# Relation between local density and density relaxation near glass transition
in a glass forming binary mixture

D. C. Thakur  Jalim Singh  Anna Varughese  Prasanth P. Jose
<EMAIL_ADDRESS>School of Basic Sciences, Indian Institute of
Technology Mandi, Kamand, Himachal Pradesh 175005, India

###### Abstract

We employ molecular dynamics simulation to elucidate the microscopic origin of
slow relaxation in supercooled Lennard-Jones liquids. These studies show a
direct correlation between the growth of the highest peak of the radial
distribution function $g(r)$, radial force from this peak, and density
relaxation. From the available free-volume of a neighboring pairs for string-
like motion, we derive a relation between surface density of the spherical
shell $\rho_{loc}$ around a reference particle at the highest peak of $g(r)$
and density relaxation $\tau_{\alpha}$, i.e.
$\tau_{\alpha}\propto\exp{(\rho_{0}/(\rho_{0}-\rho_{loc}))}$, where dynamics
diverge at $\rho_{0}$ that is akin to Vogel–Fulcher–Tammann (VFT) relation
connecting $\tau_{\alpha}$ and temperature. This relation works well in the
supercooled regime of the model systems simulated.

Fluids undergo glass transition under rapid compression or cooling at a faster
rate than structural relaxation. Formation of transient molecular-cages during
glass transition results in the slowing down of density relaxation. Recent
experiments in two-dimensional colloids show that the molecular-cages have
higher local density Li  _et al._ (2020), where cage size fluctuation
facilitates the density relaxation Pastore  _et al._ (2017). Therefore,
theories of dense or viscous liquids are useful to study the glass transition.
The perturbation theory of dense Lennard-Jones (LJ) liquids by Weeks,
Chandler, and Andersen (WCA) shows that repulsive interactions govern the
structure of dense liquids Weeks  _et al._ (1971), besides, the dynamics also
follows the same Chandler (1974). A test of WCA theory in simulations of Kob-
Andersen (KA) binary mixture Kob and Andersen (1994) with and without
attractive (KAWCA) interactions near the glass transition Berthier and Tarjus
(2009, 2011) show that the relaxation dynamics is remarkably slower with
attractive interactions at number density $\rho=$1.2, often studied for glass-
transition in the KA model Kob and Andersen (1995a, b); the dynamics of KA and
KAWCA models are nearly identical at higher density $\rho=$1.8. Studies also
show that the relaxation dynamics becomes identical when the range of
interaction include the whole first coordination shell Toxvaerd and Dyre
(2011); besides, the attractive and repulsive interactions together form a
mean fluctuating potential of the cage, irrespective of the type; identical
cage potential leads to same density relaxation Pedersen  _et al._ (2010).
Therefore, characterization of type, range, and depth of interactions in
viscous liquids are important in understanding of molecular-cages. There are
many later studies with different approaches that address the origin of this
difference in dynamics of supercooled KA and KAWCA models.

Such a study based on an explicit analysis of forces shows that the difference
between mean forces on particles in KA and KAWCA models are different when
dynamics differ Bohling  _et al._ (2013); Chattoraj and Ciamarra (2020).
Another study shows that the configurational entropy obtained from the radial
distribution function $g(r)$ Hansen and McDonald (1986) leads to difference in
dynamics, thus $g(r)$ governs relaxation dynamics Banerjee  _et al._ (2014).
Besides, machine learning studies show that $g(r)$ help in interpreting the
dynamics Landes  _et al._ (2020). Interestingly, an exponential relation
between the relaxation dynamics of the system and order parameter derived from
the relative angle between atoms in the first coordination shell, which is
similar to Vogel–Fulcher–Tammann relation Tong and Tanaka (2020). Insights
from studies that show the importance of first coordination shell Toxvaerd and
Dyre (2011); Tong and Tanaka (2020) and importance of $g(r)$ Banerjee  _et
al._ (2014); Landes  _et al._ (2020) suggest that $g(r)$ at the first
coordination shell has a decisive role in determining the relaxation dynamics.
Moreover, the schematic mode-coupling theory use the value of the structure
factor at the first peak (corresponding to the length scale of nearest
neighbor separation) Bengtzelius  _et al._ (1984); Berthier and Biroli (2011)
in calculations of relaxation time. Besides, studies show that short-time
$\beta$ relaxation and $\alpha$ relaxation have identical temperature
dependence Karmakar  _et al._ (2016) connecting short time dynamics arising
from local structure to the overall relaxation dynamics, which is also
identified by the information theory Jack  _et al._ (2014). Another study on
gas-supercooled coexistence in polymers show that there is a correlation
between density relaxation and particle distribution in the first coordination
shell Singh and Jose (2021). The transient cage formation leads to increase in
local density Li  _et al._ (2020) that reduces available free-volume for the
molecules to relax, therefore the relation between structure and dynamics can
be addressed from the free-volume theory Grest and Cohen (1981); for a hard-
sphere system the density relaxation $\tau_{\alpha}\propto\exp(Bv_{0}/v_{f})$,
where $v_{f}=v-v_{0}$ is the difference between mean free-space and self-
volume Turnbull and Cohen (1958). The free-volume have a vague definition for
continuous potentials with different representations such as Voronoi volume
Starr  _et al._ (2002) or a detailed classification of vibrationally
accessible and hardcore free volume White and Lipson (2016) etc. This study
deduces a relation connecting the surface density $\rho_{loc}$ of a spherical
shell with radius $r^{*}$ at the peak of $g(r)$ (cage density that represent
$v_{f}$) and relaxation time, from a systematic analysis of variation in
$g(r)$ with temperature.

Methods: Systems with KA Kob and Andersen (1994) and KAWCA Berthier and Tarjus
(2011) potentials at number densities $\rho=$1.2 to 1.8 in grid of
$\delta\rho=$0.2 are simulated in uneven temperature grid from high to low
temperatures. The KA potential reads:
${V_{\alpha\beta}(r)}=4\epsilon_{\alpha\beta}{\left[\left({\sigma_{\alpha\beta}/{r}}\right)^{12}-{\left({\sigma_{\alpha\beta}/r}\right)^{6}}\right]+K_{\alpha\beta}}$,
where parameters are
$\alpha,\beta\in\\{A,B\\},\epsilon_{AA}=1.0,\sigma_{AA}=1.0,\epsilon_{AB}=1.5,\sigma_{AB}=0.8,\epsilon_{BB}=0.5,$
and $\sigma_{BB}=0.88$ Kob and Andersen (1994) . $K_{\alpha\beta}$ ensures
zero potential at the cut-off. Length, temperature and time are expressed in
units of $\sigma_{AA}$, $\sigma_{AA}\sqrt{{m}/{\epsilon_{AA}}}$, and
$\epsilon_{AA}/{k_{B}}$. The KA model has potential cut-off of
$2.5\sigma_{\alpha\beta}$ whereas KAWCA model is cut-off at minima
$2^{1/6}\sigma_{\alpha\beta}$. The simulated system has $N$ = 4000 particles
with $A$ to $B$ ratio 4:1. Molecular dynamics using velocity Verlet algorithms
simulate systems in the microcanonical ensemble Allen and Tildesley (1991).
Initial configuration is prepared at $T=8$ for densities $\rho=$ 1.2, 1.4, and
1.6, and $T=$ 12 for density $\rho=$ 1.8 and quenched to the desired
temperature. The system is equilibrated till 20 $\tau_{\alpha}$ or more at all
temperature.

Figure 1:  Properties of two state points are compared in upper and lower
panels: (a) and (e) shows over all (OA) and partial radial distribution
functions $g_{\alpha\beta}(r)$ ; (b) and (f) corresponding variation in the
radial force $F_{\alpha\beta}(r)$; (c) and (g)
$\langle\Delta\mathbf{r}^{2}(t)\rangle$; (d) and (f) $F_{s}(k,t)$ in typical
supercooled states. Legends are the same in all panels.

