# Time reversal of spinal processes for linear and non-linear branching
processes near stationarity

Benoît Henry, Sylvie Méléard, Viet Chi Tran IMT Lille Douai, Institut Mines-
Télécom, Univ. Lille, F-59000 Lille, France; E-mail: benoit.henry@imt-lille-
douai.frInstitut Universitaire de France and Ecole polytechnique, CNRS,
Institut polytechnique de Paris, CMAP, route de Saclay, 91128 Palaiseau Cedex-
France; E-mail<EMAIL_ADDRESS>Univ Gustave Eiffel,
Univ Paris Est Creteil, CNRS, F-77454 Marne-la-Vallée, France; E-mail:
<EMAIL_ADDRESS>
###### Abstract

We consider a stochastic individual-based population model with competition,
trait-structure affecting reproduction and survival, and changing environment.
The changes of traits are described by jump processes, and the dynamics can be
approximated in large population by a non-linear PDE with a non-local mutation
operator. Using the fact that this PDE admits a non-trivial stationary
solution, we can approximate the non-linear stochastic population process by a
linear birth-death process where the interactions are frozen, as long as the
population remains close to this equilibrium. This allows us to derive, when
the population is large, the equation satisfied by the ancestral lineage of an
individual uniformly sampled at a fixed time $T$, which is the path
constituted of the traits of the ancestors of this individual in past times
$t\leq T$. This process is a time inhomogeneous Markov process, but we show
that the time reversal of this process possesses a very simple structure (e.g.
time-homogeneous and independent of $T$). This extends recent results where
the authors studied a similar model with a Laplacian operator but where the
methods essentially relied on the Gaussian nature of the mutations.

Keywords: stochastic individual-based models, birth-death processes,
interaction, competition, jump process, non-local mutation operator, many-to-
one formulas, ancestral path, genealogy, phylogeny.

_MSC 2000 subject classification:_ 92D25, 92D15, 60J80, 60K35, 60F99.

Acknowledgements: This work has been supported by the Chair “Modélisation
Mathématique et Biodiversité” of Veolia Environnement-École Polytechnique-
Museum National d’Histoire Naturelle-Fondation X. V.C.T. also acknowledges
support from Labex Bézout (ANR-10-LABX-58).

##  1 Introduction

We are interested in describing the ancestry of an individual sampled from a
trait-structured population whose dynamics is ruled by births, deaths,
mutations and environmental changes. More precisely, we consider as a toy
model a stochastic individual-based population model in continuous time, with
variable size, and in which each individual is characterized by its own trait
$x$ which is interpreted here as its fitness. For simplicity, the trait $x$ is
considered to be real-valued. This trait can change through time (by mutations
occurring continuously in time). The case where it is driven by a Brownian
motion has been considered in a previous paper by the authors [4]. Here, we
are interested in a non-local mutation kernel. Computations exploiting the
Gaussian nature of the mutations can not be used any more. We base our work on
duality properties satisfied by the semi-groups and generators underlying the
mutations and environmental changes.
The interest in ancestries and phylogenies (the trait values of ancestors of
the population) has developed in recent years as the phylogenies provide a new
understanding for the evolution of the biodiversity in response to the
ecological dynamics or environmental changes (e.g. [27]).

We will be interested in large population limits and the model is
parameterized by an integer $K$ (think of the carrying capacity for instance)
that we will let go to infinity. The size of the population is then
$N^{K}_{t}$ at time $t>0$. An individual of trait $x\in\mathbb{R}$ gives birth
to a new individual of same trait at rate $b(x)$ and dies at the rate
$d(x)+N_{t}^{K}/K$. In the death rate, the term $d(x)$ corresponds to the
natural death to which is added a competition term expressing the additional
death rate exerted by the interaction with the other individuals in the
population. Here, this competition is assumed of logistic type, i.e. it is
proportional to the size $N_{t}^{K}$ and does not account for the whole trait
distribution. During their life, the trait of an individual mutates according
to a kernel $\gamma m(x,y)dy$ and experiences a linear drift with
environmental velocity $\rho\in\mathbb{R}$ due to environmental changes (see
[4] for details). We assume that $\gamma>0$ is the jump rate and that
$m(x,y)dy$ is the probability measure describing the jumps (assumed to be
absolutely continuous with respect to the Lebesgue measure, for the sake of
simplicity).

Individual labels can be chosen in the Ulam-Harris-Neveu set
$\mathcal{I}=\cup_{n\in\mathbb{N}}\mathbb{N}^{n}$ (e.g. see [21]) where
offspring labels are obtained by concatenating the label of their parent with
their ranks among their siblings. This set is endowed with a partial order
$\prec$, where $i\prec j$ if there exists $i^{\prime}\in\mathcal{I}$ so that
$j$ is the concatenation of the chains of integers $i$ and $i^{\prime}$. We
denote by $V^{K}_{t}\subset\mathcal{I}$ the set of labels of individuals alive
at time $t$ (implying that $N^{K}_{t}=\mbox{Card}(V^{K}_{t})$) and by
$X^{i}_{t}$ the trait of the $i$-th individual at this time. The lineage of
the individual $i\in V^{K}_{t}$ consists in the path defined from $[0,t]$ to
$\mathbb{R}$ and that associates to $s$ the trait of the closest ancestor to
$i$ living at time $s$, and that we will denote by $X^{i}_{s}$. Such path is
càdlàg because of the mutation kernel and can be extended to a function of the
Skorokhod space $\mathbb{D}=\mathbb{D}(\mathbb{R}_{+},\mathbb{R})$ by setting
it to the constant value equal to $X^{i}_{t}$ for times $s>t$. Also, we will
say that this path $(X^{i}_{t},t>0)$ is ‘forward in time’, in opposition to
the ancestral path $(X^{i}_{T-t},t\in[0,T])$ of an individual $i\in V^{K}_{T}$
for a given $T>0$ that is considered in ‘backward in time’. The set of all
lineages for living individuals at time $t$ can be represented by the
following point measure on $\mathbb{D}$:

$H^{K}_{t}=\frac{1}{K}\sum_{i\in V^{K}_{t}}\delta_{(X^{i}_{s\wedge t},\
s\in\mathbb{R}_{+})}.$ (1)

Denoting by $\mathcal{M}_{f}(\mathbb{D})$ the set of finite measures on
$\mathbb{D}(\mathbb{R}_{+},\mathbb{R})$, the process $(H^{K}_{t})_{t>0}$ is a
càdlàg process of $\mathbb{D}(\mathbb{R}_{+},\mathcal{M}_{f}(\mathbb{D}))$
which is the historical particle system, following the terminology and concept
introduced by Dawson Perkins [9, 29], Dynkin [13] (see also [10, 16]). The
spaces $\mathbb{D}$ and
$\mathbb{D}(\mathbb{R}_{+},\mathcal{M}_{f}(\mathbb{D}))$ are equipped with the
Skorokhod topology and $\mathcal{M}_{f}(\mathbb{D})$ is equipped with the
topology of weak convergence (see e.g. [2]). Méléard and Tran [24] and Kliem
[20] have studied limits of this process under a diffusive scaling when
$K\rightarrow+\infty$. In a recent work [4], we have studied a similar
historical process in large population, without rescaling of time and with
particles undergoing Brownian motion. We obtained the distribution, backward
in time, of a typical ancestral lineage (the lineage of an individual $i\in
V^{K}_{T}$ sampled uniformly among the population living at time $T$), using
extensively explicit computation based on Brownian properties. In the present
paper, we extend these results to the case where the motion is a drifted jump
process with generator:

$L\varphi(x)=\rho\
\partial_{x}\varphi(x)+\gamma\int_{\mathbb{R}}(\varphi(y)-\varphi(x))\
m(x,y)dy,$ (2)

where $\gamma$, $\rho$ and $m(x,y)$ have been introduced above. The jump part
corresponds to mutations and the drift part corresponds to the environmental
changes. Given a collection of independent Poisson point measures
$(Q^{i}(ds,dy,d\theta),i\in\mathcal{I})$ on
$\mathbb{R}_{+}\times\mathbb{R}\times\mathbb{R}_{+}$ with common intensity
measure the Lebesgue measure, the trait dynamics $X^{i}$ of individual $i$
solves:

$X^{i}_{t}=X^{i}_{0}+\rho
t+\sum_{j\in\mathcal{I}}\int_{0}^{t}\int_{\mathbb{R}}\int_{\mathbb{R}_{+}}{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\\{j=i(s),\ \theta\leq\gamma
m(X^{i}_{s_{-}},y)\\}}\big{(}y-X^{i}_{s_{-}}\big{)}Q^{j}(ds,dy,d\theta).$ (3)

where $i(s)=\max_{\prec}\\{j\in V^{K}_{s},\ j\prec i\\}$ denotes the index of
the most recent ancestor of $i$ living at time $s$.

The process $(Z^{K}_{t})_{t\in\mathbb{R}_{+}}$ corresponding to the trait
distribution of the living individuals at time $t>0$,

$Z^{K}_{t}(dx)=\frac{1}{K}\sum_{i\in V^{K}_{t}}\delta_{X^{i}_{t}}(dx),$ (4)

converges in the limit $K\rightarrow+\infty$ in
$\mathbb{D}(\mathbb{R}_{+},\mathcal{M}_{f}(\mathbb{R}))$ to the solution of
the following partial differential equation (PDE):

$\partial_{t}f_{t}(x)=-\rho\
\partial_{x}f_{t}(x)+\gamma\int_{\mathbb{R}}(f_{t}(y)-f_{t}(x))\
m(y,x)dy+\left(h(x)-\int_{\mathbb{R}}f_{t}(y)dy\right)f_{t}(x),$ (5)

where

$h(x)=b(x)-d(x)$

is the natural growth rate. In [4], the surprising result is that the random
ancestral lineage, when reversed in time, becomes a simple Ornstein-Uhlenbeck
process whose laws is time homogeneous and independent of $T$. This phenomenon
is unusual in the setting of spinal processes theory since, in general,
processes both dependent in time and $T$ arise. Unfortunately, the method
developed in [4] essentially relies on the Brownian nature of the particles’
motions and the quadratic term in the death rate as this allows many explicit
computations. This raises the question of whether the simplicity of the time-
reversed spinal process is due to the particular context of [4].
The results of the present paper extend the ones of [4] in several directions
as we do not require the particles’ motions to be of Brownian type and the
death-rate to include a quadratic term. More importantly, the method developed
in the article is far more robust to other extensions (for instance replacing
the jump operator by a jump-diffusion operator). The generator of the
ancestral path of the randomly chosen individual (in forward time) can be
obtained by following the work of Marguet [23] and other earlier works (see
also [4, 6] and notably [18, 17] for the Feynmann-Kac formula for branching
processes): this path is a Markov process inhomogeneous in time. The time-
reversal of this process is obtained by following techniques developped by
Chung and Walsh, Nagasawa, Reinhard and Roynette [25, 26, 5, 32, 30] (see also
[12]) and we will see that it is a homogeneous Markov process. These
techniques are based on a duality theory for semigroups which are particularly
well-suited in our context as many-to-one formulas express an intrinsic
duality structure within branching processes.

Informally speaking, we prove the following theorem which characterizes the
law of the time reversed spinal process. This result is made more precise in
Theorem 4.2.

######  Theorem 1.1.

Assuming that the initial trait distribution $Z^{K}_{0}$ of the population
converges to the stationary solution $F(x)dx$ of (5), the process describing,
backward in time, the lineage of an individual sampled in the living
population at time $T>0$ converges, when $K\rightarrow+\infty$, to a time
homogeneous Markov process $Y$ whose law is independent of $T$ and
characterized by its semigroup $(P^{R}_{t})_{t}$ acting on any bounded
measurable function $\varphi$:

$P^{R}_{t}\varphi=\frac{1}{F}\widehat{P^{\ast}_{t}}(F\varphi)$ (6)

where $\widehat{P^{\ast}_{t}}$ is defined by

$\widehat{P^{\ast}_{t}}\varphi(x)=\mathbb{E}_{x}\left[\exp\left(\int_{0}^{t}(h(X^{\ast}_{s})-\lambda)\
ds\right)\varphi(X^{\ast}_{t})\right],$

with $X^{\ast}$ a Markov process whose generator is the formal adjoint
$L^{\ast}$ of $L$ and $\lambda=\int_{\mathbb{R}}F(x)\ dx$.

In terms of generator, this says that the time reversed process ${Y}^{R}$ of
the spinal process $Y$ has the infinitesimal generator (see Proposition 4.4)
given by

$\displaystyle L^{R}\varphi(x)=$
$\displaystyle\frac{L^{*}(F\varphi)(x)}{F(x)}+(h(x)-\lambda)\varphi(x)=\rho\varphi^{\prime}+\gamma\int_{\mathbb{R}}\left(\varphi(y)-\varphi(x)\right)\
\frac{F(y)}{F(x)}m(y,x)dy$ (7)

whenever this makes sense. We can see, as in the Gaussian case developed in
[4], that the ancestral lineage of a typical individual backward in time has a
very simple dynamics: here, the jump measure is biased according to the
stationary distribution $F$. Notice that the expressions (6) and (7) also hold
in the Gaussian case. In fact, these expressions are quite general and could
be generalized to mutation mechanisms other than the Gaussian setting of [4]
or the case considered here, provided we can prove that the PDE associated
with the large population approximation of the trait distribution (see here
(5)) admits a unique stationary measure $F$.

|
---|---
(a) | (b)
Figure 1: Ancestral lineages of the living individuals at two different times,
in black. The sampling time is represented by the vertical line. In gray, all
the individuals that have lived are pictured, which allow to show the traits
occupied by past lost lineages.

In a recent work [28], the semigroup $P^{R}$ is introduced and justified from
a macroscopic point of view using the so-called neutral fractions that have
been introduced by [31] to keep track of ancestries in PDEs that describe
macroscopic populations. The present work provides a rigorous mathematical
justification of these semi-groups grounded on an individual-based model and a
time inversion of the typical ancestral line.

