# Hydrodynamic limit of a stochastic model of proliferating cells with
chemotaxis.

Radosław Wieczorek  University of Silesia, ul. Bankowa 14, 40-007 Katowice,
Poland<EMAIL_ADDRESS>
###### Abstract

A hybrid stochastic individual-based model of proliferating cells with
chemotaxis is presented. The model is expressed by a branching diffusion
process coupled to a partial differential equation describing concentration of
chemotactic factor. It is shown that in the hydrodynamic limit when number of
cells goes to infinity the model converges to the solution of a
nonconservative Patlak-Keller-Segel-type system. A nonlinear mean-field
stochastic model is defined and it is proven that the movement of descendants
of a single cell in the individual model converges to this mean-field process.

###### keywords:

propagation of chaos, stochastic particles system, branchind diffusion,
chemotaxis, mean field approximation

###### MSC:

[2010] 60K35, 92C17, 60F99, 60C35, 60H30, 60G57

##  1 Introduction

This paper is concerned with a stochastic model of biological cells that
undergo chemotaxis and proliferate. The hydrodynamic limit of this model is
shown to solve the equation of Patlak-Keller-Segel type with cell
proliferation of the form

$\displaystyle[left={\empheqlbrace}]\partial_{t}p(t,\mathbf{x})$
$\displaystyle\frac{1}{2}\Delta
p(t,\mathbf{x})+\nabla\left(p(t,\mathbf{x})\mathbf{b}(\mathbf{x},\nabla\varrho)\right)+\lambda(\mathbf{x},\nabla\varrho(t,\mathbf{x}))p(t,\mathbf{x})$
(1a) $\displaystyle\partial_{t}\varrho(t,\mathbf{x})=$ $\displaystyle
D\Delta\varrho(t,\mathbf{x})-r\varrho(t,\mathbf{x})+\alpha[\kappa\\!*p(t,\cdot)](\mathbf{x}).$
(1b)

Mathematical description of biological cells undergoing chemotaxis, i.e.
moving in response to gradients of chemical factors, has a long tradition and
still attracts great interest. The seminal papers by Patlak [1] and Keller and
Segel [2] stimulated the whole branch of applied mathematics (one can find
extensive reviews [3, 4] with many citations). The limit passages between
microscopic, kinetic models of collective cell behaviour to the macroscopic
description with PDE’s has received great attention in recent four decades [5,
6, 7, 8, 9, 10, 11, 12, 13]. The first rigorous proof of the convergence of a
stochastic particle system to chemotaxis equations system was given in [14]
following the work of [15]. Later, many authors continued these ideas [16, 9,
11, 17]. Macroscopic mean-field limits of weakly interacting particles where
investigated also in [18, 19, 20, 21]. Most particle approximations of PKS
type equations consider cells and chemical particles in a similar way [14, 16,
9], while they have completely different scale. Si it is natural to consider
cells as stochastic particles and chemical factor as continuous field
described by a partial differential equation [22, 11, 23, 12]. Although
usually the equation or system describing the cell population is conservative,
i.e. it preserves total mass (the number of particles), in many models it is
natural to assume that the cell population is not constant and the cells
proliferate [24, 25, 26, 27]. It is crucial for angiogenesis models [28, 29,
22, 12], where the proliferation is responsible for vessel branching.

In this paper we use the pathwise propagation of chaos approach similar to
that of [11], but we allow for the proliferation of cells. Therefore, the
number of cells (or total mass) is not conserved and, moreover, the individual
process has noncontinuous trajectories. To retain the pathwise description and
convergence, we write the birth and death process as a solution of stochastic
equations (cf. [30]).

We construct a sequence of processes indexed by the initial number of
particles $n_{0}$. The description of our process can be divided into three
components:

Movement of cells. The cells move according to the following SDE

$d\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(t)=\mathbf{b}(\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(t),\nabla\varrho_{n_{0}}(t,\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(t)))dt+\sigma\,dW_{i,\mathbbm{j}}(t),$
(1b)

where $W_{i,\mathbbm{j}}(t)$ are independent Brownian motions, $\sigma$ is the
diffusion coefficient and $\mathbf{b}$ is the (chemotactic) drift that depends
on the position of a cell and the gradient of the concentration
$\varrho_{n_{0}}$ of some chemical factor. One can also allow for the
dependence of $\mathbf{b}$ on the concentration $\varrho_{n_{0}}$ itself, not
only on its gradient, and all facts and proofs of the paper remain true, but
for the sake of shortness and simplicity of the notation we neglect this
dependence. The indexing $i,\mathbbm{j}$ will be explained later.

Equation for chemoreactant. The concentration $\varrho_{n_{0}}$ of chemotactic
factor satisfies the following PDE

$\frac{\partial\varrho_{n_{0}}(t,\mathbf{x})}{\partial
t}=D\Delta\varrho_{n_{0}}(t,\mathbf{x})-r\varrho_{n_{0}}(t,\mathbf{x})+\alpha\,\kappa\\!*\\!\xi^{n}_{t}\,(\mathbf{x}),$
(1c)

where $D$, $r$ and $\alpha$ are diffusion, degradation and production rates.
The measure

$\xi^{n_{0}}_{t}=\frac{1}{n_{0}}\sum_{{i,\mathbbm{j}}}\delta_{X^{n_{0}}_{i,\mathbbm{j}}(t)},$

is the empirical measure of all cells and function $\kappa$ is a mollifying
kernel that represents the fact that cells are actually not points, but have
spatial size. The spatial convolution

$\displaystyle\kappa\\!*\xi^{n_{0}}_{t}\,(\mathbf{x})=$
$\displaystyle\int_{\mathbb{R}^{d}}\kappa(x-y)\xi^{n_{0}}_{t}(dy)=\frac{1}{n_{0}}\sum_{i,\mathbbm{j}}\int_{\mathbb{R}^{d}}\kappa(x-y)\delta_{X_{i,\mathbbm{j}}(t)}(dy)$
$\displaystyle=$
$\displaystyle\frac{1}{n_{0}}\sum_{i,\mathbbm{j}}\kappa(x-X_{i,\mathbbm{j}}(t))$

is a mollified version of the empirical measure describing spatial positions
of cells, responsible for the production of the chemoreactant. From the
mathematical point of view this allows us to consider classical solutions to
equation (1c).

Cell population dynamics. We assume that cells may die or proliferate with
rates depending on the chemoreactant. Death means that a cell disappears and
proliferation means that a cell dies leaving two new daughter cells at the
same place as the mother cell. The birth rate of a cell placed at $x$ at time
$t$ depend on the position and on the the concentration of chemoreactant and
is given by $\lambda_{\text{b}}(\mathbf{x},\varrho^{n_{0}}(t,\mathbf{x}))$ and
the mortality rate is
$\lambda_{\text{d}}(\mathbf{x},\varrho^{n_{0}}(t,\mathbf{x}))$.

The main goal of the paper is to prove that in hydrodynamic limit the model
converge to the solutions of (1) (Theorem 5), and, most importantly, to prove
that, if the initial number of cells tends to infinity, the trajectories of
the descendants of a single cell converge to the trajectories of the hybrid
nonlinear mean-field model defined in section 2.2 (Theorem 6). To this aim, we
also want to rigorously define the described individual processes and show
wellposedness.