The decrease of temperature results in the growth of $g(r)$ in the first
coordination shell; thus, shows an increase of local density. Various
contribution to peaks of $g(r)$ from partial $g_{\alpha\beta}$ Kob and
Andersen (1995a) in Figs. 1(a) and (e) at $(\rho,T)=$(1.2,0.45) and (1.8,3)
(two representative supercooled states) show that the first peak of $g(r)$ has
most contribution from $g_{AB}$ (peak of $g_{BB}(r)$ is weak), which is
smaller than the second peak that is mostly from $g_{AA}(r)$. The first peak
of $g_{AB}(r)$ is the tallest among $g_{\alpha\beta}(r)$ due to larger
$\epsilon_{AB}$; however, due to lesser $AB$ pairs contribute to smaller first
smaller peak in $g(r)$. The tallest peak of $g(r)$ is from $g_{AA}(r)$, which
is the correlation among major component $A$. At these two representative
state points, at $\rho=$1.2, the $g(r)$ of KA and KAWCA models differ, while
at $\rho=$1.8, they match; the relaxation dynamics follows the same trend -
see discussion on relaxation dynamics later. To confine a reference particle,
the cage formed by neighboring particles exert a fluctuating force, the mean
of this force at $r$ give a measure of the confinement - the net force on the
molecule in the radial direction is zero in equilibrium. The mean force
exerted on a particle by density at $r$ in radial direction is
$F_{\alpha\beta}(r)\hat{r}=\mathbf{f}_{\alpha\beta}(r)n_{\alpha\beta}(r)$,
where the force $\mathbf{f}_{\alpha\beta}(r)=-\frac{\partial
V_{\alpha\beta}(r)}{\partial r}\hat{r}$, $V_{\alpha\beta}(r)$ is any of the
three types of different potentials given above, $n_{\alpha\beta}$ is
unnormalized pair density of the each species. The mean normalized force
$F(r)=\frac{1}{N\rho}\sum_{\alpha\beta}|F_{\alpha\beta}(r)\hat{r}|$ gives
average radial force on a reference particle; correlation of normalized
$F_{\alpha\beta}(r)\hat{r}$ is used in mode-coupling theory studies Dell and
Schweizer (2015) to compute the memory kernel. The $F(r)$ at the position of
the highest peak in Fig. 1 at $\rho=$1.2 considerably differ for KA and KAWCA
mainly due $AB$ component, which have large negative contribution that reduces
at $\rho=$1.8 that also correlates with variation in the relaxation dynamics
Thakur  _et al._ (2020). The difference in mean force on particles in these
models show difference in relaxation in earlier studies Bohling  _et al._
(2013); Chattoraj and Ciamarra (2020). Next, we explore a relation between
features of density relaxation and static channels of relaxation in the cages
of KA and KAWCA models.

Owing to smaller $\sigma_{AB}$ and $\sigma_{BB}$ [see Figs. 2(a) and (b)], the
mean square displacements
$\langle\Delta\mathbf{r}^{2}(t)\rangle=\langle|\mathbf{r}_{i}(0)-\mathbf{r}_{i}(t)|^{2}\rangle$
($\mathbf{r}_{i}(t)$ \- position vector of $i$th particle at time $t$) of $A$
is slower than $B$ in Figs. 1(c) and (g). Relaxation of any particle is due to
motion of particles hindered by the cage, which implies higher energy barrier
from $A$ than $B$ in a cage; therefore, $B$ is a channel of relaxation in a
cage at all densities. The overall density relaxation is governed by $A$ due
to relatively slow motion. In Figs. 1 (c) and (g),
$\langle\Delta\mathbf{r}^{2}(t)\rangle$ of $A$ and $B$ for KA and KAWCA models
differ at lower density $\rho=$1.2, while they nearly matches at higher
density $\rho=$1.8, where $F(r)$ and $g(r)$ also coincide for KA and KAWCA
models. Therefore, at $\rho=$1.2 $B$ particles that act as a channel of
relaxation in a cage is different in the KA and KAWCA models due to attractive
interactions, while they are nearly the same for $\rho=$1.8 at a very high
density. Besides, effect of attraction varies with location of $r^{*}$, with
respect to $r^{*}_{\alpha\beta}=2^{1/6}\sigma_{\alpha\beta}$, which is the
cut-off distance of attractive forces in KA model; they are:
$r^{*}_{AA}\sim$1.1, $r^{*}_{AB}\sim$0.9, and $r^{*}_{BB}\sim$0.99. In Fig.
2(c), in $\rho=$1.2 at low $T$, $r^{*}\sim 1.06>r^{*}_{AB},r^{*}_{BB}$, where
forces in KA and KAWCA models differ due to attractive forces in Fig. 1(b),
while for $\rho=$1.8, $r^{*}_{BB}>r^{*}\sim 0.92\sim r^{*}_{AB}$ both models
have identical $F(r)$ in Fig. 1(f) due to reduction of the attractive forces.
The relaxation of incoherent intermediate scattering function Hansen and
McDonald (1986)
$F^{\alpha}_{s}(k,t)=\langle\rho^{\alpha}_{\mathbf{k}}(t)\rho^{\alpha}_{-\mathbf{k}}(0)\rangle$,
where
$\rho^{\alpha}_{\mathbf{k}}(t)=e^{i\mathbf{k}\cdot\mathbf{r}^{\alpha}_{i}(t)}$
at $k=$7.25 Berthier and Tarjus (2009), where $k$ is the mean wavenumber at
the first peak of $S(k)$, show similar variation of in the relaxation time.
Thus, the microscopic structure of the cage in the first coordination shell is
related to the density relaxation. Therefore, we look for a relation between
average relaxation time $\tau_{\alpha}$ (
$\tau_{\alpha}=\int_{0}^{\infty}F_{s}(k,t)dt$ with $F_{s}(k,t)$ is averaged
over $F^{\alpha}_{s}(k,t)$) and peak of $g(r)$ from the microscopic mechanism
of relaxation.

Figure 2: (a) Typical 3d of distribution of KA/KAWCA particles of a cage at
$\rho=$ 1.2 ($r_{A}\propto\sigma_{AA}$ and
$r_{B}\propto(\sigma_{AB}+\sigma_{BB})/2$ ). (b)The schematic representation
of (a) with A (large) and B (small) in 2d. In (b), the green circle shows the
reference particle; the black double arrow shows the preferred relaxation
path; the red arrow shows the unfavoured relaxation path; the green arrows
show the string-like collective motion; dotted circles are at a typical peak
of $g(r)$. (c) Variation of $r^{*}$ (radius of dashed circle in (b)) with $T$.
Figure 3:  Collapsed plot of logarithm of scaled relaxation time versus
relative variation of the local density $\rho_{loc}$ for KA (black) and
KAWCA(red) models.  Figure 4:  $\tau_{\alpha}$ in semilog (left $Y$ scale)
versus $1/T$ , by circles connected by VFT fit and $\rho_{loc}$ in linear
scale(right $Y$ scale) versus $1/T$ as (squares), lines are Eq. 2, for KA
(black) and KAWCA (red) models

Typical arrangement of atoms in Figs. 2(a) and (b) at the first coordination
shell near potential energy minima result in a dense spherical shell Li  _et
al._ (2020). The inspiration of microscopic mechanism of rearrangement in such
a system is from the string-like motion Kob  _et al._ (1997); Donati  _et al._
(1999); Aichele  _et al._ (2003); Yip  _et al._ (2020), where atoms move among
nearly identical environment with a slight expansion of the shell Pastore  _et
al._ (2017) because the energy cost to create a hole is high, the moves that
contribute predominately to the relaxation is along the black arrows in Fig.
2(b). Various energy barriers associated with a typical string-like motion
include the energy of a hole, entropy of the statistical distribution of
strings etc. Langer and Lemaitre (2005); Langer (2006). In this model, a short
move marked in a black arrow in Fig. 2(b), where the displacement of a member
of a cage along the line joining cage particle and the reference particle is
the primary mode of relaxation. Such paths connecting many particles lead to a
typical string-like collective motion (green arrows in Fig. 2(b)). The free-
energy barrier $E$ from the mean-field theory for polymers Flory (1953);
Langer and Lemaitre (2005); Langer (2006) along the string of length $n$ in
available free-volume $v_{m}$ is $E\propto n^{2}/v_{m}$; here we consider only
pairs. Such connected pairs in three dimensions generate a string-like or
ring-like collective motion along the high-density path on a two-dimensional
(2D) surface marked by the $r^{*}$ that is typical in glasses Swayamjyoti  _et
al._ (2014). All moves are along the quasi-linear path of inter-penetrating 2D
surfaces marked as dotted circles in the one-dimensional (1D) projection in
Fig. 2(b). A study on the glass transition in polymer films give details of
the barriers in typical string-like motion Salez  _et al._ (2015). Based on
the mean-field energy barrier of a pair and free-volume theory suggest that
free-volume available in the shell of radius $r^{*}$ controls the dynamics, as
every such path is through the shell around some other particle. Short strings
at higher temperature supercooled liquid become longer as temperature reduces
Stevenson and Wolynes (2010), which is considered as cooperatively rearranging
regionsStevenson  _et al._ (2006); Zhang  _et al._ (2011); Nagamanasa  _et
al._ (2015) in theories of glass transition Berthier and Biroli (2011).

The free-volume change in the spherical shell has two contributions $\delta
V=\delta\rho N+\delta N\rho$ in terms of density $\rho$ and number of
particles $N$. The reduction of free volume $\delta V=v-v_{0}$ in a spherical
shell as the system approach a transition is due to the increase of particles.
That is approximated as $(\rho-\rho_{loc})N_{m}$, where $N_{m}$ is the mean
shell occupation in the supercooled state. $\delta N$ is variation of number
of particles in a spherical-shell of infinitesimal thickness with
$r^{*}\simeq$constant (fig. 2(c)) that is negligible. Then the relaxation time
from free-volume theory modified as function of $\rho_{loc}$ reads

$\tau_{\alpha}=\tau_{\rho}e^{\left(\frac{Bv_{0}}{v-v_{0}}\right)}\simeq\tau_{\rho}e^{\left(\frac{B\rho_{0}}{\rho_{0}-\rho_{loc}}\right)}.$
(1)

A fit of this relation in KA and KAWCA models for $\rho=$ 1.2, 1.4, 1.6, and
1.8 in Fig. 3 show $B\sim$ 1, which is consistent with experiments Doolittle
(1951). The $\rho_{0}$ for respective densities in the order above are
3.9,4.5,5.2, and 5.9 for KA and 4.6,4.8,5.3 and 5.9 show a systematic increase
of $\rho_{0}$ and they both approach each other at high density. The Eq. 1 is
verified in glass forming linear flexible polymers near supercooled state
Singh  _et al._ (2020). As the Eq. 1 is similar to Vogel–Fulcher–Tammann (VFT)
relation $\tau_{\alpha}=\tau_{T}\exp(A/(T-T_{0}))$. Equating the Eq. 1 and VFT
expression yields relation between $\rho_{loc}$ and temperature difference

$\rho_{loc}=\rho_{0}-\frac{\rho_{0}(T-T_{0})}{A+\ln{({\tau_{T}}/{\tau_{\rho}})}(T-T_{0})}.$
(2)

The variation of $\rho_{loc}$ and inverse of temperature in Fig. 4 show linear
variation with concomitant logarithmic variation of $\tau_{\alpha}$. In the
supercooled liquids Eq. 2 is valid; it is clear from the KAWCA model at
$\rho=$1.2 where cages become stable at only low temperatures, where the plot
of Eq. 2 in Fig. 4(a) matches with variation of $\rho_{loc}$.

