). On the other hand, our main
phenomenological result (Theorem 1.1) is developed in section 4, which is
preceded by section 3 in which we describe some necessary preliminaries on
coalescent processes, FV processes, and Dawson-Watanabe processes. In turn,
section 4 is decomposed into three main subsections, section 4.1 providing
details and expanding upon Theorem 1.1, and sections 4.2 and 4.3 which
constitute the two main steps in the construction of an “intermediary” process
$\left(\nu_{t}\right)_{t\geq 0}$ with generator of the form
$\mathcal{G}=\mathcal{G}^{(D)}_{\kappa}+\mathcal{G}^{(B)}_{\sigma}+\mathcal{G}^{(J)}_{\Lambda}$
from which the process $\left(\mu_{t}\right)_{t\geq 0}$ can be obtained via a
Lamperti time change.

##  2 Lamperti Transformations in Normed Spaces

In this section we get inspiration from the works of [30, 1, 11] and
generalize the self-similar Lamperti transformation to processes taking values
in a conic subset $E$ of a normed space
$(\mathbb{V},\left\lVert\cdot\right\rVert)$. In particular $E$ is closed under
non-negative scalar transformations (i.e. $x\in E$ implies $ax\in E$ for all
$a\geq 0$), and we can furthermore define a polar decomposition from $E$ to
$S\times\mathbb{R}$, where $S$ is the intersection of $E$ with the unit sphere
in $\mathbb{V}$, as $x\mapsto\left({x}/{\left\lVert x\right\rVert},\left\lVert
x\right\rVert\right)$. In fact all we need are these two properties for $E$
and that the function $\left\lVert\cdot\right\rVert$ satisfies $\left\lVert
ax\right\rVert=\left\lvert a\right\rvert\left\lVert x\right\rVert$, but we
subscribe to the normed space setting in order to avoid technicalities. We
also recall that we have assumed the metric space $(E,\mathtt{d})$ to be
locally-compact and second-countable, and that $\hat{E}$ is its one-point
compactification if $E$ is not compact, and $\infty$ is an isolated point if
$E$ is compact. Furthermore, we also assume that the function
$\left\lVert\cdot\right\rVert$ is continuous on $(E,\mathtt{d})$.

Most of the arguments used in our main proofs can already be found in [1, 22],
we expand them to general state spaces and also deal with possible explosion
of the processes involved.

We assume that all the processes that we consider are standard in the sense of
Definition 9.2 in [9]; namely that they satisfy the following:

  1. i)

their respective filtrations $(\mathcal{F}_{t})_{t>0}$ are right-continuous,

  2. ii)

they are absorbed at $\infty$ at time $\zeta_{\infty}\in[0,\infty]$,

  3. iii)

they are càdlàg and quasi-left-continuous on $[0,\zeta_{\infty})$,

  4. iv)

they are strong-Markov processes with measurable probability kernels
$P_{t}(x,\cdot)$ from $\hat{E}$ to $\hat{E}$ corresponding to the law of the
process at time $t$.

Let us call $D([0,\infty),\hat{E})$ the trajectory space of such processes,
i.e. the space of trajectories on $\hat{E}$ that are càdlàg on
$[0,\zeta_{\infty})$ and have constant value $\infty$ on
$[\zeta_{\infty},\infty)$. Endow
$D([0,\infty),\hat{E})\subset{\hat{E}}^{\mathbb{R_{+}}}$ with $\mathcal{F}$,
the trace $\sigma$-algebra induced by $\sigma(\pi_{t};t\geq 0)$ where
$\pi_{t}$ is the projection at time $t$. Note that when $E$ is a locally-
compact second-countable metric space, the extended space $\hat{E}$ becomes
compact and metrizable (see e.g. Proposition VII.1.15 im [29]) and
$\mathcal{F}$ coincides with the Borel $\sigma$-algebra induced by the
Skorohod topology on $D([0,\infty),\hat{E})$ (Theorem 12.5 in [6]). Also we
will denote by $\mathbbm{P}_{x}(\cdot)$ the law of the processes on
$\hat{E}^{\mathbb{R_{+}}}$, started at $x$. Finally $\zeta_{x}$ will denote
the reaching time of a process to an absorbing state $x$ (which could be an
extra state, or the point at infinity $\infty$); whereas $\zeta$ will denote
the lifetime of the process. Typically $\zeta=\zeta_{\infty}\wedge\zeta_{0}$.
We will often decorate $\zeta,\mathcal{F}$, and $\mathbbm{P}$ with
superscripts specifying the distinct processes we will deal with, as below.

An $\hat{E}$-valued Markov process $(X_{t})_{t>0}$ that is absorbed at
$\\{\infty,0\\}$ and has lifetime
$\zeta^{X}=\zeta^{X}_{0}\wedge\zeta^{X}_{\infty}$ is said to satisfy the
_self-similarity (SS)_ property with index $\alpha\in\mathbb{R}$ ($\alpha$-SS)
if (1) holds. Alternatively, it is easily seen that the process $X$ is
$\alpha$-SS if and only if for all $t\geq 0,a>0,x\in E,A\in\mathcal{B}(E)$,
its transition kernels $P^{X}_{t}$ satisfy

$P^{X}_{t}(x,A)=P^{X}_{a^{-\alpha}t}\left(a^{-1}x,a^{-1}A\right).$

Secondly, an $\hat{E}$-valued Markov process $Y$ absorbed at $\\{\infty,0\\}$
with lifetime $\zeta^{Y}=\zeta^{Y}_{0}\wedge\zeta^{Y}_{\infty}$ is said to be
_scalar multiplicative homogeneous (SMH)_ if its transition kernels
$P^{Y}_{t}$ satisfy, for all $t\geq 0,u>0,x\in E,A\in\mathcal{B}(E)$,

(3) $P^{Y}_{t}(x,A)=P^{Y}_{t}(ux,uA),$

or, in other words, if

$\forall
u>0,\quad\\{Y_{t},\mathbbm{P}_{x}^{Y}\\}=\\{uY_{t},\mathbbm{P}_{u^{-1}x}^{Y}\\}.$

Thirdly, recalling that $S$ is the intersection of $E$ with the unit sphere in
$\mathbb{V}$, a $S\times\mathbb{R}$-valued Markov process
$\left(Z_{t}\right)_{t\geq 0}=\left(\rho_{t},\xi_{t}\right)_{t\geq 0}$ that
has lifetime $\zeta^{Z}=\zeta^{\xi}_{\infty}$ (the explosion time of $\xi$ on
the one-point compactification of $\mathbb{R}$, after which $Z$ is absorbed at
some extra state), is said to be a _Markov additive process (MAP)_ if (2)
holds. Equivalently, it is easily seen that the process
$\left(\rho_{t},\xi_{t}\right)_{t\geq 0}$ is a MAP if and only if, for all
$t\geq 0,a\in\mathbb{R},(\theta,z)\in S\times\mathbb{R}^{+}$ and
$A\in\mathcal{B}(S\times\mathbb{R}^{+})$,

(4) $P^{Z}_{t}((\theta,z),A)=P^{Z}_{t}((\theta,z+a),A+(0,a));$

where we have written $A+(0,a)\coloneqq\\{(\rho,z+a)\colon(\rho,z)\in A\\}$.
This can be interpreted as saying that $Z$ is additive-homogeneous on the
second coordinate.

We will establish transformations between SS and SMH processes on the one
hand, and between SMH processes and MAPs on the other. In fact, the latter is
simply a bijection given by the “log-polar decomposition” isomorphism
$\Phi\colon E\setminus\\{0\\}\to S\times\mathbb{R}$ defined as
$\Phi(x)=\left(x/\left\lVert x\right\rVert,\log(\left\lVert
x\right\rVert)\right)$. We have the following.

######  Proposition 2.1 (SMH $\iff$ MAP).

Let $Y$ be a SMH Markov process with trajectories in $D([0,\infty),\hat{E})$
and absorbed at $0$, and set
$\zeta^{Y}=\zeta^{Y}_{0}\wedge\zeta^{Y}_{\infty}$. Then
$\left(Z_{t}\right)_{t\geq 0}=\left(\Phi(Y_{t})\right)_{t\geq 0}$ is a MAP
with lifetime $\zeta^{Z}=\zeta^{Y}$.

Conversely, if $\left(Z_{t}\right)_{t\geq
0}=\left(\rho_{t},\xi_{t}\right)_{t\geq 0}$ is a MAP with lifetime
$\zeta^{Z}=\zeta^{\xi}_{\infty}$, then $Y=\left(\Phi^{-1}(Z_{t})\right)_{t\geq
0}\equiv\left(e^{\xi_{t}}\rho_{t}\right)_{t\geq 0}$ is a SMH Markov process
with lifetime $\zeta^{Y}=\zeta^{Z}$.

###### Proof.

Given that $\Phi$ is bijective and continuous except at the absorbing state
$0$, it is clear that the transformed processes are standard whenever the
starting process is. Thus we need only verify (4) in the first case, and (3)
in the second. We only do this for the first. We have, assuming (3) in the
second equality below, and for all $t\geq 0,a\in\mathbb{R},(\theta,z)\in
S\times\mathbb{R},A\in\mathcal{B}(S\times\mathbb{R})$,

$\displaystyle P^{Z}_{t}((\theta,z),A)$
$\displaystyle=P^{Y}_{t}(e^{z}\theta,{\Phi^{-1}(A)})=P^{Y}_{t}(e^{z+a}\theta,e^{a}{\Phi^{-1}(A)})$
$\displaystyle=P^{Z}_{t}((\theta,z+a),A+(0,a)).$

∎

The transformation between SS and SMH processes is given in terms of random
time changes. The proof follows the heuristics in the proof of Theorem 2.3 in
[1] but adapted to our setting.

######  Theorem 2.2 (Self-Similar Lamperti Time Change).

Let $\left(X_{t}\right)_{t\geq 0}$ be a standard Markov process with
trajectories in $D([0,\infty),\hat{E})$. Let $\zeta_{\infty}^{X}$ be its
explosion time, $\zeta^{X}_{0}$ be its absorption time to $0\in E$, and define
its lifetime $\zeta^{X}=\zeta_{\infty}^{X}\wedge\zeta_{0}^{X}$. Consider the
additive functional $t\to\int_{0}^{t}\left\lVert
X_{u}\right\rVert^{-\alpha}du$ for $t\in[0,\zeta^{X}]$, and its generalized
inverse

$c_{\alpha}(t)\coloneqq\inf\left\\{s>0\colon\int_{0}^{s}\left\lVert
X_{u}\right\rVert^{-\alpha}du\geq t\right\\},\quad
t\in[0,\int_{0}^{\zeta^{X}}\left\lVert X_{u}\right\rVert^{-\alpha}du].$

  1. i)

If $\left(X_{t}\right)_{t\geq 0}$ is $\alpha$-SS, then the process
$\left(Y_{t}\right)_{t\geq 0}=\left(X_{c_{\alpha}(t)}\right)_{t\geq 0}$ is a
standard SMH Markov process with lifetime
$\zeta^{Y}=\zeta_{\infty}^{Y}\wedge\zeta_{0}^{Y}\underset{a.s.}{=}\int_{0}^{\zeta^{X}}\left\lVert
X_{u}\right\rVert^{-\alpha}du$.

  2. ii)

Setting $X_{\infty}=\infty$ if $\alpha>0$, and $X_{\infty}=0$ if $\alpha\leq
0$, the process $\left(Y_{t}\right)_{t\geq 0}$ is the unique solution to

(5) $\text{a.s. }\forall t\geq 0,\quad Y_{t}=X_{\int_{0}^{t}\left\lVert
Y_{u}\right\rVert^{\alpha}du\wedge\zeta^{X}}.$

Conversely, let $\left(Y_{t}\right)_{t\geq 0}$ be a standard Markov process
with trajectories in $D([0,\infty),\hat{E})$. Let
$\zeta^{Y}=\zeta_{\infty}^{Y}\wedge\zeta^{Y}_{0}$ be its lifetime, after which
it is absorbed at some extra state. Consider, for any $\alpha\geq 0$, the
inverse additive functional

$\gamma_{\alpha}(t)\coloneqq\inf\left\\{s>0\colon\int_{0}^{s}\left\lVert
Y_{u}\right\rVert^{\alpha}du\geq t\right\\},\quad
t\in[0,\int_{0}^{\zeta^{Y}}\left\lVert Y_{u}\right\rVert^{\alpha}du].$

  1. iii)

If $\left(Y_{t}\right)_{t\geq 0}$ is SMH, then the process
$\left(X_{t}\right)_{t\geq 0}=\left(Y_{\gamma_{\alpha}(t)}\right)_{t\geq 0}$
is a standard $\alpha$-SS Markov process with lifetime
$\zeta^{X}=\zeta_{\infty}^{X}\wedge\zeta_{0}^{X}\underset{a.s.}{=}\int_{0}^{\zeta^{Y}}\left\lVert
Y_{u}\right\rVert^{\alpha}du$.

  2. iv)

The process $\left(X_{t}\right)_{t\geq 0}$ is the unique solution to

(6) $\text{a.s. }\forall t\geq 0,\quad
Y_{t}=X_{\int_{0}^{t\wedge\zeta^{Y}}\left\lVert
Y_{u}\right\rVert^{\alpha}du},$

that also satisfies $X_{t}=X_{t\wedge\int_{0}^{\zeta^{Y}}\left\lVert
Y_{s}\right\rVert^{\alpha}ds}$.

###### Proof.

To ease notations, we will omit the index $\alpha$ in $c_{\alpha}(t)$ and
$\gamma_{\alpha}(t)$.

By Propositions IV.1.6 and IV.1.13 in [9] (and the discussion around eq.
IV.1.8 therein), the mappings $t\to\int_{0}^{t}\left\lVert
X_{u}\right\rVert^{-\alpha}du$ and $t\to\int_{0}^{t}\left\lVert
Y_{u}\right\rVert^{\alpha}du$ both define continuous strong additive
functionals of $(\left(X_{t}\right)_{t\geq 0},\zeta^{X})$ and
$(\left(Y_{t}\right)_{t\geq 0},\zeta^{Y})$ respectively (see Definitions
IV.1.1 and IV.1.11 in [9]). Then, by Exercise V.2.11 iv) in [9], the time-
changed processes $\left(X_{c(t)}\right)_{t\geq 0}$ and
$\left(Y_{\gamma(t)}\right)_{t\geq 0}$ are strong Markov processes in each
case. Since $c(t)$ (resp. $\gamma(t)$) is continuous on
$[0,\int_{0}^{\zeta^{X}}\left\lVert X_{u}\right\rVert^{-\alpha}du)$ (resp.
$[0,\int_{0}^{\zeta^{Y}}\left\lVert Y_{u}\right\rVert^{\alpha}du)$), it
follows that $\left(X_{c(t)}\right)_{t\geq 0}$ (resp.
$\left(Y_{\gamma(t)}\right)_{t\geq 0}$) is quasi-left continuous. Also, by
Lemma 2.4 below, the mapping
$x\to\mathbbm{E}_{x}^{Y}[f(Y_{t})]\equiv\mathbbm{E}_{x}^{X}[f(X_{c(t)})]$ is
measurable for every $t\geq 0$ and $f\in\mathtt{\bar{B}}(\hat{E})$, the space
of bounded Borel functions on $\hat{E}$, so that the process defined by
$Y_{t}=X_{c(t)}$ in i)) is indeed a standard process; and similarly for the
process defined by $X_{t}=Y_{\gamma(t)}$ in iii)).