We call the model hybrid for three reasons: first is that in the base model
the discrete individual-based model is coupled to the continuous description
of chemoreactant by PDEs[29, 22, 12]. Second, the stochastic trajectories of
cells have jumps due to birth and death of cells [31, 32]. Third, pertains
specifically to the intermediate model described in section 2.2, which is
obtained by the convergence of fully stochastic individual based models in
Theorem 6. In this hybrid model, the stochastic particle system is coupled to
the deterministic solution of the asymptotic PDE (1).

The article is organised as follows. In the next section we introduce the
notation and present the rigorous definitions of the microscopic model and two
‘hybrid’ mean field models and write up the macroscopic equations. Section 3
is devoted to the presentation of assumptions and results concerning the
wellposedness. Section 4 contains the convergence results. The proofs are
presented in section 5.

##  2 Definitions of the processes

In this section we formally define the considered processes. We start with the
individual, fully microscopic model.

###  2.1 Definition of the microscopic processes

Note that if we use the empirical measure approach even for a simple two
Brownian particles case, then giving the initial condition
$\delta_{X_{1}^{0}}+\delta_{X_{2}^{0}}$ and two Brownian motions $W_{1}$ and
$W_{2}$ does not guarantee pathwise uniqueness — we need to know the order of
particles. The problem gets harder if the number of particles varies in time.
Since our goal is to obtain a pathwise propagation of chaos result, we need to
define a process in a more direct way.

Therefore, we construct the process in the following way. We mark particles by
means of a subtree of Ulam-Harris tree, namely let

$\mathbb{J}=\bigcup_{n\in\mathbb{N}_{0}}{\\{0,1\\}}^{n}$ (1d)

with a convention that ${\\{0,1\\}}^{0}=\emptyset$ means the root. Elements of
$\mathbb{J}$ will be written as blackboard bold lowercase letters such as
$\mathbbm{i},\mathbbm{j},\mathbbm{k}$. If we write $\mathbbm{k}=\mathbbm{j}0$,
we mean that $\mathbbm{k}$ is longer by one then $\mathbbm{j}$ and is created
from $\mathbbm{j}$ by adding $0$ at the end. Moreover, we will denote by
$\mathbbm{j}^{\mkern-2.0mu\leftarrow}$ element of $\mathbb{J}$ obtained from
$\mathbbm{j}$ by removing last number, eg. if $\mathbbm{j}=\emptyset 101$,
then $\mathbbm{j}^{\mkern-2.0mu\leftarrow}=\emptyset 10$. Moreover, let us
assume that

  1. (A.1)

$\left(W_{i,\mathbbm{j}}\right)_{i\in\mathbb{N},\mathbbm{j}\in\mathbb{J}}$ is
an infinite array of independent $d$-dimensional standard Wiener processes,
and
$\left(\mathcal{N}_{i,\mathbbm{j}}\right)_{i\in\mathbb{N},\mathbbm{j}\in\mathbb{J}}$
is an array of independent standard (i.e. such that intensity is Lebesgue
measure) Poisson point processes on $[0,\infty)\times[0,\infty)$ on a complete
probability space $(\Omega,\mathcal{F},\mathbb{P})$.

These processes will be used as a source of randomness in all processes, in
particular in Eq. (1ia) and (1ka).

We actually define a sequence of processes indexed by the initial number of
cell $n_{0}$. Let us assume that we start with $n_{0}$ cells and let each
initial cell be described by its position in $\mathbb{R}^{d}$, the number $i$,
and $\mathbbm{j}=\emptyset$ denoting that it is the first cell in its own tree
of inheritance. The initial cells are located at
$\mathbf{X}_{i,\emptyset}(0)\in\mathbb{R}^{d}$, $i=1,\dots,n_{0}$. At each
branching event, two new cells appear as daughters of a cell described by
$\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}$ at the same place as the mother cell. The
daughter cells inherit the position, the cell line number $i$ and obtain new
subsequent indices $\mathbbm{k}_{1}=\mathbbm{j}0$ and
$\mathbbm{k}_{2}=\mathbbm{j}1$. So, each cell is described by a triple
$(\mathbf{x},i,\mathbbm{j})\in{\mathbb{R}^{d}}\times\\{1,\dots,n_{0}\\}\times\mathbb{J}$
and possesses its own Brownian motion $W_{i,\mathbbm{j}}$ and a Poisson clock
$\mathcal{N}_{i,\mathbbm{j}}$. Let $\tau^{n_{0}}_{i,\mathbbm{j}}$ denote the
moment when $i,\mathbbm{j}$-th cell appears — that is
$\tau^{n_{0}}_{(i,\emptyset)}$ is always 0, and let
$\sigma^{n_{0}}_{i,\mathbbm{j}}$ be the moment when $i,\mathbbm{j}$-th cell
dies (the production of daughter cells also means death of the mother). The
time of death $\sigma^{n_{0}}_{i,\mathbbm{j}}$ of an $(i,j)$-th cell is
defined as a minimal $\sigma$ for which

$\mathcal{N}_{i,\mathbbm{j}}\biggl{(}(t,z):\,z\in\left[0,\lambda_{\text{b}}\bigl{(}\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(t),\varrho^{n_{0}}(t,\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(t))\bigr{)}+\lambda_{\text{d}}\bigl{(}\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(t),\varrho^{n_{0}}(t,\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(t))\bigr{)}\right],\\\
t\in\left[\tau^{n_{0}}_{i,\mathbbm{j}},\sigma\right]\biggr{)}=1$ (1g)

with a convention that $\min\emptyset=\infty$. The times of birth
$\tau^{n_{0}}_{(i,\mathbbm{j}0)}=\tau^{n_{0}}_{(i,\mathbbm{j}1)}$ of daughters
of $i,\mathbbm{j}$-th cell are defined as a minimal $\tau$ for which

$\mathcal{N}_{i,\mathbbm{j}}\left((t,z):\,z\in\left[0,\lambda_{\text{b}}\bigl{(}\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(t),\varrho^{n_{0}}(t,\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(t))\bigr{)}\right],t\in\left[\tau^{n_{0}}_{(i,\mathbbm{j})},\tau\right]\right)=1.$
(1h)

Clearly, not for every $\mathbbm{j}\in\mathbb{J}$ a cell will exist, e.g. if
$\mathbbm{k}$-th cell dies, no cell with index created from $\mathbbm{k}$ by
appending any zeros or ones cannot be born. In this case we have
$\tau^{n_{0}}_{i,\mathbbm{j}}=\sigma^{n_{0}}_{i,\mathbbm{j}}=\infty$.

The movement of a $i,\mathbbm{j}$-th cell between time
$\tau^{n_{0}}_{i,\mathbbm{j}}$ and $\sigma^{n_{0}}_{i,\mathbbm{j}}$ is given
by

$d\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(t)=\mathbf{b}(\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(t),\nabla\varrho^{n_{0}}(t,\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(t)))dt+\sigma\,dW_{i,\mathbbm{j}}(t).$
(1ia) with initial condition
$\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}\left(\sigma^{n_{0}}_{i,\mathbbm{j}}\right)=\mathbf{X}^{n_{0}}_{i,\mathbbm{j}^{\mkern-2.0mu\leftarrow}}\left(\sigma^{n_{0}}_{i,\mathbbm{j}}\right)$
coupled with the equation for nutrient field
$\frac{\partial\varrho^{n_{0}}(t,\mathbf{x})}{\partial
t}=D\Delta\varrho^{n_{0}}(t,\mathbf{x})-r\varrho^{n_{0}}(t,\mathbf{x})+\alpha[K*\xi^{n_{0}}_{t}](\mathbf{x})$
(1ib)

with $\varrho^{n_{0}}(0,\cdot)=\varrho_{0}$, where $\xi^{n_{0}}_{t}$, given by

$\xi^{n_{0}}_{t}=\frac{1}{n_{0}}\sum_{{i,\mathbbm{j}}}\mathds{1}_{\left[\sigma^{n_{0}}_{i,\mathbbm{j}},\tau^{n_{0}}_{i,\mathbbm{j}}\right)}(t)\delta_{X^{n_{0}}_{i,\mathbbm{j}}(t)},$
(1j)

is an empirical measure of all cells alive at time $t$.