This study based on the simulation of KA and KAWCA models at different
densities in temperature grids from high to low temperature near supercooled
state look into cage formation with and without attractive interactions. The
analysis shows that the variation in the local density and associated forces
are related to the relaxation dynamics, which is consistent with earlier
investigations Toxvaerd and Dyre (2011); Pedersen  _et al._ (2010); Tong and
Tanaka (2020); Singh and Jose (2021). This model based on a short string like
motion shows the connection between the density in the first-coordination
shell and the relaxation dynamics. The VFT-like structure of the Eq. 1
consistent with a similar relation for an order parameter derived among angle
between particles in the first coordination shell Tong and Tanaka (2020) which
is valid in 2D as well; therefore, the results of this study provide a model
for 2D and 3D nonassociated liquids Chandler (1987). The enhancement of the
first peak also found in repulsive 3D colloidal systems that are reminiscence
of molecular jamming similar to the repulsive colloids, where $g(r)$ grows
systematically near jamming Zhang  _et al._ (2009). Force transmission in
attractive bio-polymers also shows jamming Jose and Andricioaei (2012), it is
worthwhile to have more detailed investigations on this aspect. The effect of
attractive force is the modification of caging potential; thus, $\rho_{loc}$
are different in KA and KAWCA models with attractive forces. Therefore, the
conclusion of scaling of relaxation time for $T_{g}$ Chattoraj and Ciamarra
(2020) makes the relaxation time-invariant is consistent with these studies.
There are many studies that use an alternative representation of the free-
volume as a function of Debye-Waller factor $\langle u\rangle^{2}$ Hall and
Wolynes (1987); Starr  _et al._ (2002); Simmons  _et al._ (2012); Larini  _et
al._ (2008), which shows an exponential variation of relaxation which is
comparable to Eq.1 Singh  _et al._ (2020); thus suggest a relation between
$\rho_{loc}$ and $\langle u\rangle^{2}$, which warrant further investigations.

###### Acknowledgements.

We thank HPC facility at IIT Mandi for computational support.

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#  On SDE associated with continuous-state branching processes conditioned to
never be extinct

M.C. Fittipaldi DIM–CMM, UMI 2807 UChile-CNRS, Universidad de Chile, Casilla
170-3, Correo 3, Santiago, Chile ; Supported by Basal-CONICTY;
<EMAIL_ADDRESS>J. Fontbona DIM–CMM, UMI 2807 UChile-CNRS,
Universidad de Chile, Casilla 170-3, Correo 3, Santiago, Chile ; Partially
supported by Basal-CONICTY<EMAIL_ADDRESS>
###### Abstract

We study the pathwise description of a (sub-)critical continuous-state
branching process (CSBP) conditioned to be never extinct, as the solution to a
stochastic differential equation driven by Brownian motion and Poisson point
measures. The interest of our approach, which relies on applying Girsanov
theorem on the SDE that describes the unconditioned CSBP, is that it points
out an explicit mechanism to build the immigration term appearing in the
conditioned process, by randomly selecting jumps of the original one. These
techniques should also be useful to represent more general $h$-transforms of
diffusion-jump processes.

Key words Stochastic Differential Equations; Continuous-state branching
processes;

Non-extinction; Immigration.

AMS 2010 Subject Classification 60J80, 60H20,60H10.

##  1 Introduction and preliminaries

Stochastic differential equations (SDE) representing continuous-state
branching processes (CSBP) or CSBP with immigration (CBI) have attracted
increasing attention in the last years, as powerful tools for studying
pathwise and distributional properties of these processes as well as some
scaling limits, see e.g. Dawson and Li [5], [6] , Lambert [17], Fu and Li [10]
and Caballero et al. [4].

In this note, we are interested in SDE representations for (sub)-critical CSBP
conditioned to never be extinct. It is well known that such conditioned CSBP
correspond to CBIs with particular immigration mechanisms (see [25]). Thus, it
is possible to obtain SDE representations for them by using general results
and techniques developed in some of the aforementioned works, see [5] and
[10]. However, our goal is to directly obtain such representation by rather
using the fact that the law of the conditioned CSBP is obtained from the one
of the non conditioned process, by means of an explicit $h-$transform. In
accordance with that relation between the laws and to the “spine” or immortal
particle picture of the conditioned process ([25], [9]), one should expect to
identify, after measure change, copies of the original driving random
processes and an independent subordinator accounting for immigration. Our
proof will show how to obtain these processes by using Girsanov theorem and an
enlargement of the probability space in order to select by a suitable marking
procedure those jumps of the original (non conditioned) process that will
constitute (or will not) the immigrants. The enlargement of the probability
space and the marking procedure are both inspired in a construction of Lambert
[17] on stable Lévy processes. They are also reminiscent of the sized biased
tree representation of measure changes for Galton-Watson trees (Lyons et. al
[22]) or for branching Brownian motions (see e.g. Kyprianou [15] and Englänger
and Kyprianou [8]), but we do not aim at fully developing those ideas in the
present framework. In a related direction, in a recently posted article [12]
Hénard obtains the same SDE description of the conditioned CSBP, using the
look-down particle representation of CSBP of Donnelly and Kurtz [7].

We start by recalling some definitions and classic results about CSBPs and
Lévy processes along the lines of [16, Chap. 1,2 and 10], in particular the
relationship between them through the Lamperti transform. (We also refer the
reader to Le Gall [20] and Li [21] for further background on CSBP).

###  1.1 Continuous-state branching processes

Continuous-state branching processes (CSBP) were introduced by Jirina [13] in
1958. Later, Lamperti [19] showed that they can be obtained as scaling limits
of a sequence of Galton-Watson processes. A CSBP with probability laws given
the initial state $\\{\mathbb{P}_{x}:x\geq 0\\}$ is a càdlàg
$[0,\infty)$-valued strong Markov processes $Z=\\{Z_{t}:t\geq 0\\}$ satisfying
the branching property. That is, for any $t\geq 0$ and
$z_{1},z_{2}\in[0,\infty)$, $Z_{t}$ under $\mathbb{P}_{z_{1}+z_{2}}$ has the
same law as the independent sum $Z_{t}^{(1)}+Z_{t}^{(2)}$, where the
distribution of $Z_{t}^{(i)}$ is equal to that of $Z_{t}$ under
$\mathbb{P}_{z_{i}}$ for $i=1,2$. Usually, $Z_{t}$ represents the population
at time $t$ descending from an initial population $x$. The law of $Z$ is
completely characterized by its Laplace transform

$\mathbb{E}_{x}(e^{-\theta
Z_{t}})\,=\,e^{-xu_{t}(\theta)},\,\,\forall\,x>0,\,t\geq 0,$

where $u$ is a differentiable function in $t$ satisfying

$\left\\{\begin{array}[]{ll}\displaystyle{\frac{\partial u_{t}}{\partial
t}(\theta)+\psi(u_{t}(\theta))}=0\\\ \\\
u_{0}(\theta)=\theta,\end{array}\right.$ (1)

and $\psi$ is called the branching mechanism of $Z$, which has the form

$\psi(\lambda)=-q-a\lambda+\frac{1}{2}\sigma^{2}\lambda^{2}+\int_{(0,\infty)}(e^{-\lambda
x}-1+\lambda x\textbf{1}_{(x<1)})\Pi(dx)\quad\lambda\geq 0,$ (2)

for some $q\geq 0,\,a\in\mathbb{R},\,\sigma\geq 0$ and $\Pi$ a measure
supported in $(0,\infty)$ such that
$\int_{(0,\infty)}(1\wedge x^{2})\Pi(dx)\,<\,\infty.$ In particular, $\psi$ is
the characteristic exponent of a spectrally positive Lévy process, i.e. one
with no negative jumps. Since clearly,
$\mathbb{E}_{x}(Z_{t})=xe^{-\psi^{\prime}(0+)t}$, defining
$\rho:=\psi^{\prime}(0+)$ one has the following classification of CSBPs :

  * (i)

subcritical, if $\rho>0$,

  * (ii)

critical, if $\rho=0$ and

  * (iii)

supercritical, if $\rho<0$,

according to whether the process will, on average, decrease, remain constant
or increase.