The article is organized as follows. Section 2 aims to describe the setting of
this work: Section 2.1 introduces the process describing the motion of the
trait of the individuals and its dual process which plays an important role in
the following. Section 2.2 provides the deterministic equations approximating
the dynamics of $Z^{K}$ and $H^{K}$, and whose stationary solutions are
studied in Section 2.4. These stationary solutions are central in our
approach. Indeed, using these solutions as limiting initial conditions for the
historical process allows its approximation by a linear branching process. The
coupling of the processes $Z^{K}$ and $H^{K}$ with linear births and deaths
processes $\widetilde{Z}^{K}$ and $\widetilde{H}^{K}$ is presented in Section
2.5. For the processes $\widetilde{H}^{K}$, the branching property holds and
we can use many-to-one formulas to obtain asymptotic representations of the
ancestral lineages. In Section 3, the spine of $\widetilde{H}^{K}$, i.e. the
ancestral lineage of a typical individual chosen at time $T$, is studied and
in particular its time reversal (Section 3.2.2). We then conclude in Section 4
and (7) is established.

Notations: In the sequel, we will denote by
$\mathbb{L}^{\infty}=\mathbb{L}^{\infty}(\mathbb{R})$ the set of measurable
bounded functions on $\mathbb{R}$ and by
$\mathbb{L}^{1}=\mathbb{L}^{1}(\mathbb{R})$ the set of functions that are
integrable with respect to the Lebesgue measure on $\mathbb{R}$.
$C_{b}=C_{b}(\mathbb{R})\subset\mathbb{L}^{\infty}(\mathbb{R})$ is the set of
bounded continuous functions and $C^{1}_{b}=C^{1}_{b}(\mathbb{R})$ the subset
of bounded differentiable functions such that the derivative $f^{\prime}\in
C_{b}$. From now, for any two measurable functions $f$ and $g$, $\langle
f,g\rangle$ stands for $\int_{\mathbb{R}}f(x)g(x)\ dx$ whenever this last
expression makes sense. Similarly, for a finite measure $\mu$ and a measurable
function $f$, $\langle\mu,f\rangle=\int_{\mathbb{R}}f(x)\mu(dx)$ whenever this
integral is well defined.

##  2 Models and settings

###  2.1 Individual based model and hypotheses

Recall that the population at time $t$ can be represented by the point measure
$Z^{K}_{t}$ defined in (4) and that the ancestries of the living individuals
at time $t$ are given by $H^{K}_{t}$ defined in (1). The trait of an
individual evolves during its life according to the drifted jump process with
generator $L$ defined in (2). We will denote by $(X_{t})_{t\in\mathbb{R}_{+}}$
the Markov process with infinitesimal generator $(L,D(L))$, with
$C_{b}^{1}\subset D(L)$.

Before going further let us precise the hypotheses that we need and that will
be assumed satisfied throughout this work.

Assumptions $(H)$

  * (a)

$(x,A)\in\mathbb{R}\times\mathcal{B}(R)\to\int_{A}m(x,y)dy$ is weakly
continuous in the first variable and satisfies

$\exists\varepsilon>0,\kappa_{0}>0,\forall x\in\mathbb{R},\
m(x,y)\geq\kappa_{0}\mathds{1}_{(x-\varepsilon,x+\varepsilon)}(y)$

  * (b)

$h$ is continuous, $h(0)>0$ and there exists $c\in\mathbb{R}$ such that $\
\forall x\in\mathbb{R}$, $h(x)\leq c$, and
$\lim\limits_{x\to\pm\infty}h(x)=-\infty$.

  * (c)

There exist $q\geq 1$ and $x_{0}>0$ such that for all $|x|\geq x_{0}$,
$h(x)\leq-|x|^{q}$, and

$\sup_{x\in\mathbb{R}}\int_{\mathbb{R}}y^{2q}m(x,y)dy<+\infty.$ (8)

  * (d)

$\forall y\in\mathbb{R},\ \int_{\mathbb{R}}m(x,y)\
dx=1=\int_{\mathbb{R}}m(y,x)\ dx.$

Let us comment on these assumptions. Assumptions $(H.a)-(H.b)$ are meant to
provide the existence and uniqueness of a non-trivial stationary solution to
Equation (5). These are borrowed from [7] where Cloez and Gabriel studied a
related eigen-problem. The first part of Assumption $(H.a)$ and Assumption
$(H.c)$ are made to prove the convergence of the particle systems $Z^{K}$ to
the limiting PDE (5) (see the hypothesis of Theorem 2.2). Assumption $(H.c)$
is not so restrictive, as shown in the following examples, and could be easily
changed for other growth rates $h$ provided that $(H.b)$ holds. Whether
$(H.a)$ and $(H.b)$ can be further weakened is a difficult problem (see [7,
8]). Assumption $(H.d)$ allows us to give a simple and straightforward
definition of the dual process of $X$. Our work can be extended to the case
where $\int_{\mathbb{R}}m(x,y)dx<+\infty$ provided we add the correct
renormalizations. These hypotheses can certainly be weakened as the adjoint
process always exists [12], but we choose to use these assumptions as a trade-
off between simplicity and generality.

######  Example 2.1.

1\. In [4], the following birth and death rates are used $b(x)=1$,
$d(x)=x^{2}/2$, so that $h(x)=1-x^{2}/2$ satisfies $(H.b)$ and $(H.c)$. The
assumption $(H.c)$ is satisfied provided

$\sup_{x\in\mathbb{R}}\int_{\mathbb{R}}y^{4}m(x,y)dy<+\infty.$

2\. The case of a convolution operator, i.e. where $m(x,y)=\widetilde{m}(x-y)$
for some continuous probability density $\widetilde{m}$ satisfying
$\widetilde{m}(0)>0$ and (8), also enters the Assumptions (H).
For instance, a Gaussian mutation kernel and a polynomial growth rate $h$
satisfy these Assumptions.

###  2.2 Limiting PDE

Let $T>0$. The processes $(Z^{K}_{t})_{t\in[0,T]}$ are solutions of stochastic
differential equations driven by Poisson point measures (see Appendix A). In
this section, we study their asymptotic behavior when $K\rightarrow+\infty$,
assuming that the initial conditions $Z^{K}_{0}$ converge to a non-trivial
measure $\xi_{0}$, assumed to be deterministic for sake of simplicity. The
dynamics of measures is described with respect to test functions: for finite
measures on $\mathbb{R}$, such as the $Z^{K}_{t}$’s, we consider $\varphi\in
C^{1}_{b}(\mathbb{R})$ which is a dense space of $C_{b}(\mathbb{R})$ for the
uniform norm.

######  Theorem 2.2.

Assume that the hypotheses $(H)$ are satisfied. Let us assume that the initial
conditions $(Z^{K}_{0}(dx))_{K}$ satisfy

$\sup_{K\in\mathbb{N}^{*}}\mathbb{E}\big{(}\langle
Z^{K}_{0},1\rangle^{2+\epsilon}\big{)}<+\infty\qquad\mbox{ and
}\qquad\sup_{K\in\mathbb{N}^{*}}\mathbb{E}\big{(}\langle
Z^{K}_{0},x^{2q}\rangle^{1+\epsilon}\big{)}<+\infty.$ (9)

and that the sequence $(Z^{K}_{0}(dx))_{K}$ converges in probability (and
weakly as measures) to the deterministic finite measure $\,\xi_{0}(dx)$. Let
$T>0$ be given. Then the sequence of processes $\,(Z^{K}_{t})_{t\in[0,T]}$
converges in $\mathbb{L}^{2}$, in
$\mathbb{D}([0,T],\mathcal{M}_{f}(\mathbb{R}))$ to a deterministic continuous
function $\,(\xi_{t})_{t\in[0,T]}$ of $C([0,T],\mathcal{M}_{f}(\mathbb{R}))$
that is the unique solution of the weak equation: $\forall\varphi\in
C^{1}_{b}(\mathbb{R})$,

$\langle\xi_{t},\varphi\rangle=\langle\xi_{0},\varphi\rangle+\int_{0}^{t}\int_{\mathbb{R}}\left\\{\left(h(x)-\langle\xi_{s},1\rangle\right)\varphi(x)+L\varphi(x)\right\\}\xi_{s}(dx)\,ds.$
(10)

More precisely:

$\lim_{K\to\infty}\mathbb{E}(\sup_{t\leq T}|\langle
Z^{K}_{t},\varphi\rangle-\langle\xi_{t},\varphi\rangle|^{2})=0.$ (11)

Moreover, we have that $\
\sup_{t\in[0,T]}\langle\xi_{t},1+x^{2}\rangle<+\infty$.

Idea of the proof: Under the assumptions (9), we can adapt the proof of Lemma
B.1 in [4] to obtain that

$\sup_{K\in\mathbb{N}^{*}}\mathbb{E}\big{(}\sup_{t\in[0,T]}\langle
Z^{K}_{t},1\rangle^{2+\epsilon}\big{)}<+\infty\qquad\mbox{ and
}\qquad\sup_{K\in\mathbb{N}^{*}}\mathbb{E}\big{(}\sup_{t\in[0,T]}\langle
Z^{K}_{t},x^{2q}\rangle^{1+\epsilon/2}\big{)}<+\infty.$ (12)

Then, Theorem 2.2 is a straightforward adaptation of Theorem 2.2 and
Proposition 3.4 of [4], and we refer to this paper for the proof. Notice that
(9) could be weakened if one can ensure that this implies

$\sup_{K\in\mathbb{N}^{*}}\mathbb{E}\big{(}\sup_{t\in[0,T]}\langle
Z^{K}_{t},|h|^{2}\rangle^{1+\epsilon/2}\big{)}<+\infty.$

The more general Assumption (9) is made because we stick here with a general
growth rate $h$ satisfying ($H.c$). Particular cases where $h$ is explicit and
differentiable can be treated directly.

###  2.3 Duality properties for the infinitesimal generator and the transition
semigroup of the underlying path process

Before considering the stationary solutions of PDE (10) and the ancestral path
of an individual chosen at random in the population at a given time $T>0$, we
study the stochastic process $(X_{t})_{t\geq 0}$ describing the change in time
of the trait of a given individual with initial value $x\in\mathbb{R}$. The
process $(X_{t})_{t\geq 0}$ follows the stochastic differential equation

$\displaystyle X_{t}=$ $\displaystyle x+\rho
t+\int_{0}^{t}\int_{\mathbb{R}_{+}}\int_{\mathbb{R}}(y-X_{s_{-}}){\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\theta\leq\gamma m(X_{s_{-}},y)}Q(ds,d\theta,dy),$ (13)

where $Q$ is a Poisson point measure on
$\mathbb{R}\times\mathbb{R}_{+}\times\mathbb{R}$ with intensity the Lebesgue
measure. We define on the same probability space the stochastic process
$(X^{*}_{t})_{t\geq 0}$ as solution of

$\displaystyle X^{*}_{t}=$ $\displaystyle x-\rho
t+\int_{0}^{t}\int_{\mathbb{R}_{+}}\int_{\mathbb{R}}(y-X^{*}_{s_{-}}){\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\theta\leq\gamma m(y,X^{*}_{s_{-}})}Q(ds,d\theta,dy).$ (14)

Note that the transport terms are opposite and that the jump kernels are dual
in some $\mathbb{L}^{1}$-setting. These processes are both Markov processes,
with transition semigroups respectively $(P_{t},t\geq 0)$ and
$(P^{*}_{t},t\geq 0)$. For bounded and measurable functions $f$, they are
given by

$P_{t}f(x)=\mathbb{E}_{x}\big{(}f(X_{t})\big{)},\qquad\mbox{ and }\qquad
P^{*}_{t}f(x)=\mathbb{E}_{x}\big{(}f(X^{*}_{t})\big{)}.$ (15)

The number of jumps of the process $X$ between $0$ and $t$ follows a Poisson
distribution of parameter $\gamma$. Conditionally on this number, the jump
times are distributed as the order statistic of a vector of independent
uniform random variables on $[0,t]$. Summing over these jumps, we can write

$\displaystyle P_{t}f(x)$
$\displaystyle=\mathbb{E}_{x}\big{(}f(X_{t})\big{)}=e^{-\gamma t}\,\sum_{k\geq
0}\frac{(\gamma t)^{k}}{k!}\mathbb{E}\big{(}f(x+\rho
t+U_{1}+\ldots+U_{k})\big{)},$ (16)

where $U_{1},\ldots,U_{k}$ are the jump steps of the $k$ jumps (whose laws
depend on $x$). A similar expression with a drift $-\rho$ is available for the
process $(X^{*}_{t})_{t\geq 0}$.

The aim of this part is to prove the following theorem.

######  Theorem 2.3.

We can extend $P$ and $P^{*}$ respectively to $\mathbb{L}^{\infty}$ and
$\mathbb{L}^{1}$. They satisfy the duality relation

$\displaystyle\langle P_{t}f,g\rangle=\langle
f,P^{*}_{t}g\rangle,\qquad\forall f\in\mathbb{L}^{\infty},g\in\mathbb{L}^{1}.$
(17)

The proof will be deduced from a succession of lemmas and remarks.

Using Itô’s formula for jump processes with drift [19, Th. 5.1, page 66], it
is easy to prove that the domain of the infinitesimal generators of $X$ and
$X^{*}$ contains at least the functions of $C^{1}_{b}$. Further, we have that
for $f\in C^{1}_{b}$, $g\in C^{1}_{b}$,

$Lf(x)=\rho\ f^{\prime}(x)+\gamma\int_{\mathbb{R}}(f(y)-f(x))\ m(x,y)dy,$ (18)

and

$L^{*}g(x)=-\rho\ g^{\prime}(x)+\gamma\int_{\mathbb{R}}(g(y)-g(x))\ m(y,x)dy.$
(19)

Using the integration by parts formula allows to prove easily that these
generators are in duality for functions of $C^{1}_{b}$. We easily extend
$L^{*}$ to the functions of $\mathbb{L}^{1}$.