Note that the generating processes $W_{i,\mathbbm{j}}$ and
$\mathcal{N}_{i,\mathbbm{j}}$ have intentionally no index $n_{0}$. In order to
obtain pathwise convergence, they are shared by processes with all $n_{0}$.
They are defined for all $i\in\mathbb{N}$, but the definition of $n_{0}$-th
process uses only those with $i=1,\dots,n_{0}$.

######  Remark 1.

Note, that equation (1ia) can be written in a form

$\displaystyle\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(t)=$
$\displaystyle\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}\left(\sigma^{n_{0}}_{i,\mathbbm{j}}\right)+\int_{\sigma^{n_{0}}_{i,\mathbbm{j}}}^{t}\mathbf{b}(\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s),\nabla\varrho^{n_{0}}(s,\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s)))\,ds$
$\displaystyle+\sigma\left(W_{i,\mathbbm{j}}(t)-W_{i,\mathbbm{j}}(\sigma^{n_{0}}_{i,\mathbbm{j}})\right),$
$\displaystyle\text{for
}t\in\left[\sigma^{n_{0}}_{i,\mathbbm{j}},\tau^{n_{0}}_{i,\mathbbm{j}}\right).$

It does not demand using Ito integral and has a pathwise unique solution.

###  2.2 Hybrid mean-field model

The next model considered will be the limit of the individual-based model (cf.
Theorems 5 and 6). We consider one initial cell at position
$\mathbf{\bar{X}_{\emptyset}}(0)=\mathbf{\bar{X}_{1,\emptyset}}(0)$ with the
same population dynamics as before. Its descendants will be denoted by
$\mathbf{\bar{X}_{\mathbbm{j}}}$ with $\mathbbm{j}\in\mathbb{J}$ and their
birth and death times are $\bar{\sigma}_{\mathbbm{j}}$ and
$\bar{\tau}_{\mathbbm{j}}$, respectively, defined in analogous way to
(1g)-(1h) with $\varrho^{n_{0}}$ replaced by $\varrho$ and the same Poisson
clocks $\mathcal{N}_{1,\mathbbm{j}}$, as for the first cell line in each
microscopic model. The movement of $\mathbbm{j}$-th cell during its life is
given by

$d\mathbf{\bar{X}_{\mathbbm{j}}}(t)=F\bigl{(}\mathbf{\bar{X}}_{\mathbbm{j}},\nabla\varrho(t,\mathbf{\bar{X}}_{\mathbbm{j}})\bigr{)}dt+\sigma\,dW_{1,\mathbbm{j}}(t),$
(1ka) coupled with the mean-field chemoreactant equation
$\dfrac{\partial\varrho(t,\mathbf{x})}{\partial
t}=D\Delta\varrho(t,\mathbf{x})-r\varrho(t,\mathbf{x})+\alpha(K\\!*\bar{\mu}_{t})(\mathbf{x}),$
(1kb)

with $\varrho(0,\cdot)=\varrho_{0}$, where $\bar{\mu}_{t}$ is the mean of the
empirical measure of
$\left(\mathbf{\bar{X}_{\mathbbm{j}}}(t)\right)_{\mathbbm{j}\in\mathbb{J}}$,
namely

$\bar{\mu}_{t}(A)=\operatorname{\mathbb{E}}\bar{\xi}_{t}(A)=\operatorname{\mathbb{E}}\left[\sum_{{i,\mathbbm{j}}}\mathds{1}_{\left[\bar{\tau}_{\mathbbm{j}},\bar{\sigma}_{\mathbbm{j}}\right)}(t)\delta_{X^{n_{0}}_{i,\mathbbm{j}}(t)}(A)\right],\text{
for }A\in\mathcal{B}(\mathbb{R}^{d}),$ (1l)

where $\bar{\xi}_{t}$, $\bar{\tau}_{\mathbbm{j}}$ and
$\bar{\sigma}_{\mathbbm{j}}$ are defined analogously to $\xi^{n_{0}}_{t}$,
$\tau^{n_{0}}_{i,\mathbbm{j}}$ and $\sigma^{n_{0}}_{i,\mathbbm{j}}$.

######  Remark 2.

Note that (1kb) differs from (1ib) only by replacing $\xi^{n_{0}}_{t}$ by
$\bar{\mu}_{t}$.

###  2.3 Macroscopic model: Patlak-Keller-Segel type equation with
proliferation

The limit macroscopic model is given by the system of equations

$\displaystyle[left={\empheqlbrace}]\partial_{t}p(t,\mathbf{x})=$
$\displaystyle\frac{1}{2}\Delta
p(t,\mathbf{x})+\nabla\left(p(t,\mathbf{x})\mathbf{b}(\mathbf{x},\nabla\varrho)\right)+\lambda(\mathbf{x},\nabla\varrho(t,\mathbf{x}))p(t,\mathbf{x})$
(1ma) $\displaystyle\partial_{t}\varrho(t,\mathbf{x})=$ $\displaystyle
D\Delta\varrho(t,\mathbf{x})-r\varrho(t,\mathbf{x})+\alpha[\kappa\\!*p(t,\cdot)](\mathbf{x})$
(1mb)

with $\lambda=\lambda_{\text{b}}-\lambda_{\text{d}}$ where
[$\kappa\\!*p(t,\cdot)](\mathbf{x})=\int_{\mathbb{R}^{d}}\kappa(x-y)p(t,y)dy$.

###  2.4 Second hybrid model

One of the motivations of the convergence result of this paper is to show the
possibility of replacing in simulations a multiparticle model (1g)-(1ib) by a
one with smaller number cells. However, the hybrid model (1ka)-(1kb) still
includes the number of proliferating cells. Therefore, we present here
alternative, much simpler version of the hybrid model that is related to the
limit of the inidivilual-based one. Namely, let us consider a single cell
which which moves according to the same rule as the particles in previous
models, and endow it with a variable $M(t)$ denoting its mass:

$\displaystyle[left={\empheqlbrace}]$ $\displaystyle
d\mathbf{X}(t)=\mathbf{b}(\mathbf{X}(t),\nabla\varrho(t,\mathbf{X}(t)))dt+\sigma\,dW(t),$
(1na) $\displaystyle
dM(t)=\lambda\bigl{(}\mathbf{X}(t),\varrho(t,\mathbf{X}(t))\bigr{)}M(t),$
(1nb) $\displaystyle\dfrac{\partial\varrho(t,\mathbf{x})}{\partial
t}=D\Delta\varrho(t,\mathbf{x})-r\varrho(t,\mathbf{x})+\alpha(K*\mu_{t})(\mathbf{x}),$
(1nc)

where

$\mu_{t}(A)=\operatorname{\mathbb{E}}\left[M(t)\mathds{1}_{A}(X(t))\right]=\int_{A\times[0,\infty)}m\,\mathbb{P}_{(\mathbf{X},M)}(d\mathbf{x},dm)\text{
for }A\in\mathcal{B}(\mathbb{R}^{d})$ (1o)

is the average mass in the area $A$.