In the following, we will assume that $Z$ is conservative, i.e. $\forall$
$t>0$, $\mathbb{P}_{x}(Z_{t}<\infty)=1$. By Grey (1974), this is true if and
only if $\int_{0^{+}}\frac{d\xi}{|\psi(\xi)|}=\infty$, so it is sufficient to
asume $\psi(0)=0$ and $|\psi^{\prime}(0+)|<\infty$.

###  1.2 Lévy Processes and their connection with CSBP

Let $X=\\{X_{t}:t\geq 0\\}$ be a spectrally positive Lévy process with
characteristic exponent $\psi$ given by (2) with $q=0$, and initial state
$x\geq 0$ . By the Lévy-Ito decomposition it is well known that it can be
written as the following sum of independent processes

$X_{t}=x+at+\sigma
B_{t}^{X}+\int_{0}^{t}\int_{1}^{\infty}rN^{X}(ds,dr)+\int_{0}^{t}\int_{0}^{1}r\tilde{N}^{X}(ds,dr),$

where $a$ is a real number, $\sigma\geq 0$, $B^{X}$ is a Brownian motion,
$N^{X}$ is an independent Poisson measure on $[0,\infty)\times(0,\infty)$ with
intensity measure $dt\times\Pi(dr)$ and
$\tilde{N}^{X}(dt,dr):=N^{X}(dt,dr)-dt\Pi(dr)$ denotes the compensated measure
associated to $N^{X}$ (the last integral thus being a square integrable
martingale of compensated jumps of magnitude less than unity).

Lamperti [18] established a one-to-one correspondence between CSBPs and
spectrally positive Lévy processes via a random time change. More precisely,
for a Lévy process $X$ as above the process

$Z:=\\{Z_{t}=X_{\theta_{t}\wedge T_{0}}:t\geq 0\\},$

where $T_{0}=\inf\\{t>0:X_{t}=0\\}$ and
$\theta_{t}=\inf\left\\{s>0:\int_{0}^{s}\frac{du}{X_{u}}>t\right\\}$, is a
continuous-state branching process with branching mechanism $\psi$ and initial
value $Z_{0}=x$. Conversely, given $Z=\\{Z_{t}:t\geq 0\\}$ a CSBP with
branching mechanism $\psi$, such that $Z_{0}=x>0$, we have that

$X:=\\{X_{t}=Z_{\varphi_{t}\wedge T}:t\geq 0\\},$

where $T=\inf\\{t>0:Z_{t}=0\\}$ and
$\varphi_{t}=\inf\left\\{s>0:\int_{0}^{s}Z_{u}du>t\right\\}$, is a Lévy
process with no negative jumps, stopped at $T_{0}$ and satisfying
$\psi(\lambda)=\log\mathrm{I\\!E}(e^{-\lambda X_{1}})$, with initial position
$X_{0}=x$.

Relying on this relationship, Caballero et al. [4, Prop 4] provide a pathwise
description of the dynamics of a CSBP: for $(Z_{t},t\geq 0)$ there exist a
standard Brownian motion $B^{Z}$, and an independent Poisson measure $N^{Z}$
on $[0,\infty)\times(0,\infty)\times(0,\infty)$ with intensity measure
$dt\times d\nu\times\Pi(dr)$ in an enlarged probability space such that

$\begin{split}Z_{t}=&x+a\int_{0}^{t}Z_{s}ds+\sigma\int_{0}^{t}\sqrt{Z_{s}}dB_{s}^{Z}+\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{1}^{\infty}rN^{Z}(ds,d\nu,dr)\\\
&+\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{0}^{1}r\tilde{N}^{Z}(ds,d\nu,dr),\end{split}$
(3)

where $\tilde{N}^{Z}$ is the compensated Poisson measure associated with
$N^{Z}$. Pathwise properties of stochastic differential equations driven by
Brownian motion and Poisson point processes have been studied in more general
settings in [5], [10] and [6]. In particular, strong existence and pathwise
uniqueness for (3) is established [10]. Related SDE have also been considered
in Bertoin and Le Gall [2], [3].

##  2 CSBPs conditioned to be never extinct as solutions of SDEs

###  2.1 CSBP conditioned to be never extinct

We assume from now on that $Z$ is a (sub-)critical CSBP such that
$\psi(\infty)=\infty$ and $\int^{\infty}\frac{d\xi}{\psi(\xi)}<\infty$. Under
these and the previous conditions, the process does not explode and there is
almost surely extinction in finite time. Branching processes conditioned to
stay positive were first studied in the continuous-state framework by Roelly
and Rouault [25], who proved that for $Z$ as before,

$\mathbb{P}_{x}^{\uparrow}(A):=\lim_{s\uparrow\infty}\mathbb{P}_{x}(A|T>t+s),\quad
A\in\sigma(Z_{s}:s\leq t)$ (4)

is a well defined probability measure which satisfies

$\mathbb{P}_{x}^{\uparrow}(A)=\mathbb{E}(\textbf{1}_{A}e^{\rho
t}\frac{Z_{t}}{x}).$

In particular, $\mathbb{P}_{x}^{\uparrow}(T<\infty)=0$, and $\\{e^{\rho
t}Z_{t}:t\geq 0\\}$ is a martingale under $\mathbb{P}_{x}$. Note that
$\mathbb{P}_{x}^{\uparrow}$ is the law of the so-called $Q$-process (for in-
depth looks at this type of processes, we refer the reader to [17], [23] and
references therein). They also proved that $(Z,\mathbb{P}^{\uparrow})$ has the
same law as a CBI with branching mechanism $\psi$ and immigration mechanism
$\phi(\theta)=\psi^{\prime}(\theta)-\rho,$ $\theta\geq 0.$ This means that
$(Z,\mathbb{P}^{\uparrow})$ is a càdlàg $[0,\infty)$-valued process, and for
all $x,t>0$ and $\theta\geq 0$

$\mathbb{E}_{x}^{\uparrow}(e^{-\theta
Z_{t}})=\exp\\{-xu_{t}(\theta)-\int_{0}^{t}\phi(u_{t-s}(\theta))ds\\},$

where $u_{t}(\theta)$ is the unique solution to (1). Note also that $\phi$ is
the Laplace exponent of a subordinator.

###  2.2 Main Result

The above result is the key for the study of CSBP conditioned on non-
extinction, but we seek a more explicit description for the paths of $Z$ under
$\mathbb{P}^{\uparrow}$. To this end, we shall prove that
$(Z,\mathbb{P}^{\uparrow})$ has a SDE representation, which agrees with the
interpretation of a CSBP conditioned on non-extinction as a CBI, but also
gives us a pathwise description for the conditioned process. In particular,
this result extends Lambert’s results for the stable case [17, Theorem 5.2]
(see below for details) as well as equation (3).

###### Theorem 2.1

Under $\mathbb{P}^{\uparrow}$, the process $Z$ is the unique strong solution
of the following stochastic differential equation:

$\begin{array}[]{lcl}Z_{t}&=&x+a\int_{0}^{t}Z_{s}ds+\sigma\int_{0}^{t}\sqrt{Z_{s}}dB^{\uparrow}_{s}+\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{1}^{\infty}rN^{\uparrow}(ds,d\nu,dr)+\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{0}^{1}r\tilde{N}^{\uparrow}(ds,d\nu,dr)\\\
\\\ &&+\int_{0}^{t}\int_{0}^{\infty}rN^{\star}(ds,dr)+\sigma^{2}t\end{array}$
(5)

where $\\{B^{\uparrow}_{t}:t\geq 0\\}$ is a Brownian motion, $N^{\uparrow}$
and $N^{\star}$ are Poisson measures on $[0,\infty)\times(0,\infty)^{2}$ and
$[0,\infty)\times(0,\infty)$ with intensities measures $ds\times
d\nu\times\Pi(dr)$ and $ds\times r\Pi(dr)$, respectively, and these objects
are mutually independent (as usual, $\tilde{N}^{\uparrow}$ stands for the
compensated measure associated with $N^{\uparrow}$). Moreover, the point
processes $N^{\uparrow}$ and $N^{\star}$ can be constructed by change of
measure and a marking procedure on an enlargement of the probability space
where $B^{Z}$ and $N^{Z}$ in (3) are defined, which supports and independent
i.i.d. sequence of uniform random variables in the unit interval.

This result implies that we can recover $Z$ conditioned on non-extinction as
the solution of a SDE driven by a copy of $B^{Z}$, a copy of $N^{Z}$, and a
Poisson random measure with intensity $ds\times r\Pi(dr)$, plus a drift.
(Notice that taking out the last line, corresponding to a subordinator with
drift, one again obtains equation (3).)

##  3 Relations to previous results

###  3.1 Stable processes

We will show that, as pointed out before, Lambert’s SDE representation of
stable branching processes given in [17, Theorem 5.2] can be seen as a special
case of Theorem 2.1.