We now show that the process $Y$ in i)) is SMH. Let $\hat{c}(t)$ be the
functional $c(t)$ applied to the process $\left(\hat{X}_{t}\right)_{t\geq
0}\coloneqq\left(aX_{a^{-\alpha}t}\right)_{t\geq 0}$. Observe that the change
of variable $v=a^{-\alpha}u$ yields

$\displaystyle a^{-\alpha}\hat{c}(t)$
$\displaystyle=a^{-\alpha}\inf\left\\{s\geq 0\colon\int_{0}^{s}\left\lVert
aX_{a^{-\alpha}u}\right\rVert^{-\alpha}du\geq t\right\\}$
$\displaystyle=\inf\left\\{a^{-\alpha}s\geq
0\colon\int_{0}^{a^{-\alpha}s}\left\lVert X_{v}\right\rVert^{-\alpha}dv\geq
t\right\\}$ $\displaystyle=\inf\left\\{s\geq 0\colon\int_{0}^{s}\left\lVert
X_{v}\right\rVert^{-\alpha}dv\geq t\right\\}$ $\displaystyle=c(t).$

Thus $\hat{X}_{\hat{c}(t)}=aX_{a^{-\alpha}\hat{c}(t)}=aX_{c(t)}$. This,
together wit the $\alpha$-SS property of $X$ in the second equality below
give, for $A\in\mathcal{B}(\hat{E}),x\in\hat{E}$, and $a>0$,

$\displaystyle\mathbbm{P}_{x}^{Y}(Y_{t}\in A)$
$\displaystyle=\mathbbm{P}_{x}^{X}(X_{c(t)}\in
A)=\mathbbm{P}_{a^{-1}x}^{X}(\hat{X}_{\hat{c}(t)}\in
A)=\mathbbm{P}_{a^{-1}x}^{X}(aX_{c(t)}\in A)$
$\displaystyle=\mathbbm{P}_{a^{-1}x}^{Y}(Y_{t}\in a^{-1}A).$

We proceed similarly to prove iii). For $a>0$ let $\hat{\gamma}(t)$ be the
functional $\gamma(t)$ applied to the process $\left(\hat{Y}_{t}\right)_{t\geq
0}\coloneqq\left(aY_{t}\right)_{t\geq 0}$. Observe that

$\displaystyle\hat{\gamma}(t)$ $\displaystyle=\inf\left\\{s\geq
0\colon\int_{0}^{s}\left\lVert aY_{u}\right\rVert^{\alpha}du\geq t\right\\}$
$\displaystyle=\inf\left\\{s\geq 0\colon\int_{0}^{s}\left\lVert
Y_{u}\right\rVert^{\alpha}du\geq a^{-\alpha}t\right\\}$
$\displaystyle=\gamma(a^{-\alpha}t).$

The above yields $\hat{Y}_{\hat{\gamma}(t)}=aY_{\gamma(a^{-\alpha}t)}$. This,
together with the SMH property of $Y$ in the second equality below give

$\displaystyle\mathbbm{P}_{x}^{X}(X_{t}\in A)$
$\displaystyle=\mathbbm{P}_{x}^{Y}(Y_{\gamma(t)}\in
A)=\mathbbm{P}_{a^{-1}x}^{Y}(\hat{Y}_{\hat{\gamma}(t)}\in
A)=\mathbbm{P}_{a^{-1}x}^{Y}(aY_{\gamma(a^{-\alpha}t)}\in A)$
$\displaystyle=\mathbbm{P}_{a^{-1}x}^{X}(aX_{a^{-\alpha}t}\in A).$

We now show ii). Write $\psi(t)\coloneqq\int_{0}^{t}\left\lVert
X_{s}\right\rVert^{-\alpha}ds$. If $\alpha\geq 0$ note that, since
$\left(X_{t}\right)_{t\geq 0}$ is absorbed at $\infty$, we have
$\zeta^{X}=\zeta_{\infty}^{X}$ whenever $\psi(\infty)<\infty$, and
$\zeta_{\infty}^{X}=\infty$ whenever $\psi(\infty)=\infty$. In any case $\psi$
is monotonic and absolutely continuous on compact time intervals contained in
$[0,\zeta^{X}]$ with inverse function $c(t)$ and a.e. derivative $\left\lVert
X_{t}\right\rVert^{-\alpha}$. It follows that for any $\psi(ds)$-integrable
function $\beta_{s}$ on $[0,\zeta^{X}]$, we have

$\int_{0}^{t}\beta_{s}\left\lVert
X_{s}\right\rVert^{-\alpha}ds=\int_{0}^{\psi(t)}\beta_{c(s)}ds.$

If $t<\psi(\infty)$, in particular if $\psi(\infty)=\infty$, we have
$0\underset{a.e.}{<}\left\lVert
X_{s}\right\rVert^{\alpha}\underset{a.e.}{<}\infty$ on $s\in[0,c(t)]$ so that

$\displaystyle c(t)$ $\displaystyle=\int_{0}^{c(t)}\left\lVert
X_{s}\right\rVert^{\alpha}\left\lVert
X_{s}\right\rVert^{-\alpha}ds=\int_{0}^{t}\left\lVert
X_{c(s)}\right\rVert^{\alpha}ds;$

whereas if $\infty>t\geq\psi(\infty)$ then

$\displaystyle\int_{0}^{t}\left\lVert
X_{c(s)}\right\rVert^{\alpha}ds\geq\lim_{u\uparrow\psi(\infty)}\int_{0}^{u}\left\lVert
X_{c(s)}\right\rVert^{\alpha}ds=\lim_{u\uparrow\psi(\infty)}c(u)=c(\psi(\infty))\equiv\zeta^{X}_{\infty}.$

In any case

$c(t)\equiv c(t)\wedge\zeta^{X}=\int_{0}^{t}\left\lVert
Y_{s}\right\rVert^{\alpha}ds\wedge\zeta^{X}.$

In particular, recalling that we have set $X_{\infty}=\infty$ if $\alpha>0$
and $X_{\infty}=0$ if $\alpha\leq 0$, we have

$Y_{t}=X_{c(t)\wedge\zeta^{X}}=X_{\int_{0}^{t}\left\lVert
X_{c(s)}\right\rVert^{\alpha}ds\wedge\zeta^{X}}=X_{\int_{0}^{t}\left\lVert
Y_{s}\right\rVert^{\alpha}ds\wedge\zeta^{X}}.$

To see that this is the unique solution note that if $Y^{\prime}_{t}$
satisfies (5), then the function

$\tilde{c}(t)=\int_{0}^{t}\left\lVert
Y^{\prime}_{s}\right\rVert^{\alpha}ds=\int_{0}^{t}\left\lVert
X_{\tilde{c}(s)}\right\rVert^{\alpha}ds$

is continuous, strictly increasing, and has derivative $\left\lVert
X_{\tilde{c}(s)}\right\rVert^{\alpha}$ a.e. on $[0,t]$ whenever
$\tilde{c}(t)<\zeta^{X}$. Then, in this case we have

$t=\int_{0}^{t}\left\lVert X_{\tilde{c}(s)}\right\rVert^{\alpha}\left\lVert
X_{\tilde{c}(s)}\right\rVert^{-\alpha}ds=\int_{0}^{\tilde{c}(t)}\left\lVert
X_{s}\right\rVert^{-\alpha}ds$

which implies $\tilde{c}(t)=c(t)$ whenever $\tilde{c}(t)<\zeta^{X}$ and
$\tilde{c}(\psi(\infty))=\lim_{t\uparrow\psi(\infty)}c(t)=\zeta^{X}$. Then
since $Y^{\prime}_{t}=X_{\tilde{c}(t)\wedge\zeta^{X}}$, we conclude
$Y^{\prime}_{t}=X_{c(t)\wedge\zeta^{X}}=Y_{t}$ for all $t\geq 0$.

Finally, we prove iv). Write $\varphi(t)=\int_{0}^{t}\left\lVert
Y_{s}\right\rVert^{\alpha}ds$. Note that
$\zeta^{Y}\leq\gamma\left(\varphi(\infty)\right)$ and $\gamma(t\wedge
s)=\gamma(t)\wedge\gamma(s)$ for every $s,t\geq 0$. Then, the fact that
$(X_{t})_{t\geq 0}$ solves (6) follows from $X_{t}\equiv
Y_{\gamma(t\wedge\varphi(\zeta^{Y})\wedge\varphi(\infty))}$ and the fact that
$\varphi$ is strictly increasing and continuous on $[0,\zeta^{Y}]$ with left
inverse $\gamma$. This yields

$X_{\varphi(t\wedge\zeta^{Y})}=Y_{\gamma\left(\varphi(t\wedge\zeta^{Y})\wedge\varphi(\zeta^{Y})\right)}=Y_{t\wedge\zeta^{Y}}.$

The fact that $(X_{t})_{t\geq 0}$ is the unique solution follows from the fact
that $\varphi(\gamma(t))=t$ on $[0,\zeta^{Y}]$ which gives, for any other
solution $\left(X^{\prime}_{t}\right)_{t\geq 0}$ to the functional equation,

$Y_{t\wedge\zeta^{Y}}=X^{\prime}(\varphi(\gamma(t)\wedge\zeta^{Y}))=X^{\prime}(t\wedge\varphi(\zeta^{Y})).$

∎

The composition of the above two transformations between MAP and SMH
processes, and between SMH and $\alpha$-SS processes respectively, leads to
Theorem 1.2 which for $\mathbb{R}^{d}$-valued processes is the Lamperti
transformation of [1]; and for $\mathbb{R}^{+}$-valued processes is the
original transformation of [30]. We also refer the reader to the result in
[11] and [21].

The following proposition gives a characterization of $\alpha$-SS and SMH
processes in terms of their generators. For $a\geq 0$, let $\mathcal{S}_{a}$
be the operator that scales space by a factor of $a$, i.e. that takes
$f\in\mathtt{\bar{B}}(E)$ to $\mathcal{S}_{a}f(x)=f(ax)$.

######  Proposition 2.3.

Let $\left(X_{t}\right)_{t\geq 0}$ be a Markov process with generator having a
core $\mathcal{A}\subset\mathtt{\bar{B}}(E)\times\mathtt{\bar{B}}(E)$ such
that $\mathcal{S}_{a}\mathcal{D}(\mathcal{A})\subset\mathcal{D}(\mathcal{A})$.
Then the following are equivalent.

  1. i)

The process $X$ is $\alpha$-SS.

  2. ii)

For all $a\geq 0$,
$\mathcal{A}=a^{-\alpha}\mathcal{S}_{a^{-1}}\mathcal{A}\mathcal{S}_{a}$.

Similarly, let $\left(Y_{t}\right)_{t\geq 0}$ be a Markov process with
generator having a core
$\mathcal{A}\subset\mathtt{\bar{B}}(E)\times\mathtt{\bar{B}}(E)$ such that
$\mathcal{S}_{a}\mathcal{D}(\mathcal{A})\subset\mathcal{D}(\mathcal{A})$. Then
the following are equivalent.

  1. iii)

The process $Y$ is SMH.

  2. iv)

For all $a\geq 0$,
$\mathcal{A}=\mathcal{S}_{a^{-1}}\mathcal{A}\mathcal{S}_{a}$.

###### Proof.

We only prove equivalence between i) and ii), the proof of the SMH case being
analogous. Assuming i) we obtain
$\mathbbm{E}^{X}_{x}[f(X_{t})]=\mathbbm{E}^{X}_{a^{-1}x}\left[\mathcal{S}_{a}f(X_{a^{-\alpha}t})\right]$,
so that taking time derivatives,

$\displaystyle\mathcal{A}f(x)$
$\displaystyle=\frac{d}{dt}\big{|}_{t=0}\mathbbm{E}^{X}_{x}[f(X_{t})]$
$\displaystyle=\frac{d}{dt}\big{|}_{t=0}\mathbbm{E}^{X}_{a^{-1}x}[\mathcal{S}_{a}f(X_{a^{-\alpha}t})]$
$\displaystyle=a^{-\alpha}\mathcal{A}\mathcal{S}_{a}f(a^{-1}x)$
$\displaystyle=a^{-\alpha}\mathcal{S}_{a^{-1}}\mathcal{A}\mathcal{S}_{a}f(x).$

Then, ii) follows. The converse implication follows from the fact that
$a^{-\alpha}\mathcal{S}_{a^{-1}}\mathcal{A}\mathcal{S}_{a}$ coincides with the
generator of the Markov process
$\\{aX_{a^{-\alpha}t};\mathbbm{P}^{X}_{a^{-1}x},x\in\hat{E}\\}$ on
$\mathcal{D}(\mathcal{A})$. Then ii) implies that its semigroup coincides with
that of $\\{X_{t};\mathbbm{P}^{X}_{x},x\in\hat{E}\\}$, and the two processes
are equal in distribution. ∎

We end this section with the following technical lemma that was used in the
proof of Theorem 2.2.

######  Lemma 2.4.

  1. We have the following measurability of mappings.

  2. i)

The mappings $\left(x_{t}\right)_{t\geq 0}\to\left(c(t)\right)_{t\geq 0}$ and
$\left(y_{t}\right)_{t\geq 0}\to\left(\gamma(t)\right)_{t\geq 0}$ are
measurable from $D([0,\infty),\hat{E})$ to $D^{0}([0,\infty),[0,\infty])$, the
non-decreasing elements of $D([0,\infty),[0,\infty])$, endowed with the
relative $\sigma$-algebra inherited from the Skorohod $\sigma$-algebra in
$D([0,\infty),[0,\infty])$.

  3. ii)

The mappings $\left(x_{t}\right)_{t\geq 0}\to\left(x_{c(t)}\right)_{t\geq 0}$
and $\left(z_{t}\right)_{t\geq 0}\to\left(z_{\gamma(t)}\right)_{t\geq 0}$ are
measurable from $D([0,\infty),\hat{E})$ to $D([0,\infty),\hat{E})$.

###### Proof.

Recall that, $E$ being locally compact and second countable, the Borel
$\sigma$-algebra on $D([0,\infty),\hat{E})$ generated by the Skorohod
topology, and the trace $\sigma$-algebra generated by the finite-dimensional
projections on $D([0,\infty),\hat{E})$, coincide. Then, i) follows from the
fact that, for each $t_{0}$, the map $\left(x_{t}\right)_{t\geq 0}\to
c(t_{0})$ is measurable from $\mathcal{B}(D([0,\infty),\hat{E}))$ to
$\mathcal{B}(\hat{[}0,\infty])$, where $\mathcal{B}(\cdot)$ stands for the
Borel $\sigma$-algebra in any topological space. Since
$\mathcal{B}(D([0,\infty),[0,\infty]))$ is generated by the finite dimensional
projections (see section 12 in [6]), the latter implies that the map
$\left(x_{t}\right)_{t\geq 0}\to\left(c(t)\right)_{t\geq 0}$ is measurable
from $D([0,\infty),\hat{E})$ to $D([0,\infty),[0,\infty])$.

On the other hand, ii) follows from i) plus Appendix M16 in [6]. ∎

##  3 Preliminary objects of study

In the following $\mathtt{C}(\cdot)$ (resp. $\mathtt{B}(\cdot)$) refers to the
space of $\mathbb{R}$-valued continuous (resp. measurable) functions defined
on some toplogical (resp. measurable) space; whereas $\mathtt{\bar{C}}(\cdot)$
(resp. $\mathtt{\bar{B}}(\cdot)$) refers to its bounded counterpart. Also
$\mathtt{C^{k}_{0}}$ refers to the space of continuous functions vanishing at
infinity, that have a continuous $k$-th derivative. Additionally,
$\mathcal{D}(\cdot)$ will refer to the domain of an operator.