######  Lemma 2.4.

The generator $L^{*}$ can been extended in such a way that $L$ and $L^{*}$ are
in duality as follows: for $f\in C^{1}_{b}$ and $g\in\mathbb{L}^{1}$,

$\langle Lf,g\rangle=\langle f,L^{*}g\rangle.$ (20)

Proof 9 By integration by part, the generator $L$ is associated to the adjoint
$L^{\prime}$ by the following relation: for any $f\in C^{1}_{b}$ and
$g\in\mathbb{L}^{1}$, $\langle Lf,g\rangle=\langle f,L^{\prime}g\rangle$, with

$L^{\prime}g(x)=-\rho\ {\partial\over\partial
x}g(x)+\gamma\int_{\mathbb{R}}(g(y)-g(x))\ m(y,x)dy$

where ${\partial\over\partial x}g$ is understood in the distribution sense.
Indeed since $g\in\mathbb{L}^{1}$, it converges to $0$ at infinity. In
particular $L^{\prime}$ and $L^{*}$ coincide on $C^{1}_{b}$ and we will keep
the notation $L^{*}$ for the operator defined on $\mathbb{L}^{1}$. $\square$

Let us now prove Theorem 2.3.
Proof 9[Proof of Theorem 2.3]We proceed in two steps. First, we define the
adjoint $P^{\prime}_{t}$ of $P_{t}$, and then we prove that it is $P^{*}_{t}$.

Step 1: Let $t>0$. From (16), we can check that the semigroup $P_{t}$ defines
an operator from $\mathbb{L}^{\infty}$ into $\mathbb{L}^{\infty}$. It is known
(e.g. [3, IV.3.C page 65]) that the dual of $\mathbb{L}^{\infty}$ is strictly
larger (for the inclusion) than $\mathbb{L}^{1}$. Let $P^{\prime}_{t}$ be the
adjoint of $P_{t}$ on the dual space $(\mathbb{L}^{\infty})^{\prime}$ of
$\mathbb{L}^{\infty}$. The domain of $P^{\prime}_{t}$ is

$D(P^{\prime}_{t}):=\\{\mu\in(\mathbb{L}^{\infty})^{\prime},\ \exists c\geq
0,\ \forall f\in\mathbb{L}^{\infty},\ |\langle\mu,P_{t}f\rangle\leq
c\|f\|_{\infty}\\}.$

Since for any $g\in\mathbb{L}^{1}$, $|\langle
g,P_{t}f\rangle|\leq\|g\|_{1}\|f\|_{\infty}$, we see that
$\mathbb{L}^{1}\subset D(P^{\prime}_{t})$ and $P^{\prime}_{t}g$ is well
defined. For any $f\in\mathbb{L}^{\infty}$,

$\langle f,P^{\prime}_{t}g\rangle=\langle P_{t}f,g\rangle.$

Choosing $f=\mbox{sign}(P^{\prime}_{t}g)$, we obtain that

$\int_{\mathbb{R}}|P^{\prime}_{t}g(x)|\ dx=\langle
P_{t}f,g\rangle\leq\|g\|_{1},$ (21)

so that $P^{\prime}_{t}g\in\mathbb{L}^{1}$ and $\||P^{\prime}_{t}\||\leq 1$.

Step 2: Now, our purpose is to prove that $P^{\prime}_{t}=P^{*}_{t}$ where
$P^{*}_{t}$ has been defined in (15). Let us first prove that $P^{*}_{t}$
sends $\mathbb{L}^{1}$ to $\mathbb{L}^{1}$. Let $g\in\mathbb{L}^{1}$. Summing
on the number of jumps for the process $X^{*}$ between $0$ and $t$, we can
write that

$\displaystyle\int_{\mathbb{R}}|P_{t}^{*}g(x)|dx$
$\displaystyle=\int_{\mathbb{R}}|\mathbb{E}_{x}\big{(}g(X^{*}_{t})\big{)}|dx$
$\displaystyle\leq e^{-\gamma t}\,\sum_{k\geq 0}\frac{(\gamma
t)^{k}}{k!}\int_{\mathbb{R}}|\mathbb{E}_{x}\big{(}g(x-\rho
t+U_{1}+\ldots+U_{k})\big{)}|\,dx,$

where $U_{1},\ldots,U_{k}$ are the jump steps of the $k$ jumps. To simplify
notation, let us consider the case of one jump. Using the fact that,
conditionally on the number of jumps in the time interval $[0,t]$, the jump
times are uniformly distributed on $[0,t]$, we obtain

$\displaystyle\int_{\mathbb{R}}\Big{|}\mathbb{E}_{x}\big{(}g(x-\rho
t+U_{1}\big{)}\Big{|}dx$
$\displaystyle=\int_{\mathbb{R}}\Big{|}\frac{1}{t}\int_{0}^{t}\int_{\mathbb{R}}g(y-\rho(t-t_{1}))m(y,x-\rho
t_{1})\,dy\,dt_{1}\Big{|}\,dx$
$\displaystyle\leq\frac{1}{t}\int_{0}^{t}\int_{\mathbb{R}}\Big{|}g(y-\rho(t-t_{1}))\Big{|}\bigg{(}\int_{\mathbb{R}}m(y,x-\rho
t_{1})\,dx\bigg{)}\,dy\,dt_{1}$
$\displaystyle\leq\frac{1}{t}\int_{0}^{t}\int_{\mathbb{R}}\Big{|}g(y-\rho(t-t_{1}))\Big{|}\,dy\,dt_{1}$
$\displaystyle\leq\,\|g\|_{1},$

where we have used Assumption (H.d), i.e. that $\int_{\mathbb{R}}m(y,x)dx=1$.
The same estimate can be obtained for the other terms
$\int_{\mathbb{R}}|\mathbb{E}_{x}\big{(}g(x-\rho
t+U_{1}+\ldots+U_{k})\big{)}|\,dx$, which implies that

$\|P_{t}^{*}g\|_{1}\leq\,\|g\|_{1}$

so that $\||P^{*}_{t}\||\leq 1$.

Step 3: Let us now consider $f\in C^{1}_{b}$, $g\in\mathbb{L}^{1}\cap
C^{1}_{b}$ and $t>0$. We define the function $\varphi(x,s)=g(x-\rho(t-s))$. To
ease the following computation, let us give a name to the jump operator in
$L^{*}$ (19):

$Jg(x)=\gamma\int_{\mathbb{R}}\big{(}g(y)-g(x)\big{)}m(y,x)dy.$

It is easy to prove using ($H.d$) that if $g\in\mathbb{L}^{1}$, then
$Jg\in\mathbb{L}^{1}$ and

$\|Jg\|_{1}\leq 2\gamma\|g\|_{1}.$

On the one side,
$P_{t}^{*}g(x)=\mathbb{E}_{x}\big{(}g(X^{*}_{t})\big{)}=\mathbb{E}_{x}\big{(}\varphi(X^{*}_{t},t)\big{)}$,
and using Itô’s formula:

$\displaystyle\mathbb{E}_{x}\big{(}\varphi(X^{*}_{t},t)\big{)}=$
$\displaystyle g(x-\rho
t)+\int_{0}^{t}\mathbb{E}_{x}\Big{(}L^{*}\varphi(.,s)(X^{*}_{s})+\rho
g^{\prime}(X^{*}_{s}-\rho(t-s))\Big{)}ds$ $\displaystyle=$ $\displaystyle
g(x-\rho t)+\int_{0}^{t}P^{*}_{s}\big{(}J\varphi(.,s)\big{)}(x)ds.$ (22)

On the other side, using the definition of $P^{\prime}_{t}$, $\langle
f,P^{\prime}_{t}g\rangle=\langle P_{t}f,g\rangle$. Our purpose is to develop
the right hand side to have an expression similar to (22). Let us introduce
$\psi(s)=P_{s}f(x)g(x-\rho(t-s))$. Notice that $P_{t}f(x)g(x)=\psi(t)$. By the
Kolmogorov formula:

$P_{t}f(x)=f(x)+\int_{0}^{t}LP_{s}f(x)\ ds,$ (23)

and the fact that $g(x)=g(x-\rho t)+\int_{0}^{t}\rho g^{\prime}(x-\rho(t-s))\
ds$, we obtain:

$\displaystyle\langle P_{t}f,g\rangle=$
$\displaystyle\int_{\mathbb{R}}f(x)g(x-\rho t)dx$ $\displaystyle\hskip
28.45274pt+\int_{0}^{t}\int_{\mathbb{R}}\Big{(}LP_{s}f(x)g(x-\rho(t-s))+\rho
P_{s}f(x)g^{\prime}(x-\rho(t-s))\Big{)}dx\ ds$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}}f(x)g(x-\rho
t)dx+\int_{0}^{t}\int_{\mathbb{R}}P_{s}f(x)\big{(}L^{*}\varphi(.,s)(x)+\rho
g^{\prime}(x-\rho(t-s))\big{)}dx\ ds$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}}f(x)g(x-\rho
t)dx+\int_{0}^{t}\int_{\mathbb{R}}f(x)P^{\prime}_{s}\big{(}J\varphi(.,s)\big{)}(x)dx\
ds.$ (24)

Using (22), and since (24) holds for every $f\in C^{1}_{b}$, we obtain for
almost every $x$,

$\displaystyle P^{*}_{t}g(x)=$ $\displaystyle g(x-\rho
t)+\int_{0}^{t}P^{*}_{s}\big{(}J\varphi(.,s)\big{)}(x)\ ds$ $\displaystyle
P^{\prime}_{t}g(x)=$ $\displaystyle g(x-\rho
t)+\int_{0}^{t}P^{\prime}_{s}\big{(}J\varphi(.,s)\big{)}(x)\ ds.$

From this, we deduce that:

$\displaystyle\|P^{\prime}_{t}g-P^{*}_{t}g\|_{1}=$
$\displaystyle\int_{\mathbb{R}}\big{|}P^{\prime}_{t}g(x)-P^{*}_{t}g(x)\big{|}\
dx$ $\displaystyle\leq$
$\displaystyle\int_{0}^{t}\int_{\mathbb{R}}\big{|}P^{\prime}_{s}\big{(}J\varphi(.,s)\big{)}(x)-P^{*}_{s}\big{(}J\varphi(.,s)\big{)}(x)\big{|}\
dx\ ds$ $\displaystyle\leq$
$\displaystyle\int_{0}^{t}\||P^{\prime}_{s}-P^{*}_{s}\||\
\|J\varphi(.,s)\|_{1}\ ds$ $\displaystyle\leq$
$\displaystyle\int_{0}^{t}2\gamma\ \||P^{\prime}_{s}-P^{*}_{s}\||\ \|g\|_{1}\
ds.$ (25)

This implies that

$\||P^{\prime}_{t}-P^{*}_{s}\||\leq
2\gamma\int_{0}^{t}\||P^{\prime}_{s}-P^{*}_{s}\||\ ds.$

Now applying Gronwall’s inequality, we have that
$\||P^{\prime}_{t}-P^{*}_{t}\||=0$. As a consequence, we have for all $f\in
C^{1}_{b}$ and all $g\in\mathbb{L}^{1}\cap C^{1}_{b}$ that:

$\langle P_{t}f,g\rangle=\langle f,P_{t}^{*}g\rangle.$ (26)

Is is now standard to extend this identity to $f\in\mathbb{L}^{\infty}$ and
$g\in\mathbb{L}^{1}$. The theorem is proved. $\square$

###  2.4 Duality and stationary solution of the limiting PDE

First, we start with a sufficient condition ensuring that the solution of the
limiting PDE (10) is a function $\xi$ whose values $\xi_{t}$ are absolutely
continuous measures with respect to the Lebesgue measure on $\mathbb{R}$.
Exploiting the duality relations, we show that the densities $f_{t}$ of the
measures $\xi_{t}$ solve a PDE with $L^{*}$.

######  Proposition 2.5.

If the measure $\xi_{0}$ admits a non negative density $f_{0}$ with respect to
the Lebesgue measure on $\mathbb{R}$, then so does $\xi_{t}$ for all $t>0$.
The densities $f_{t}$ for $t>0$ define a solution in
$C([0,T],L^{1}(\mathbb{R}))$ of:

$\partial_{t}f_{t}(x)=L^{\ast}f_{t}(x)+\left(h(x)-\int_{\mathbb{R}}f_{t}(y)dy\right)f_{t}(x).$
(27)

Proof 9 The idea of the proof is the following: if (27) possesses a solution
$f_{t}$ in $C([0,T],\mathbb{L}^{1})$, then $f_{t}(x)dx$ is solution of (10),
and the identification $\xi_{t}=f_{t}(x)dx$ follows from the uniqueness of the
solution of (10). Thus, we only have to prove that (27) possesses a solution
with initial condition $f_{0}$. To prove this, we follow closely the
computation in [14]. The proof is detailed in Appendix C. $\square$

Now, let us discuss on the stationary solutions of (10). If we assume to have
in hand a non-negative stationary solution $F\in\mathbb{L}^{1}$ of (10), it
necessarily satisfies

$L^{\ast}F+hF=\lambda F,$

with $\lambda=\int_{\mathbb{R}}F(x)\ dx$. Thus, $F$ is an eigenvector of the
operator $\mathcal{A}$ defined by

$\mathcal{A}f=L^{\ast}f+hf.$ (28)

Reciprocally, we can state the following result.

######  Proposition 2.6.

Under the assumptions $(H.a)$ and $(H.b)$, there exists a unique (up to a
multiplicative constant) positive eigenvector $G\in L^{1}(\mathbb{R})$ for
$\mathcal{A}$ associated to an eigenvalue $\lambda$. If $\lambda$ is positive,
then, $F=\lambda G/\|G\|_{1}$ is the unique non-trivial positive stationary
solution to (10).

Proof 9 Existence and uniqueness of a positive eigenvector $G$ have been
proven by Cloez and Gabriel [7]. The positiveness of the eigenvalue $\lambda$
associated with the non-negative eigenvector is not stated in the result of
[7]. If $\lambda>0$, the stationary solution of (10) can be obtained by
defining $F=\lambda G/\|G\|_{1}$: we have that $\|F\|_{1}=\lambda$ and by
linearity $L^{*}F+hF=\lambda F$. $\square$

In the Gaussian case [4], the condition for the positiveness of $\lambda$
could be expressed in terms of the model parameters. This was feasible as the
stationary distribution $F$ was an explicit Gaussian distribution. In the
present situation, only sufficient conditions can be derived such as the one
stated in the following lemma.