Let $X$ be a spectrally positive $\alpha$-stable process with characteristic
exponent $\psi$ and characteristic measure $\Pi(dr)=kr^{-(\alpha+1)}dr$, where
$k$ is some positive constant and $1<\alpha\leq 2$. Let $Z$ be the branching
process with branching mechanism $\psi$. Thanks to Theorem 2.1 we know that,
under $\mathbb{P}^{\uparrow}$, $Z$ satisfies the following stochastic
differential equation:

$Z_{t}=\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{1}^{\infty}rN^{\uparrow}(ds,d\nu,dr)+\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{0}^{1}r\tilde{N}^{\uparrow}(ds,d\nu,dr)+\int_{0}^{t}\int_{0}^{\infty}rN^{\star}(ds,dr),$
(6)

where $N^{\uparrow}$ is a Poisson random measure with intensity $ds\times
d\nu\times\Pi(dr)$ and $N^{\star}$ is an independent Poisson random measure
with intensity $ds\times r\Pi(dr)$. Now, we define

$\theta_{n}=\displaystyle{\frac{r^{\uparrow}_{n}\textbf{1}_{(\nu^{\uparrow}_{n}\leq
Z_{t_{n}-})}}{Z_{t_{n}-}^{1/\alpha}}},$

where $\\{(t_{n},r_{n}^{\uparrow},\nu_{n}^{\uparrow}):n\in\mathbb{N}\\}$ are
the atoms of $N^{\uparrow}$. We claim that, under $\mathbb{P}^{\uparrow}$,
$\\{(t_{n},\theta_{n}):n\in\mathbb{N}\\}$ are atoms of a Poisson random
measure $N^{\prime}$ with intensity $ds\times\Pi(du)$. Indeed, for any bounded
non-negative predictable process $H$, and any positive bounded function $f$
vanishing at zero,

$M_{t}:=\sum_{t_{n}\leq
t}H_{t_{n}}f(\theta_{n})-\int_{0}^{t}H_{s}ds\int_{0}^{\infty}\int_{0}^{\infty}f\left(\frac{r}{Z_{s}^{1/\alpha}}\right)\textbf{1}_{(\nu\leq
Z_{s})}d\nu\Pi(dr)$

is a martingale. If we change variables, the particular form of $\Pi$ implies
that

$M_{t}=\sum_{t_{n}\leq
t}H_{s}f(\theta_{n})-\int_{0}^{t}H_{s}ds\int_{0}^{\infty}f(u)\Pi(du).$

Taking expectations, our claim follows thanks to Lemma 4.2 below. Since
$\sum_{t_{n}\leq t}r^{\uparrow}_{n}\textbf{1}_{(\nu^{\uparrow}_{n}\leq
Z_{t_{n}-})}=\sum_{t_{n}\leq t}Z^{1/\alpha}_{t_{n}-}\theta_{n},$ we can
rewrite (6) as

$Z_{t}=\int_{0}^{t}\int_{1}^{\infty}Z^{1/\alpha}_{s-}uN^{\prime}(ds,du)+\int_{0}^{t}\int_{0}^{1}Z^{1/\alpha}_{s-}u\tilde{N}^{\prime}(ds,du)+\int_{0}^{t}\int_{0}^{\infty}rN^{\star}(ds,dr).$

Defining

$X_{t}:=\int_{0}^{t}\int_{1}^{\infty}uN^{\prime}(ds,du)+\int_{0}^{t}\int_{0}^{1}u\tilde{N}^{\prime}(ds,du),$

by the Lévy-Ito decomposition it is easy to see that $X$ is an $\alpha$-stable
Lévy process with characteristic exponent $\psi$. Similarly,

$S_{t}:=\int_{0}^{t}\int_{0}^{\infty}rN^{\star}(ds,dr)$

is seen to be an $(\alpha-1)$-stable subordinator. Independence of $X$ and $S$
is granted by construction, because the two processes do not have simultaneous
jumps. Thus, we have

$dZ_{t}=Z^{1/\alpha}_{t}dX_{t}+dS_{t},$

which corresponds to Lambert’s result.

###  3.2 CSBP flows as SDE solutions

A family of CSBP processes $Z=\\{Z_{t}(a):t\geq 0,a\geq 0\\}$ allowing the
initial population size $Z_{0}(a)=a$ to vary, can be constructed
simultaneously as a two parameter process or stochastic flow satisfying the
branching property. This was done by Bertoin and Le-Gall [1] by using families
of subordinators. In [2], [3] they later used Poisson measure driven SDE to
formulate such type of flows in related contexts, including equations close to
(3). In the same line, Dawson and Li [6] proved the existence of strong
solutions for stochastic flows of continuous-state branching processes with
immigration, as SDE families driven by white noise processes and Poisson
random measures with joint regularity properties. The stochastic equations
they study (in particular equation (1.5) ) are close to equation (5), the main
difference being the immigration behavior which in their case only covers
linear drifts. For simplicity reasons Theorem 2.1 is presented in the case of
a Brownian motion and Poisson measure driven SDE, but our arguments can be
extended to the white-noise and Poisson measure driven stochastic flow
considered in [6] (in absence of immigration).

##  4 Proof of the main theorem

In [17], a suitable marking of Poisson point processes was used to firstly
construct a stable Lévy process, conditioned to stay positive, out of the
realization of the unconditioned one. After time-changing the author takes
advantage of the scaling property of $\alpha$-stable processes to derive an
SDE for the branching process. Our proof is inspired in his marking argument
but in turn it is carried out directly in the time scale of the CSBP. We will
need the following version of Girsanov’s theorem (c.f. Theorem 37 in Chapter
III.8 of [24]):

###### Theorem 4.1

Let $(\varOmega,\mathcal{F},(\mathcal{F}_{t}),\mathbb{P})$ be a filtered
probability space, and let $M$ be a $\mathbb{P}$-local martingale with
$M_{0}=0$. Let $\mathbb{P}^{\star}$ be another probability measure absolutely
continuous with respect to $\mathbb{P}$, and let
$D_{t}=\mathbb{E}(\frac{d\mathbb{P}^{\star}}{d\mathbb{P}}|\mathcal{F}_{t})$.
Assume that $\langle M,D\rangle$ exists for $\mathbb{P}$. Then
$A_{t}=\int_{0}^{t}\frac{1}{Z_{s^{-}}}d\langle M,Z\rangle_{s}$ exists a.s. for
the probability $\mathbb{P}^{\star}$, and $M_{t}-A_{t}$ is a
$\mathbb{P}^{\star}$-local martingale.

The following well-known characterization of Poisson point processes will also
be useful:

###### Lemma 4.2

Let $(\Omega,\mathcal{F},(\mathcal{F}_{t}),\mathbb{P})$ be a filtered
probability space, $(S,\mathcal{S},\eta)$ an arbitrary $\sigma$-finite measure
space, and $\\{(t_{n},\delta_{n})\in\mathbb{R_{+}}\times S\\}$ a countable
family of random variables such that $\\{t_{n}\leq t,\delta_{n}\in
A\\}\in{\cal F}_{t}$ for all $n\in\mathbb{N}$, $t\geq 0$ and $A\in{\cal S}$,
and moreover

$\mathbb{E}\sum\limits_{n:t_{n}\leq
t}F_{t_{n}}g(\delta_{n})=\mathbb{E}\int\limits_{0}^{t}F_{s}ds\int\limits_{S}g(x)m(dx)$
(7)

for any nonnegative predictable process $F_{s}$ and any nonnegative function
$g:S\rightarrow\mathrm{I\\!R}$. Then, $(t_{n},\delta_{n})_{n\in\mathbb{N}}$
are the atoms of a Poisson random measure $N$ on $\mathbb{R_{+}\times S}$ with
intensity $dt\times m(dx)$.

###### Proof.

Writing

$\displaystyle{e^{\left\\{\sum\limits_{t_{n}\leq
t}f(\delta_{n}))\right\\}}}=\sum\limits_{n:t_{n}\leq
t}\left[\prod\limits_{k:t_{k}<t_{n}}e^{f(\delta_{k})}\right](e^{f(\delta_{n})}-1)=\sum\limits_{n:t_{n}\leq
t}\left[e^{\sum\limits_{k:t_{k}\leq
s}f(\delta_{k})}\right](e^{f(\delta_{n})}-1)$

we get from (7) that

$\mathbb{E}\left[e^{\sum\limits_{n:t_{n}\leq
t}f(\delta_{n})}\right]=\int\limits_{0}^{t}\mathbb{E}\left[e^{\sum\limits_{k:t_{k}\leq
s}f(\delta_{k})}\right]ds\int_{S}(e^{f(x)}-1)m(dx)$

since $F_{s}:=\prod\limits_{t_{k}<s}e^{f(\delta_{k})}$ is a predictable
process. Solving this differential equation yields

$\mathbb{E}\left[e^{\sum\limits_{t_{n}\leq
t}f(\delta_{n})}\right]=e^{-t\int\limits_{S}(1-e^{f(x)})m(dx)},$

and the statement follows by Campbell’s formula (see e.g. [14]) ∎

###### Proof of Theorem 3.1.

We will prove that under the laws $\mathbb{P}_{x}^{\uparrow}$ the process $Z$
in equation (3) is a weak solution of (5). Pathwise uniqueness, which then
classically implies also strong existence, can be shown as in [10].