###  3.1 $\Lambda$-coalescents

We expose the construction of coalescents with multiple merger from the
seminal works of [34, 35]. For a positive integer $p$, let
$[p]=\\{1,\cdots,p\\}$ and $\mathscr{P}_{[p]}$ be the space of partitions of
$[p]$ endowed with the discrete topology. We call the elements of any
partition $\pi\in\mathscr{P}_{[p]}$ the blocks of $\pi$ and denote its number
by $\\#\pi$. Let $\Lambda$ be a finite measure on $[0,1]$ which can be
decomposed as

$\Lambda=\Lambda(\\{0\\})\delta_{0}+\mathbbm{1}_{(0,1]}\Lambda.$

The $(p,\Lambda)$-coalescent process $\left(\Pi_{t}\right)_{t\geq 0}$ is a
Markov jump process with values in $\mathscr{P}_{[p]}$ that evolves through
“coagulations” or mergers. The latter consists of constructing a new coarser
partition of $[p]$ from an initial $\pi\in\mathscr{P}_{[p]}$ by taking the
union of a collection of blocks that are present in $\pi$. The coagulations of
$\left(\Pi_{t}\right)_{t\geq 0}$ are directed by the measure $\Lambda$ via the
following rules; at time $t\geq 0$:

Pairwise coagulations:

Any pair of blocks of $\Pi_{t}$ coagulate at rate $\Lambda(\\{0\\})$.

Coin-flip coagulations:

Any collection of $2\leq i\leq j=\\#\Pi_{t}$ blocks of $\Pi_{t}$, coagulate
into a single block at rate
$\beta_{j,i}^{(\Lambda)}\coloneqq\int_{(0,1]}\zeta^{i-2}(1-\zeta)^{j-i}\Lambda(d\zeta)$.

The first dynamics correspond to those of Kingman’s coalescent [27]. The
second dynamics have the following well-known interpretation: at rate
$\zeta^{-2}{\Lambda(d\zeta)}$ a value $\zeta\in(0,1]$ is drawn; then, each
block of $\Pi_{t}$ decides to participate in the coagulation event with
probability $\zeta$. This representation for the rates implies that those
processes are consistent according to $p$ and can thus be extended to
$p=\infty$. In this case, we will talk about $\Lambda$-coalescents.

A famous and important example of $\Lambda$-coalescent processes is the family
of Beta coalescents [37, 19] in which
$\Lambda(d\zeta)=c{\zeta^{1-\beta}(1-\zeta)^{\beta-1}}d\zeta$ for $\beta\in
0,2)$. See [2] for a thorough exposition of general coalescent processes.

###  3.2 The $\Lambda$-Fleming-Viot processes

We begin with a few remarks on the space $\mathtt{M}(\mathcal{T})$ endowed
with the topology of weak convergence. By Theorem 1.14 in [31] the space
$\mathtt{M}(\mathcal{T})$ is locally-compact whenever $\mathcal{T}$ is
compact. In fact, following the proof of this theorem, the set
$\mathtt{M}_{r}(\mathcal{T})=\\{\mu\in\mathtt{M}(\mathcal{T})\colon\mu(\mathcal{T})\leq
r\\}$, for $r\geq 0$, is compact.

Let us write

$\left\langle f,\mu\right\rangle\coloneqq\int\mu(da)f(a).$

An important class of functions in $\mathtt{C}(\mathtt{M}(\mathcal{T}))$ is
the algebra of polynomials $\mathtt{Pol}\left(\mathtt{M}(\mathcal{T})\right)$
which is the linear span of monomials of the form

$F_{\phi,p}(\rho)=\left\langle\phi,\rho^{\otimes
p}\right\rangle,\quad\phi\in\mathtt{\bar{C}}(\mathcal{T}^{p}).$

By a straightforward extension of Lemma 2.1.2 in [12] (extending the arguments
therein to $\mathtt{\bar{C}}(\mathtt{M}(\mathcal{T}))$), the polynomials
$\mathtt{Pol}\left(\mathtt{M}(\mathcal{T})\right)$ are dense in the topology
of uniform convergence on compact sets on
$\mathtt{C}(\mathtt{M}(\mathcal{T}))$, and convergence determining for the
topology of weak convergence in $\mathtt{M}(\mathtt{M}_{r}(\mathcal{T}))$ for
every $r\geq 0$.

A function $F\in\mathtt{C}(\mathtt{M}(\mathcal{T}))$ is said to be
differentiable if its derivative in the direction of $a\in\mathcal{T}$ (more
precisely of $\delta_{a}$) given by

$F^{\prime}(\mu;a)\coloneqq\lim_{\epsilon\to
0}\frac{F(\mu+\epsilon\delta_{a})-F(\mu)}{\epsilon},$

exists and is continuous as a function of $a\in\mathcal{T}$. We denote by
$F^{\prime\prime}(\mu;a,b)$ the second derivative of $F$, first in the
direction of $a$ and then in the direction of $b$; whereas for higher
derivatives we write $F^{(\ell)}(\mu;a_{1},\cdots,a_{\ell})$ for the
corresponding sequential derivatives in the directions of
$a_{1},\cdots,a_{\ell}$. By Lemma 2.1.2 in [12], the polynomials
$\mathtt{Pol}\left(\mathtt{M}(\mathcal{T})\right)$ are infinitely
differentiable. Their derivatives are given by

$F_{\phi,p}^{\prime}(\mu;a)=\sum_{i=1}^{p}\left\langle\phi,\mu^{\otimes
i-1}\otimes\delta_{a}\otimes\mu^{p-i-1}\right\rangle,$

and recursively, for $\bm{a}=(a_{1},\cdots,a_{\ell})$,

(7)
$F^{(\ell)}_{\phi,p}(\mu;\bm{a})=\begin{cases}\sum_{\sigma\in\mathcal{P}(\bm{a},\mu,p)}\left\langle\phi,\otimes_{i=1}^{p}\sigma_{i}\right\rangle&\text{
if }\ell\leq p\\\ 0&\text{ otherwise},\end{cases}$

where the sum is over all the permutations
$\sigma\in\mathcal{P}(\bm{a},\mu,p)$, say
$\sigma=(\sigma_{1},\cdots,\sigma_{p})$, of the atomic measures
$\delta_{a_{1}},\cdots,\delta_{a_{\ell}}$ and $p-\ell$ copies of $\mu$. Note
that in particular

$\left\lvert
F_{\phi,p}(\mu)+F_{\phi,p}^{\prime}(\mu;a)+F_{\phi,p}^{\prime\prime}(\mu;a,b)\right\rvert\leq
C\left\lVert\phi\right\rVert(1+\cdots+\left\lVert\mu\right\rVert^{p})$

for some $C$ depending on $p$, so that the polynomials
$\mathtt{Pol}(\mathtt{M}(\mathcal{T}))$ are contained in the set

$\displaystyle D\coloneqq\Big{\\{}$ $\displaystyle
F\in\mathtt{C}(\mathtt{M}(\mathcal{T}))\colon\exists C>0,p\geq 1\text{ s.t. }$
(8) $\displaystyle\left\lvert F(\mu)\right\rvert+\left\lvert
F^{\prime}(\mu;a)\right\rvert+\left\lvert
F^{\prime\prime}(\mu;a,a)\right\rvert\leq
C(1+\dots+\left\lvert\left\lVert\mu\right\rVert\right\rvert^{p})\Big{\\}},$

which will appear in the domain of generators of measure-valued processes
further ahead.

Now, let $\Lambda$ and $\beta_{j,i}^{(\Lambda)}$ be as in section 3.1. The
$\Lambda$-Fleming-Viot process [4] is the process with values in the space
$\mathtt{PM}(\mathcal{T})$ of probability measures on $\mathcal{T}$ and
generator of the form

$\displaystyle\mathcal{R}F_{\phi,p}(\rho)$
$\displaystyle=\int_{\mathcal{T}}\rho(da)\sum_{\ell=2}^{p}\beta_{p,\ell}^{(\Lambda)}\sum_{\sigma\in\mathcal{P}(\bm{a},\mu,p)}\left\\{\left\langle\phi,\otimes_{i=1}^{p}\sigma_{i}\right\rangle-\left\langle\phi,\rho^{\otimes
p}\right\rangle\right\\}$ (9)
$\displaystyle=\int_{\mathcal{T}}\rho(da)\sum_{\ell=2}^{p}\beta_{p,\ell}^{(\Lambda)}\left(F^{(\ell)}_{\phi,p}(\rho;a)-\binom{p}{\ell}F_{\phi,p}(\rho)\right);$

where we have made a slight abuse of notation by writing
$F^{(\ell)}_{\phi,p}(\mu;a)$ for the $\ell$-times derivative of $F$, all in
the direction of $a$. The above form of the generator yields the following
well-known duality relation between $\Lambda$-Fleming-Viot processes and
$\Lambda$-coalescents. This relation can be extended to a path-wise duality
relation via a coupling of both processes that is based on the lookdown
construction of [14], see section 2 in [8] and also [7] for details.

######  Lemma 3.1 ([4];[13]).

For fixed $\phi\in\mathtt{\bar{B}}(\mathcal{T}^{p})$, $p\geq 1$, let
$H_{\phi}(\pi,\rho)\colon\mathscr{P}_{[p]}\times\mathtt{PM}(\mathcal{T})\to\mathbb{R}$
be given by

$H_{\phi}(\pi,\rho)\coloneqq\langle\phi_{\pi},\rho^{\otimes p}\rangle$

where $\phi_{\pi}\in\mathtt{\bar{B}}(\mathcal{T}^{\\#\pi})$ is constructed
from $\phi$ by identifying its input coordinates, say $(a_{1},\cdots,a_{p})$,
according to the blocks of $\pi$. Then we have

$\mathbbm{E}_{\rho_{0}}[\langle\phi_{\pi_{0}},\rho_{t}^{\otimes
p}\rangle]=\tilde{}\mathbbm{E}_{\pi_{0}}[\langle\phi_{\Pi_{t}},\rho_{0}^{\otimes
p}\rangle]$

whenever $\left(\rho_{t}\right)_{t\geq 0}$ is a $\Lambda$-Fleming-Viot process
under $\mathbbm{P}$ and $\left(\Pi_{t}\right)_{t\geq 0}$ is a
$(p,\Lambda)$-coalescent process under $\tilde{}\mathbbm{P}$.

The case $\Lambda=\frac{\sigma^{2}}{2}\delta_{0}$ corresponds to the standard
Flemming-Viot process without mutation and of parameter $\sigma$ which is dual
to Kingman’s coalescent and in which (3.2) becomes

$\mathcal{R}F(\rho)=\frac{\sigma^{2}}{2}\int_{\mathcal{T}}\rho(da)(F^{\prime\prime}(\rho;a,a)-F(\rho)).$

When $\Lambda(0)=0$, the generator (3.2) can be written as, see [8],

$\mathcal{R}F(\rho)=\int_{(0,1]}\frac{\Lambda(\hskip
3.0ptd\zeta)}{\zeta^{2}}\int_{\mathcal{T}}\rho(\hskip
3.0ptda)(F(\rho(1-\zeta)+\zeta\delta_{a})-F(\rho)).$

Combining both cases we obtain the following form of the generator

$\displaystyle\mathcal{R}F(\rho)=$
$\displaystyle\frac{\sigma^{2}}{2}\int_{\mathcal{T}}\rho(da)(F^{\prime\prime}(\rho;a,a)-F(\rho))+$
(10) $\displaystyle\int_{(0,1]}\frac{\Lambda(\hskip
3.0ptd\zeta)}{\zeta^{2}}\int_{\mathcal{T}}\rho(\hskip
3.0ptda)(F(\rho(1-\zeta)+\zeta\delta_{a})-F(\rho)).$

When $\zeta^{-2}\Lambda(\hskip 3.0ptd\zeta)$ is finite, the above form of the
generator gives the following picture for the dynamics of the process. It has
jumps of the form $\rho_{t-}\to\rho_{t-}(1-\zeta)+\zeta\delta_{a}$ at the
atoms $(t,\zeta)$ of a Poisson point process on $\mathbb{R_{+}}\times[0,1]$
with intensity $dt\times\zeta^{-2}\Lambda(\hskip 3.0ptd\zeta)$; here the
position $a$ of the new atom of size $\zeta$ is chosen randomly according to
$\rho_{t-}$. After each jump, the process starts as an independent copy of a
standard Flemming-Viot process of parameter $\Lambda(\\{0\\})$ started at the
new state $\rho_{t}=\rho_{t-}(1-\zeta)+\zeta\delta_{a}$. An adaptation of the
proof of Theorem 3 in [4] (to include the case of “pre-limit”
$\Lambda$-measures not vanishing at $\\{0\\}$) yields the following weak-limit
construction of the process for general finite measures $\Lambda$.

######  Proposition 3.2.

For a sequence of measures
$\Lambda_{n}\overset{n\to\infty}{\underset{weakly}{\longrightarrow}}\Lambda$
such that $\zeta^{-2}\Lambda_{n}(d\zeta)$ are finite measures in $(0,1]$, let
$\left(\rho^{(n)}_{t}\right)_{t\geq 0}$ be corresponding
$\Lambda_{n}$-Flemming-Viot processes started at $\rho^{(n)}_{0}$. If
$\rho^{(n)}_{0}\overset{n\to\infty}{\underset{d}{\longrightarrow}}\rho_{0}$
for the weak topology on $\mathtt{PM}(\mathcal{T})$, then
$\left(\rho^{(n)}_{t}\right)_{t\geq
0}\overset{n\to\infty}{\underset{d}{\longrightarrow}}\left(\rho_{t}\right)_{t\geq
0}$ for the Skorohod topology on $D([0,\infty),\mathtt{PM}(\mathcal{T}))$,
where $\left(\rho_{t}\right)_{t\geq 0}$ is the $\Lambda$-Flemming-Viot process
started at $\rho_{0}$.

###  3.3 The Dawson-Watanabe process and its Lamperti transformation

Here we introduce the Dawson-Watanabe process without mutation/spatial motion.
This will suffice our applications further ahead; the interested reader can
refer to [33] for a more general setting.

For fixed $\sigma\in\mathbb{R}$, the Dawson-Watanabe process without mutation
can be defined as the unique continuous process $\left(\mu_{t}\right)_{t\geq
0}$ on $\mathtt{M}(\mathcal{T})$ such that for all
$\phi\in\mathtt{\bar{C}}(\mathcal{T})$ the process

$\quad
M_{t}(\phi)=\left\langle\phi,\mu_{t}\right\rangle-\left\langle\phi,\mu_{0}\right\rangle$

is a martingale with quadratic variation

$\left[M(\phi)\right]_{t}=\frac{\sigma^{2}}{2}\int_{0}^{t}\left\langle\phi^{2},\mu_{s}\right\rangle
ds.$

Alternatively, it can be defined as the unique solution to the martingale
problem for the operator

$\mathcal{W}F(\mu)=\frac{\sigma^{2}}{2}\int_{\mathcal{T}}\mu(da)F^{\prime\prime}(\mu;a,a),$

with domain $\mathcal{D}(\mathcal{W})$ satisfying $\\{f(H(\mu))\colon
H\in\mathtt{Pol}(\mathtt{M}(\mathcal{T})),f\in\mathtt{C^{\infty}_{0}}(\mathbb{R})\\}\subset\mathcal{D}(\mathcal{W}).$
(see Corollary 2.23 in [17]).

Taking functions of the form $H(\mu)=\left\langle 1,\mu\right\rangle$, and
$f\in\mathtt{C^{\infty}_{0}}(\mathbb{R})$ in this generator, it is easy to see
that the total mass process
$\left(\left\lVert\mu_{t}\right\rVert\right)_{t\geq 0}$ has generator of the
form $\mathcal{K}f(x)=\frac{\sigma^{2}}{2}xf^{\prime\prime}(x)$ on the set
$\\{f\in\mathtt{C^{2}_{0}}(\mathbb{R})\colon
xf^{\prime\prime}\in\mathtt{C}([0,\infty])\\}\subset\mathcal{D}(\mathcal{K})$
which conforms with the general form of the generator of a continuous positive
1-SS Markov process. By Theorem 5.1 in [30] the process
$\left(\left\lVert\mu_{t}\right\rVert\right)_{t\geq 0}$ is then a uniquely
determined diffusion (Feller’s diffusion), it is absorbed at $0$ with
probability 1 and, furthermore, by Theorem 4.1 therein the time-changed
process $\left(\log(\left\lVert\mu_{c_{1}(t)}\right\rVert)\right)_{t\geq 0}$
is a continuous Lévy process with diffusion parameter $\sigma$ and drift
parameter $-\sigma$. The latter will be denoted $d$ to be tracked as it will
appear in the next sections.