######  Lemma 2.7.

Assume that in addition to (H.a) and (H.b), there exists $x_{1}>0$ such that
$\inf_{x\in(-x_{1},x_{1})}h(x)>\gamma$ and that we can choose the constants
$x_{1}$, $\varepsilon$ and $\kappa_{0}$ such that
$\gamma\kappa_{0}\varepsilon^{3}\geq 12\rho x_{1}$. Then, $\lambda>0$.

Proof 9 Following [7], let us consider the function
$\psi_{0}(x)=\left(1-\frac{x^{2}}{x_{1}^{2}}\right)_{+}^{2}.$ We have:

$(L+h)\psi_{0}(x)\\\ \begin{aligned}
=&-4\rho\frac{x}{x_{1}^{2}}\left(1-\frac{x^{2}}{x_{1}^{2}}\right)_{+}+\gamma\int_{R}\big{(}\psi_{0}(y)-\psi_{0}(x)\big{)}m(x,y)dy+h(x)\psi_{0}(x)\\\
=&-4\rho\frac{x}{x_{1}^{2}}\left(1-\frac{x^{2}}{x_{1}^{2}}\right)_{+}+\gamma\int_{-x_{1}}^{x_{1}}\left(1-\frac{y^{2}}{x_{1}^{2}}\right)^{2}m(x,y)dy+\big{(}h(x)-\gamma\big{)}\psi_{0}(x)\\\
\geq&\big{(}h(x)-\gamma)\psi_{0}(x)-\frac{4\rho}{x_{1}}\left(1-\frac{x^{2}}{x_{1}^{2}}\right)_{+}+\gamma\kappa_{0}x_{1}\int_{-1}^{1}\big{(}1-z^{2}\big{)}^{2}{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\big{(}\frac{x-\varepsilon}{x_{1}},\frac{x+\varepsilon}{x_{1}}\big{)}}(z)dz\end{aligned}$
(29)

by using Assumption $(H.a)$. Without loss of generality, it can be assumed
that $0<\varepsilon<x_{1}$. When $x\in[-x_{1},x_{1}]$, the integral in the
third term is lower bounded by

$\displaystyle\int_{1-\varepsilon/x_{1}}^{1}(1-z^{2})^{2}dz=$
$\displaystyle\frac{\varepsilon}{x_{1}}\Big{[}\frac{8}{15}-\frac{7}{15}\Big{(}\big{(}1-\frac{\varepsilon}{x_{1}}\big{)}+\big{(}1-\frac{\varepsilon}{x_{1}}\big{)}^{2}\Big{)}+\frac{1}{5}\Big{(}\big{(}1-\frac{\varepsilon}{x_{1}}\big{)}^{3}+\big{(}1-\frac{\varepsilon}{x_{1}}\big{)}^{4}\Big{)}\Big{]}$
$\displaystyle=$
$\displaystyle\frac{20}{15}\frac{\varepsilon^{3}}{x_{1}^{3}}-\frac{\varepsilon^{4}}{x_{1}^{4}}+\frac{1}{5}\frac{\varepsilon^{5}}{x_{1}^{5}}\geq\frac{1}{3}\frac{\varepsilon^{3}}{x_{1}^{3}}.$
(30)

Gathering (29) and (30), we obtain that

$(L+h)\psi_{0}\geq\inf_{x\in(-x_{1},x_{1})}h(x)\psi_{0}-\gamma\psi_{0}-\frac{4\rho}{x_{1}}\left(1-\frac{x^{2}}{x_{1}^{2}}\right)_{+}+\frac{\gamma\kappa_{0}\varepsilon^{3}}{3x_{1}^{2}}.$

Under our hypotheses,

$-\frac{4\rho}{x_{1}}\left(1-\frac{x^{2}}{x_{1}^{2}}\right)_{+}+\frac{4\rho}{x_{1}}\geq
0,$ (31)

so that $(L+h)\psi_{0}\geq\beta_{0}\psi_{0}$ with

$\beta_{0}=\inf_{x\in(-x_{1},x_{1})}h(x)-\gamma>0.$ (32)

It follows from differentiating the semigroup $S_{t}$ associated with $L+h$
acting on $C_{0}(\mathbb{R})$ that

$S_{t}\psi_{0}\geq\psi_{0}e^{\beta_{0}t}.$

Now, according to our application of Theorem 2.1 of [7], we have that for the
semigroup $S^{\ast}$ associated to $\mathcal{A}$ (28),

$e^{-\lambda t}S^{\ast}_{t}f\xrightarrow[t\to\infty]{}F\text{ in
}\mathbb{L}^{1}.$

Hence, for any positive $f\in\mathbb{L}^{1}$,

$\langle\psi_{0},F\rangle=\limsup_{t\to\infty}\ \langle\psi_{0},e^{-\lambda
t}S^{\ast}_{t}f\rangle=\limsup_{t\to\infty}e^{-\lambda t}\langle
S_{t}\psi_{0},f\rangle\geq\limsup_{t\to\infty}e^{(\beta_{0}-\lambda)t}\langle\psi_{0},f\rangle.$

Thus, $\beta_{0}-\lambda\leq 0$, which gives the result since $\beta_{0}>0$.
$\square$

It is important to note that $F$ is also a solution of a linearized version of
(27)

$\partial_{t}f_{t}(x)=L^{\ast}f_{t}(x)+h^{\lambda}(x)f_{t}(x),$ (33)

where

$h^{\lambda}(x)=h(x)-\lambda.$ (34)

Note that this notation $h^{\lambda}$ will be extensively used in what
follows.

Let us prove in the next lemma that Assumptions ($H.a$) to ($H.c$) ensure that
$F$ has a finite $2q$-moment, compatible with the assumptions (9) for the
initial conditions of the population process.

######  Lemma 2.8.

Assume ($H.a$) to ($H.c$) and assume that $\lambda>0$. Then, $F$ has a finite
moment of order $2q$, i.e.

$\int_{\mathbb{R}}x^{2q}F(x)\ dx<+\infty.$

Proof 9 Set, for any $x\in\mathbb{R}$ and any $n\in\mathbb{N}$,

$g_{n}(x)=\frac{1}{1+\frac{x^{2q+2}}{n}}.$

Note that $g_{n}$ is non-negative, and that there exists a constant
$C\in\mathbb{R}_{+}$ (that can be chosen independent of $n$), such that
$|g^{\prime}_{n}|\leq Cg_{n}$. Obviously, $x^{2q}g_{n}\in D(L)$, which implies
that

$\lambda\langle x^{2q}g_{n},F\rangle=\langle
x^{2q}g_{n},L^{\ast}F+hF\rangle=\langle L(x^{2q}g_{n})+hx^{2q}g_{n},F\rangle.$

Hence,

$\displaystyle\lambda\langle x^{2q}g_{n},F\rangle\leq$
$\displaystyle\rho\langle 2q|x|^{2q-1}g_{n}(x),F\rangle+\rho\left|\langle
x^{2q}g^{\prime}_{n}(x),F\rangle\right|+\gamma\,m_{2q}-\gamma\langle
x^{2q}g_{n}(x),F\rangle+\langle hx^{2q}g_{n},F\rangle$ $\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}}\left(|x|^{2q-1}g_{n}(x)\left(2q\rho+|x|\left(C\rho+\gamma+h(x)\right)\right)F(x)\right)\
dx+\gamma m_{2q},$ (35)

where

${m_{2q}=\sup_{x\in\mathbb{R}}\int_{\mathbb{R}}y^{2q}\ m(x,y)\ dy<+\infty,}$

by Assumption (8). Now, using Assumption ($H.c$), there exists a compact set
$K$ of $\mathbb{R}$ such that

$\int_{\mathbb{R}\setminus
K}\left(|x|^{2q-1}g_{n}(x)\left(2q\rho+|x|\left(C\rho+\gamma+h(x)\right)\right)F(x)\right)\
dx\leq 0,$

which implies, with the constant $c$ appearing in ($H.b$):

$\lambda\langle
x^{2q}g_{n},F\rangle\leq\int_{K}\left(|x|^{2q-1}g_{n}(x)\left(2q\rho+|x|\left(C\rho+\gamma+c\right)\right)F(x)\right)\
dx+\gamma m_{2q}.$ (36)

Now, Fatou’s lemma and Lebesgue convergence theorem imply that

$\displaystyle\lambda\int_{\mathbb{R}}x^{2q}F(x)\ dx\leq$
$\displaystyle\liminf_{n\to+\infty}\lambda\int_{\mathbb{R}}x^{2q}g_{n}(x)F(x)\
dx$ $\displaystyle\leq$
$\displaystyle\liminf_{n\to+\infty}\int_{K}\left(|x|^{2q-1}g_{n}(x)\left(2q\rho+|x|\left(C\rho+\gamma+c\right)\right)F(x)\right)\
dx+\gamma m_{2q}$ $\displaystyle=$
$\displaystyle\int_{K}\left(|x|^{2q-1}\left(2q\rho+|x|\left(C\rho+\gamma+c\right)\right)F(x)\right)\
dx+\gamma m_{2q}<+\infty,$

by (8). $\square$

###  2.5 Coupled population processes

When we start from the initial condition $F$, the non-linear competition term
$\langle\xi_{t},1\rangle=\|F\|_{1}=\lambda$ remains constant in time.
Considering initial conditions close to $F$ for the process will lead to non-
linear terms close to $\lambda$. This remark is the basis of the coupling with
two other individual-based random processes
$(\widetilde{Z}^{K}_{t})_{t\in\mathbb{R}_{+}}$ and
$(\widetilde{H}^{K}_{t})_{t\in\mathbb{R}_{+}}$. These processes are pathwisely
defined by similar equations as (60) and (63) (Appendix A) for $Z^{K}$ and
$H^{K}$ but where the non-linear competition term $N^{K}_{t}/K$ has been
replaced by the expected limiting competition rate $\lambda$. When $K$ tends
to infinity and if the initial conditions of both processes converge to $F$,
then $Z^{K}$ and $\widetilde{Z}^{K}$, and $H^{K}$ and $\widetilde{H}^{K}$ will
be close. The next proposition states a precise approximation result linking
$\widetilde{H}^{K}$ and $H^{K}$ when $K$ goes to infinity and $Z^{K}_{0}$
converges to $F$.

######  Proposition 2.9.

Assume that (9) holds and that $\,{Z}^{K}_{0}\xrightarrow[K\to\infty]{w}F\,$.
Then for any continuous and bounded function $\Phi$ on $\mathbb{D}$,

$\lim_{K\to+\infty}\mathbb{E}(\sup_{t\leq T}|\langle
H^{K}_{t},\Phi\rangle-\langle\widetilde{H}^{K}_{t},\Phi\rangle|^{2})=0.$

The proof is similar to the proof of Proposition 3.4 in [4], to which we
refer.

The the processes $\widetilde{Z}^{K}$ and $\widetilde{H}^{K}$ are linear birth
and death processes and satisfy the branching property. Therefore, the spinal
techniques as developed by [1, 17, 18, 22, 23] can be used. Note also that it
is sufficient to consider the processes started from a unique individual. In
the sequel, we will denote by $\widetilde{Z}$ the branching process
$K\widetilde{Z}^{K}$ started from a single individual of trait $x$:
$\widetilde{Z}_{0}=\delta_{x}$.

##  3 Linear case : Feynman-Kac formula and spinal process

In this part we only focus on the processes $\widetilde{Z}^{K}$ and
$\widetilde{H}^{K}$. It is possible to summarize their intensity measures with
a single process by mean of many-to-one formulas [23]. We will next use these
results to approximate the distribution of a typical lineage for the original
population process.

###  3.1 Feynman-Kac formula and law of the spinal process

######  Lemma 3.1.

Let $\varphi$ in $C_{b}(\mathbb{R})$. Then, for any positive time $t$, for any
$x\in\mathbb{R}$, we have

$\mathbb{E}_{\delta_{x}}\left[\langle\widetilde{Z}_{t},\varphi\rangle\right]=\mathbb{E}_{x}\left[\exp\left(\int_{0}^{t}h^{\lambda}(X_{s})\
ds\right)\varphi(X_{t})\right]=:\widehat{P}_{t}\varphi(x),$ (37)

where $X$ is the process defined in (13).

Proof 9 Let us give a simple proof based on Itô’s formula. Let us first note
that the intensity measure of $\widetilde{Z}_{t}$,
$\,\nu_{t}(dy)=\mathbb{E}_{\delta_{x}}\left[\widetilde{Z}_{t}(dy)\right]\,$
defined for any $\varphi$ in $C_{b}(\mathbb{R})$ by

$\langle\nu_{t},\varphi\rangle=\mathbb{E}_{\delta_{x}}\left[\langle\tilde{Z}_{t},\varphi\rangle\right]$

is the unique weak solution of

$\begin{cases}\partial_{t}\nu_{t}=L^{\prime}\nu_{t}+h^{\lambda}(x)\nu_{t}(dx),\\\
\nu_{0}=\delta_{x},\end{cases}$ (38)

where $L^{\prime}$ is the adjoint of $L$. Indeed, taking expectation in (65)
immediately shows that $\nu$ is weak solution of (38). Uniqueness of such a
solution is proven as in Theorem 2.2 (see Th.2.2 in [4]).