We write $B=B^{Z}$ and $N=N^{Z}$, and we denote by $\\{\mathcal{F}_{t}\\}$ the
filtration

$\mathcal{F}_{t}:=\sigma(B_{s},(r_{n},\nu_{n})\textbf{1}_{t_{n}\leq
s};n\in\mathbb{N},s\leq t),$

where
$\\{(t_{n},r_{n},\nu_{n})\in[0,\infty)\times(0,\infty)\times(0,\infty)\\}_{n\in\mathbb{N}}$
are the atoms of the Poisson point process $N$. We will use the absolute
continuity of $\mathbb{P}^{\uparrow}$ w.r.t. $\mathbb{P}$ and the Radon-
Nikodym density $D_{t}=\frac{e^{\rho t}Z_{t}}{x}$ applying the previous
theorem to the process $\\{B_{t}:t\geq 0\\}$ and, indirectly, to the Poisson
random measure $N$ and its compensated measure.

Dealing with the diffusion part is standard since $d\langle
D,B\rangle_{t}=\frac{e^{\rho t}}{x}\sigma\sqrt{Z_{t}}dt,$ so that

$B_{t}^{\uparrow}:=B_{t}-\int_{0}^{t}\frac{d\langle
D,B\rangle_{s}}{D_{s}}=B_{t}-\sigma\int_{0}^{t}Z_{s}^{-\frac{1}{2}}ds$

is a Brownian motion under $\mathbb{P}^{\uparrow}$ by Girsanov theorem.

We next study the way the Poisson random measure $N$ is affected by the change
of probability, which is the main part of the proof. Enlarging the probability
space and filtration if needed, we may and shall assume that there is a
sequence $(u_{n})_{n\geq 1}$ of independent random variables uniformly
distributed on $[0,1]$, independent of $B$ and $N$ and such that
$u_{n}\textbf{1}_{t_{n}\leq t}$ is $\mathcal{F}_{t}$-measurable. Define random
variables $(\Delta_{n},\delta_{n})\in[0,\infty)^{2}\times[0,\infty)$ by

$(\Delta_{n},\delta_{n}):=\left\\{\begin{array}[]{lcl}((0,0),r_{n}\textbf{1}_{(\nu_{n}\leq
Z_{t_{n}^{-}})})&\mbox{ if
}&\displaystyle{u_{n}>\frac{D_{t_{n}-}}{D_{t_{n}}}\,=\frac{Z_{t_{n}-}}{Z_{t_{n}}}}\mbox{
and }Z_{t_{n}}>0,\\\ ((r_{n},\nu_{n}),0)&\mbox{ if
}&\displaystyle{u_{n}\leq\frac{D_{t_{n}-}}{D_{t_{n}}}}\mbox{ and
}Z_{t_{n}}>0,\\\ ((0,0),0)&\mbox{ if }&Z_{t_{n}}=0.\\\ \end{array}\right.$

Let $f_{R,\epsilon}$ be a nonnegative function such that for all $(r,\nu,s)$

  * -

$f_{R,\epsilon}((r,\nu),s)=0$ when $\nu\geq R$, for some fixed $R\geq 0$,

  * -

$f_{R,\epsilon}((r,\nu),s)=0$ when $r<\epsilon$, for some fixed
$0<\epsilon\leq 1$, and

  * -

$f_{R,\epsilon}((0,0),0)=0$.

For any non-negative predictable process $F$, we have

$\begin{array}[]{lcl}\sum\limits_{t_{n}\leq
t}F_{t_{n}}f_{R,\epsilon}(\Delta_{n},\delta_{n})&=&\sum\limits_{t_{n}\leq
t}F_{t_{n}}f_{R,\epsilon}((0,0),r_{n}\textbf{1}_{\\{\nu\leq
Z_{t_{n}-}\\}})\textbf{1}_{\\{u_{n}>\frac{Z_{t_{n}-}}{Z_{t_{n}}}\\}}\\\ \\\
&&+\sum\limits_{t_{n}\leq
t}F_{t_{n}}f_{R,\epsilon}((r_{n},\nu_{n}),0)\textbf{1}_{\\{u_{n}\leq\frac{Z_{t_{n}-}}{Z_{t_{n}}}\\}}.\end{array}$

Therefore, since
$1-\frac{Z_{t_{n}-}}{Z_{t_{n}}}=\frac{r_{n}\textbf{1}_{\\{\nu_{n}\leq
Z_{t_{n}-}\\}}}{Z_{t_{n}}}$, the process

$\begin{array}[]{lcl}S_{t}&:=&\sum\limits_{t_{n}\leq
t}F_{t_{n}}f_{R,\epsilon}(\Delta_{n},\delta_{n})-\int_{0}^{t}dsF_{s}\int_{0}^{\infty}\int_{0}^{\infty}f_{R,\epsilon}((0,0),r\textbf{1}_{\\{\nu\leq
Z_{s}\\}})\displaystyle{\frac{r\textbf{1}_{(\nu\leq
Z_{s})}}{Z_{s}+r\textbf{1}_{(\nu\leq Z_{s})}}}\Pi(dr)d\nu\\\ \\\
&&-\int_{0}^{t}dsF_{s}\int_{0}^{\infty}\int_{0}^{\infty}f_{R,\epsilon}((r,\nu),0)\displaystyle{\frac{Z_{s}}{Z_{s}+r\textbf{1}_{(\nu\leq
Z_{s})}}}\Pi(dr)d\nu\end{array}$

is a martingale under $\mathbb{P}$. The quadratic covariation of $S$ and $D$
is given by

$\begin{array}[]{lcl}[S,D]_{t}&=&\sum\limits_{t_{n}\leq
t}F_{t_{n}}f_{R,\epsilon}(\Delta_{n},\delta_{n})\frac{e^{\rho
t_{n}}}{x}r_{n}\textbf{1}_{(\nu_{n}\leq Z_{t_{n}-})}\\\ \\\
&=&\sum\limits_{t_{n}\leq
t}F_{t_{n}}f_{R,\epsilon}((0,0),r_{n}\textbf{1}_{\\{\nu\leq
Z_{t_{n}-}\\}})\frac{e^{\rho t_{n}}}{x}r_{n}\textbf{1}_{\\{\nu_{n}\leq
Z_{t_{n}-}\\}}\textbf{1}_{\left\\{u_{n}>\frac{Z_{t_{n}-}}{Z_{t_{n}}}\right\\}}\\\
\\\ &&+\sum\limits_{t_{n}\leq
t}F_{t_{n}}f_{R,\epsilon}((r_{n},\nu_{n}),0)\frac{e^{\rho
t_{n}}}{x}r_{n}\textbf{1}_{\\{\nu_{n}\leq
Z_{t_{n}-}\\}}\textbf{1}_{\left\\{u_{n}\leq\frac{Z_{t_{n}-}}{Z_{t_{n}}}\right\\}}\,,\par\end{array}$

since $S$ is a jump process. Thus, the conditional quadratic covariation is

$\begin{array}[]{lcl}\langle D,S\rangle_{t}&=&\int_{0}^{t}\frac{e^{\rho
s}}{x}F_{s}ds\int_{0}^{\infty}\int_{0}^{\infty}f_{R,\epsilon}((0,0),r\textbf{1}_{\\{\nu\leq
Z_{s}\\}})\displaystyle{\frac{r\textbf{1}_{(\nu\leq
Z_{s})}}{Z_{s}+r\textbf{1}_{(\nu\leq Z_{s})}}r}d\Pi(dr)d\nu\\\ \\\
&&+\int_{0}^{t}\frac{e^{\rho
s}}{x}F_{s}ds\int_{0}^{\infty}\int_{0}^{\infty}f_{R,\epsilon}((r,\nu),0)\displaystyle{\frac{Z_{s}}{Z_{s}+r\textbf{1}_{(\nu\leq
Z_{s})}}r\textbf{1}_{(\nu\leq Z_{s})}}d\Pi(dr)d\nu.\end{array}$

Then, using Girsanov’s theorem, we see that the process

$\begin{array}[]{lcl}S_{t}^{\uparrow}:&=&S_{t}-\int_{0}^{t}\int_{0}^{\infty}\int_{0}^{\infty}F_{s}f_{R,\epsilon}((0,0),r\textbf{1}_{(\nu\leq
Z_{s})})\displaystyle{\frac{r\textbf{1}_{(\nu\leq
Z_{s})}}{Z_{s}+r\textbf{1}_{(\nu\leq Z_{s})}}\frac{r}{Z_{s}}}\Pi(dr)d\nu ds\\\
\\\
&&-\int_{0}^{t}\int_{0}^{\infty}\int_{0}^{\infty}F_{s}f_{R,\epsilon}((r,\nu),0)\displaystyle{\frac{Z_{s}}{Z_{s}+r\textbf{1}_{(\nu\leq
Z_{s})}}\frac{r\textbf{1}_{(\nu\leq Z_{s})}}{Z_{s}}}\Pi(dr)d\nu ds\end{array}$

is a $(\mathcal{F}_{t})$-martingale under $\mathbb{P}^{\uparrow}$. By the
definition of $S$,