######  Remark 3.

The measure $\mu_{t}$ is equal to $\bar{\mu}_{t}$ given by (1l). Note that
equations (1kb) and (1nc) are then the same. These facts will be proven and
used in the proof of wellposedness of the hybrid model

######  Remark 4.

Note moreover, that if $\mu_{0}$ is absolutely continuous, then the density of
$\mu_{t}$ (and therefore $\bar{\mu}_{t}$) satisfies (1ma). In that case all
equations (1kb), (1mb) and (1nc) coincide.

###  2.5 Other remarks

Usually, the aim of a rigorous proof of convergence of individual models to
macroscopic ones is twofold: from one hand side, it is convenient to be able
to derive the macroscopic model of population from the primitive rules that
govern the individuals and to know if and why the macroscopic model properly
approximates the collective behaviour. On the other hand side, often the
individual based models are easier to simulate and the proofs of convergence
play significant role in justifying the use of individual-based model
simulations as Monte Carlo methods for complicated PDEs. We have also the
third motivation that justifies the hybrid mean-field model. Sometimes (cf.
[12, 23, 33]) simulating the individual-based model is costly, as in our case
where it demands simulating branching diffusion coupled to PDE. On the other
hand side, solving only the macroscopic PDE can be insufficient, because we do
not follow the geometry of trajectories. In such a case we can use the hybrid
mean-field model in the following way: firstly solve numerically the non-
stochastic macroscopic model 1m for density of cells and concentration of
chemoreactant, and then simulate the branching diffusion 1s with already given
evolution of chemoreactant $\varrho$.

##  3 Assumptions and wellposedness

One of main goals of this paper is to make the definition of the individual
model as strict as possible while keeping it readable. To that aim, besides
the description in section 2.1, we need to define the state space of the
process, which can be done in various ways. We add to the space of positions
$\mathbb{R}^{d}$ an additional state $\phi$ denoting a nonexisting cell and we
describe the state of all particles as an infinite array of points from
$\mathbb{R}^{d}\cup\\{\phi\\}$ indexed by
$(i,\mathbbm{j})\in\mathbb{N}\times\mathbb{J}$ such that only finite numbers
of elements are different then $\phi$, that is

$\mathbb{X}=\left\\{\left(\mathbf{x}_{i,\mathbbm{j}}\right)_{(i,\mathbbm{j})\in\mathbb{N}\times\mathbb{J}}:\mathbf{x}_{i,\mathbbm{j}}\in\mathbb{R}^{d}\cup\\{\phi\\},\text{
such that
}\\#\\{\mathbf{x}_{i,\mathbbm{j}}:\mathbf{x}_{i,\mathbbm{j}}\neq\phi\\}\text{
is finite}\right\\}$ (1p)

with a natural metrics

$d_{\mathbb{X}}(\mathbbm{x},\mathbbm{y})=\max_{(i,\mathbbm{j})\in\mathbb{N}\times\mathbb{J}}|\mathbf{x}_{i,\mathbbm{j}}-\mathbf{y}_{i,\mathbbm{j}}|,\quad\text{
for
}\mathbbm{x}=\left(\mathbf{x}_{i,\mathbbm{j}}\right)_{(i,\mathbbm{j})\in\mathbb{N}\times\mathbb{J}},\mathbbm{y}=\left(\mathbf{y}_{i,\mathbbm{j}}\right)_{(i,\mathbbm{j})\in\mathbb{N}\times\mathbb{J}}\in\mathbb{X},$
(1q)

with a convention that $|\mathbf{x}-\phi|=1$ for any
$\mathbf{x}\in\mathbb{R}^{d}$. Because of the birth and death process the
trajectories of the microscopic process are not continuous. We apply the
standard convention to use cádlág paths, so the space of trajectories will be
the Skorokhod space $D_{\mathbb{X}}[0,\infty)$.

Now, we can formally describe the branching diffusion component of the
microscopic model as the solution to the following system of SDEs

$\displaystyle\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}\left(t\right)=$
$\displaystyle\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}\left(0\right)+\int_{0}^{t}\mathbf{b}(\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s),\nabla\varrho(t,\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s)))\,ds+\sigma\int_{0}^{t}\mathds{1}_{\mathbb{R}^{d}}\left(\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s)\right)\,dW_{i,\mathbbm{j}}(s),$
$\displaystyle+\int_{0}^{t}\chi^{n_{0}}_{\rm
b}(\mathbf{X}^{n_{0}}_{i,\mathbbm{j}^{\mkern-2.0mu\leftarrow}}(s^{-}),z)\mathcal{N}_{i,\mathbbm{j}^{\mkern-2.0mu\leftarrow}}(ds,dz)+\int_{0}^{t}\chi^{n_{0}}_{\rm
d}(\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s^{-}),z)\mathcal{N}_{i,\mathbbm{j}}(ds,dz)$
(1r)

for $i\in\\{1,\dots,n_{0}\\}$, $\mathbbm{j}\in\mathbb{J}$ with

$\displaystyle\chi^{n_{0}}_{\rm b}(\mathbf{x},z)=$
$\displaystyle\begin{cases}-\phi+\mathbf{x},\text{ if
}\mathbf{x}\neq\phi,\;z\leq\lambda_{\text{b}}(\mathbf{x},\varrho^{n_{0}}(\mathbf{x})),\\\
0,\text{ otherwise, }\end{cases}$ $\displaystyle\chi^{n_{0}}_{\rm
d}(\mathbf{x},z)=$ $\displaystyle\begin{cases}\phi,\text{ if
}\mathbf{x}\neq\phi,\;z\leq\lambda_{\text{b}}(\mathbf{x},\varrho^{n_{0}}(\mathbf{x}))+\lambda_{\text{d}}(\mathbf{x},\varrho^{n_{0}}(\mathbf{x})),\\\
0,\text{ otherwise, }\end{cases}$

where we use a convention that $\phi-\phi=0\in\mathbb{R}^{d}$ and
$\mathbf{x}+\phi=\phi$ for any $\mathbf{x}\in\mathbb{R}^{d}$ and
$\mathbf{b}(\phi,x)=0$ for $x\in\mathbb{R}$.

In a similar way, to fully describe the branching diffusion of the hybrid
model we add birth and death events to the equation (1ka) obtaining

$\displaystyle\mathbf{\bar{X}}_{\mathbbm{j}}\left(t\right)=$
$\displaystyle\mathbf{\bar{X}}_{\mathbbm{j}}\left(0\right)+\int_{0}^{t}\mathbf{b}(\mathbf{\bar{X}}_{\mathbbm{j}}(s),\nabla\varrho(t,\mathbf{\bar{X}}_{\mathbbm{j}}(s)))\,ds+\sigma\int_{0}^{t}\mathds{1}_{\mathbb{R}^{d}}\left(\mathbf{\bar{X}}_{\mathbbm{j}}(s)\right)\,dW_{1,\mathbbm{j}}(s),$
(1s) $\displaystyle+\int_{0}^{t}\chi_{\rm
b}(\mathbf{\bar{X}}_{\mathbbm{j}^{\mkern-2.0mu\leftarrow}}(s^{-}),z)\mathcal{N}_{1,\mathbbm{j}^{\mkern-2.0mu\leftarrow}}(ds,dz)+\int_{0}^{t}\chi_{\rm
d}(\mathbf{\bar{X}}_{\mathbbm{j}}(s^{-}),z)\mathcal{N}_{1,\mathbbm{j}}(ds,dz),$

for $\mathbbm{j}\in\mathbb{J}$, where $\chi_{\rm b}$ and $\chi_{\rm d}$ are
defined like $\chi^{n_{0}}_{\rm b}$ and $\chi^{n_{0}}_{\rm d}$ with
$\varrho^{n_{0}}$ replaced by $\varrho$.