Let us now show that the r.h.s. term of (37) also satisfies (38). Uniqueness
will yield the result. Let $\varphi$ in $C^{1}_{b}(\mathbb{R})$. It is known
since $\varphi$ is in the extended domain of $L$ that

$M_{t}=\varphi(X_{t})-\varphi(X_{0})-\int_{0}^{t}L\varphi(X_{s})\ ds$

is a martingale. Thus applying Itô’s formula with jumps (e.g. [19, Th.5.1]) to
the semimartingale $\
\exp\left(\int_{0}^{t}h^{\lambda}(X_{s})ds\right)\varphi(X_{t})$, we have

$\exp\left(\int_{0}^{t}h^{\lambda}(X_{s})ds\right)\varphi(X_{t})=\varphi(X_{0})+\int_{0}^{t}\exp\left(\int_{0}^{s}h^{\lambda}(X_{u})du\
\right)dM_{s}\\\
+\int_{0}^{t}\exp\left(\int_{0}^{s}h^{\lambda}(X_{u})du\right)L\varphi(X_{s})\
ds+\int_{0}^{t}\varphi(X_{s})h^{\lambda}(X_{s})\exp\left(\int_{0}^{s}h^{\lambda}(X_{u})du\right)\
ds$ (39)

By Assumption $(H.b)$, the stochastic integral with respect to $dM_{s}$
defines a square integrable martingale and by taking the expectation, we
obtain that

$\mathbb{E}_{x}\left[\exp\left(\int_{0}^{t}h^{\lambda}(X_{s})ds\right)\varphi(X_{t})\right]=\varphi(x)\\\
+\mathbb{E}_{x}\bigg{[}\int_{0}^{t}\exp\left(\int_{0}^{s}h^{\lambda}(X_{u})du\right)\bigg{\\{}h^{\lambda}(X_{s})\,\varphi(X_{s})+L\varphi(X_{s})\bigg{\\}}ds\bigg{]}.$
(40)

If we define the measure $\mu_{t}$ for any test function $\varphi\in
C^{1}_{b}(\mathbb{R})$ by

$\langle\mu_{t},\varphi\rangle=\mathbb{E}_{x}\left[\exp\left(\int_{0}^{t}h^{\lambda}(X_{s})ds\right)\varphi(X_{t})\right],$

we obtain from (40) that

$\langle\mu_{t},\varphi\rangle=\langle\delta_{x},\varphi\rangle+\int_{0}^{t}\langle\mu_{s},h^{\lambda}\varphi+L\varphi\rangle
ds.$

That proves that the flow $(\mu_{t},t\geq 0)$ is a weak solution of (38) and
the conclusion follows by uniqueness of Theorem 2.2. $\square$

The previous many-to-one formula characterizes the law of $\tilde{Z}_{t}$. It
can be extended to the whole trajectory (see [4, 22]).

######  Lemma 3.2.

We have that for $T>0$, $\Phi:\mathbb{D}([0,T],\mathbb{R})\to\mathbb{R}$ a
continuous and bounded function and $x\in\mathbb{R}$:

$\displaystyle\mathbb{E}_{\delta_{x}}\left[\langle\widetilde{H}_{T},\Phi\rangle\right]$
$\displaystyle=\mathbb{E}_{\delta_{x}}\left[\sum_{i\in\widetilde{V}_{T}}\Phi({X}^{i}_{s},\
s\leq T)\right]=\mathbb{E}_{x}\left[\exp\left(\int_{0}^{T}h^{\lambda}(X_{s})\
ds\right)\Phi(X_{s},\ s\leq T)\right].$ (41)

This trajectorial Feynman-Kac formula can be used to characterize the law of
an auxiliary process that will help us to understand the typical lineage
later. Let us introduce, for all $x\in\mathbb{R}$ and $t\geq 0$, the expected
population mass, defined by

$m_{t}(x)=\mathbb{E}_{\delta_{x}}\big{[}\langle\widetilde{Z}_{t},1\rangle\big{]}=\mathbb{E}_{\delta_{x}}\left[\langle\widetilde{H}_{t},1\rangle\right]=\mathbb{E}_{x}\left[\exp\left(\int_{0}^{t}h^{\lambda}(X_{s})\
ds\right)\right].$ (42)

Thus, we use the r.h.s. of (41) to define a family of probability measures
$\mu_{x}^{T}$ on $\mathbb{D}([0,T],\mathbb{R})$ by

$\mu_{x}^{T}(A)=\frac{\mathbb{E}_{\delta_{x}}\left[\langle\widetilde{H}_{T},{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{A}\rangle\right]}{\mathbb{E}_{\delta_{x}}\left[\langle\widetilde{H}_{T},1\rangle\right]}=\frac{1}{m_{T}(x)}\mathbb{E}_{x}\left[\exp\left(\int_{0}^{T}h^{\lambda}(X_{s})\
ds\right)\mathds{1}_{X_{.}\in A}\right]$ (43)

for any measurable subset $A$ of $\mathbb{D}([0,T],\mathbb{R})$. For a
probability measure $\nu$, let us also define

$\mu^{T}_{\nu}(A)=\int_{\mathbb{R}}\mu^{T}_{x}(A)\ \nu(dx).$ (44)

Our next proposition characterizes the law of the underlying process (forward
in time).

######  Proposition 3.3.

The distribution $\mu^{T}_{x}$ is the one of a time inhomogeneous Markov
process $Y$ issued from $x$ and with semigroup $(\widetilde{P}_{s,t})_{t\geq
s\geq 0}$ given for a bounded continuous function $\varphi$ by

$\widetilde{P}_{s,t+s}\varphi(x)=\cfrac{\widehat{P}_{t}(\varphi
m_{T-t-s})(x)}{m_{T-s}(x)}.$ (45)

Proof 9 Let $\varphi$ be a some test function, and assume that $s\leq t$ are
such that $s+t\leq T$. Denoting by $\mathcal{F}=(\mathcal{F}_{s})_{s\geq 0}$
and $\mathcal{F^{\prime}}=(\mathcal{F^{\prime}}_{s})_{s\geq 0}$ the natural
filtrations associated respectively to $Y$ and $X$, our aim is to prove that:

$\mathbb{E}_{x}\left[\varphi(Y_{t+s})\mid\mathcal{F}_{s}\right]=\widetilde{P}_{s,t+s}\varphi(Y_{s}).$
(46)

Now, for a $\mathcal{F}_{s}$-measurable random variable $\Psi(Y_{u},\ u\leq
s)$, we have

$\mathbb{E}\left[\Psi(Y_{u},\ u\leq s)\varphi(Y_{t+s})\right]\\\
\begin{aligned} =&\frac{1}{m_{T}(x)}\mathbb{E}_{x}\left[\Psi(X_{u},\ u\leq
s)\varphi(X_{t+s})e^{\int_{0}^{T}h^{\lambda}(X_{u})du}\right]\\\
=&\frac{1}{m_{T}(x)}\mathbb{E}_{x}\Big{[}\Psi(X_{u},\ u\leq
s)e^{\int_{0}^{s}h^{\lambda}(X_{u})du}\\\ &\hskip
56.9055pt\times\mathbb{E}\left[\varphi(X_{t+s})e^{\int_{s}^{t+s}h^{\lambda}(X_{u})du}\
\mathbb{E}\left(e^{\int_{t+s}^{T}h^{\lambda}(X_{u})du}\ |\
\mathcal{F}^{\prime}_{t+s}\right)\ |\
\mathcal{F}^{\prime}_{s}\right]\Big{]}\\\
=&\frac{1}{m_{T}(x)}\mathbb{E}_{x}\left[\Psi(X_{u},\ u\leq
s)e^{\int_{0}^{s}h^{\lambda}(X_{u})du}\
\mathbb{E}\left[\varphi(X_{t+s})e^{\int_{s}^{t+s}h^{\lambda}(X_{u})du}m_{T-(t+s)}(X_{t+s})\
|\ \mathcal{F}^{\prime}_{s}\right]\right]\\\
=&\frac{1}{m_{T}(x)}\mathbb{E}_{x}\left[\Psi(X_{u},\ u\leq
s)e^{\int_{0}^{s}h^{\lambda}(X_{u})du}\widehat{P}_{t}\big{(}\varphi\
m_{T-t-s}\big{)}(X_{s})\right].\end{aligned}$ (47)

For the first equality of (47), we have use the definition of the distribution
$\mu^{T}_{x}$ of $Y$ (see (43)). For the third inequality, we have use the
strong Markov property for $X$ at time $t+s$ with the definition (42) of
$m_{T-(t+s)}(x)$. For the fourth inequality, we have use again the strong
Markov property for $X$ at time $s$ and the definition (48) of
$\widehat{P}_{t}$.

Using again the strong Markov property, we have that

$m_{T-s}(X_{s})=\mathbb{E}_{X_{s}}\Big{(}e^{\int_{0}^{T-s}h^{\lambda}(X_{u})du}\Big{)}=\mathbb{E}_{x}\Big{(}e^{\int_{s}^{T}h^{\lambda}(X_{u})du}\
|\ \mathcal{F}^{\prime}_{s}\Big{)}$

so that the last term of Equation (47) gives

$\displaystyle\begin{split}\mathbb{E}\left[\Psi(Y_{u},\ u\leq
s)\varphi(Y_{t+s})\right]&=\frac{1}{m_{T}(x)}\mathbb{E}_{x}\left[e^{\int_{0}^{T}h^{\lambda}(X_{u})du}\
\Psi(X_{u},\ u\leq s)\frac{\widehat{P}_{t}\left(\varphi
m_{T-t-s}\right)(X_{s})}{m_{T-s}(X_{s})}\right]\\\
&=\mathbb{E}_{x}\left[\Psi(Y_{u},\ u\leq s)\frac{\widehat{P}_{t}\left(\varphi
m_{T-t-s}\right)(Y_{s})}{m_{T-s}(Y_{s})}\right]\end{split}$

by using again (43) for the last equality.

$\square$

###  3.2 Duality properties and time reversal of the process $Y$

In this section, the purpose is to show that the time-reversal of the process
$Y$ is the homogeneous Markov process as announced in Theorem 1.1. To do so,
we use very general general results for time reversal of Markov processes (see
[12, Chapter XVIII.46], and reference therein). We need to prove some duality
relations.

####  3.2.1 Duality between the Feynman-Kac semigroups $\widehat{P}$ and
$\widehat{P}^{*}$

In Lemma 3.1, we have proved that the expectation of the branching process is
related to a (non Markovian) semigroup based on the multiplicative functional
$\exp\big{(}\int_{0}^{t}h^{\lambda}(X_{s})\ ds\big{)}$ which is bounded by
$e^{ct}+1$ (Assumption $(H.b)$). For any function $f\in\mathbb{L}^{\infty}$,
we can define

$\displaystyle\widehat{P}_{t}f(x)=\mathbb{E}_{x}\left(\exp\left(\int_{0}^{t}h^{\lambda}(X_{s})\
ds\right)f(X_{t})\right),$ (48)

and
$\|\widehat{P}_{t}f\|_{\infty}\leq(e^{ct}+1)\|{P}_{t}f\|_{\infty}\leq(e^{ct}+1)\|f\|_{\infty}$.
In an analogous way, we may also define the Feynman-Kac semigroup associated
with the process $X^{*}$: for any function $g\in\mathbb{L}^{1}$,

$\displaystyle\widehat{P}^{*}_{t}g(x)=\mathbb{E}_{x}\left(\exp\left(\int_{0}^{t}h^{\lambda}(X^{*}_{s})\
ds\right)g(X^{*}_{t})\right),$ (49)

and $\
\|\widehat{P}^{*}_{t}g\|_{1}\leq(e^{ct}+1)\|{P}^{*}_{t}g\|_{1}\leq(e^{ct}+1)\|g\|_{1}$.

######  Remark 3.4.

Note that the semigroups $\widehat{P}_{t}$ and $\widehat{P}^{*}_{t}$ are not
conservative: for example, $\widehat{P}_{t}{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}(x)=m_{t}(x)$, see (42). The function $h^{\lambda}$ does not
necessarily have constant sign, but for the constant $c$ defined in Hypothesis
($H.b$), the rescaled semigroups $e^{-ct}\widehat{P}_{t}$ and
$e^{-ct}\widehat{P}^{\ast}_{t}$ are sub-Markovian. Hence, in the following
proof we will work up to this scaling $e^{-ct}$, then assuming that the
semigroups $\widehat{P}_{t}$ and $\widehat{P}^{*}_{t}$ are sub-Markovian.

Let us now prove that the duality relation between $P$ and $P^{*}$ extends to
$\widehat{P}$ and $\widehat{P}^{*}$.

######  Lemma 3.5.

The semigroups $\widehat{P}$ and $\widehat{P}^{\ast}$ satisfy the duality
relation:

$\langle\widehat{P}_{t}f,g\rangle=\langle
f,\widehat{P}^{\ast}_{t}g\rangle,\quad\forall t\geq 0,\quad\forall
f\in\mathbb{L}^{\infty},g\in\mathbb{L}^{1}.$ (50)

Proof 9 For any positive integer $n$ and any $x\in\mathbb{R}$, we define
$h_{n}(x)=h^{\lambda}(x)\vee(-n)$. We consider, the sequences of semigroups
$\widehat{P}^{n}$ and $\widehat{P}^{\ast,n}$ defined similarly as (48) and
(49) but using $h_{n}$ instead of $h^{\lambda}$.

We now show that for any positive integer $n$, the semigroups
$\widehat{P}^{n}$ and $\widehat{P}^{\ast,n}$ are in duality (with respect to
the Lebesgue measure).