On the other hand, it is well known from Theorem 1.1 i) in [8] (see also [32])
that the time-changed frequency process $\left(\rho_{t}\right)_{t\geq 0}$
defined by
$\rho_{t}=\frac{\mu_{c_{1}(t)}}{\left\lVert\mu_{c_{1}(t)}\right\rVert}$ is a
standard Fleming-Viot process of parameter $\sigma$.

We end this section with our first application of Theorem 2.2, which
complements the latter observations, and which we will then generalize in
Theorem 4.2.

######  Proposition 3.3.

The process $\left(\mu_{t}\right)_{t\geq 0}$ is a $1$-SS Markov process. The
time changed process $\left(\nu_{t}\right)_{t\geq
0}=\left(\mu_{c_{1}(t)}\right)_{t\geq 0}$ is SMH and its generator
$\mathcal{G}^{(B)}_{\sigma}$ has the form

(11)
$\mathcal{G}^{(B)}_{\sigma}F(\nu)\coloneqq\left\lVert\nu\right\rVert\mathcal{W}F(\nu)$

in the set

(12)
${D}_{1}\coloneqq\\{F\in\mathcal{D}(\mathcal{W})\colon\left\lVert\cdot\right\rVert\mathcal{W}F(\cdot)\in\mathtt{\bar{B}}(\mathtt{M}(\mathcal{T}))\\}\cup\\{D\cap\mathcal{D}(\mathcal{W})\\}.$

Furthermore, the process $\left(\rho_{t},\xi_{t}\right)_{t\geq
0}=\left(\frac{\nu_{t}}{\left\lVert\nu_{t}\right\rVert},\log(\left\lVert\nu_{t}\right\rVert)\right)_{t\geq
0}$ is a MAP with $\left(\rho_{t}\right)_{t\geq 0}$ being a standard Fleming-
Viot process of parameter $\sigma$, and $\left(\xi_{t}\right)_{t\geq 0}$ a
continuous Lévy process with diffusion parameter $\sigma$ and drift parameter
$d=-\sigma$.

###### Proof.

We first check that $\left(\nu_{t}\right)_{t\geq 0}$ is characterized by (11).
Since $\left(\left\lVert\mu_{t}\right\rVert\right)_{t\geq 0}$ is continuous
and absorbed at 0, we get that

$\inf_{s>0}\left\\{s\colon\left\lVert\mu_{s}\right\rVert=0\right\\}\underset{a.s.}{=}\inf_{s>0}\left\\{s\colon\int_{0}^{s}\frac{1}{\left\lVert\mu_{s}\right\rVert}ds=\infty\right\\},$

which implies that $\left(\nu_{t}\right)_{t\geq 0}$ is a solution to the
martingale problem for $\mathcal{G}^{(B)}_{\sigma}$ on the domain
$\\{F\in\mathcal{D}(\mathcal{W})\colon\left\lVert\cdot\right\rVert\mathcal{W}F(\cdot)\in\mathtt{\bar{B}}(\mathtt{M}(\mathcal{T}))\\}$
through a direct application of Theorem VI.1.3 in [16]. Thanks to the fact
that the time-changed process
$\left(\left\lVert\nu_{t}\right\rVert\right)_{t\geq 0}$ is the exponential of
the continuous Lévy process given by $\xi_{t}=dt+\sigma B_{t}$, the domain can
easily be extended to contain the set $D\cap\mathcal{D}(\mathcal{W})$ by a
mild adaptation of the proof of Theorem VI.1.3 in [16]. Indeed, we first note
that $N_{t}=F(\mu_{t})-\int_{0}^{t}\mathcal{W}F(\mu_{u})du$ is a martingale.
Furthermore, for $F\in D\cap\mathcal{D}(\mathcal{W})$, we have the bound

$\displaystyle\sup_{0\leq s\leq t}\left\lvert
N_{c_{1}(s)}\right\rvert^{2}=\sup_{0\leq s\leq t}\left\lvert
F(\mu_{c_{1}(s)})-\int_{0}^{c_{1}(s)}\mathcal{W}F(\mu_{c_{1}(u)})du\right\rvert^{2}$
$\displaystyle=\sup_{0\leq s\leq t}\left\lvert
F(\nu_{s})-\int_{0}^{s}\left\lVert\nu_{u}\right\rVert\mathcal{W}F(\nu_{u})du\right\rvert^{2}\leq
C(\sup_{0\leq s\leq t}1+\cdots+e^{p\xi_{s}})^{2}$

for some $C>0,p\geq 1$. By Doob’s $\mathtt{L}^{2p}$ inequality applied to the
submartingale $\left(e^{2p\xi_{t}}\right)_{t\geq 0}$, the last line above is
integrable. Using Hölder’s inequality we also obtain

$\displaystyle\lim_{T\to\infty}\mathbbm{E}[\left\lvert
N_{T}\right\rvert;c_{1}(t)>T]$ $\displaystyle\leq\mathbbm{E}[\sup_{0\leq s\leq
t}\left\lvert
N_{c_{1}(s)}\right\rvert^{2}]^{1/2}\mathbbm{P}(c_{1}(t)>T)^{1/2}\to 0$

where we have used $c_{1}(t)\underset{a.s.}{<}\infty$. The optional sampling
theorem (e.g. Theorem II.2.13 [16]) then implies that
$\left(N_{c_{1}(t)}\right)_{t\geq 0}$ is a
$\mathscr{M}_{c_{1}(t)}$-martingale.

On the other hand, the stated properties for
$\left(\rho_{t},\xi_{t}\right)_{t\geq 0}$ will all follow from Theorem 2.2
once we prove that $\left(\mu_{t}\right)_{t\geq 0}$ is 1-SS. The latter
follows from Proposition 2.3 and the computation

$\mathcal{W}\mathcal{S}_{a}F(\mu)=\frac{\sigma^{2}}{2}\int_{\mathcal{T}}\mu(db)(\mathcal{S}_{a}F)^{\prime\prime}(\mu;b,b)=\frac{\sigma^{2}}{2}\int_{\mathcal{T}}\mu(db)a^{2}F^{\prime\prime}(a\mu;b,b)$

so that

$\mathcal{S}_{a^{-1}}\mathcal{W}\mathcal{S}_{a}F(\mu)=a\frac{\sigma^{2}}{2}\int_{\mathcal{T}}\mu(db)F^{\prime\prime}(\mu;b,b)=a\mathcal{W}F(\mu).$

∎

##  4 The genealogy of self-similar populations

###  4.1 Main results and comments

We will now apply the generalized Lamperti transformation developed in section
2 for the conic subspace ${E=}\mathtt{M}(\mathcal{T})$. The first result is
showing the existence of a Feller process with values in ${E}$ enjoying the
SMH property and having a generator of the form

(13) $\displaystyle\mathcal{G}F(\nu)=$
$\displaystyle\mathcal{G}^{(D)}_{\kappa}F(\nu)+\mathcal{G}^{(B)}_{\sigma}F(\nu)+\mathcal{G}^{(J)}_{\Lambda}F(\nu)$

where $\mathcal{G}^{(D)}_{\kappa},\mathcal{G}^{(B)}_{\sigma}$ and
$\mathcal{G}^{(J)}_{\Lambda}$ are as in Theorem 1.1. The domain of the
generator will be proved to contain the set

$D_{2}=\\{F\in D\cap
D_{1}\colon\mathcal{G}F\in\mathtt{C}(\mathtt{M}(\mathcal{T}))\\},$

with $D$ as in (3.2) and $D_{1}$ as in (12).

######  Theorem 4.1.

There exists a Feller process $\left(\nu_{t}\right)_{t\geq 0}$ on
$\mathtt{M}(\mathcal{T})$ with generator of the form (13) in $D_{2}$.
Furthermore, the process $\left(\nu_{t}\right)_{t\geq 0}$ is SMH and is
absorbed at $0$.

The proof of the above theorem is split into several intermediary results. In
section 4.2 we provide a Poissonian construction of the process
$(\nu_{t})_{t\geq 0}$ when the measure $\zeta^{-2}\Lambda(d\zeta)$ is finite;
while in section 4.3 we extend the construction to any finite measure
$\Lambda$ through a weak limit and the convergence of the generators.

Having constructed the SMH process $\left(\nu_{t}\right)_{t\geq 0}$, our main
result of this section is the following theorem which, on the one hand,
generalizes Proposition 2 in [28] to $\beta$-stable measure-valued processes
$(\alpha>1)$ and, on the other, re-frames Theorem 1 in [8] as a Lamperti
transformation at least in the case $\beta\in(1,2]$ which comprises the
intersection between branching and self-similar processes of index
$\alpha=\beta-1$ (see Remark 2 below). Finally, the following theorem also
describes the genealogy of populations whose total size evolves as a positive
SS Markov process with non-negative jumps, through the well-known duality
relationship between $\Lambda$-Fleming-Viot processes and
$\Lambda$-coalescents [4] in the same spirit as [8] (see Lemma 3.1 and Remark
1). The core of the proof of the following Theorem 4.2 is already furnished by
section 2.

######  Theorem 4.2.

Let $\left(\nu_{t}\right)_{t\geq 0}$ be the SMH process constructed in Theorem
4.1.

  1. i)

The process $(\rho_{t},\xi_{t})_{t\geq 0}$ where
$\rho_{t}=\nu_{t}/\left\lVert\nu_{t}\right\rVert$ and
$\xi_{t}=\log(\left\lVert\nu_{t}\right\rVert)$ is a MAP. Furthermore,
$\left(\rho_{t}\right)_{t\geq 0}$ is a $\Lambda$-Fleming-Viot process, whereas
$\left(\xi_{t}\right)_{t\geq 0}$ is the Lévy process with triplet
$(d+\kappa,{\sigma},\Pi)$ where $\Pi(d\zeta)$ is the pushforward of the
measure $\zeta^{-2}\Lambda(d\zeta)$ under the transformation
$\zeta\to-\log(1-\zeta)$ on $(0,1]$.

  2. ii)

Let $\alpha\in\mathbb{R}$ and recall the random time change
$\gamma_{\alpha}(t)$ of Theorem 2.2. The time-changed process
$\left(\mu_{t}\right)_{t\geq 0}=\left(\nu_{\gamma_{\alpha}(t)}\right)_{t\geq
0}$ is the unique solution to

(14) $\mu_{t}=\nu_{\int_{0}^{t}\left\lVert\mu_{s}\right\rVert^{-\alpha}ds}.$

Furthermore, it is a $\alpha$-SS standard Markov process with generator of the
form

(15) $\displaystyle\mathcal{F}F(\mu)=$
$\displaystyle\frac{1}{\left\lVert\mu\right\rVert^{\alpha}}\mathcal{G}F(\mu)$

with domain containing $\\{F\in
D_{2}\colon\left\lVert\cdot\right\rVert^{-\alpha}\mathcal{G}{F}(\cdot)\in\mathtt{\bar{B}}(\mathcal{M}(\mathcal{T}))\\}\subset\mathcal{D}(\mathcal{F}).$

  3. iii)

Conversely, if $\left(\mu_{t}\right)_{t\geq 0}$ is a standard Markov process
with generator of the form (15). Then the process
$\left(\rho_{t},\xi_{t}\right)_{t\geq 0}$ given by
$\rho_{t}=\mu_{c_{\alpha}(t)}/\left\lVert\mu_{c_{\alpha}(t)}\right\rVert$ and
$\xi_{t}=\log(\left\lVert\mu_{c_{\alpha}(t)}\right\rVert)$ is a MAP as in i).
Furthermore, $\nu_{t}=\rho_{t}e^{\xi_{t}}$ is the unique solution to

$\nu_{t}=\mu_{\int_{0}^{t}\left\lVert\nu_{s}\right\rVert^{\alpha}ds}$

and it is a standard Markov process with generator of the form (13).

Looking at $\mathcal{G}^{(J)}_{\Lambda}$ in the generator of the process
$\left(\nu_{t}\right)_{t\geq 0}$ we observe that it has jumps of the form

$\nu\to\nu+\left\lVert\nu\right\rVert x\delta_{a}$

for some weight $x>0$ and a location $a$ for the new atom that is chosen
according to the empirical distribution $\nu/\left\lVert\nu\right\rVert$
before the jump. Since $\left\lVert\nu\right\rVert x$, the size of the new
atom, depends on the total mass of the population, these processes will not be
branching processes in general, nor will be their time-changed counterparts
$\left(\mu_{t}\right)_{t\geq 0}$. However, the above form for the jumps of
$\left(\nu_{t}\right)_{t\geq 0}$ ensures that its corresponding frequency
process is Markov with jumps

$\displaystyle\rho=\frac{\nu}{\left\lVert\nu\right\rVert}$
$\displaystyle\to\frac{\nu}{\left\lVert\nu\right\rVert}\left(\frac{1}{1+x}\right)+\frac{x}{1+x}\delta_{a}$
$\displaystyle=\rho\left(\frac{1}{1+x}\right)+\frac{x}{1+x}\delta_{a}.$

The particular choice $x=\zeta/(1-\zeta)$ ensures that the resulting dynamics
are exactly those of (3.2) for FV processes. On the other hand, the dynamics
of the time-changed process $\left(\mu_{t}\right)_{t\geq 0}$ incorporate an
additional dependency on the total mass by modulating the overall reproduction
rate of the population according to $\left\lVert\mu_{t}\right\rVert^{-\alpha}$
as seen in (15). By picking the right combination of the parameters $\alpha$
and $\Lambda\equiv\Lambda_{\alpha}$ one can show, via a simple computation on
the generator (15), that these two sources of dependency on the total mass in
fact cancel out. The resulting process $\left(\mu_{t}\right)_{t\geq 0}$ is
then a branching process, in particular a stable branching process (see Remark
2 below).

###### Proof of Theorem 4.2.

Item i) will be proved along with the construction of the process
$\left(\nu_{t}\right)_{t\geq 0}$ in sections 4.2 and 4.3.

Item ii) follows from first applying Theorem 2.2 to obtain that
$\left(\mu_{t}\right)_{t\geq 0}$ is a standard $\alpha$-SS Markov process.
Now, observe that
$\left(\left\lVert\nu_{t}\right\rVert^{-\alpha}\right)_{t\geq 0}$ is a.s.
bounded on bounded time intervals since $\left(\xi_{t}\right)_{t\geq 0}$ is
a.s. bounded away from $-\infty$ in bounded time intervals. The latter also
implies that
$\inf\\{t\colon\left\lVert\nu_{t}\right\rVert^{-\alpha}=0\\}\underset{a.s.}{=}\infty=\gamma_{\alpha}(\infty)$.
Then by Theorem VI.1.1 in [16] the process $\left(\mu_{t}\right)_{t\geq 0}$ is
the unique solution to (14) and, furthermore, by Theorem VI.1.3 therein, it is
a solution to the martingale problem for (15) on $\\{F\in
D_{2}\colon\left\lVert\cdot\right\rVert^{-\alpha}\mathcal{G}{F}(\cdot)\in\mathtt{\bar{B}}(\mathcal{M}(\mathcal{T}))\\}$.