The second component of the microscopic process is the solution to the PDE
(1ib). We use here a semigroup notation, so the solution will be denoted as
$\varrho^{n_{0}}_{t}(\mathbf{x})$ instead of $\varrho^{n_{0}}(t,\mathbf{x})$,
where $\rho^{n_{0}}_{t}\in C^{2}_{b}(\mathbb{R}^{d})$, and the space of
trajectories will be $C_{C^{2}_{b}(\mathbb{R}^{d})}{[0,\infty)}$ — the space
of continuous functions from $[0,\infty)$ to the space of bounded twice
continuously differentiable functions with bounded second derivatives
$C^{2}_{b}(\mathbb{R}^{d})$.

Let $S_{t}$ be a semigroup generated by the operator $Af(\mathbf{x})=D\Delta
f(\mathbf{x})-r\,f(\mathbf{x})$, $f\in C^{2}_{b}(\mathbb{R}^{d})$. Then we can
rewrite equation (1ib) as

$\varrho^{n_{0}}_{t}(\mathbf{x})=S_{t}\varrho_{0}(\mathbf{x})+\alpha\int_{0}^{t}S_{t-s}[K*\xi^{n_{0}}_{s}](\mathbf{x})\,ds$
(1t)

and, analogously, equation (1kb) as

$\varrho_{t}(\mathbf{x})=S_{t}\varrho_{0}(\mathbf{x})+\alpha\int_{0}^{t}S_{t-s}[K*\bar{\mu}_{s}](\mathbf{x})\,ds,$
(1u)

and similarly (1nc) with $\bar{\mu}_{s}$ replaced by $\mu_{s}$.

We will use the following assumptions:

  1. (A.2)

let $\sigma$, $D$, $r$, and $\alpha$ be positive constants.

  1. (A.3)

let $\lambda_{\text{b}},\lambda_{\text{d}}\in
C^{1}_{b}\left(\mathbb{R}^{d}\times\mathbb{R}_{+}\right)$ be nonnegative
functions and $\lambda_{\text{b}}+\lambda_{\text{d}}<\bar{\lambda}$ for some
constant $\bar{\lambda}>0$; let
$\mathbf{b}:\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}$ be bounded and
boundedly differentiable; let $\kappa\in C^{2}_{b}\left(\mathbb{R}^{d}\right)$
such that $\int_{\mathbb{R}^{d}}\kappa(\mathbf{x})d\mathbf{x}=1$ and
$\Delta\kappa$ is Lipschitz and let $L_{\kappa}$ be maximum of Lipschitz
coefficients for $\kappa$ and $\Delta\kappa$.

Now we can state the well-posedness theorems. Their proofs will be given in
section 5.1.

######  Theorem 1.

Suppose that $\varrho_{0}\in C^{2}_{b}(\mathbb{R}^{d})$ and
$\mathbf{X}^{n_{0}}_{i,\emptyset}(0)$ for $i=1,\dots,n_{0}$ are independent
random variables with probability law $\mu_{0}\in\mathcal{P}(\mathbb{R}^{d})$.
Let assumptions (A.1)-(A.3) be satisfied. For any $T>0$ there exists a process
$(\mathbbm{x}^{n_{0}},\varrho^{n_{0}})=\left(\bigl{(}\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}\bigr{)}_{i\in\\{1,\dots,n_{0}\\},\mathbbm{j}\in\mathbb{J}},\varrho^{n_{0}}\right)\in
D_{\mathbb{X}}{[0,T]}\times C_{C^{2}_{b}(\mathbb{R}^{d})}{[0,T]}$ described by
(1ia) and (1ib) with population dynamics given by (1g) and (1h) and it is
pathwise uniquely defined.

######  Theorem 2.

Suppose that $\varrho_{0}\in C^{2}_{b}(\mathbb{R}^{d})$ and
$\bar{\mathbf{X}}_{\emptyset}(0)$ is a random variable with probability law
$\mu_{0}\in\mathcal{P}(\mathbb{R}^{d})$. Let assumptions (A.1)-(A.3) be
satisfied. For any $T>0$ there exists a unique function $\varrho\in
C_{C^{2}_{b}(\mathbb{R}^{d})}{[0,T]}$ and the hybrid mean-field process
$\mathbbm{\bar{x}}=\bigl{(}\mathbf{\bar{X}}_{\mathbbm{j}}\bigr{)}_{\mathbbm{j}\in\mathbb{J}}$
with trajectories in $D_{\mathbb{X}}[0,T]$ described by (1k) with (1s) and it
is pathwise uniquely defined.

For completeness we state also the existence-uniqueness theorem for the
macroscopic model. We do not present its proof, which is straightforward
thanks to the regularization by $\kappa$. It goes by simple fixed point
argument.

######  Theorem 3.

Let assumptions (A.1)-(A.3) be satisfied. If $\varrho_{0}$ and $p_{0}$ are in
$C^{2}_{b}(\mathbb{R}^{d})$, then there exists a unique classical solution to
the system (1m).

Now, we have the wellposedness theorem for the second mean-field model.

######  Theorem 4.

Let assumptions (A.2)-(A.3) be satisfied and $W$ be a $d$-dimensional standard
Wiener process. Suppose that $\varrho(0,\cdot)=\varrho_{0}\in
C^{2}_{b}(\mathbb{R}^{d})$, ${\mathbf{X}}(0)$ is a random variable with
probability law $\mu_{0}\in\mathcal{P}(\mathbb{R}^{d})$ and $M(0)=1$. Then
system (1n)-(1o) has a unique solution.

##  4 Convergence results

The approach to the convergence of particle systems is based on the
propagation of chaos results from [11] and [34, 35], namely, the proof of
Theorem 6 is based on the fact that processes for all $n_{0}$ and the limit
are coupled by use of the same probability space and the same processes
$\left(W_{i,\mathbbm{j}}\right)_{i\in\mathbb{N},\mathbbm{j}\in\mathbb{J}}$ and
$\left(\mathcal{N}_{i,\mathbbm{j}}\right)_{i\in\mathbb{N},\mathbbm{j}\in\mathbb{J}}$.
Since the processes have not continuous trajectories it demands more delicate
approach. We will use the following notation:

  * 1.

denote by
$\mathbbm{x}^{n_{0}}(t)={\bigl{(}\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(t)\bigr{)}_{i\in\\{1,\dots,n_{0}\\},\mathbbm{j}\in\mathbb{J}}}$,
coupled with $\varrho^{n_{0}}_{t}$, $t\in[0,T]$, the solution of microscopic
model defined by (1g)-(1j).