$\begin{array}[]{lcl}S_{t}^{\uparrow}&=&\sum\limits_{t_{n}\leq
t}F_{t_{n}}f_{R,\epsilon}(\Delta_{n},\delta_{n})-\int_{0}^{t}F_{s}ds\int_{0}^{\infty}\int_{0}^{\infty}\left[f_{R,\epsilon}((0,0),r\textbf{1}_{(\nu\leq
Z_{s})})\displaystyle{\frac{r\textbf{1}_{(\nu\leq
Z_{s})}}{Z_{s}}}+f_{R,\epsilon}((r,\nu),0)\right]\Pi(dr)d\nu\\\ \\\
&=&\sum\limits_{t_{n}\leq
t}F_{t_{n}}f_{R,\epsilon}(\Delta_{n},\delta_{n})-\int_{0}^{t}F_{s}ds\int_{0}^{\infty}\int_{0}^{\infty}\left[f_{R,\epsilon}((0,0),r)\displaystyle{\frac{r}{Z_{s}}\textbf{1}_{(\nu\leq
Z_{s})}}+f_{R,\epsilon}((r,\nu),0)\right]\Pi(dr)d\nu\\\ \\\ \end{array}$

since $f_{R,\epsilon}((0,0),0)=0$. Recalling that $S^{\uparrow}$ is a
$(\mathcal{F}_{t})$-martingale on $\mathbb{P}^{\uparrow}$ starting from $0$,
we deduce that

$\begin{array}[]{lcl}\mathbb{E}^{\uparrow}\left[\sum\limits_{t_{n}\leq
t}F_{t_{n}}f_{R,\epsilon}(\Delta_{n},\delta_{n})\right]&=&\mathbb{E}^{\uparrow}\left[\int_{0}^{t}F_{s}ds\int_{0}^{\infty}f_{R,\epsilon}((0,0),r)r\Pi(dr)\right]\\\
\\\
&&+\mathbb{E}^{\uparrow}\left[\int_{0}^{t}F_{s}ds\int_{0}^{\infty}\int_{0}^{\infty}f_{R,\epsilon}((r,\nu),0)\Pi(dr)d\nu\right].\end{array}$

By standard arguments, this formula is also true for any nonnegative function
$f$ such that $f((0,0),0)=0$. By Lemma 4.2 we see that, under
$\mathbb{P}^{\uparrow}$, $(t_{n},\Delta_{n})_{n\geq 0}$ and
$(t_{n},\delta_{t})_{n\geq 0}$ are atoms of two Poisson point processes
$N^{\uparrow}$ and $N^{\star}$ with intensity measures $dt\times
d\nu\times\Pi(dr)$ and $dt\times r\Pi(dr)$ on
$[0,\infty)\times(0,\infty)\times(0,\infty)$ and $[0,\infty)\times(0,\infty)$
respectively. By construction, $N^{\uparrow}$ and $N^{\star}$ are independent
because they never jump simultaneously. Now set

$J_{t}:=\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{1}^{\infty}rN(ds,d\nu,dr)=\sum\limits_{t_{n}\leq
t}r_{n}\textbf{1}_{(\nu_{n}\leq Z_{t_{n}^{-}})}\textbf{1}_{(r_{n}\geq 1)}.$

From above, we have

$J_{t}=\sum_{t_{n}\leq t}\Delta_{n}^{(1)}\textbf{1}_{(\Delta^{(2)}_{n}\leq
Z_{t_{n}^{-}})}\textbf{1}_{(\Delta_{n}^{(1)}\geq 1)}+\sum_{t_{n}\leq
t}\delta_{n}\textbf{1}_{(\delta_{n}\geq 1)},$

where $\Delta_{n}^{(i)}$ is the $i-$th coordinate of $\Delta_{n}$, $i=1,2$.
Therefore

$J(t)=\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{1}^{\infty}rN^{\uparrow}(ds,d\nu,dr)+\int_{0}^{t}\int_{1}^{\infty}rN^{\star}(ds,dr).$

Finally, given $0<\varepsilon<1$, let $\\{\tilde{M}_{t}^{(\varepsilon)},t\geq
0\\}$ be the $\mathbb{P}$-martingale

$\begin{array}[]{lcl}\tilde{M}_{t}^{(\varepsilon)}&:=&\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{\varepsilon}^{1}rN^{Z}(ds,d\nu,dr)-\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{\varepsilon}^{1}r\,dsd\nu\Pi(dr)\\\
\\\ &=&\sum\limits_{t_{n}\leq t}r_{n}\textbf{1}_{(\nu_{n}\leq
Z_{t_{n}^{-}})}\textbf{1}_{(\varepsilon<r_{n}<1)}-\int_{0}^{t}\int_{0}^{Z_{s}}\int_{\varepsilon}^{1}r\,dsd\nu\Pi(dr),\end{array}$

which converges in the $L^{2}(\mathbb{P})$ sense when $\varepsilon\rightarrow
0$ to
$\tilde{M}_{t}:=\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{0}^{1}r\tilde{N}^{Z}(ds,d\nu,dr).$
In terms of $(\Delta_{n})$ and $(\delta_{n})$, we can write

$\begin{array}[]{lcl}\tilde{M}^{(\varepsilon)}&=&\left(\sum\limits_{t_{n}\leq
t}\Delta^{(1)}_{n}\textbf{1}_{(\Delta^{(2)}_{n}\leq
Z_{t_{n}^{-}})}\textbf{1}_{(\varepsilon<\Delta^{(1)}_{n}<1)}-\int_{0}^{t}\int_{0}^{Z_{s}}\int_{\varepsilon}^{1}rdsd\nu\Pi(dr)\right)\\\
\\\ &&+\sum\limits_{t_{n}\leq
t}\delta_{n}\textbf{1}_{(\varepsilon<\delta_{n}<1)}\\\ \\\
&=&\left(\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{\varepsilon}^{1}rN^{\uparrow}(ds,d\nu,dr)-\int_{0}^{t}\int_{0}^{Z_{s}}\int_{\varepsilon}^{1}rdsd\nu\Pi(dr)\right)\\\
\\\ &&+\int_{0}^{t}\int_{\varepsilon}^{1}rN^{\star}(ds,dr).\end{array}$

Thanks to [16, Theorem 2.10], the limit as $\varepsilon\to 0$ in the
$L^{2}(\mathbb{P}^{\uparrow})$ sense of the $\mathbb{P}^{\uparrow}$-martingale
given by the first term on the right hand side exists, and it is equal to the
martingale
$\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{0}^{1}r\tilde{N}^{\uparrow}(ds,d\nu,dr)$,
where $\tilde{N}^{\uparrow}$ is the compensated measure associated with
$N^{\uparrow}$. Also, as $\int_{0}^{\infty}(1\wedge x^{2})\Pi(dx)<\infty$, by
[16, Theorem 2.9] the second term on the right hand side converges
$\mathbb{P}^{\uparrow}$-a.s., so we have

$\tilde{M}_{t}=\int_{0}^{t}\int_{0}^{Z_{s^{-}}}\int_{0}^{1}r\tilde{N}^{\uparrow}(ds,d\nu,dr)+\int_{0}^{t}\int_{0}^{1}rN^{\star}(ds,dr).$

Bringing all parts together, we have shown that $Z$ satisfies under
$\mathbb{P}^{\uparrow}$ the desired SDE, except for the independence of the
processes $B^{\uparrow}$ and $(N^{\uparrow},N^{\star}$), which we shall
establish in what follows.

Since $N^{\uparrow}$ and $N^{\star}$ have $\sigma$-finite intensities and
thanks to the Markov property of the three processes with respect to the
filtration $(\mathcal{F}_{t})$, it is enough to show that for every $t>s\geq
0$, $\zeta\in\mathbb{R}$, $\lambda_{k},\gamma_{k}\in\mathbb{R}_{+}$,
$k\in\\{1,...,m\\}$, and $m\in\mathbb{N}$

$\begin{array}[]{lcl}\mathbb{E}^{\uparrow}\left[\displaystyle{e^{-\zeta(B^{\uparrow}_{t}-B^{\uparrow}_{s})}e^{-\sum\limits_{k=1}^{m}\lambda_{k}N^{\uparrow}((s,t]\times
W_{k})}e^{-\sum\limits_{k=1}^{m}\gamma_{k}N^{\star}((s,t]\times
V_{k})}}\bigg{|}\mathcal{F}_{s}\right]\\\ \\\
=\displaystyle{e^{-\frac{\zeta^{2}}{2}(t-s)}e^{\sum\limits_{k=1}^{m}\int_{s}^{t}\int_{W_{k}}(e^{-\lambda_{k}}-1)\Pi(dr)d\nu
du}e^{\sum\limits_{k=1}^{m}\int_{s}^{t}\int_{V_{k}}(e^{-\gamma_{k}}-1)r\Pi(dr)du}},\end{array}$

where $\\{W_{k}\\}_{k=1}^{m}$ and $\\{V_{k}\\}_{k=1}^{m}$ are disjoint sets of
$(0,\infty)\times(0,\infty)$ and $(0,\infty)$ such that
$\int_{0}^{t}\int_{W_{k}}\Pi(dr)d\nu du$ and
$\int_{0}^{t}\int_{V_{K}}r\Pi(dr)du$ are finite. To that end, set

$F(x,y_{1},..,y_{m},z_{1},..,z_{m}):=e^{-\zeta
x}e^{-\sum_{k=1}^{m}\lambda_{k}y_{k}}e^{-\sum_{k=1}^{m}\gamma_{k}z_{k}}.$

Applying Itô’s formula to the semimartingale

$X(t)=\left(B^{\uparrow}(t),N^{\uparrow}((0,t]\times
W_{1}),..,N^{\uparrow}((0,t]\times W_{m}),N^{\star}((0,t]\times
V_{1}),..,N^{\star}((0,t]\times V_{m})\right),$

we obtain:

$\begin{array}[]{lcl}F(X(t))&=&F(X(s))-\int_{s}^{t}\zeta
F(X(u))dB^{\uparrow}_{u}-\sum\limits_{j=1}^{m}\int_{s}^{t}\int_{W_{j}}\lambda_{j}F(X(u))N^{\uparrow}(du,d\nu,dr)\\\
\\\
&&-\sum\limits_{j=1}^{m}\int_{s}^{t}\int_{V_{j}}\gamma_{j}F(X(u))N^{\star}(du,dr)+\frac{\zeta^{2}}{2}\int_{s}^{t}F(X(u))du+\sum\limits_{s<u\leq
t}F(X(u))-F(X(u^{-}))\\\ \\\ &&+\sum\limits_{s<t_{n}\leq
t}\sum\limits_{j=1}^{m}\left[\lambda_{j}F(X(t_{n}))\textbf{1}_{\\{\Delta_{n}\in
W_{j}\\}}+\gamma_{j}F(X(t_{n}))\textbf{1}_{\\{\delta_{n}\in
V_{j}\\}}\right].\end{array}$

From above, we deduce that

$\begin{array}[]{lcl}F(X(t))-F(X(s))&=&\bar{M}_{t}-\bar{M}_{s}+\frac{\zeta^{2}}{2}\int_{s}^{t}F(X(u))du\\\
\\\ &&+\sum\limits_{s<t_{n}\leq
t}\left[F\left(X(t_{n}-)+(0,\textbf{1}_{\\{\Delta_{n}\in
W_{1}\\}},..,\textbf{1}_{\\{\delta_{n}\in
V_{m}\\}})\right)-F(X(t_{n}-))\right],\end{array}$

where $(\bar{M}_{t})$ is a $(\mathcal{F}_{t})$-martingale. Defining
$f(\Delta_{n},\delta_{n}):=e^{-\sum\limits_{k=1}^{m}\lambda_{k}\textbf{1}_{\\{\Delta_{n}\in
W_{k}\\}}-\sum\limits_{k=1}^{m}\lambda_{k}\textbf{1}_{\\{\delta_{n}\in
V_{k}\\}}}-1,$ we have

$F(X(t))-F(X(s))=\bar{M}_{t}-\bar{M}_{s}+\frac{\zeta^{2}}{2}\int_{s}^{t}F(X(u))du+\sum\limits_{s<t_{n}\leq
t}F(X(t_{n}-))[f(\Delta_{n},\delta_{n})].$

Let now $A\in\mathcal{F}_{s}$. Multiplying both sides by
$F(-(X(s)))\textbf{1}_{A}$, yields:

$\begin{array}[]{lcl}\mathbb{E}^{\uparrow}[F(X(t-s))\textbf{1}_{A}]-\mathbb{P}^{\uparrow}(A)&=&\frac{\zeta^{2}}{2}\int_{s}^{t}\mathbb{E}^{\uparrow}\left[F(X(u-s))\textbf{1}_{A}\right]du\\\
\\\
&&+\int_{s}^{t}\mathbb{E}^{\uparrow}\left[F(X(u-s))\textbf{1}_{A}\right]du\sum\limits_{k=1}^{m}\int_{W_{k}}(e^{-\lambda_{k}}-1)\Pi(dr)d\nu\\\
\\\
&&+\int_{s}^{t}\mathbb{E}^{\uparrow}\left[F(X(u-s))\textbf{1}_{A}\right]du\sum\limits_{k=1}^{m}\int_{V_{k}}(e^{-\gamma_{k}}-1)r\Pi(dr).\par\end{array}$

Thus,

$\mathbb{E}^{\uparrow}\left[F(X(t-s))\textbf{1}_{A}\right]=\mathbb{P}^{\uparrow}(A)e^{-\frac{\zeta^{2}}{2}(s-t)}e^{\sum\limits_{k=1}^{m}\int_{s}^{t}\int_{W_{k}}(e^{-\lambda_{k}}-1)\Pi(dr)d\nu
du}e^{\sum\limits_{k=1}^{m}\int_{s}^{t}\int_{V_{k}}(e^{-\gamma_{k}}-1)r\Pi(dr)du}$

which means that the three processes are mutually independent, which ends the
proof of weak existence.

As concerns pathwise uniqueness, we just remark that the proof of Theorem 3.2
in [10] covers the case of equation (5). Indeed, if $B^{\uparrow}$,
$N^{\uparrow}$ and $N^{\star}$ are independent processes as before driving two
solutions $\\{Z_{t}^{(1)}\\}$ and $\\{Z_{t}^{(2)}\\}$ of (5), setting
$\zeta_{t}:=Z_{t}^{(1)}-Z_{t}^{(2)}$ one gets that

$\begin{array}[]{lcl}\zeta_{t}&=&\zeta_{0}+\int_{0}^{t}a\left(Z_{s}^{(1)}-Z_{s}^{(2)}\right)ds+\int_{0}^{t}\sigma\left(\sqrt{Z_{s}^{(1)}}-\sqrt{Z_{s}^{(2)}}\right)dB^{\uparrow}_{s}\\\
\\\
&&+\int_{0}^{t}\int_{U_{0}}r\left(\textbf{1}_{(\nu<Z_{s}^{(1)})}-\textbf{1}_{(\nu<Z_{s}^{(2)})}\right)N^{\uparrow}(ds,d\nu,dr)\\\
\\\
&&+\int_{0}^{t}\int_{U_{1}}r\left(\textbf{1}_{(\nu<Z_{s}^{(1)})}-\textbf{1}_{(\nu<Z_{s}^{(2)})}\right)\tilde{N}^{\uparrow}(ds,d\nu,dr),\end{array}$
(8)

where $U_{0}=[0,\infty)\times[1,\infty)$ and $U_{1}=[0,\infty)\times(0,1)$.
From this point on, the proof of Theorem 3.2 in [10] applies, since conditions
(2.a,b) and (3.a,b) therein are satisfied. Indeed, in their notations, we have
the intensity measure $\mu(du)=\Pi(dr)d\nu$ for $N^{0}=N^{\uparrow}|_{U_{0}}$
and $N^{1}=N^{\uparrow}|_{U_{1}}$ (where $u=(r,\nu)$), continuous functions on
$\mathbb{R}$ given by $b(x):=ax\textbf{1}_{0\leq x}$ and
$\sigma(x):=\sigma\sqrt{x}\textbf{1}_{0\leq x}$, and Borel functions on
$\mathbb{R}\times U_{i}$, $i=\\{0,1\\}$ given by
$g(x,u)=g_{0}(x,u)=g_{1}(x,u)=r\textbf{1}_{\nu<x}$ such that $g(x,u)+x\geq 0$
for $x>0$ and $g(x,u)=0$ for $x\leq 0$. Moreover,

  1. 1.

there is a constant $K:=|a|+M\geq 0$ , where
$\int_{1}^{\infty}r\Pi(dr)=M<\infty$, such that

$|ax|+\int_{0}^{\infty}\int_{1}^{\infty}r\textbf{1}_{\nu<x}\Pi(dr)d\nu\leq
K(x+1)\,;$

  2. 2.

there is a non-negative and non-decreasing function $L(x)=(\sigma^{2}+I)(x)$
on $\mathbb{R}_{+}$, with $I=\int_{0}^{1}r^{2}\Pi(dr)$, so that

$\sigma^{2}x+\int_{0}^{\infty}\int_{0}^{1}r^{2}\textbf{1}_{\nu<x}\Pi(dr)d\nu\leq
L(x);$

  3. 3.

there is a continuous non-decreasing function $x\rightarrow b_{2}(x):=x$ on
$\mathbb{R}_{+}$ such that for $b_{1}(x)=b(x)+b_{2}(x)$, on has

$|(a+1)(b_{1}(x)-b_{1}(y))|+\int_{0}^{\infty}\int_{1}^{\infty}r\textbf{1}_{y<\nu<x}\Pi(dr)d\nu\leq
r(|x-y|)\,;$

where $r$ is the non-decreasing and concave function $r(z)=:(|a+1|+M)z$ on
$\mathbb{R}_{+}$ satisfying $\int_{0_{+}}r(z)^{-1}dz=\infty$; and

  4. 4.

for every fixed $u\in U_{0}$ the function $x\rightarrow g(x,u)$ is non-
decreasing, and there is a non-negative and non-decreasing function
$\rho(z):=[\sigma^{2}+I]\sqrt{z}$ on $\mathbb{R}_{+}$ so that
$\int_{0_{+}}\rho(z)^{-2}dz=\infty$ and

$(\sigma\sqrt{x}-\sigma\sqrt{y})^{2}+\int_{0}^{\infty}\int_{0}^{1}r^{2}\textbf{1}_{y<\nu<x}\Pi(dr)d\nu\leq\rho(|x−y|)^{2}.$

Conditions 1,2,3 and 4 respectively ensure that hypotheses (2.a,b) and (3.a,b)
in [10] hold, and pathwise uniqueness follows. ∎

Acknowledgements We would like to thank Julien Berestycki for pointing out to
us relevant references and for several remarks that helped us to improve
earlier versions of this work.

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