Finally, iii) follows by first noting that
$\left(\left\lVert\mu_{t}\right\rVert\right)_{t\geq 0}$ is a positive
$\alpha$-SS Markov process with non-negative jumps (see Remark 1). This
implies that if $\zeta^{\mu}_{0}$ is its absorption time at $0$, then
$\mu_{\zeta^{\mu-}_{0}}\underset{a.s.}{=}0$; i.e. $\left(\mu_{t}\right)_{t\geq
0}$ is absorbed continuously at $0$. Then Lemmas 3.1 and 3.3 in [30] imply
that $c_{\alpha}(\infty)\underset{a.s.}{=}\infty.$ In any case we have

$\inf\\{t\colon\left\lVert\mu_{t}\right\rVert^{\alpha}=0\\}\underset{a.s.}{=}c_{\alpha}(\infty).$

Then, a second application of Theorems VI.1.1 and VI.1.3 in [16] as before
yields the result. ∎

######  Remark 1.

As in section 3.3, when taking functions of the form $F(\mu)=f(\left\langle
1,\mu\right\rangle)$ in (15), we obtain that the total mass process
$\left(\left\lVert\mu_{t}\right\rVert\right)_{t\geq 0}$, is a positive SS
Markov process with generator

$\displaystyle\mathcal{K}f(x)=$
$\displaystyle{(d+\kappa)}x^{1-\alpha}f^{\prime}(x)+\frac{\sigma^{2}}{2}x^{2-\alpha}f^{\prime\prime}(x)$
(16) $\displaystyle+$ $\displaystyle
x^{-\alpha}\int_{1}^{\infty}[f(x\zeta)-f(x)-f^{\prime}(x)\log(\zeta)\mathbbm{1}_{\left\lvert\log(\zeta)\right\rvert<1/2}]\Theta(d\zeta)$

where $\Theta(d\zeta)$ is the pushforward of the measure
$\zeta^{-2}{\Lambda(d\zeta)}$ under the transformation
$\zeta\to{1}/({1-\zeta})$ (c.f. Theorem 6.1 in [30]).

######  Remark 2.

Taking $\sigma=0$, and
$\Lambda(d\zeta)=c{\zeta^{1-\beta}(1-\zeta)^{\beta-1}}d\zeta$ with $c>0$ and
$\beta\in(1,2)$; and also

$\displaystyle\kappa$
$\displaystyle=-\int_{0}^{1}\frac{\Lambda(d\zeta)}{\zeta^{2}}\left\\{\frac{\zeta}{1-\zeta}-(\left\lvert\log(1-\zeta)\right\rvert\mathbbm{1}_{\zeta<1/2})\right\\},$

in (15), we obtain

$\displaystyle\mathcal{F}(\nu)=\frac{1}{\left\lVert\nu\right\rVert^{\beta-1}}\int_{\mathcal{T}}\frac{\nu(da)}{\left\lVert\nu\right\rVert}\int_{0}^{1}\frac{\Lambda(d\zeta)}{\zeta^{2}}\bigg{\\{}$
$\displaystyle
F\left(\nu+\left\lVert\nu\right\rVert\frac{\zeta}{1-\zeta}\delta_{a}\right)-F(\nu)$
$\displaystyle-\left\lVert\nu\right\rVert\frac{\zeta}{1-\zeta}F^{\prime}(\nu;a)\bigg{\\}}.$

The latter, after the change of variable
$h=\left\lVert\nu\right\rVert\frac{\zeta}{1-\zeta}$, becomes

$\mathcal{F}(\nu)=c\int_{\mathcal{T}}\nu(da)\int_{0}^{\infty}h^{-1-\beta}dh\bigg{\\{}F\left(\nu+h\delta_{a}\right)-F(\nu)-hF^{\prime}(\nu;a)\bigg{\\}},$

the generator of a $\beta$-stable measure-valued branching process with
$\beta>1$. The case $\beta=2$ is covered by picking
$\Lambda(d\zeta)=c\delta_{0}(d\zeta)$. Thus, Theorem 4.2 ii) generalizes
Proposition 2 [28] to the measure-valued setting, while Theorem 4.2 iii)
recovers Theorem 1 i) and ii) (for $\beta>1$) in [8].

######  Remark 3.

Theorems 4.1 and 4.2 can be generalized in the two following directions using
the same arguments in the proof, although details will not appear in this
paper to avoid such burden in notation in our main results.

  * i)

A general positive SS Markov process, see [3], can be attained for the total
mass process $\left(\left\lVert\mu_{t}\right\rVert\right)_{t\geq 0}$ by adding
negative jumps of the form $x\to x\zeta$, $0<\zeta<1,$ with intensity
$\Theta^{-}(d\zeta)$, the pushforward of a Lévy measure $\Pi$ on $(-\infty,0)$
(i.e. a measure satisfying $\int_{-\infty}^{0}(\zeta^{2}\wedge
1)\Pi(d\zeta)<\infty$) under the transformation $\zeta\to e^{\zeta}$. A simple
way to do this is to allow for the underlying SMH process
$\left(\nu_{t}\right)_{t\geq 0}$ to have jumps of the form $\nu\to\nu\zeta$
with intensity $\Theta^{-}(d\zeta)$. In this sense, a term of the form

$\mathcal{G}^{(J-)}_{\Theta^{-}}F(\nu)=\int_{(0,1)}\Theta^{-}(d\zeta)\left\\{F\left(\nu\zeta\right)-F(\nu)\right\\}$

is added to its generator. Adding negative jumps in this way does not change
the dynamics of the corresponding frequency process, so that the arguments
leading to the construction of the SMH process $\left(\nu_{t}\right)_{t\geq
0}$ can be easily generalized in this direction.

  * ii)

It is possible to extend our results to the $\Xi$-Fleming-Viot case with
$\int_{[0,1]^{\mathbb{N}}}\sum_{k=1}^{\infty}\zeta_{k}^{2}\Xi(d\zeta)<\infty$.
In this case the generator of $\left(\nu_{t}\right)_{t\geq 0}$ is given by
updating the jumping part $\mathcal{G}^{(J)}$ as

$\displaystyle\mathcal{G}^{(J)}_{\Xi}F(\nu)$
$\displaystyle=\int_{\left(\mathcal{T}\right)^{\mathbb{N}}}\left(\frac{\nu}{\left\lVert\nu\right\rVert}\right)^{\otimes\mathbb{N}}(da)\int_{(0,1]^{\mathbb{N}}}\Xi(d\zeta)\bigg{\\{}F\left(\nu+\left\lVert\nu\right\rVert\sum_{i=1}^{\infty}\frac{\zeta_{i}}{1-\left\lVert\zeta\right\rVert_{\mathtt{l}_{1}}}\delta_{a_{i}}\right)-F(\nu)$
$\displaystyle\qquad\qquad\qquad-\left\lVert\nu\right\rVert\left\lvert\log(1-\left\lVert\zeta\right\rVert_{\mathtt{l}_{1}})\right\rvert\mathbbm{1}_{\left\lVert\zeta\right\rVert_{\mathtt{l}_{1}}<1/2}F^{\prime}(\nu;a)\bigg{\\}}.$

Heuristically, jumps are of the form

$\nu\to\nu+\sum_{k=1}^{\infty}\frac{\zeta_{i}}{1-\left\lVert\zeta\right\rVert_{\mathtt{l}_{1}}}\delta_{a_{k}}$

where the positions of the new atoms $(a_{k})_{k\geq 0}$ are i.i.d with
distribution $\frac{\nu_{t-}}{\left\lVert\nu_{t-}\right\rVert}$ and the atom
sizes $\zeta=(\zeta_{1},\zeta_{2},\dots)$, satisfying
$\left\lVert\zeta\right\rVert_{\mathtt{l}_{1}}<1$, arrive with intensity
$dt\otimes\Xi(d\zeta)$.

The process $\left(\rho_{t}\right)_{t\geq 0}$ becomes the $\Xi$-Flemming-Viot
process and $\left(\xi_{t}\right)_{t\geq 0}$ is a Lévy process with
compensated jumps given by

$\int_{0}^{t}\int_{0}^{1}\left\lvert\log(1-\left\lVert\zeta\right\rVert_{\mathtt{l}_{1}})\right\rvert\tilde{\mathcal{P}}(ds,d\zeta)$

where
$\tilde{\mathcal{P}}(ds,d\zeta)=\mathcal{P}(ds,\zeta)-\mathbbm{1}_{\left\lVert\zeta\right\rVert_{\mathtt{l}_{1}}<1/2}ds\otimes\Xi(d\zeta)$
and $\mathcal{P}$ is a Poisson point process on
$\mathbb{R}^{+}\times(0,1)^{\mathbb{N}}$ with intensity
$ds\otimes\Xi(d\zeta)$.

###  4.2 Poissonian construction for finite $\zeta^{-2}\Lambda(d\zeta)$

A simple way to think of the process $\left(\nu_{t}\right)_{t\geq 0}$ when
$\zeta^{-2}\Lambda(d\zeta)$ is finite is as a time-changed and mass-scaled
Dawson-Watanabe process to which atoms are added at times of an independent
Poison point process (PPP). The size of such atoms will in general depend on
the total population size at jump times.

Formally, for an initial condition $\nu$, let
$\left(\hat{\nu}_{t}\right)_{t\geq
0}=\left(\mu^{(DW)}_{\gamma_{1}(t)}\right)_{t\geq 0}$ where
$\left(\mu^{(DW)}_{t}\right)_{t\geq 0}$ is a standard $\sigma$-Dawson-Watanabe
process started at $\nu$; and $\gamma_{1}(t)=\inf\left\\{s\geq
0\colon\int_{0}^{s}\frac{1}{\left\lVert\mu^{(DW)}_{u}\right\rVert}du>t\right\\}$
is the 1-SS Lamperti time change. We will work with several independent copies
of $\left(\hat{\nu}_{t}\right)_{t\geq 0}$ started at different initial
measures $\nu$; let us denote them by $\left(\hat{\nu}_{t}(\nu)\right)_{t\geq
0}$ when necessary.

Let

$\hat{\kappa}\coloneqq\kappa-\int_{(0,1/2]}\left\lvert\log(1-\zeta)\right\rvert\frac{\Lambda(d\zeta)}{\zeta^{2}}$

and let $\mathcal{P}$ be a PPP with intensity $\hskip
3.0ptdt\otimes\zeta^{-2}{\Lambda{(\hskip 3.0ptd\zeta)}}$ on
$\mathbb{R_{+}}\times(0,1)$. For $t>0$ let
$(t_{1},\zeta_{1}),\dots,(t_{K},\zeta_{k})$ be the atoms of $\mathcal{P}$ such
that their first coordinate is less than $t$, ordered increasingly along the
first (time) coordinate and set $t_{K+1}=t$. Then, conditionally on
$(t_{i},\zeta_{i})_{1\leq i\leq K}$, define $\nu_{s}$, $s\in[0,t)$,
recursively as

(17)
$\displaystyle\nu_{s}\coloneqq\begin{cases}e^{\hat{\kappa}s}\hat{\nu}_{s}\left(\nu\right)&\text{
if }0\leq s<t_{1},\\\
e^{\hat{\kappa}(s-t_{i})}\hat{\nu}_{s-t_{i}}\left(\nu_{t_{i}-}+\left\lVert\nu_{t_{i}-}\right\rVert\frac{\zeta_{i}}{1-\zeta_{i}}\delta_{a_{i}}\right)&\text{
if }t_{i}\leq s<t_{i+1},\end{cases}$

where the processes $\left(\hat{\nu}_{t}(\cdot)\right)_{t\geq 0}$ that appear
in each time interval $[t_{i},t_{i+1}]$ are all independent and the points
$(a_{i})_{1\leq i\leq K}$ are chosen independently according to
$\nu_{t_{i}-}/\left\lVert\nu_{t_{i}-}\right\rVert$.

######  Lemma 4.3.

The process $\left(\nu_{t}\right)_{t\geq 0}$ is a standard SMH process with
generator given by $\mathcal{G}$ in $\eqref{eq:generatormvMp}$ and domain
$\mathcal{D}(\mathcal{G}^{(B)}_{\sigma})$ containing $D_{1}$.

###### Proof.

First, when $\Lambda\equiv 0$, i.e. for the Markov process
$\left(e^{\hat{\kappa}t}\hat{\nu}_{t}\right)_{t\geq 0}$, a simple computation
yields

$\frac{d}{dt}\bigg{|}_{t=0}\mathbbm{E}_{\nu}[F(e^{\hat{\kappa}t}\hat{\nu}_{t})]=\mathcal{G}^{(D)}_{\hat{\kappa}}F(\nu)+\mathcal{G}^{(B)}_{\sigma}F(\nu).$

To add the jumps, and compute the resulting generator, we use Theorem 2.4 in
[36]. For this, we set (in their notation) $\phi_{t}=C_{\Lambda}t$ with
$C_{\Lambda}\coloneqq\int_{[0,1]}\zeta^{-2}\Lambda(d\zeta)$ and the transition
kernel $K(\nu,\cdot)$ given by, for
$F\in\mathtt{\bar{B}}(\mathtt{M}(\mathcal{T}))$,

$K(\nu,F)=\begin{cases}\int_{(0,1]}\zeta^{-2}\frac{\Lambda(d\zeta)}{C_{\Lambda}}\int_{\mathcal{T}}\frac{\nu}{\left\lVert\nu\right\rVert}(da)F(\nu+\left\lVert\nu\right\rVert\frac{\zeta}{1-\zeta}\delta_{a})&\text{
whenever }\nu\neq 0,\\\ F(0)&\text{ otherwise.}\end{cases}$

Then, $\phi_{t}$ being of “Kac type”, according to Theorem 2.4 (and Example 1)
in [36], the generator of the process in (17) is given by

$\displaystyle\frac{d}{dt}\bigg{|}_{t=0}\mathbbm{E}[F(e^{\hat{\kappa}t}\hat{\nu}_{t})]$
$\displaystyle=\mathcal{G}^{(D)}_{\hat{\kappa}}F(\nu)+\mathcal{G}^{(B)}_{\sigma}F(\nu)$
$\displaystyle+\int_{(0,1]}\zeta^{-2}\Lambda(d\zeta)\int_{\mathcal{T}}\frac{\nu}{\left\lVert\nu\right\rVert}(da)\left\\{F(\nu+\left\lVert\nu\right\rVert\frac{\zeta}{1-\zeta}\delta_{a})-F(\nu)\right\\}$
$\displaystyle=\mathcal{G}^{(D)}_{\kappa}F(\nu)+\mathcal{G}^{(B)}_{\sigma}F(\nu)+\mathcal{G}^{(J)}_{\Lambda}F(\nu)$

with domain $\mathcal{D}(\mathcal{G}^{(B)}_{\sigma})$. ∎

######  Proposition 4.4.

The process $(\rho_{t},\xi_{t})_{t\geq 0}$ where
$\rho_{t}=\nu_{t}/\left\lVert\nu_{t}\right\rVert$ and
$\xi_{t}=\log(\left\lVert\nu_{t}\right\rVert)$ is a MAP whose first coordinate
is a $(\frac{\sigma^{2}}{2}\delta_{0}+\Lambda)$-Fleming-Viot process, and its
second coordinate is the Lévy process with triplet $(d+\kappa,{\sigma},\Pi)$
where $\Pi(d\zeta)$ is the pushforward of the measure
$\zeta^{-2}\Lambda(d\zeta)$ under the transformation $\zeta\to-\log(1-\zeta)$.
In particular $\nu_{t}=\rho_{t}e^{\xi_{t}},$ defines a Feller process.

We only provide a sketch of proof whose arguments can be formalized by further
applications of Theorems 2.1 and 2.2 in [36]. We only provide the
corresponding sample path analysis.