  * 2.

let
$\mathbbm{x}^{n_{0}}_{1}(t)={\bigl{(}\mathbf{\bar{X}}^{n_{0}}_{1,\mathbbm{j}}(t)\bigr{)}_{\mathbbm{j}\in\mathbb{J}}}$,
$t\in[0,T]$ denote the branch of process $\mathbbm{x}^{n_{0}}$ starting from
the first cell $\mathbf{\bar{X}}^{n_{0}}_{1,\mathbbm{j}}(0)$ driven by
processes $\left(W_{1,\mathbbm{j}}\right)_{\mathbbm{j}\in\mathbb{J}}$ and
$\left(\mathcal{N}_{1,\mathbbm{j}}\right)_{\mathbbm{j}\in\mathbb{J}}$.

  * 3.

now we define a mean field processes
$\bar{\mathbbm{x}}(t)={\bigl{(}\mathbf{\bar{X}}_{\mathbbm{j}}(t)\bigr{)}_{\mathbbm{j}\in\mathbb{J}}}$
and $\varrho_{t}$, $t\in[0,T]$, defined by (1s)-(1kb) driven by processes
$\left(W_{1,\mathbbm{j}}\right)_{\mathbbm{j}\in\mathbb{J}}$ and
$\left(\mathcal{N}_{1,\mathbbm{j}}\right)_{\mathbbm{j}\in\mathbb{J}}$.

  * 4.

$\xi^{n_{0}}_{t}$ is a process given by (1j) which can be written as

$\xi^{n_{0}}_{t}=\frac{1}{n_{0}}\sum_{i=1}^{n_{0}}\sum_{\mathbbm{j}\in\mathbb{J}}\mathds{1}_{\mathbb{R}^{d}}(X^{n_{0}}_{i,\mathbbm{j}}(t))\delta_{X^{n_{0}}_{i,\mathbbm{j}}(t)}.$

Now we can state two convergence theorems.

######  Theorem 5.

Let assumptions (A.1)-(A.3) be satisfied. The for each $T>0$

  1. (i)

the sequence of processes $\xi^{n_{0}}_{t}$ converges in distribution to
$\bar{\mu}_{t}$ defined by (1l) on $D_{\mathcal{M}}[0,T]$ with Skorokhod
topology. The space $\mathcal{M}$ is considered here with a topology of vague
convergence.

  2. (ii)

the sequence of processes $\varrho^{n_{0}}_{t}$ converges to $\varrho_{t}$
given by (1u) in distribution on $C_{C_{b}(\mathbb{R}^{d})}[0,T]$.

Our next goal it to obtain pathwise convergence of $\mathbbm{x}^{n_{0}}_{1}$
to $\mathbbm{\bar{x}}$. We use the approach of [11], but since both processes
$\mathbbm{x}^{n_{0}}_{1}$ and $\mathbbm{\bar{x}}$ have discontinuous
trajectories the result is not as strong as there. The reason is, that jumps
can occur not only in different moments of time, but also in different
directions, and after such a jump the processes are irrevocably not more close
to each other. Nevertheless, we have the following fact.

######  Theorem 6.

Let assumptions (A.1)-(A.3) be satisfied. Then for each $T>0$

$\sup_{t\in[0,T]}d_{\mathbb{X}}\left(\mathbbm{x}^{n_{0}}_{1}(t),\mathbbm{\bar{x}}(t)\right)$
(1v)

converges to $0$ in probability.

##  5 Proofs

###  5.1 Wellposedness

In this section we prove firstly Theorem 1, then Theorem 4 and eventually
Theorem 2, since its proof depends on fragments of two previous proofs.

We will repeatedly use the following fact

######  Lemma 1.

For any $T>0$ and all $n_{0}\in\mathbb{N}$

$\operatorname{\mathbb{E}}\left[\sup_{t\in[0,T]}\langle
1,\xi^{n_{0}}_{t}\rangle\right]=\operatorname{\mathbb{E}}\left[\sup_{t\in[0,T]}\frac{1}{n_{0}}\langle
1,\mathbbm{x}^{n_{0}}(t)\rangle\right]<e^{\bar{\lambda}t}$

###### Proof.

Since the birth rate is $\lambda_{\text{b}}\leq\bar{\lambda}$, we know that
$\langle 1,\mathbbm{x}^{n_{0}}(t)$ (which is the number of cells) is for any
$t$ less (in the sense of distribution) than the number of particles in a
simple birth only (Yule) process with branching rate $\bar{\lambda}$. ∎

###### Proof of Theorem 1.

For notational convenience we assume $n_{0}=1$ and we omit indices $i$ and
$n_{0}$, besides $\varrho^{n_{0}}$ to avoid confusion with $\varrho$ from eqn.
(1k). The prove for any other $n_{0}$ follows analogously with $\mathbbm{j}$
replaced by $(i,\mathbbm{j})$.

Fix $T>0$. We will prove the existence and uniqueness in $[0,T]$. Since we
think about the solution to (1i) pathwise, the prove will be done for fixed
$\omega$. But firstly, we can neglect the set of probability 0. By Lemma 1,
the number of cells at any finite $t$ is this number is finite with
probability one. Therefore, if $\Omega_{0}$ is the set of those
$\omega\in\Omega$, that the microscopic model has infinite number of cells
before time $T$, then $\mathbb{P}(\Omega_{0})=0$. Let $\Omega_{1}$ be the zero
measure set for which any of the Poisson point processes
$\mathcal{N}_{\mathbbm{j}}$ has infinite number of points in
$[0,T]\times[0,\bar{\lambda}]$.

Let us now fix $\omega\in\Omega\setminus(\Omega_{0}\cup\Omega_{1})$ and
consider fixed trajectories of
$\left(W_{\mathbbm{j}}\right)_{\mathbbm{j}\in\mathbb{J}}$ and
$\left(\mathcal{N}_{\mathbbm{j}}\right)_{\mathbbm{j}\in\mathbb{J}}$ for this
$\omega$. For $\omega\not\in(\Omega_{0}\cup\Omega_{1})$ there is a finite
number of particles born before time $T$ and for any of those particles its
Poisson point process has finite number of points in
$[0,T]\times[0,\bar{\lambda}]$, so there is a finite number, say $\bar{n}$, of
points in all those Poisson point processes. Let us denote those times by
$\left(t_{k},z_{k}\right)_{k=1,\dots,\bar{n}}$ in the order of increasing
times and $t_{0}=0$. In between times $t_{k}$ the number of cells is constant,
so we solve recursively in the intervals $[t_{k},t_{k+1})$, $k=0,1,2,\dots$ a
deterministic system

$\begin{cases}\mathbf{X}_{\mathbbm{j}}(t)=\mathbf{X}_{\mathbbm{j}}(t_{k})+\int_{t_{k}}^{t}\mathbf{b}(\mathbf{X}_{\mathbbm{j}}(s),\nabla\varrho^{n_{0}}(s,\mathbf{X}_{\mathbbm{j}}(s)))ds+\sigma(W_{\mathbbm{j}}(t)-W_{\mathbbm{j}}(t_{k})),\text{
for }\mathbbm{j}\in\mathbb{J}_{k},\\\
\frac{\partial\varrho^{n_{0}}(t,\mathbf{x})}{\partial
t}=D\Delta\varrho^{n_{0}}(t,\mathbf{x})-r\varrho^{n_{0}}(t,\mathbf{x})+\alpha[K*\xi_{t}](\mathbf{x}),\end{cases}$
(1w)

with
$\xi_{t}=\sum_{{\mathbbm{j}\in\mathbb{J}_{k}}}\delta_{X_{\mathbbm{j}}(t)}$
where $\mathbb{J}_{k}$ is the set of indices of cells alive in interval
$(t_{k},t_{k+1})$, i.e. $\mathbb{J}_{0}=\\{\emptyset\\}$ and