Let $f$ be a measurable positive and bounded function and $g$ a function in
$\mathbb{L}^{1}$. it is straightforward to check that $\widehat{P}^{n}$ and
$\widehat{P}^{\ast,n}$ respectively satisfies

$\widehat{P}^{n}_{t}f(x)=\int_{0}^{t}\widehat{P}^{n}_{s}\left(h_{n}P_{t-s}f\right)(x)\
ds+P_{t}f(x)$ (51)

and

$\widehat{P}^{\ast,n}_{t}g(x)=\int_{0}^{t}P^{\ast}_{t-s}\left(h_{n}\widehat{P}^{\ast,n}_{s}g\right)(x)\
ds+P^{\ast}_{t}g(x).$ (52)

Indeed, for instance, the r.h.s of Equation (51) rewrites using the Markov
property

$\int_{0}^{t}\mathbb{E}_{x}\left[\exp\left(\int_{0}^{s}h_{n}(X_{u})\
du\right)h_{n}(X_{s})\mathbb{E}_{X_{s}}\left[f(X_{t-s})\right]\right]\
ds+\mathbb{E}_{x}\left[f(X_{t})\right]\\\
=\mathbb{E}_{x}\left[\int_{0}^{t}h_{n}(X_{s})\exp\left(\int_{0}^{s}h_{n}(X_{u})\
du\right)ds\
f(X_{t})\right]+\mathbb{E}_{x}\left[f(X_{t})\right]=\widehat{P}^{n}_{t}f(x)$

Since $\ \widehat{P}^{\ast,n}_{t}g\in\mathbb{L}^{1}$, we can apply [11] Chap
IX no 14, to justify the existence of a submarkovian semigroup $Q$ in duality
with $\widehat{P}^{\ast,n}$. Hence, for
$f\in\mathbb{L}^{\infty},g\in\mathbb{L}^{1}$, we have

$\langle Q_{t}f,\ g\rangle=\langle f,\ \widehat{P}^{\ast,n}_{t}g\rangle$

from which we deduce, since $h_{n}$ is bounded, that
$\||Q_{t}\||\leq\|h_{n}\|_{\infty}$. Further, we have

$\displaystyle\langle Q_{t}f,\ g\rangle=\langle f,\
\widehat{P}^{\ast,n}_{t}g\rangle$
$\displaystyle=\int_{\mathbb{R}}\int_{0}^{t}{P}^{\ast}_{t-s}\left(h_{n}\widehat{P}^{\ast,n}_{s}g\right)(x)\
ds\ f(x)\ dx+\langle f,\ P^{\ast}_{t}g\rangle$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}}{P}^{\ast}_{t-s}\left(h_{n}\widehat{P}^{\ast,n}_{s}g\right)(x)\
f(x)\ dx\ ds+\langle P_{t}f,\ g\rangle$
$\displaystyle=\int_{0}^{t}\langle{P}_{t-s}f,h_{n}\widehat{P}^{\ast,n}_{s}g\rangle\
ds+\langle P_{t}f,\ g\rangle$ $\displaystyle=\int_{0}^{t}\langle
Q_{s}(h_{n}P_{t-s}f),\ g\rangle\ ds+\langle P_{t}f,\ g\rangle.$

Hence, for Lebesgue almost every $x\in\mathbb{R}$, $Qf$ satisfies the equation

$Q_{t}f(x)=\int_{0}^{t}Q_{s}(h_{n}P_{t-s}f)(x)\ ds+P_{t}f(x).$

Finally, using (51), we obtain that

$\left|\widehat{P}^{n}_{t}f(x)-Q_{t}f(x)\right|\leq\int_{0}^{t}\left|\widehat{P}^{n}_{s}(h_{n}P_{t-s})f(x)-Q_{s}(h_{n}P_{t-s}f)(x)\right|\
ds\\\ \leq\int_{0}^{t}\||\widehat{P}^{n}_{s}-Q_{s}\||\
\|h_{n}P_{t-s}f\|_{\infty}ds\leq\|h_{n}\|_{\infty}\|f\|_{\infty}\int_{0}^{t}\||\widehat{P}^{n}_{s}-Q_{s}\||\
ds.$

Thus, for any $t\leq T$,

$\||\widehat{P}^{n}_{t}-Q_{t}\||\leq\|h_{n}\|_{\infty}\int_{0}^{t}\||\widehat{P}^{n}_{s}-Q_{s}\||\
ds,$

and Gronwall’s lemma allows to conclude that for all $s\leq T$ and all
$f\in\mathbb{L}^{\infty}$,

$Q_{s}f=\widehat{P}^{n}_{s}f.$

It follows that the semigroups $\widehat{P}^{n}$ and $\widehat{P}^{\ast,n}$
are in duality. The extension to $\widehat{P}$ and $\widehat{P}^{\ast}$
follows from a monotone convergence argument as $h_{n}$ converges in a
monotonous way to $h^{\lambda}$. $\square$

####  3.2.2 Time-reversal of the process $Y$

The next results are consequences of the duality relationship between
$\widehat{P}$ and $\widehat{P}^{\ast}$. The first shows that
$\lambda^{-1}m_{t}(x)F(x)$ is a probability density. The second result
determines the law of $Y$ at any time when this process is started from
$m_{T}(x)F(x)dx$. This initial condition will naturally appear in the next
section for the non linear problem.

######  Lemma 3.6.

Let $t$ be a fixed positive time. Then, the function $x\in\mathbb{R}\mapsto
m_{t}(x)F(x)$ is positive with

$\int_{\mathbb{R}}m_{t}(x)F(x)\ dx=\lambda.$

Proof 9 Using the duality between $\widehat{P}_{t}$ and $\widehat{P}^{*}_{t}$,
we have:

$\displaystyle\langle m_{t},F\rangle=\langle\widehat{P}_{t}1,F\rangle=\langle
1,\widehat{P}^{\ast}_{t}F\rangle=\langle 1,F\rangle=\lambda.$

The third equality comes from the fact that almost everywhere

$\widehat{P}^{*}_{t}F=F.$ (53)

Indeed, for any function $\varphi\in C_{b}$, we have:

$\displaystyle\langle\widehat{P}^{*}_{t}F,\varphi\rangle=\langle
F,\widehat{P}_{t}\varphi\rangle=\int_{\mathbb{R}}\mathbb{E}_{x}\Big{[}\langle\widetilde{Z}_{t},\varphi\rangle\Big{]}F(x)dx=\mathbb{E}_{F}\Big{[}\langle\widetilde{Z}_{t},\varphi\rangle\Big{]}=\langle
F,\varphi\rangle,$

by (37) and since $F$ is a stationary distribution for (38). $\square$

######  Proposition 3.7.

For any bounded measurable real-valued function $\varphi$ on $\mathbb{R}$, we
have

$\mathbb{E}_{m_{T}F}\left[\varphi(Y_{t})\right]=\int_{\mathbb{R}}\varphi(x)m_{T-t}(x)F(x)\
dx,$

In other words, the law of the process $Y_{t}$ at time $t$, when the initial
condition is $m_{T}F$, is given by $m_{T-t}(x)F(x)\ dx$.

Proof 9 We have that, using Equation (45),

$\mathbb{E}_{m_{T}F}\left[\varphi(Y_{t})\right]=\Big{\langle}m_{T}F,\
\frac{1}{m_{T}}\widehat{P}_{t}(\varphi m_{T-t})\Big{\rangle}=\langle F,\
\widehat{P}_{t}(\varphi m_{T-t})\rangle.$

Now, Lemma 3.5 entails that

$\langle F,\ \widehat{P}_{t}(\varphi
m_{T-t})\rangle=\langle\widehat{P}^{\ast}_{t}F,\ m_{T-t}\varphi\rangle,$

but $\widehat{P}^{\ast}_{t}F=F$. This gives the result. $\square$

It remains now to identify the law of the time-reversal of the process $Y$
started from $m_{T}F$. To do this, we use the following lemma obtained by
applying [12, Theorem 47] in our very specific setting. The proof is given in
Appendix B.

######  Lemma 3.8.

Let $R=(R_{t})_{t}$ be a positive semigroup which is in duality with a
positive semigroup $R^{\ast}$ with respect to the Lebesgue measure. Let
$f:\mathbb{R}_{+}\times\mathbb{R}\to\mathbb{R}_{+}$ such that
$R_{t}f_{t+s}=f_{s}$ for any non-negative real numbers $s$ and $t$. Let
$(V_{t})_{t\in\mathbb{R}_{+}}$ be a Markov process with semigroup given by

$\mathbb{E}(\varphi(V_{t+s}\mid\mathcal{F}_{s})=\cfrac{1}{f_{s}(V_{s})}R_{t}(f_{s+t}\varphi)(V_{s})$

and a given initial distribution $\mu$ such that, for any $t\geq 0$, the law
of $V_{t}$ is absolutely continuous with respect to the Lebesgue measure with
density $F\times f_{t}$. Then, the time reversed process at a time $T$ of $X$
is time-homogeneous and has semigroup given by

$\varphi\to\frac{R_{t}^{\ast}(F\varphi)}{F}.$

We will apply this result stated with general notations, with
$f_{t}(x)=m_{T-t}(x)$, $R_{t}=\widehat{P}_{t}$ and $Y$ in place of $V$.

Now, Propositions 3.3 and 3.7 together with Lemma 3.5 allow to apply Lemma 3.8
in our particular situation. This leads to

######  Corollary 3.9.

The time-reversal of the process $Y$ with initial condition $m_{T}F$ is a
Markov process $Y^{R}$ whose semigroup $P^{R}$ acting on bounded measurable
functions is given by

$P^{R}_{t}\varphi=\cfrac{\widehat{P}^{\ast}_{t}(\varphi
F)}{F},\quad\forall\varphi\in B_{b}(\mathbb{R}).$

######  Remark 3.10.

Let us point out that as $h^{\lambda}$ may be positive, it is always possible
that $\ \exp\left(\int_{0}^{t}h^{\lambda}(X_{s})\ ds\right)\geq 1.$ However,
taking any positive constant $C\geq c-\lambda$, we can consider the process
$X^{C}$ defined by

$X^{C}_{t}=\begin{cases}X_{t}\text{ if }\xi>t\\\ \partial\text{ if }\xi\leq
t,\end{cases}$

where $\partial$ is a dummy cemetery state and $\xi$ is a killing time
characterized by

$\mathbb{P}\left(\xi>t\mid\mathcal{F}_{t}\right)=\exp\left(\int_{0}^{t}\left(h^{\lambda}(X_{s})-C\right)\right).$

Thus, it is easily checked that $\
\mathbb{P}\left(\xi>T\right)=e^{-CT}m_{T}(x)\ $ and

$\mathbb{E}_{x}\left[\varphi({X}^{C}_{t})\mathds{1}_{\xi>T}\right]=e^{-CT}\mathbb{E}_{x}\left[\varphi(X_{t})\exp\left(\int_{0}^{t}h^{\lambda}(X_{s})\
ds\right)m_{T-t}(X_{t})\right]=e^{-CT}\widehat{P}_{t}(m_{T-t}\varphi)(x).$

In particular, we have, for $t\leq T$

$\displaystyle\mathbb{E}_{x}\left[\varphi({X}^{C}_{t})\mid\xi>T\right]$
$\displaystyle=\frac{\mathbb{E}\left[\varphi(X_{t})\mathds{1}_{\xi>t}\right]}{\mathbb{P}(\xi>T)}=\cfrac{e^{-CT}\widehat{P}_{t}(m_{T-t}\varphi)(x)}{e^{-CT}m_{T}(x)}=\cfrac{\widehat{P}_{t}(m_{T-t}\varphi)(x)}{m_{T}(x)}.$

Hence, the law of $X^{C}$ conditioned to $\xi>T$ is the law of the spinal
process and is independent of $C$. In that sense, $\mu_{x}^{T}$ can be
interpreted as the law of $X$ under $\mathbb{P}$ killed at rate $h^{\lambda}$
and conditioned not to be killed. In addition, since
$\widehat{P}^{\ast}_{t}F=F$, we have, by Lemma 3.5, that

$\mathbb{E}_{F}\left[\varphi(X^{C}_{T})\mid\xi>T\right]=\cfrac{\langle F,\
\widehat{P}_{T}\varphi\rangle}{\langle F,\ \widehat{P}_{t}1\rangle}=\langle
F,\ \varphi\rangle.$

Thus, $F$ is a quasi-stationary distribution for the killed process.

##  4 Return to the initial population process

Let $T>0$ be the time at which we consider the population state $Z^{K}_{T}$.
We want to characterize the lineage of an individual chosen uniformly in this
population of individuals alive at time $T$. Then we associate to a past time
$s<T$, the trait of the most recent ancestor at this time of a uniformly
sampled individual at time $T$. Formally, if $U^{K}_{T}$ is a uniform random
variable on $V^{K}_{T}$, conditionally on $H^{K}_{T}$, the spinal process
$Y^{K}$ is defined by

$Y^{K}_{t}=X^{U^{K}_{T}}_{t},\quad\forall t\in[0,T],$

using the notation in (3). In particular, the law of $Y^{K}$ can be
characterized with $H^{K}$ using

$\mathbb{E}_{x}\left[\Phi\left(Y^{K}_{t},\
t\in[0,T]\right)\right]=\mathbb{E}_{\delta_{x}}\left[\cfrac{\langle
H^{K}_{T},\Phi\rangle}{\langle H^{K}_{T},1\rangle}\right]$ (54)

with $\Phi:\mathbb{D}([0,T],\mathbb{R})\to\mathbb{R}$ continuous and bounded.

######  Proposition 4.1.

Assume that $(Z^{K}_{0}(dx))_{K}$ converges in probability (and weakly as
measures) to the deterministic finite measure $\,F(x)dx$. Let $T>0$ be given.
Then,

$\lim_{K\rightarrow+\infty}\mathbb{E}_{Z^{K}_{0}}\left[\Phi\big{(}Y^{K}_{s},\
s\leq T\big{)}\right]=\int_{\mathbb{R}}\langle\mu^{T}_{x},\Phi\rangle\
\frac{m_{T}(x)F(x)}{\lambda}\ dx.$ (55)

Thus the typical lineage $Y^{K}$ is asymptotically distributed as $Y$ started
from a biased initial distribution $\lambda^{-1}m_{T}(x)F(x)\ dx$.

Proof 9From (54), we have:

$\displaystyle\lim_{K\rightarrow+\infty}\mathbb{E}_{Z^{K}_{0}}\left[\Phi\big{(}Y^{K}_{s},\
s\leq T\big{)}\right]=$
$\displaystyle\mathbb{E}_{Z^{K}_{0}}\left[\frac{\langle{H}^{K}_{T},\Phi\rangle}{\langle{H}^{K}_{T},1\rangle}\right]=\lim_{K\rightarrow+\infty}\mathbb{E}_{Z^{K}_{0}}\left[\frac{\langle\widetilde{H}^{K}_{T},\Phi\rangle}{\langle\widetilde{H}^{K}_{T},1\rangle}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{\lambda}\int_{\mathbb{R}}\Big{(}m_{T}(x)\langle\mu_{x}^{T},\Phi\rangle\Big{)}\
F(x)\ dx$ $\displaystyle=$
$\displaystyle\big{\langle}\mu^{T}_{\lambda^{-1}m_{T}F},\Phi\big{\rangle}.$
(56)

The second equality can be obtained by using Proposition 2.9 and following the
lines of the proof of Proposition 4.6 in [4]. The third equality is a
consequence of Theorem 2.2 and (43). This gives the announced result.
$\square$

The asymptotic behavior of the time reversal of the spinal process $Y^{K}$
(when $K$ tends to infinity) is obtained using Lemma 3.8. That is summarized
in the following theorem. We refer to Appendix C (67) for a precise definition
of the map $\mathcal{R}$ that returns time.