###### Sketch of proof.

From the Poissonian construction of $\left(\nu_{t}\right)_{t\geq 0}$, and by
direct calculations on its sample paths, we see that the normalized process
defined by $\rho_{t}=\nu_{t}/\left\lVert\nu_{t}\right\rVert$ has jumps of the
form

$\displaystyle\rho_{t-}=\frac{\nu_{t-}}{\left\lVert\nu_{t-}\right\rVert}\to$
$\displaystyle\frac{\nu_{t-}+\left\lVert\nu_{t-}\right\rVert\left(\frac{\zeta}{1-\zeta}\right)\delta_{a}}{\left\lVert\nu_{t-}\right\rVert+\left\lVert\nu_{t-}\right\rVert\left(\frac{\zeta}{1-\zeta}\right)}=\frac{\nu_{t-}+\left\lVert\nu_{t-}\right\rVert\left(\frac{\zeta}{1-\zeta}\right)\delta_{a}}{\left\lVert\nu_{t-}\right\rVert\left(\frac{1}{1-\zeta}\right)}$
$\displaystyle=$ $\displaystyle\rho_{t-}(1-\zeta)+\zeta\delta_{a}$

with intensity $\zeta^{-2}{\Lambda(d\zeta)}\otimes\rho_{t-}(da)$; whereas, by
Proposition 3.2, in-between jumps it evolves as a standard Fleming-Viot
process of parameter $\sigma$.

Similarly, $(\left\lVert\nu_{t}\right\rVert)_{t\geq 0}$ has jumps of the form

$\left\lVert\nu_{t-}\right\rVert\to\left\lVert\nu_{t-}\right\rVert\left(1+\frac{\zeta}{1-\zeta}\right),$

so that $\xi_{t}\coloneqq\log(\left\lVert\nu_{t}\right\rVert)$ has jumps of
the form

$\xi_{t-}\to\xi_{t-}+\log\left(\frac{1}{1-\zeta}\right)$

with intensity $\zeta^{-2}{\Lambda(d\zeta)}$. Also, in-between jumps, it has a
drift $\kappa$ and, by Proposition 3.2, an additional drift $d$ and a Brownian
component of parameter $\sigma$.

That $\left(\rho_{t},\xi_{t}\right)_{t\geq 0}$ is a MAP follows from
$\left(\nu_{t}\right)_{t\geq 0}$ being SMH (Lemma 4.3) and Proposition 2.1.
Finally, the process $\left(\rho_{t},\xi_{t}\right)_{t\geq 0}$ is Feller since
by Proposition 3.3 in [18] the $\Lambda$-Fleming-Viot process
$\left(\rho_{t}\right)_{t\geq 0}$ is Feller, as well as the Lévy process
$\left(\xi_{t}\right)_{t\geq 0}$. The Feller property extends easily to the
pair $\left(\rho_{t},\xi_{t}\right)_{t\geq 0}$. Indeed, note that if
$\rho^{(k)}\to\rho$ implies
$\\{\rho_{t},\mathbbm{P}_{\rho^{(k)}}\\}\underset{d}{\longrightarrow}\\{\rho_{t},\mathbbm{P}_{\rho}\\}$,
and also $\xi^{(k)}\to\xi$ implies
$\\{\xi_{t},\mathbbm{P}_{\xi^{(k)}}\\}\underset{d}{\longrightarrow}\\{\xi_{t},\mathbbm{P}_{\xi^{(k)}}\\}$,
then $(\rho^{(k)},\xi^{(k)})\to(\rho,\xi)$ implies
$\\{(\rho_{t},\xi_{t}),\mathbbm{P}_{(\rho^{(k)},\xi^{(k)})}\\}\underset{d}{\longrightarrow}\\{(\rho_{t},\xi_{t}),\mathbbm{P}_{(\rho^{(k)},\xi^{(k)})}\\}$.
Similarly, for the continuity in probability at $t=0$, note that
$\rho_{t}\overset{t\to
0}{\underset{{\operatorname{P}}}{\longrightarrow}}\rho_{0}$ and
$\xi_{t}\overset{t\to
0}{\underset{{\operatorname{P}}}{\longrightarrow}}\xi_{0}$ imply
$(\rho_{t},\xi_{t})\overset{t\to
0}{\underset{{\operatorname{P}}}{\longrightarrow}}(\rho_{0},\xi_{0})$. That
$\left(\nu_{t}\right)_{t\geq 0}$ is Feller now follows from the continuity of
the mapping $\phi^{-1}(\rho,\xi)=e^{\xi}\rho$ from
$\mathtt{PM}(\mathcal{T})\times\mathbb{R}$ to $\mathtt{M}(\mathcal{T})$. ∎

###  4.3 A Weak Limit and Martingale Problem

In this section we choose a particular sequence of processes
$\left\\{\left(\nu^{(n)}_{t}\right)_{t\geq 0}\right\\}_{n}$ that will weakly
approximate a process $\left(\nu_{t}\right)_{t\geq 0}$ with generator
$\mathcal{G}$ as in (13), the latter will be proven to be a Feller process.
Having proved that the limit process $\left(\nu_{t}\right)_{t\geq 0}$ exists
and is Feller, a general approximation theorem in terms of the convergence of
the respective generators can be derived via Theorem 17.25 in [26]. In the
following Proposition 4.5, and in order to simplify technical arguments, we
only focus on the construction of the process $\left(\nu_{t}\right)_{t\geq 0}$
for which we impose stronger conditions than just the convergence of the
generators.

In the following we set $\Lambda(\\{0\\})=\frac{\sigma^{2}}{2}$. Also,
throughout this section, we will work with the jumping measure
$\mathbbm{1}_{0<\zeta\leq 1/2}\Lambda(d\zeta)$ instead of $\Lambda(d\zeta)$
for the limit process; while the approximating processes
$\left(\nu^{(n)}_{t}\right)_{t\geq 0}$ will have jumping measure

(18) $\Lambda_{n}(d\zeta)=\mathbbm{1}_{1/n<\zeta\leq 1/2}\Lambda(d\zeta).$

Removing the atoms in which $\zeta>1/2$ ensures that the jumps of
$\left(\left\lVert\nu^{(n)}_{t}\right\rVert\right)_{t\geq 0}$ are bounded,
which in turn simplifies the proof of tightness for the family
$\left\\{\left(\nu^{(n)}_{t}\right)_{t\geq 0}\right\\}_{n}$. This imposes no
loss of generality for the construction of the process
$\left(\nu_{t}\right)_{t\geq 0}$ with generator of the general form
$\mathcal{G}$ since the finitely-many large jumps that occur with intensity
$\mathbbm{1}_{\zeta>1/2}\Lambda(d\zeta)$ can always be “added back” via the
same argument as in Lemma 4.3; whilst ensuring that the resulting frequency
process $\left(\rho_{t}\right)_{t\geq 0}$ is the corresponding
$\Lambda$-Felming-Viot process and the logarithm of the norm
$\left(\xi_{t}\right)_{t\geq 0}$ is the corresponding Lévy process as in
Proposition 4.4.

######  Proposition 4.5.

Let $\left(\nu^{(n)}_{t}\right)_{t\geq 0}$ be constructed as in section 4.2
with $\Lambda_{n}$ as in (18) where, furthermore,
$\nu^{(n)}_{0}\underset{\mathtt{d}}{\overset{n\to\infty}{\longrightarrow}}\nu_{0}$
and $\sup_{n}\mathbbm{E}[\left\lVert\nu^{(n)}_{0}\right\rVert^{p}]<\infty$ for
every $p\geq 1$. Let $\left(\rho^{(n)}_{t},\xi^{(n)}_{t}\right)_{t\geq 0}$ be
the corresponding MAPs.

Then, as $n\to\infty$,

$\left(\rho^{(n)}_{t},\xi^{(n)}_{t},\nu^{(n)}_{t}\right)_{t\geq
0}\Rightarrow\left(\rho_{t},\xi_{t},\nu_{t}\right)_{t\geq 0}$

in the Skorohod topology on
$D([0,\infty);\mathtt{PM}(\mathcal{T})\times\mathbb{R}\times\mathtt{M}\left(\mathcal{T}\right))$,
where

$\left(\nu_{t}\right)_{t\geq 0}=\left(e^{\xi_{t}}\rho_{t}\right)_{t\geq 0}$

and $\left(\rho_{t}\right)_{t\geq 0}$ is a $\Lambda$-Fleming-Viot process, and
$\left(\xi_{t}\right)_{t\geq 0}$ a Lévy process with triplet
$(d+\kappa,\sigma,\Pi)$ where $\Pi(d\zeta)$ is the pushforward of the measure
$\zeta^{-2}\Lambda(d\zeta)$ under the transformation $\zeta\to-\log(1-\zeta).$
Furthermore, the process $\left(\nu_{t}\right)_{t\geq 0}$ is Feller with
generator of the form $\mathcal{G}$ in $D_{2}$.

###### Proof.

We proceed in multiple steps: proving existence of solutions to the
$\mathcal{G}$-martingale problem through tightness of the family of processes
$\left\\{\left(\nu^{(n)}_{t}\right)_{t\geq 0}\right\\}_{n}$ in the Skorohod
topology for $D([0,\infty);\mathtt{M}\left(\mathcal{T}\right))$, and the
convergence of the approximating generators $\mathcal{G}_{n}$ to
$\mathcal{G}$; and then identifying the limit and showing that it is Feller.
The proof of tightness rests on the well-known Aldous-Rebolledo criterion
(e.g. Theorem 1.17 in [15]), whereas the properties of the limiting process
follow through an analysis of $\left(\rho_{t}\right)_{t\geq 0}$ and
$\left(\xi_{t}\right)_{t\geq 0}$ separately.

Step 1: Tightness. The proof rests on Theorem III.9.1 in [16]. Let us first
prove that

(19) $\lim_{r\to\infty}\limsup_{n\geq 1}\mathbbm{P}\left(\sup_{0\leq t\leq
T}\left\lVert\nu^{(n)}_{t}\right\rVert\geq r\right)=0$

which implies the compact containment condition (eq. (9.1) therein) for our
processes since, $\mathcal{T}$ being compact, the space
$\mathtt{M}\left(\mathcal{T}\right)$ is locally compact. In fact the balls of
radius $r$,
$\mathtt{M}_{r}\left(\mathcal{T}\right)=\left\\{\mu\in\mathtt{M}\left(\mathcal{T}\right)\colon\left\langle
1,\mu\right\rangle\leq r\right\\},$ are compact (see e.g. Theorem 1.14 in
[31]).

By Proposition 4.4 we have
$\left\lVert\nu^{(n)}_{t}\right\rVert=e^{\xi^{(n)}_{t}}$ where
$\left(\xi^{(n)}_{t}\right)_{t\geq 0}$ is the Lévy process

$\xi^{(n)}_{t}=\log(\left\lVert\nu^{(n)}_{0}\right\rVert)+(\kappa+d)t+M^{(n)}_{t}.$

Here $\left(M^{(n)}_{t}\right)_{t\geq 0}$ is its martingale component, given
by

$M^{(n)}_{t}=\sigma
B^{(n)}_{t}+\int_{0}^{t}\int_{1/n<\zeta<1/2}\left\lvert\log(1-\zeta)\right\rvert\tilde{}\mathcal{P}^{(n)}(ds,d\zeta),$

where $\left(B^{(n)}_{t}\right)_{t\geq 0}$ is a brownian motion and
$\tilde{}\mathcal{P}^{(n)}$ is a compensated Poisson random measure on
$\mathbb{R_{+}}\times[1/n,1/2]$ of intensity
$ds\times\zeta^{-2}\Lambda(d\zeta)$. Markov’s inequality followed by Hölder’s
inequality yield, for any $p,q$ with $1/p+1/q=1$,

$\displaystyle\mathbbm{P}\left(\sup_{0\leq t\leq
T}\left\lVert\nu^{(n)}_{t}\right\rVert\geq
r\right)\leq\frac{e^{\left\lvert\kappa+d\right\rvert
T}}{r}\mathbbm{E}\left[\left\lVert\nu^{(n)}_{0}\right\rVert\sup_{0\leq t\leq
T}e^{M^{(n)}_{t}}\right]$ (20)
$\displaystyle\leq\frac{e^{\left\lvert\kappa+d\right\rvert
T}}{r}\mathbbm{E}\left[\left\lVert\nu^{(n)}_{0}\right\rVert^{q}\right]^{1/q}\mathbbm{E}\left[\sup_{0\leq
t\leq T}e^{pM^{(n)}_{t}}\right]^{1/p}.$

The $\mathtt{L}^{q}$-norm of $\left\lVert\nu^{(n)}_{0}\right\rVert$ above is
uniformly bounded on $n$ by hypothesis. We now bound, also uniformly on $n$,
the second expectation in the r.h.s. above. Then taking $r\to\infty$ in the
last line above will yield (19). By Doob’s maximal inequality applied to the
submartingale $\left(e^{pM^{(n)}_{t}}\right)_{t\geq 0}$ we have for some
constant $C_{p}$, and plugging Campbell’s formula in the second inequality
below,

(21) $\displaystyle\mathbbm{E}[\sup_{0\leq t\leq T}e^{pM^{(n)}_{t}}]\leq
C_{p}\mathbbm{E}[e^{pM^{(n)}_{T}}]$ $\displaystyle\leq
C_{p}e^{Tp^{2}\frac{\sigma^{2}}{2}}\exp\left\\{T\int_{1/n}^{1/2}\frac{\Lambda(d\zeta)}{\zeta^{2}}\left\\{e^{-p\log(1-\zeta)}-1+p\log(1-\zeta)\right\\}\right\\}.$

Since the integrand above is of order $\mathcal{O}(\zeta^{2})$ as $\zeta\to
0$, we have

$\int_{0}^{1/2}\frac{\Lambda(d\zeta)}{\zeta^{2}}\left\\{e^{-p\log(1-\zeta)}-1+p\log(1-\zeta)\right\\}<\infty,$

which gives a bound that is uniform on $n$ for (21).

In view of Theorem III.9.1 in [16], and the fact that the polynomials
$\mathtt{Pol}(\mathtt{M}(\mathcal{T}))$ are dense in
$\mathtt{\bar{C}}(\mathtt{M}(\mathcal{T}))$ for the topology of uniform
convergence in compact sets; it remains to prove that
$\left\\{\left(F(\nu^{(n)}_{t})\right)_{t\geq 0}\right\\}_{n}$ is relatively
compact for every $F\in\mathtt{Pol}(\mathtt{M}(\mathcal{T}))\subset D$, where
we recall that $D$ is as in (3.2). For the latter, in turn, we will prove the
conditions of the Aldous-Rebolledo criterion.

First, the tightness of $\\{F(\nu^{(n)}_{t})\\}_{n}$ for fixed $t$ follows
directly from (19).