$\displaystyle\mathbb{J}_{k}=$
$\displaystyle\mathbb{J}_{k-1}\cup\\{\mathbbm{j}i:\mathbbm{j}\in\mathbb{J}_{k-1},\,i=0,1,\,z_{k}<\lambda_{\text{b}}\bigl{(}X_{\mathbbm{j}}(t_{k}^{-}),\varrho^{n_{0}}(t_{k}^{-},X_{\mathbbm{j}}(t_{k}^{-}))\bigr{)}\\}$
$\displaystyle\setminus\\{\mathbbm{j}:\mathbbm{j}\in\mathbb{J}_{k-1},\,z_{k}<\lambda_{\text{b}}\bigl{(}X_{\mathbbm{j}}(t_{k}^{-}),\varrho^{n_{0}}(t_{k}^{-},X_{\mathbbm{j}}(t_{k}^{-}))\bigr{)}+\lambda_{\text{d}}\bigl{(}X_{\mathbbm{j}}(t_{k}),\varrho^{n_{0}}(t_{k},X_{\mathbbm{j}}(t_{k}))\bigr{)}\\}.$

The initial conditions at zero are
$\mathbf{X}_{\mathbbm{j}}(t_{0})=\mathbf{X}^{1}_{1,\emptyset}(0)$ and
$\varrho^{n_{0}}(0,\cdot)=\varrho_{0}$, and for $t_{k}>0$ recursively
$\varrho^{n_{0}}(t_{k},\cdot)=\varrho^{n_{0}}(t_{k}^{-},\cdot)$ and for
$\mathbbm{j}\in\mathbb{J}_{k}$

$\mathbf{X}_{\mathbbm{j}}(t_{k})=\begin{cases}\mathbf{X}_{\mathbbm{j}}(t_{k}^{-}),\text{
if }\mathbbm{j}\in\mathbb{J}_{k-1},\\\
\mathbf{X}_{\mathbbm{j}^{\mkern-2.0mu\leftarrow}}(t_{k}^{-})\text{ otherwise.
}\end{cases}$

The proof of existence and uniqueness of solutions to (1w) on
$(t_{k},t_{k+1})$ is straightforward and goes e.g. by Piccard type argument. ∎

The next proof uses classical methods from [34] and is similar to the proof of
Proposition 2.3. in [11].

###### Proof of Theorem 4.

We prove the existence on the interval $[0,T]$. The scheme of the proof is the
following: given a fixed function $\varrho:[0,T]\to C^{2}(\mathbb{R}^{d})$ we
solve the SDE (1na)-(1nb). Let $\tilde{\varrho}$ be a solution to (1u) with
$\mu$ given by (1o). Then we show that operator
$P:\varrho\mapsto\tilde{\varrho}$ is a contraction, so there is a unique
$\varrho$ satisfying (1n), and therefore a unique $X$ and $M$. We consider the
operator $P$ on the space $E=\\{\varrho\in
C_{[0,T]}(C^{2}_{b}(\mathbb{R}^{d})):\mathop{\rm
Lip}\varrho_{t}\wedge\mathop{\rm Lip}\nabla\varrho_{t}\leq L\text{ for
}t\in[0,T]\\}$ where $L$ is the maximum of $L_{\kappa}$ and the Lipschitz
coefficient of $\varrho_{0}$, with the Bielecki norm
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\varrho\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\gamma}=\max_{t\in[0,T]}e^{-\gamma
t}\left(\|\varrho_{t}\|_{\infty}+\|\nabla\varrho_{t}\|_{\infty}\right)$. Note
that, thanks to properties of heat kernel, $\varrho_{t}$ given by (1u) and
$\nabla\varphi_{t}$ are Lipschitz with coefficient $L$, so $P(E)\subset E$.

To prove that $P$ is contractive, take $\varrho^{(1)},\varrho^{(2)}\in
C_{[0,T]}\left(C^{2}(\mathbb{R}^{d})\right)$ and let $(X^{(i)},M^{(i)})$ be
the pathwise unique solutions to

$\begin{cases}\mathbf{X}^{(i)}(t)=\mathbf{X}(0)+\int_{0}^{t}\mathbf{b}(\mathbf{X}^{(i)}(s),\nabla\varrho^{(i)}(s,\mathbf{X}^{(i)}(s)))ds+\sigma\,W(t),\\\
M^{(i)}(t)=M(0)+\int_{0}^{t}\lambda\bigl{(}\mathbf{X}^{(i)}(s),\varrho^{(i)}(s,\mathbf{X}^{(i)}(s))\bigr{)}M^{(i)}(s)\,ds,\end{cases}$
(1x)

for $i=1,2$. Note that $M^{(i)}(t)\leq
M(0)e^{\bar{\lambda}t}=e^{\bar{\lambda}t}$. Thus we have

$\displaystyle|M^{(1)}(t)-M^{(2)}(t)|\leq$
$\displaystyle\int_{0}^{t}\Bigl{[}\bar{\lambda}|M^{(1)}(s)-M^{(2)}(s)|+e^{\bar{\lambda}t}L_{\lambda}(1+L)\|\mathbf{X}^{(1)}(s)-\mathbf{X}^{(2)}(s)\|$
$\displaystyle+e^{\bar{\lambda}t}L_{\lambda}\|\varrho_{s}^{(1)}-\varrho_{s}^{(2)}\|_{\infty}\Bigr{]}\,ds,$
$\displaystyle\|\mathbf{X}^{(1)}(t)-\mathbf{X}^{(2)}(t)\|\leq$
$\displaystyle\int_{0}^{t}\Bigl{[}e^{\bar{\lambda}}L_{\mathbf{b}}(1+L)\|\mathbf{X}^{(1)}(s)-\mathbf{X}^{(2)}(s)\|+L_{\mathbf{b}}\|\nabla\varrho_{s}^{(1)}-\nabla\varrho_{s}^{(2)}\|_{\infty}\Bigr{]}\,ds.$

So by Gronwall’s Lemma we have

$|M^{(1)}(t)-M^{(2)}(t)|+\|\mathbf{X}^{(1)}(t)-\mathbf{X}^{(2)}(t)\|\leq\\\
c_{1}\int_{0}^{t}\left(\|\varrho_{s}^{(1)}-\varrho_{s}^{(2)}\|_{\infty}+\|\nabla\varrho_{s}^{(1)}-\nabla\varrho_{s}^{(2)}\|_{\infty}\right)ds.$
(1y)

where $c_{1}$ (and similarly $c_{2}$ to $c_{4}$ below) is a constant depend
only on the bounds and Lipschitz coefficients of the functions $K$, $\nabla
K$, $\Delta K$, $\lambda$ and $\mathbf{b}$ and time $T$. Moreover

$\bigl{|}[\kappa(\mathbf{y}-X^{(1)}(t))M^{(1)}(t)-\kappa(\mathbf{y}-X^{(2)}(t))M^{(2)}(t)]\bigr{|}\leq
c_{2}(|M^{(1)}(t)-M^{(2)}(t)|+\|\mathbf{X}^{(1)}(t)-\mathbf{X}^{(2)}(t)\|).$