######  Theorem 4.2.

Under the Assumptions (H) and (9), and if the sequence $(Z^{K}_{0})_{K}$
converges in probability and weakly to the deterministic measure $F(x)dx$,
then we have, for any bounded measurable functions $\Phi$,

$\lim\limits_{K\to\infty}\mathbb{E}_{Z^{K}_{0}}\left[\frac{\langle
H^{K}_{T},\Phi\circ\mathcal{R}\rangle}{\langle
H^{K}_{T},1\rangle}\right]=\mathbb{E}_{F}\left[\Phi\left({Y}^{R}_{s},\,s\in[0,T]\right)\right]$
(57)

where ${Y}^{R}$ is a Markov process with semigroup given by

$P^{R}_{t}\varphi=\cfrac{\widehat{P}_{t}^{\ast}\big{(}F\varphi\big{)}}{F}.$
(58)

######  Remark 4.3.

Theorem 4.2 says that, considering $U_{K}$ to be a random variable whose
conditional distribution with respect to $H^{K}_{T}$ is uniform on
$V^{K}_{T}$, then the process $(X^{U_{K}}_{(T-s)-},\ s\in\leavevmode\nobreak\
[0,T])$ converges in $\mathbb{D}([0,T],\mathbb{R})$ to ${Y}^{R}$ started from
the initial distribution $F$.
For a continuous bounded function $\Phi$, the limit in the left hand side of
(57) equals

$\lim\limits_{K\to\infty}\mathbb{E}_{Z^{K}_{0}}\left[\frac{\langle
H^{K}_{T},\Phi\circ\mathcal{R}\rangle}{\langle
H^{K}_{T},1\rangle}\right]=\frac{1}{\lambda}\mathbb{E}_{F}\left[\langle\Xi_{T},\Phi\circ\mathcal{R}\rangle\right].$

Thus, the theorem tells us that the distribution of a typical ancestral
lineage (backward in time) in the historical measure $\Xi_{T}$ has
distribution $Y^{R}$.

We can conclude with the computation of the generator $L^{R}$ of the process
$Y^{R}$.

######  Proposition 4.4.

The infinitesimal generator $(L^{R}D(L^{R}))$ of the Markov process ${Y}^{R}$
is such that $C^{1}_{b}\subset D(L^{R})$ and for $\varphi\in C^{1}_{b}$, we
have

$L^{R}\varphi(x)=\rho\varphi^{\prime}(x)+\gamma\int_{\mathbb{R}}\big{(}\varphi(y)-\varphi(x)\big{)}\frac{F(y)}{F(x)}m(y,x)\
dy.$ (59)

Proof 9 The infinitesimal generator of the Markov process ${Y}^{R}$ associated
with the semi-group $P^{R}$ is formally given by

$L^{R}\varphi=\cfrac{L^{\ast}(F\varphi)}{F}+h^{\lambda}\varphi.$

Using (19), we obtain for $\varphi\in C^{1}_{b}$,that

$\displaystyle L^{R}\varphi(x)=$
$\displaystyle\rho\varphi^{\prime}(x)+\gamma\int_{\mathbb{R}}\big{(}\frac{F(y)}{F(x)}\varphi(y)-\varphi(x)\big{)}m(y,x)\
dy+\big{(}\rho\frac{F^{\prime}(x)}{F(x)}+h(x)-\lambda\big{)}\varphi(x)$
$\displaystyle=$
$\displaystyle\rho\varphi^{\prime}(x)+\gamma\int_{\mathbb{R}}\big{(}\varphi(y)-\varphi(x)\big{)}\frac{F(y)}{F(x)}m(y,x)\
dy$ $\displaystyle\gamma$
$\displaystyle+\frac{1}{F(x)}\Big{(}\gamma\int_{\mathbb{R}}(F(y)-F(x))m(y,x)\
dx+\rho F^{\prime}(x)+h(x)F(x)-\lambda F(x)\Big{)}\varphi(x).$

Recall the definition of the operator $\mathcal{A}$ in (28). The parenthesis
in the last term of the right hand side equals to $L^{\ast}F+hF-\lambda
F=\mathcal{A}F-\lambda F=0$ since $F$ is an eigenvector of $\mathcal{A}$ for
the eigenvalue $\lambda$. This provides the announced (59). The backward
ancestral lineage of a typical individual follows the drift $\rho$ and has
biased jumps with the jump kernel $F(y)m(y,x)/F(x)$. $\square$

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##  Appendix A Stochastic differential equations for $Z^{K}$, $H^{K}$ and
their couplings

Recall the birth and death rates $b(x)$ and $d(x)+N^{K}_{t}/K$ explained in
the Introduction. Let us consider a Poisson point process $N(ds,di,d\theta)$
on $\mathbb{R}_{+}\times\mathcal{I}\times\mathbb{R}_{+}$ with intensity
measure $ds\otimes n(di)\otimes d\theta$ where $ds$ and $d\theta$ are Lebesgue
measures on $\mathbb{R}_{+}$ and where $n(di)$ is the counting measure on
$\mathcal{I}$. Using the Poisson point processes
$(Q^{i}(ds,dy,d\theta),i\in\mathcal{I})$ on
$\mathbb{R}_{+}\times\mathbb{R}\times\mathbb{R}_{+}$ and with intensity
measures the Lebesgue measures that have been defined in the Introduction (see
(3)), we can write SDEs satisfied by $Z^{K}$ and $H^{K}$ defined in (4) and
(4).

Let us consider a test function $\varphi\in
C^{1}_{b}(\mathbb{R}_{+}\times\mathbb{R},\mathbb{R})$, then:

$\displaystyle\langle
Z^{K}_{t},\varphi(t,.)\rangle=\int_{\mathbb{R}}\varphi(t,x)Z^{K}_{t}(dx)=\frac{1}{K}\sum_{i\in
V^{K}_{t}}\varphi\big{(}t,X^{i}_{t}\big{)}$ $\displaystyle=$
$\displaystyle\langle Z^{K}_{0},\varphi(0,.)\rangle+\int_{0}^{t}\langle
Z^{K}_{s},\partial_{s}\varphi(s,.)-\rho\partial_{x}\varphi(s,.)\rangle\ ds$
$\displaystyle+$
$\displaystyle\int_{0}^{t}\int_{\mathcal{I}}\int_{\mathbb{R}_{+}}{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{i\in
V^{K}_{s_{-}}}\frac{\varphi(s,X^{i}_{s_{-}})}{K}\Big{(}{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\theta\leq b(X^{i}_{s_{-}})}-{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{b(X^{i}_{s_{-}})<\theta\leq
b(X^{i}_{s_{-}})+d(X^{i}_{s_{-}})+\frac{N^{K}_{t}}{K}}\Big{)}N(ds,di,d\theta)$
$\displaystyle+$
$\displaystyle\sum_{i\in\mathcal{I}}\int_{0}^{t}\int_{\mathbb{R}}\int_{\mathbb{R}_{+}}\frac{1}{K}{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{i\in V^{K}_{s_{-}},\theta\leq\gamma
m(X^{i}_{s_{-}},y)}\Big{(}\varphi(s,y)-\varphi(s,X^{i}_{s_{-}})\Big{)}Q^{i}(ds,dy,d\theta).$
(60)

Using standard Itô calculus (see [19] and [15]), we obtain that:

$\displaystyle\langle Z^{K}_{t},\varphi(t,.)\rangle=$ $\displaystyle\langle
Z^{K}_{0},\varphi(0,.)\rangle+\int_{0}^{t}\langle
Z^{K}_{s},\partial_{s}\varphi(s,.)-\rho\partial_{x}\varphi(s,.)+(h-\langle
Z^{K}_{s},1\rangle\varphi(s,.)\rangle\ ds$ $\displaystyle+$
$\displaystyle\int_{0}^{t}\int_{\mathbb{R}}\gamma\int_{\mathbb{R}}\big{(}\varphi(s,y)-\varphi(s,x)\big{)}m(x,y)dy\
Z^{K}_{s}(dx)\ ds+M^{K,\varphi}_{t}$ (61)

where $M^{K,\varphi}$ is a squre integrable martingale with predictable
quadratic variation process:

$\displaystyle\langle M^{K,\varphi}\rangle_{t}=$
$\displaystyle\frac{1}{K}\int_{0}^{t}\int_{\mathbb{R}}\big{(}b(x)+d(x)+\langle
Z^{K}_{s},1\rangle\big{)}\varphi^{2}(s,x)\ Z^{k}_{s}(dx)\ ds$ $\displaystyle+$
$\displaystyle\frac{1}{K}\int_{0}^{t}\int_{\mathbb{R}}\int_{\mathbb{R}}\gamma\big{(}\varphi(s,y)-\varphi(s,x)\big{)}^{2}m(x,y)dy\
Z^{K}_{s}(dx)\ ds.$ (62)

We proceed similarly for $H^{K}$. For finite measures on $\mathbb{D}$, we will
consider test functions $\Phi_{\varphi}$ defined for $\Phi\in
C^{1}_{b}(\mathbb{R},\mathbb{R})$ and $\varphi\in
C^{1}(\mathbb{R}_{+}\times\mathbb{R},\mathbb{R})$ by $\
\Phi_{\varphi}(y)=\Phi\Big{(}\int_{0}^{T}\varphi(t,y_{t})\ dt\Big{)}\ $ for
$y\in\mathbb{D}$. Recall that the paths $y$ in the support of $H^{K}_{t}$ for
$t\in[0,T]$ are constant after time $t$.

For such test function, time $t\in[0,T]$ and path
$y\in\mbox{supp}(H^{K}_{t})\subset\mathbb{D}$, let us define the derivative

$\mathcal{D}\Phi_{\varphi}(t,y)=\Phi^{\prime}\Big{(}\int_{0}^{T}\varphi(s,y_{s})\
ds\Big{)}\int_{t}^{T}\partial_{x}\varphi(s,y_{t})\ ds.$

Also, for $y\in\mathbb{D}$, $s\in[0,T]$ and $x\in\mathbb{R}$, we will denote
by $(y|s|x)$ the càdlàg path defined as

$(y|s|x)(t)=\begin{cases}y(t)&\mbox{ if }t<s,\\\ x&\mbox{ if }t\geq
s.\end{cases}$

For such test function $\Phi_{\varphi}$,

$\displaystyle\langle H^{K}_{t},\Phi_{\varphi}\rangle=\langle
H^{K}_{0},\Phi_{\varphi}\rangle+\int_{0}^{t}\langle
H^{K}_{s},-\rho\mathcal{D}\Phi_{\varphi}(s,.)\rangle\ ds$ $\displaystyle+$
$\displaystyle\int_{0}^{t}\int_{\mathcal{I}}\int_{\mathbb{R}_{+}}{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{i\in
V^{K}_{s_{-}}}\frac{\Phi_{\varphi}(X^{i})}{K}\Big{(}{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\theta\leq b(X^{i}_{s_{-}})}-{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{b(X^{i}_{s_{-}})<\theta\leq
b(X^{i}_{s_{-}})+d(X^{i}_{s_{-}})+\frac{N^{K}_{t}}{K}}\Big{)}N(ds,di,d\theta)$
$\displaystyle+$
$\displaystyle\sum_{i\in\mathcal{I}}\int_{0}^{t}\int_{\mathbb{R}}\int_{\mathbb{R}_{+}}\frac{1}{K}{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{i\in V^{K}_{s_{-}},\theta\leq\gamma
m(X^{i}_{s_{-}},y)}\Big{(}\Phi_{\varphi}(X^{i}|s|y)-\Phi_{\varphi}(X^{i})\Big{)}Q^{i}(ds,dy,d\theta)$
(63) $\displaystyle=$ $\displaystyle\langle
H^{K}_{0},\Phi_{\varphi}\rangle+\int_{0}^{t}\langle
H^{K}_{s},-\rho\mathcal{D}\Phi_{\varphi}(s,.)\rangle\ ds$ $\displaystyle+$
$\displaystyle\int_{0}^{t}\int_{\mathbb{D}}\Big{(}h(y_{s})-\langle
H^{K}_{s},1\rangle\Big{)}\Phi_{\varphi}(y)\ H^{K}_{s}(dy)\ ds$
$\displaystyle+$
$\displaystyle\int_{0}^{t}\int_{\mathbb{D}}\gamma\int_{\mathbb{R}}\Big{(}\Phi_{\varphi}(y|s|x)-\Phi_{\varphi}(y)\Big{)}m(y_{s},x)\
dx\ H^{K}_{s}(dy)\ ds+M^{K,\Phi,\varphi}_{t},$ (64)

where $M^{K,\Phi,\varphi}$ is a square integrable martingale with predictable
quadratic variation process:

$\displaystyle\langle M^{K,\Phi,\varphi}\rangle_{t}=$
$\displaystyle\frac{1}{K}\int_{0}^{t}\int_{\mathbb{D}}\big{(}b(y_{s})+d(y_{s})+\langle
H^{K}_{s},1\rangle\big{)}\Phi_{\varphi}^{2}(x)\ H^{k}_{s}(dy)\ ds$
$\displaystyle+$
$\displaystyle\frac{1}{K}\int_{0}^{t}\int_{\mathbb{D}}\int_{\mathbb{R}}\gamma\big{(}\Phi_{\varphi}(y|s|x)-\Phi_{\varphi}(y)\big{)}^{2}m(y_{s},x)dx\
H^{K}_{s}(dy)\ ds.$

The processes $\widetilde{Z}^{K}$ and $\widetilde{H}^{K}$ are constructed
similarly to (60) and (63) with $N^{K}_{t}/K$ replaced by $\lambda$, and with
the same initial conditions, Poisson point processes and motion processes.
Therefore they are solutions of the following equations.