Second, by Lemma 4.3, $N_{t}^{(n)}\coloneqq
F(\nu^{(n)}_{t})-\int_{0}^{t}\mathcal{G}_{n}F(\nu^{(n)}_{s})ds$ defines a
martingale so that the finite variation process of
$\left(F(\nu^{(n)}_{t})\right)_{t\geq 0}$ is given by

$V^{(n)}_{t}=\int_{0}^{t}\mathcal{G}_{n}F(\nu^{(n)}_{s})ds.$

On the other hand, the compensator of the process
$\left((N_{t}^{(n)})^{2}\right)_{t\geq 0}$ is given by

$\left[N^{(n)}\right]_{t}=\int_{0}^{t}\mathbf{\Gamma}_{n}F(\nu^{(n)}_{s})ds$

where $\mathbf{\Gamma}_{n}$ is the carré du champ operator associated to
$\mathcal{G}_{n}$. The latter is given by

$\mathbf{\Gamma}_{n}F=\mathcal{G}_{n}F^{2}-2F\mathcal{G}_{n}F$

(see e.g. section 1.2.2 in [24]). The remaining conditions of the Aldous-
Rebolledo criterion on $\left(V^{(n)}_{t}\right)_{t\geq 0}$ and
$\left(\left[N^{(n)}\right]_{t}\right)_{t\geq 0}$ (which are given by (23) and
(24) below) will follow once we prove that for any
$F\in\mathtt{Pol}(\mathtt{M}(\mathcal{T}))$, $\delta>0$, and any stopping time
$0\leq\tau_{n}\leq T$,

(22)
$\limsup_{n\to\infty}\sup_{0\leq\theta\leq\delta}\mathbbm{E}\left[\int_{\tau_{n}}^{\tau_{n}+\theta}(\left\lvert
F(\nu^{(n)}_{s})\right\rvert\vee
1)\left\lvert\mathcal{G}_{n}F(\nu^{(n)}_{s})\right\rvert ds\right]\leq\delta
C$

for some $C>0$ depending only on $F$. Indeed, by Markov’s inequality we
obtain, on the one hand,

$\displaystyle\limsup_{n\to\infty}\sup_{0\leq\theta\leq\delta}\mathbbm{P}\left(\left\lvert
V^{(n)}_{\tau_{n}+\theta}-V^{(n)}_{\tau_{n}}\right\rvert>\epsilon\right)$
$\displaystyle\leq\limsup_{n\to\infty}\sup_{0\leq\theta\leq\delta}\mathbbm{E}\left[\int_{\tau_{n}}^{\tau_{n}+\theta}\left\lvert\mathcal{G}_{n}F(\nu^{(n)}_{s})\right\rvert
ds\right]\leq\frac{\delta C}{\epsilon}.$

So that, taking $\delta=\epsilon^{2}/C$, (22) implies

(23) $\sup_{n\geq
n_{0}}\sup_{0\leq\theta\leq\delta}\mathbbm{P}\left[\left\lvert
V^{(n)}_{\tau_{n}+\theta}-V^{(n)}_{\tau_{n}}\right\rvert>\epsilon\right]\leq\epsilon.$

On the other hand, for any $\delta>0$,

$\displaystyle\mathbbm{E}\left[\left\lvert\left[N^{(n)}\right]_{\tau_{n}+\delta}-\left[N^{(n)}\right]_{\tau_{n}}\right\rvert\right]$
$\displaystyle\leq\mathbbm{E}\left[\int_{\tau_{n}}^{\tau_{n}+\delta}\left\lvert\mathcal{G}_{n}F^{2}(\nu^{(n)}_{s})\right\rvert
ds\right]+2\mathbbm{E}\left[\int_{\tau_{n}}^{\tau_{n}+\delta}\left\lvert
F(\nu^{(n)}_{s})\mathcal{G}_{n}F(\nu^{(n)}_{s})\right\rvert ds\right],$

so that now (22) applied to each term in the r.h.s. above, and choosing
$\delta$ adequately, yields

(24) $\sup_{n\geq
n_{0}}\sup_{0\leq\theta\leq\delta}\mathbbm{P}\left[\left\lvert\left[N^{(n)}\right]_{\tau_{n}+\theta}-\left[N^{(n)}\right]_{\tau_{n}}\right\rvert>\epsilon\right]\leq\epsilon$

again through Markov’s inequality.

Let us then prove (22). On the one hand, by (3.2), the diffusion and drift
terms of $\mathcal{G}_{n}F$ are uniformly bounded on $n\geq 1$ by

$\displaystyle\left\lvert\mathcal{G}^{(B)}_{\sigma}F(\nu)\right\rvert+\left\lvert\mathcal{G}^{(D)}_{\kappa}F(\nu)\right\rvert$
$\displaystyle\leq\left\lVert\nu\right\rVert\int_{\mathcal{T}}\nu(da)\frac{\sigma^{2}}{2}\left\lvert
F^{\prime\prime}(\nu;a,a)\right\rvert+\int_{\mathcal{T}}\nu(da)\kappa\left\lvert
F^{\prime}(\nu;a)\right\rvert$ (25) $\displaystyle\leq
C_{1}\left(1+\dots+\left\lVert\nu\right\rVert^{p_{1}}\right)$

for some $C_{1}>0$ and $p_{1}\geq 1$. On the other hand, for the jump
component $\mathcal{G}^{(J)}_{\Lambda_{n}}F$ built from the measure
$\Lambda_{n}$ defined in (18), we have

(26) $\left\lVert\Lambda_{n}(d\zeta)-\mathbbm{1}_{0<\zeta\leq
1/2}\Lambda(d\zeta)\right\rVert_{TV}=\Lambda([0,\frac{1}{n}))\overset{n\to\infty}{\longrightarrow}0.$

Fix a measure $\nu$ and regard
$F(\nu+\left\lVert\nu\right\rVert\frac{\zeta}{1-\zeta}\delta_{a})$ as a
function of $\zeta$. Then writing $\psi(\zeta)=\frac{\zeta}{1-\zeta}$ so that
$\psi^{\prime}(\zeta)=\frac{1}{(1-\zeta)^{2}}$ and
$\psi^{\prime\prime}(\zeta)=\frac{2}{(1-\zeta)^{3}}$, we have

$\displaystyle\frac{d}{d\zeta}F(\nu+\left\lVert\nu\right\rVert\psi(\zeta)\delta_{a})$
$\displaystyle=\lim_{h\to
0}\frac{F(\nu+\left\lVert\nu\right\rVert\psi(\zeta)\delta_{a}+\left\lVert\nu\right\rVert(\psi(\zeta+h)-\psi(\zeta))\delta_{a})-F(\nu+\left\lVert\nu\right\rVert\psi(\zeta)\delta_{a})}{\left\lVert\nu\right\rVert(\psi(\zeta+h)-\psi(\zeta))}$
$\displaystyle\times\lim_{h\to
0}\left\lVert\nu\right\rVert\frac{\psi(\zeta+h)-\psi(\zeta)}{h}$
$\displaystyle=\left\lVert\nu\right\rVert
F^{\prime}(\nu+\left\lVert\nu\right\rVert\psi(\zeta)\delta_{a};a)\psi^{\prime}(\zeta),$

and, similarly,

$\displaystyle\frac{d}{d\zeta^{2}}F(\nu+\left\lVert\nu\right\rVert\psi(\zeta)\delta_{a})$
$\displaystyle=\left\lVert\nu\right\rVert^{2}F^{\prime\prime}(\nu+\psi(\zeta)\delta_{a};a,a)(\psi^{\prime}(\zeta))^{2}$
$\displaystyle+\quad\left\lVert\nu\right\rVert
F^{\prime}(\nu+\psi(\zeta)\delta_{a};a)\psi^{\prime\prime}(\zeta).$

Taylor’s expansion for $\zeta$ near $0$ and the assumption $F\in D$ give, for
some constant $C_{2}>0$ and some $p_{2}\geq 1$,

$\left\lvert
F\left(\nu+\left\lVert\nu\right\rVert\frac{\zeta}{1-\zeta}\delta_{a}\right)-F(\nu)-\left\lVert\nu\right\rVert\zeta
F^{\prime}\left(\nu;a\right)\right\rvert\leq
C_{2}\left(1+\dots+\left\lVert\nu\right\rVert^{p_{2}}\right)\frac{\zeta^{2}}{(1-\zeta)^{4}}.$

Furthermore, using that

$\forall
0<\zeta<1,\quad\left\lvert\zeta-\left\lvert\log(1-\zeta)\right\rvert\right\rvert=\sum_{k=2}^{\infty}\frac{\zeta^{k}}{k}\leq\frac{\zeta^{2}}{1-\zeta}$

which is continuous and bounded on $\zeta\in[0,1/2)$, we see that the function
of $\zeta$

$\mathbbm{1}_{\zeta\in(0,1]}\left\langle\frac{F\left(\nu+\left\lVert\nu\right\rVert\frac{\zeta}{1-\zeta}\delta_{a}\right)-F(\nu)-\left\lVert\nu\right\rVert\left\lvert\log(1-\zeta)\right\rvert\mathbbm{1}_{\zeta<1/2}F^{\prime}(\nu;a)}{\zeta^{2}},\frac{\nu(da)}{\left\lVert\nu\right\rVert}\right\rangle$

is bounded uniformly on $[0,1/2]$ by
$C_{3}(1+\dots+\left\lVert\nu\right\rVert^{p_{2}})$ for some $C_{3}>0$
depending only on $F$. Thus (26) yields

(27)
$\displaystyle\left\lvert\mathcal{G}^{(J)}_{{\Lambda_{n}}}F(\nu)-\mathcal{G}^{(J)}_{\Lambda}F(\nu)\right\rvert\leq
C_{3}\left(1+\dots+\left\lVert\nu\right\rVert^{p_{2}}\right)\left\lVert\Lambda_{n}(d\zeta)-\Lambda(d\zeta)\right\rVert_{TV}$

and, in particular,

(28) $\left\lvert\mathcal{G}^{(J)}_{{\Lambda_{n}}}F(\nu)\right\rvert\leq
C_{4}\left(1+\dots+\left\lVert\nu\right\rVert^{p_{2}}\right)$

for some $C_{4}$. Putting together (4.3) and (28), and also using that
$F(\nu)$ satisfies the same type of inequality (by (3.2)), we obtain for some
$C>0$ and $p\geq 1$, and for any stopping time $0\leq\tau_{n}\leq T$ and
$0\leq\theta\leq\delta<1$,

(29) $\mathbbm{E}\left[\int_{\tau_{n}}^{\tau_{n}+\theta}(\left\lvert
F(\nu^{(n)}_{s})\right\rvert\vee
1)\left\lvert\mathcal{G}_{n}F(\nu^{(n)}_{s})\right\rvert ds\right]\leq
C\delta\mathbbm{E}\left[\sup_{t\in[0,T+\delta]}1+\dots+e^{p\xi^{(n)}_{t}}\right]$

where we have again used
$\left(\left\lVert\nu^{(n)}_{t}\right\rVert\right)_{t\geq
0}=\left(e^{\xi^{(n)}_{t}}\right)_{t\geq 0}$ from Proposition 4.4. It remains
to prove that the expectation in the r.h.s. above is uniformly bounded on
$n\geq 1$. But this follows from (4.3) and (21).

Step 2: Limiting martingale problem. Assume that, along a subsequence
$\left\\{n_{i}\right\\}_{i\geq 1}$ (that we will denote by $n$ to ease
notations) we have

$\left(\nu^{(n)}_{t}\right)_{t\geq
0}\overset{n\to\infty}{\underset{d}{\longrightarrow}}\left(\nu_{t}\right)_{t\geq
0}$

in $D([0,\infty),\mathtt{M}\left(\mathcal{T}\right))$ for some process
$\left(\nu_{t}\right)_{t\geq 0}$. We now prove that
$\left(\nu_{t}\right)_{t\geq 0}$ must solve the martingale problem for (13) in
the domain $\\{F\in
D\colon\mathcal{G}F\in\mathtt{C}(\mathtt{M}(\mathcal{T}))\\}$.

We prove the latter via a mild adaptation of the proof of Lemma IV.5.1 in [16]
to the case where $F$ and $\mathcal{G}F$ may not be bounded (but still
continuous), and also where the convergence of the generators
$\mathcal{G}_{n}$ to $\mathcal{G}$ is uniform but only on compact sets (taking
advantage of the fact that we have already proved the compact containment
condition (19)). We only prove that the key equation (IV.5.3) therein (eq.
(4.3) below) also holds in our setting.

Let $h_{i},1\leq i\leq k,$ be functions in
$\mathtt{\bar{C}}(\mathtt{M}(\mathcal{T}))$. Also let $0\leq s_{i}\leq s<t$
all belong to the set $\\{u\colon\nu_{u}=\nu_{u-}\text{ a.s.}\\}$. Then, for
$F\in D$ such that $\mathcal{G}F\in\mathtt{C}(\mathtt{M}(\mathcal{T}))$, we
have, by the continuous mapping theorem,

$\displaystyle(F(\nu^{(n)}_{t})-F(\nu^{(n)}_{s})-\int_{s}^{t}\mathcal{G}F(\nu^{(n)}_{u})du)\prod_{i}h_{i}(\nu^{(n)}_{s_{i}})$
$\displaystyle\overset{n\to\infty}{\underset{d}{\longrightarrow}}(F(\nu_{t})-F(\nu_{s})-\int_{s}^{t}\mathcal{G}F(\nu_{u})du)\prod_{i}h_{i}(\nu_{s_{i}}).$

The corresponding convergence in expectation follows from Theorem 3.5 in [6]
and the fact that the random variables in the first line above are uniformly
integrable. To verify the uniform integrability, observe that, since $F\in D$,
the first line above is bounded by

$\displaystyle\left\lvert(F(\nu^{(n)}_{t})-F(\nu^{(n)}_{s})-\int_{s}^{t}\mathcal{G}F(\nu^{(n)}_{u})du)\prod_{i=1}^{k}h_{i}(\nu^{(n)}_{s_{i}})\right\rvert$
(30) $\displaystyle\leq C(\sup_{s\leq u\leq
t}1+\cdots+\left\lVert\nu^{(n)}_{u}\right\rVert^{p})\prod_{i=1}^{k}\left\lVert
h_{i}\right\rVert_{{\infty}}$

for some $C,p\geq 0$ (where we have used also (28)). That the r.h.s. is
uniformly integrable, in turn, is a simple consequence of Theorem 3.6 in [6]
together with the convergence

$\sup_{s\leq u\leq
t}\left\lVert\nu^{(n)}_{u}\right\rVert^{p}\overset{n\to\infty}{\underset{d}{\longrightarrow}}\sup_{s\leq
u\leq t}\left\lVert\nu_{u}\right\rVert^{p},$

and the bound in (21). Indeed, the corresponding a.s. convergence ensured by
Skorohod representation theorem, and the bound in (21), together yield the
required convergence in expectation

$\mathbbm{E}\left[\sup_{s\leq u\leq
t}\left\lVert\nu^{(n)}_{u}\right\rVert^{p}\right]\overset{n\to\infty}{\longrightarrow}\mathbbm{E}\left[\sup_{s\leq
u\leq t}\left\lVert\nu_{u}\right\rVert^{p}\right].$

We thus conclude

$\displaystyle\mathbbm{E}[(F(\nu_{t})-F(\nu_{s})-\int_{s}^{t}\mathcal{G}F(\nu_{u})du)\prod_{i=1}^{k}h_{i}(\nu_{s_{i}})]$
(31)
$\displaystyle=\lim_{n\to\infty}\mathbbm{E}[(F(\nu^{(n)}_{t})-F(\nu^{(n)}_{s})-\int_{s}^{t}\mathcal{G}F(\nu^{(n)}_{u})du)\prod_{i=1}^{k}h_{i}(\nu^{(n)}_{s_{i}})].$

We now approximate $\mathcal{G}$ by $\mathcal{G}_{n}$ in the above limit.
Observe that, by (27), and since the other two components of
${\mathcal{G}}_{n}$ are constant on $n$, we have

$\left\lvert{\mathcal{G}}_{n}F(\nu)-\mathcal{G}F(\nu)\right\rvert\leq
C_{3}\left(1+\dots+\left\lVert\nu\right\rVert^{p_{2}}\right)\Lambda((0,1/n))\overset{n\to\infty}{\longrightarrow}0$

uniformly for $\nu$ on the compact sets $\mathtt{M}_{r}$, $r\geq 0$. Plugging
the above bound into (4.3), together with the bound in (4.3) and the uniform
integrability of $\sup_{s\leq u\leq
t}\left\lVert\nu^{(n)}_{u}\right\rVert^{p}$, easily yield

$\displaystyle\mathbbm{E}[(F(\nu_{t})-F(\nu_{s})-\int_{s}^{t}\mathcal{G}F(\nu_{u})du)\prod_{i=1}^{k}h_{i}(\nu_{s_{i}})]$
(32)
$\displaystyle=\lim_{n\to\infty}\mathbbm{E}[(F(\nu^{(n)}_{t})-F(\nu^{(n)}_{s})-\int_{s}^{t}\mathcal{G}_{n}F(\nu^{(n)}_{u})du)\prod_{i=1}^{k}h_{i}(\nu^{(n)}_{s_{i}})]=0,$

where we have used Lemma 4.3 in the last equality.