For $\varphi\in D(L)$ and $\Phi\in C^{1}_{b}(\mathbb{R},\mathbb{R}),$

$\langle\widetilde{Z}^{K}_{t},\varphi\rangle=\langle{Z}^{K}_{0},\varphi\rangle+\int_{0}^{t}\int_{\mathbb{R}}\left\\{h^{\lambda}(x)\,\varphi(x)+L\varphi(x)\right\\}\widetilde{Z}^{K}_{s}(dx)\
ds+\widetilde{M}^{K,\varphi}_{t},$ (65)

where $\widetilde{M}^{K,\varphi}$ is a square integrable martingale, and for
the historical process,

$\langle\widetilde{H}^{K}_{t},\Phi_{\varphi}\rangle=\langle
H^{K}_{0},\Phi_{\varphi}\rangle+\int_{0}^{t}\int_{\mathbb{D}}-\rho\mathcal{D}\Phi_{\varphi}(s,y)\\\
+\Big{(}h^{\lambda}(y_{s})+\gamma\int_{\mathbb{R}}\big{(}\Phi_{\varphi}(y|s|x)-\Phi_{\varphi}(y)\big{)}m(y_{s},x)\
dx\Big{)}\Phi_{\varphi}(y)\ \widetilde{H}^{K}_{s}(dy)\
ds+\widetilde{M}^{K,\Phi,\varphi}_{t}$ (66)

where $\widetilde{M}^{K,\Phi,\varphi}$ is a square integrable martingale.

##  Appendix B Time Reversal of Markov processes

In this section, we consider the problem of reversing time for Markov
processes. Let $T>0$ be fixed and let us consider the linear map
$\mathcal{R}:\mathbb{D}([0,T],\mathbb{R})\to\mathbb{D}([0,T],\mathbb{R})$
defined by

$\mathcal{R}(\varphi)(s)=\begin{cases}\lim\limits_{\varepsilon\to 0,\
\varepsilon>0}\varphi(T-s-\varepsilon)&\text{ if }s\neq T\\\ \varphi(0)&\text{
if }s=T\end{cases}$ (67)

Let us start with the following lemma.

######  Lemma B.1.

The linear map $\mathcal{R}$ is $1$-Lipschitz continuous for the Skorokhod
topology.

Proof 9 Let $\varphi,\Psi\in\mathbb{D}([0,T],\mathbb{R})$, and
$\lambda:[0,T]\to[0,T]$ which is increasing continuous and satisfying
$\lambda(0)=0$ and $\lambda(T)=T$. Then, we have

$\sup_{s\in[0,T]}\left|\mathcal{R}\left(\varphi-\Psi\right)(\lambda(s))\right|\\\
=\max\left\\{\sup_{s\in(0,T]}\left|\lim\limits_{\varepsilon\to 0,\
\varepsilon>0}\left(\varphi(T-\lambda(s)-\varepsilon)-\Psi(T-\lambda(s)-\varepsilon)\right)\right|,\
|\varphi(0)-\psi(0)|\right\\}.$

Now, choosing $s\in(0,T]$ and $\varepsilon>0$, we define

$\theta(s)=\inf\\{t>s\mid\lambda(t)\geq\lambda(s)+\varepsilon\\}$

which is defined for all $s\in\lambda^{-1}([0,T-\varepsilon])$ with values in
$[\theta(0),T]$. Thus, for all $s\in\lambda^{-1}([0,T-\varepsilon])$, we have

$\left|\left(\varphi(T-\lambda(s)-\varepsilon)-\Psi(T-\lambda(s)-\varepsilon)\right)\right|=\left|\left(\varphi(T-\lambda(\theta(s)))-\Psi(T-\lambda(\theta(s)))\right)\right|\\\
\leq\sup_{s\in[0,T]}\left|\left(\varphi(T-\lambda(s))-\Psi(T-\lambda(s))\right)\right|$

leading to

$\sup_{s\in[0,T]}\left|\mathcal{R}\left(\varphi-\Psi\right)(\lambda(s))\right|\leq\sup_{s\in[0,T]}\left|\left(\varphi(T-\lambda(s))-\Psi(T-\lambda(s))\right)\right|.$

The result easily follows. $\square$

The proof of the following proposition is based on [12, Theorem 47].

######  Lemma B.2.

Let $R=(R_{t})_{t}$ be a positive semigroup which is in duality with a
positive semigroup $R^{\ast}$ with respect to the Lebesgue measure. Let
$f:\mathbb{R}_{+}\times\mathbb{R}\to\mathbb{R}_{+}$ such that
$R_{t}f_{t+s}=f_{s}$ for any non-negative real numbers $s$ and $t$. Let
$(V_{t})_{t\in\mathbb{R}_{+}}$ be a Markov process with semigroup given by

$\mathbb{E}(\varphi(V_{t+s}\mid\mathcal{F}_{s})=\cfrac{1}{f_{s}(V_{s})}R_{t}(f_{s+t}\varphi)(V_{s})$

and a given initial distribution $\mu$ such that, for any $t\geq 0$, the law
of $V_{t}$ is absolutely continuous with respect to the Lebesgue measure with
density $F\times f_{t}$. Then, the time reversed process $\mathcal{R}(X)$ at a
time $T$ of $X$ is time-homogeneous and has semigroup given by

$\varphi\to\frac{R_{t}^{\ast}(F\varphi)}{F}.$

Proof 9 Let $(X_{t})_{t\in\mathbb{R}_{+}}$ be a time-inhomogeneous Markov
process with evolution family $(Q_{s,t})_{0\leq s\leq t}$ which characterized
the dynamics of $X$ in the sense that

$\mathbb{E}\left[\varphi(X_{s+t})\mid X_{s}\right]=Q_{s,s+t}\varphi(X_{s}).$

We consider the space-time-ification of $Q$ as a semigroup $\tilde{P}$ defined
for all bounded measurable function
$\varphi:\mathbb{R}_{+}\times\mathbb{R}\to\mathbb{R}_{+}$ by

$\widetilde{P}_{t}\varphi(s,x)=(Q_{s,t+s}\varphi(t+s,\cdot))(x)=\cfrac{1}{f_{s}}P_{t}(f_{t+s}\varphi_{t+s}).$

The semigroup

$\widetilde{P}^{\ast}_{t}\varphi(s,x)=\mathds{1}_{s>t}(s,x)(P^{\ast}_{t}\varphi(s-t,\cdot))(x)$

is in duality with $P$ with respect to the measure $f_{s}(x)\lambda(ds)\ dx$.
Indeed, for any bounded measurable functions $\varphi$ and $g$ on
$\mathbb{R}_{+}\times\mathbb{R}$, we have

$\displaystyle\int_{\mathbb{R}_{+}\times\mathbb{R}}g(x,s)P_{t}(f_{t+s}\varphi_{t+s})(x)\
ds\ dx$
$\displaystyle=\int_{\mathbb{R}_{+}}\int_{\mathbb{R}}f_{t+s}(x)\varphi_{t+s}(x)P_{t}^{\ast}g(s,x)\
dx\ ds$
$\displaystyle=\int_{[t,\infty)}\int_{\mathbb{R}}f_{s}(x)\varphi(s,x)P^{\ast}_{t}g(s-t,x)\
dx\ ds$
$\displaystyle=\int_{\mathbb{R}_{+}}\int_{\mathbb{R}}\varphi(s,x)\widetilde{P}^{\ast}_{t}g(s,x)\
f_{s}(x)dx\ ds.$

The semigroup $\widetilde{P}$ induces a Markov process on
$\mathbb{R}_{+}\times\mathbb{R}$ for which we assume the initial condition to
be $\mu^{\prime}=\delta_{0}\otimes\mu$. The hypothesis of [12, Chapter
XVIII.46] is that the potential measure $\mu^{\prime}U^{\prime}$ defined, for
a measurable $\mathcal{A}\subset\mathbb{R}_{+}\times\mathbb{R}$, by

$\mu^{\prime}U^{\prime}(\mathcal{A})=\int_{\mathbb{R}_{+}\times\mathbb{R}}\int_{0}^{\infty}\widetilde{P}_{t}\mathds{1}_{\mathcal{A}}(e)\
dt\ \mu^{\prime}(de)$

has a density $k:\mathbb{R}_{+}\times\mathbb{R}\to\mathbb{R}_{+}$ with respect
to $f_{s}(x)dsdx$. To see this, let $A$ and $B$ two measurable subset of
respectively $\mathbb{R}_{+}$ and $\mathbb{R}$. First, we have

$\widetilde{P}_{t}\mathds{1}_{A\times
B}(s,x)=\mathds{1}_{A}(s+t)Q_{s,t+s}\mathds{1}_{B}(x).$

Thus,

$\displaystyle\mu^{\prime}U^{\prime}(A\times B)$
$\displaystyle=\int_{\mathbb{R}_{+}\times\mathbb{R}}\int_{0}^{\infty}\mathds{1}_{A}(s+t)Q_{s,t+s}\mathds{1}_{B}(x)\
dt\ \ \delta_{0}(ds)\ \mu(dx)$
$\displaystyle=\int_{\mathbb{R}_{+}\times\mathbb{R}}\int_{s}^{\infty}\mathds{1}_{A}(t)Q_{s,t}\mathds{1}_{B}(x)\
dt\ \ \delta_{0}(ds)\ \mu(dx)$
$\displaystyle=\int_{\mathbb{R}}\int_{0}^{\infty}\mathds{1}_{A}(t)Q_{0,t}\mathds{1}_{B}(x)\
dt\ \mu(dx)=\int_{A\times B}f_{t}(x)F(x)\ dt\ dx.$

Thus, the density of $\mu^{\prime}U^{\prime}$ with respect to $f_{s}(x)\ ds\
dx$ is given by $F$ and is time independent. Then, the time reversed process
$\widetilde{X}^{R}$ of $\widetilde{X}$ is given by

$\widetilde{X}^{R}_{t}=\widetilde{X}_{(T-t)-}=(T-t,X_{(T-t)-})$

and has semigroup given by

$\widetilde{P}^{\prime}_{t}\varphi(s,x)=\cfrac{\widetilde{P}^{\ast}_{t}(F\varphi)(s,x)}{F}.$

Now, let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be a bounded measurable maps and
set $\varphi(s,x)=\phi(x)$. Thus, for $t+s<T$,

$\displaystyle\mathbb{E}\left[\phi(X_{(T-(t+s))-})\bigg{|}X_{(T-s)-}\right]$
$\displaystyle=\mathbb{E}\left[\varphi(\widetilde{X}^{\prime}_{t+s})\bigg{|}\widetilde{X}^{\prime}_{s}\right]=\widetilde{P}^{\prime}_{t}\varphi(\widetilde{X}^{\prime}_{s})=\widetilde{P}^{\prime}_{t}\varphi(T-s,X_{(T-s)-})$
$\displaystyle=\cfrac{P^{\ast}_{t}(F\varphi)(X_{s})}{F(X_{s})}.$

$\square$

##  Appendix C Absolute continuity of the solution of (10)

Let us prove Proposition 2.5.

Recall the idea of the proof. If (27) possesses a solution $f_{t}$ in
$C([0,T],\mathbb{L}^{1})$, then $f_{t}(x)dx$ is solution of (10), and the
identification $\xi_{t}=f_{t}(x)dx$ follows from the uniqueness of the
solution of (10). Thus, we only have to prove that $f_{0}\in\mathbb{L}^{1}$
yields a solution $f_{t}$ in $C([0,T],\mathbb{L}^{1})$. To prove this, we
follow closely the computation in [14].

Consider the semigroup $S_{t}$ acting on $\mathbb{L}^{1}$ with generator
$L^{\ast}+h$. The set

$W=\left\\{F\in C([0,T],\mathbb{R})\mid F\geq 0\text{ and }\sup_{0\leq t\leq
T}|F(t)|\leq\sup_{0\leq t\leq T}\|S_{t}f_{0}\|_{1}\right\\}.$

is convex, closed and bounded. Now, consider the operator $K$ acting on
$C([0,T],\mathbb{R})$ and defined by for all $F\in C([0,T],\mathbb{R})$ by

$KF(t)=\exp\left(-\int_{0}^{t}F(u)\ du\right)\|S_{t}f_{0}\|_{1}.$

Let us prove that $K$ is a compact operator. Take $G\in K(B(0,1))$ so that
$G=KF$ for some $F\in B(0,1)$ where $B(0,1)$ is the open unit ball of
$C([0,T],\mathbb{R})$. So, we have, with $t>s$,

$|G(t)-G(s)|\\\ \begin{aligned} &\leq
e^{t}\|S_{t}f_{0}-S_{s}f_{0}\|_{1}+e^{t}\|S_{s}f_{0}\|_{1}\left|\exp\left(-\int_{0}^{t}(F(u)+1)\
du\right)-\exp\left(-\int_{0}^{s}(F(u)+1)\ du\right)\right|\\\ &\leq
e^{T}\|S_{t}f_{0}-S_{s}f_{0}\|_{1}+\|S_{s}f_{0}\|_{1}e^{T}\left|\exp\left(-\int_{s}^{t}(F(u)+1)\
du\right)-1\right|\\\ &\leq
e^{T}\|S_{t}f_{0}-S_{s}f_{0}\|_{1}+\|S_{s}f_{0}\|_{1}e^{T}\left|\int_{s}^{t}(F(u)+1)\
du\right|\\\ &\leq
e^{T}\|S_{t}f_{0}-S_{s}f_{0}\|_{1}+2\|S_{s}f_{0}\|_{1}e^{T}(t-s).\end{aligned}$

So the family $K(B(0,1))$ is equibounded and equicontinuous, and Arzelà–Ascoli
theorem entails the compactness of $K$. It is easy to check that $K(W)\subset
W$ and thus Leray-Schauder fixed point theorem gives the existence of a fixed
point $F^{\ast}$ for $K$ in $W$.
The function $f$ defined for $x\in\mathbb{R}$ by

$f_{t}(x)=\exp\left(-\int_{0}^{t}F^{\ast}(u)\ du\right)S_{t}f_{0}(x),$

provides the desired solution and ends the proof of Proposition 2.5. $\Box$