Step 3: Identification of the limit and Feller property. We first show that
the sequence
$\left\\{\left(\log(\left\lVert\nu^{(n)}_{t}\right\rVert)\right)_{t\geq
0}\right\\}_{n}$ converges in distribution towards the Lévy process
$\left(\xi_{t}\right)_{t\geq 0}$, and that the sequence
$\left\\{\left(\rho^{(n)}_{t}\right)_{t\geq 0}\right\\}_{n}$ converges in
distribution towards the $\Lambda$-Fleming-Viot process
$\left(\rho_{t}\right)_{t\geq 0}$. Indeed, this follows by the tightness of
the family $\left\\{\left(\nu^{(n)}_{t}\right)_{t\geq 0}\right\\}_{n}$ and the
continuous mapping theorem on the one hand, and on the other by identifying
the limiting processes along subsequences through martingale problem
characterizations. The latter is done by taking functions of the form
$F(\nu)=f(\log(\left\lVert\nu\right\rVert))$ with
$f\in\mathtt{C^{2}_{0}}(\mathbb{R})$, and
$F_{\phi}(\nu)=\left\langle\phi,\left(\frac{\nu}{\left\lVert\nu\right\rVert}\right)^{\otimes
p}\right\rangle$ with $\phi\in\mathtt{\bar{B}}(\mathcal{T}^{p})$, respectively
in (13) and using the uniqueness of solutions to the respective martingale
problems of the Lévy and the $\Lambda$-Fleming-Viot processes (one could also
use classical Lévy processes criteria for weak convergence such as Theorem
VII.3.4 in [23] on the one hand, and on the other Proposition 3.2 for the
Fleming-Viot part). Indeed, we check the computations only for the generator
of the frequency process, which will coincide with that of the
$\Lambda$-Fleming-Viot process. For

$\displaystyle
F_{\phi}(\nu)=G_{\phi}\left(\frac{\nu}{\left\lVert\nu\right\rVert}\right),\quad
G_{\phi}(\rho)=\left\langle\phi,\rho^{\otimes p}\right\rangle,$

we have, on the one hand,

$\displaystyle F_{\phi}^{\prime}(\nu;a)$
$\displaystyle=\frac{1}{\left\lVert\nu\right\rVert^{2p}}\left(\sum_{i=1}^{p}\left\\{\left\langle\phi,\nu^{\otimes
i}\delta_{a}\nu^{\otimes
p-i-1}\right\rangle\left\lVert\nu\right\rVert^{p}\right\\}-p\left\lVert\nu\right\rVert^{p-1}\left\langle\phi,\nu^{\otimes
p}\right\rangle\right)$
$\displaystyle=\frac{1}{\left\lVert\nu\right\rVert}\sum_{i=1}^{p}\left\\{\frac{\left\langle\phi,\nu^{\otimes
i}\delta_{a}\nu^{\otimes
p-i-1}\right\rangle}{\left\lVert\nu\right\rVert^{p-1}}-\frac{\left\langle\phi,\nu^{\otimes
p}\right\rangle}{\left\lVert\nu\right\rVert^{p}}\right\\}$ (33)
$\displaystyle=\frac{1}{\left\lVert\nu\right\rVert}\sum_{i=1}^{p}\left\\{F_{\phi^{(a)}_{i}}(\nu)-F_{\phi}(\nu)\right\\}$

where we have written

$F_{\phi^{(a)}_{i}}(\nu)=\frac{\left\langle\phi^{(a)}_{i},\nu^{\otimes
p}\right\rangle}{\left\lVert\nu\right\rVert^{p}}=\frac{\left\langle\phi,\nu^{\otimes
i}\delta_{a}\nu^{\otimes
p-i-1}\right\rangle}{\left\lVert\nu\right\rVert^{p-1}}$

with
$\phi^{(a)}_{i}(a_{1},\cdots,a_{p})=\phi(a_{1},\cdots,a_{i-1},a,a_{i+1},\cdots,a_{p})$.
More generally, for a subset $J\subset[p]$ the function
$\phi^{(a)}_{J}\in\mathtt{\bar{B}}(\mathcal{T}^{p})$ will denote the function
constructed from $\phi$ by forcing all the coordinates in $J$ to have the
value $a\in\mathcal{T}$, and evaluating $\phi$ at the new location.

From the above we obtain

(34)
$\int_{\mathcal{T}}\frac{\nu(da)}{\left\lVert\nu\right\rVert}\left\lVert\nu\right\rVert
F_{\phi}^{\prime}(\nu;a)=\sum_{i=1}^{p}\left\\{\frac{\left\langle\phi,\nu^{\otimes
p}\right\rangle}{\left\lVert\nu\right\rVert^{p}}-\frac{\left\langle\phi,\nu^{\otimes
p}\right\rangle}{\left\lVert\nu\right\rVert^{p}}\right\\}=0.$

On the other hand, writing $\mathcal{P}(a,\ell,\nu,p)$ for the set of all
permutations of $\ell$ copies of the measure $\delta_{a}$ together with
$p-\ell$ copies of $\nu$, we have

$\displaystyle
F_{\phi}(\nu+\left\lVert\nu\right\rVert\frac{\zeta}{1-\zeta}\delta_{a})-F_{\phi}(\nu)$
$\displaystyle=\frac{\left\langle\phi,\nu^{\otimes
p}\right\rangle}{\left\lVert\nu\right\rVert^{p}}\left((1-\zeta)^{p}-1\right)+\sum_{\ell=1}^{p}\sum_{\sigma\in\mathcal{P}(a,\ell,\nu,p)}\frac{\left\langle\phi,\otimes_{i=1}^{p}\sigma_{i}\right\rangle}{\left\lVert\nu\right\rVert^{p}(1-\zeta)^{-p}}\left(\frac{\left\lVert\nu\right\rVert\zeta}{1-\zeta}\right)^{\ell}$
$\displaystyle=\frac{\left\langle\phi,\nu^{\otimes
p}\right\rangle}{\left\lVert\nu\right\rVert^{p}}\left((1-\zeta)^{p}-1\right)+\sum_{\ell=1}^{p}\sum_{\sigma\in\mathcal{P}(a,\ell,\nu,p)}\frac{\left\langle\phi,\otimes_{i=1}^{p}\sigma_{i}\right\rangle}{\left\lVert\nu\right\rVert^{p-\ell}}(1-\zeta)^{p-\ell}\zeta^{\ell}$
$\displaystyle=\sum_{\ell=0}^{p}(1-\zeta)^{p-\ell}\zeta^{\ell}\sum_{\sigma\in\mathcal{P}(a,\ell,\nu,p)}\left\\{\frac{\left\langle\phi,\otimes_{i=1}^{p}\sigma_{i}\right\rangle}{\left\lVert\nu\right\rVert^{p-\ell}}-\frac{\left\langle\phi,\nu^{\otimes
p}\right\rangle}{\left\lVert\nu\right\rVert^{p}}\right\\}.$

Observe that for $\ell=0$ we have

$\int_{\mathcal{T}}\frac{\nu(da)}{\left\lVert\nu\right\rVert}\frac{\left\langle\phi,\nu^{\otimes
p}\right\rangle}{\left\lVert\nu\right\rVert^{p}}-\frac{\left\langle\phi,\nu^{\otimes
p}\right\rangle}{\left\lVert\nu\right\rVert^{p}}=0,$

whereas, by (4.3) and (34), the $\ell=1$ term similarly vanishes in
$\mathcal{G}^{(J)}_{\Lambda}$. Then, substituting above, we obtain

$\displaystyle\int_{\mathcal{T}}\frac{\nu(da)}{\left\lVert\nu\right\rVert}\left\\{F_{\phi}(\nu+\left\lVert\nu\right\rVert\frac{\zeta}{1-\zeta}\delta_{a})-F_{\phi}(\nu)\right\\}$
$\displaystyle=\sum_{\ell=2}^{p}(1-\zeta)^{p-\ell}\zeta^{\ell}\sum_{\sigma\in\mathcal{P}(a,\ell,\nu,p)}\left\\{\frac{\left\langle\phi,\otimes_{i=1}^{p}\sigma_{i}\right\rangle}{\left\lVert\nu\right\rVert^{p-\ell}}-\frac{\left\langle\phi,\nu^{\otimes
p}\right\rangle}{\left\lVert\nu\right\rVert^{p}}\right\\}.$

Comparing with (7), integrating with respect to $\zeta^{-2}{\Lambda(d\zeta)}$,
and putting all together, we obtain

$\displaystyle\mathcal{G}^{(D)}_{\kappa}F_{\phi}(\nu)+\mathcal{G}^{(J)}_{\Lambda}F_{\phi}(\nu)=$
$\displaystyle\int_{\mathcal{T}}\frac{\nu(da)}{\left\lVert\nu\right\rVert}\sum_{\ell=2}^{p}\beta^{(\Lambda)}_{p,\ell}\left(G^{(\ell)}_{\phi}\left(\frac{\nu}{\left\lVert\nu\right\rVert};a\right)-\binom{p}{\ell}G_{\phi}\left(\frac{\nu}{\left\lVert\nu\right\rVert}\right)\right).$

This is the form of the generator of the $\Lambda$-Fleming-Viot process with
$\Lambda(\\{0\\})=0$.

Finally, for the diffusion term we compute

$\displaystyle\left\lVert\nu\right\rVert F_{\phi}^{\prime\prime}(\nu;a,a)$
$\displaystyle=\left\lVert\nu\right\rVert\sum_{i=1}^{p}\frac{\left\lVert\nu\right\rVert(F^{\prime}_{\phi^{(a)}_{i}}(\nu;a)-F^{\prime}_{\phi}(\nu;a))-(F_{\phi^{(a)}_{i}}(\nu)-F_{\phi}(\nu))}{\left\lVert\nu\right\rVert^{2}}$
$\displaystyle=\sum_{i=1}^{p}\left\\{F^{\prime}_{\phi^{(a)}_{i}}(\nu;a)-F^{\prime}_{\phi}(\nu;a)-\frac{F_{\phi^{(a)}_{i}}(\nu)-F_{\phi}(\nu)}{\left\lVert\nu\right\rVert}\right\\},$

where, by (4.3) and (34), we have

$\int_{\mathcal{T}}\nu(da)\sum_{i=1}^{p}\left\\{F^{\prime}_{\phi}(\nu;a)-\frac{F_{\phi^{(a)}_{i}}(\nu)-F_{\phi}(\nu)}{\left\lVert\nu\right\rVert}\right\\}=0.$

Thus

$\int_{\mathcal{T}}\nu(da)\left\lVert\nu\right\rVert
F_{\phi}^{\prime\prime}(\nu;a,a)=\int_{\mathcal{T}}\nu(da)\sum_{i=1}^{p}F^{\prime}_{\phi^{(a)}_{i}}(\nu;a)$

where, using (4.3) again, for each term in the r.h.s. above we have

$F^{\prime}_{\phi^{(a)}_{i}}(\nu;a)=\frac{1}{\left\lVert\nu\right\rVert}\sum_{j=1}^{p}\left\\{F_{\phi^{(a)}_{i,j}}(\nu)-F_{\phi^{(a)}_{i}}(\nu)\right\\}$

where we note that $\phi^{(a)}_{i,i}=\phi^{(a)}_{i}$. Substituting above and
noting also that
$\int_{\mathcal{T}}\frac{\nu(da)}{\left\lVert\nu\right\rVert}G_{\phi^{(a)}_{i}}(\frac{\nu}{\left\lVert\nu\right\rVert})=G_{\phi}(\frac{\nu}{\left\lVert\nu\right\rVert})$,

$\displaystyle\frac{\sigma^{2}}{2}\int_{\mathcal{T}}\nu(da)\left\lVert\nu\right\rVert
F_{\phi}^{\prime\prime}(\nu;a,a)$
$\displaystyle=\frac{\sigma^{2}}{2}\int_{\mathcal{T}}\frac{\nu(da)}{\left\lVert\nu\right\rVert}\sum_{1\leq
i\neq j\leq
p}G_{\phi^{(a)}_{i,j}}(\frac{\nu}{\left\lVert\nu\right\rVert})-G_{\phi^{(a)}_{i}}(\frac{\nu}{\left\lVert\nu\right\rVert})$
$\displaystyle=\frac{\sigma^{2}}{2}\sum_{1\leq i\neq j\leq
p}\int_{\mathcal{T}}\frac{\nu(da)}{\left\lVert\nu\right\rVert}\left\\{G_{\phi^{(a)}_{i,j}}(\frac{\nu}{\left\lVert\nu\right\rVert})\right\\}-G_{\phi}(\frac{\nu}{\left\lVert\nu\right\rVert}).$

This is the form of the generator of the standard Fleming-Viot process (3.2).
This finishes the proof of the characterization of the limiting process
$\left(\rho_{t}\right)_{t\geq 0}$ of the sequence
$\left(\rho^{(n)}_{t}\right)_{t\geq 0}$.

Now, from the convergence of each of the coordinates in
$(\rho^{(n)}_{t},\xi^{(n)}_{t})$ separately, we obtain
$\left(\rho^{(n)}_{t},\xi^{(n)}_{t}\right)_{t\geq
0}\underset{d}{\longrightarrow}\left(\rho_{t},\xi_{t}\right)_{t\geq 0}$ in
$D([0,\infty),\mathtt{PM}(\mathcal{T}))\times D([0,\infty),\mathbb{R})$. Since
the latter identifies the law in the limit, the corresponding convergence in
$D([0,\infty),\mathtt{PM}(\mathcal{T})\times\mathbb{R})$ follows from the
precompactness of the family of processes
$\left\\{\Phi(\left(\nu^{(n)}_{t}\right)_{t\geq 0})\right\\}_{n}$ which, in
turn, follows from the tightness of the family
$\left\\{\left(\nu^{(n)}_{t}\right)_{t\geq 0}\right\\}_{n}$, the continuity of
the map
$\Phi(\nu)=\left(\nu/\left\lVert\nu\right\rVert,\log(\left\lVert\nu\right\rVert)\right)$
on $\mathtt{M}(\mathcal{T})\setminus\\{0\\}$, and the fact that
$\left(\xi_{t}\right)_{t\geq 0}\subset(-\infty,\infty)$ with probability one.
Finally, the function $\Phi^{-1}$ being continuous, another application of the
mapping theorem gives the convergence of the entire sequence
$\left(\nu^{(n)}_{t}\right)_{t\geq
0}\underset{d}{\longrightarrow}\left(\nu_{t}\right)_{t\geq 0}$ in
$D([0,\infty),\mathtt{M}(\mathcal{T}))$.

By Proposition 3.3 in [18] the $\Lambda$-Fleming Viot process
$\left(\rho_{t}\right)_{t\geq 0}$ is Feller, as well as the Lévy process
$\left(\xi_{t}\right)_{t\geq 0}$. The Feller property extends easily to the
pair $\left(\rho_{t},\xi_{t}\right)_{t\geq 0}$ as in the proof of Proposition
4.4.

∎

Acknowledgements
Arno Siri-Jégousse was supported by DGAPA-PAPIIT-UNAM grant IN-10282.
Alejandro H. Wences was supported by the ANR LabEx CIMI (grant
ANR-11-LABX-0040) within the French State Programme “Investissements
d’Avenir.”

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