If $\mu^{(i)}$ is given by (1o) for $(\mathbf{X}^{(i)},M^{(i)})$ then

$\kappa*\mu_{t}^{(i)}(\mathbf{y})=\operatorname{\mathbb{E}}\left[\kappa(\mathbf{y}-\mathbf{X}^{(i)}(t))M^{(i)}(t)\right].$

Therefore, if
$\tilde{\varrho}^{(i)}_{t}(\mathbf{x})=S_{t}\varrho_{0}(\mathbf{x})+\alpha\int_{0}^{t}S_{t-s}[K*\mu^{(i)}_{s}](\mathbf{x})\,ds$
then

$|\tilde{\varrho}^{(1)}_{t}(\mathbf{x})-\tilde{\varrho}^{(2)}_{t}(\mathbf{x})|=\left|\alpha\int_{0}^{t}\int_{\mathbb{R}^{d}}p(t-s,\mathbf{x},\mathbf{y})[K*\mu^{(1)}_{s}-K*\mu^{(2)}_{s}](\mathbf{y})\,d\mathbf{y}\,ds\right|\\\
\leq
c_{3}\int_{0}^{t}\int_{0}^{r}\left(\|\varrho_{s}^{(1)}-\varrho_{s}^{(2)}\|_{\infty}+\|\nabla\varrho_{s}^{(1)}-\nabla\varrho_{s}^{(2)}\|_{\infty}\right)\,ds\,dr$

and likewise

$|\nabla\tilde{\varrho}^{(1)}_{t}(\mathbf{x})-\nabla\tilde{\varrho}^{(2)}_{t}(\mathbf{x})|\leq
c_{4}\int_{0}^{t}\int_{0}^{r}\left(\|\varrho_{s}^{(1)}-\varrho_{s}^{(2)}\|_{\infty}+\|\nabla\varrho_{s}^{(1)}-\nabla\varrho_{s}^{(2)}\|_{\infty}\right)\,ds\,dr.$

Therefore,

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\tilde{\varrho}^{(1)}-\tilde{\varrho}^{(2)}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}\leq(c_{3}+c_{4})\int_{0}^{t}\int_{0}^{r}e^{-\gamma(t-s)}{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\tilde{\varrho}^{(1)}-\tilde{\varrho}^{(2)}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}dsdr=\\\
(c_{3}+c_{4})\frac{e^{-\gamma t}(-\gamma t+e^{\gamma
t}-1)}{\gamma^{2}}{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\tilde{\varrho}^{(1)}_{s}-\tilde{\varrho}^{(2)}_{s}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|},$

so, for sufficiently large $\gamma$, $P$ is contractive in
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\cdot\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\gamma}$.
∎

Now we are ready to prove the well-posedness of the mean-field model.

###### Proof of Theorem 2.

The scheme will be the following. For any function $\varrho\in
C_{C^{1}_{b}(\mathbb{R}^{d})}[0,T]$ we notice the existence and uniqueness of
branching diffusion process given by (1s) (a proof can be done as in the proof
of Theorem 1). Then we show that $\bar{\mu}$ given by (1l) is equal to $\mu$
given by (1o) for $(\mathbf{X},M)$ obtained as solution to (1x) with the same
$\varrho$. That means that, by Theorem 4, there exists a unique $\varrho$ such
that (1k) is satisfied.

To this aim, fix $\varrho:[0,T]\to C^{2}_{b}(\mathbb{R}^{d})$ continuous in
time and let process
$\bar{\mathbbm{x}}(t)={\bigl{(}\mathbf{\bar{X}}_{\mathbbm{j}}(t)\bigr{)}_{\mathbbm{j}\in\mathbb{J}}}$,
$t\in[0,T]$ be the solution to (1s) with this fixed $\varrho$. For
$\varphi:\mathbb{R}^{d}\cup\\{\phi\\}\to\mathbb{R}$ such that
$\varphi(\phi)=0$ and $\varphi|_{\mathbb{R}^{d}}\in C^{2}_{b}(\mathbb{R}^{d})$
denote

$\langle\varphi,\mathbbm{\bar{x}}(t)\rangle=\sum_{\mathbbm{j}\in\mathbb{J}}\varphi(\mathbf{\bar{X}}_{\mathbbm{j}}(t)).$

Note that this sum is finite. By Itô’s Lemma we have

$\displaystyle\langle\varphi,\mathbbm{\bar{x}}(t)\rangle=$
$\displaystyle\langle\varphi,\mathbbm{x}(0)\rangle+\int_{0}^{t}\left\langle\mathbf{b}(\cdot,\nabla\varrho(t,\cdot))\nabla\varphi(\cdot)+\frac{\sigma^{2}}{2}\Delta\varphi(\cdot),\mathbbm{\bar{x}}(s)\right\rangle
ds+\sigma\sum_{\mathbbm{j}\in\mathbb{J}}\int_{0}^{t}\nabla\varphi(\mathbf{\bar{X}}_{\mathbbm{j}}(s))dW_{1,\mathbbm{j}}(s)$
$\displaystyle-\sum_{\mathbbm{j}\in\mathbb{J}}\int_{0}^{t}\varphi(X_{\mathbbm{j}}(s^{-}))\mathds{1}_{[0,\lambda_{\text{b}}(X_{\mathbbm{j}}(s^{-}),\rho(s^{-},\mathbf{\bar{X}}_{\mathbbm{j}}(s^{-})))+\lambda_{\text{d}}(X_{\mathbbm{j}}(s^{-}),\rho(s^{-},\mathbf{\bar{X}}_{\mathbbm{j}}(s^{-})))]}(z)\mathcal{N}_{1,\mathbbm{j}}(ds,dz)$
$\displaystyle+\sum_{\mathbbm{j}\in\mathbb{J}}\int_{0}^{t}2\varphi(\mathbf{\bar{X}}_{\mathbbm{j}}(s^{-}))\mathds{1}_{[0,\lambda_{\text{b}}(X_{\mathbbm{j}}(s^{-}),\rho(s^{-},\mathbf{\bar{X}}_{\mathbbm{j}}(s^{-})))]}(z)\mathcal{N}_{1,\mathbbm{j}}(ds,dz)$

and thus

$\operatorname{\mathbb{E}}\langle\varphi,\mathbbm{x}(t)\rangle=\operatorname{\mathbb{E}}\langle\varphi,\mathbbm{x}(0)\rangle+\operatorname{\mathbb{E}}\int_{0}^{t}\left\langle
B_{\varrho,s}\varphi,\mathbbm{x}(s)\right\rangle+\left\langle\lambda(\cdot,\varrho(s,\cdot))\varphi(\cdot),\mathbbm{x}(s)\right\rangle
ds,$

where
$B_{\varrho,s}\varphi(\mathbf{x})=\frac{\sigma^{2}}{2}\Delta\varphi(\mathbf{x})+\mathbf{b}(\mathbf{x},\nabla\varrho(s,\mathbf{x}))\nabla\varphi(\mathbf{x})$
for $\mathbf{x}\in\mathbb{R}^{d}$ and $B_{\varrho,s}\varphi(\phi)=0$, and
$\lambda=\lambda_{\text{b}}-\lambda_{\text{d}}$.

Let
$\bar{\xi}_{t}=\sum_{{\mathbbm{j}\in\mathbb{J}}}\mathds{1}_{\mathbb{R}^{d}}(\mathbb{\bar{X}}_{\mathbbm{j}}(t))\delta_{\mathbb{\bar{X}}_{\mathbbm{j}}(t)}$
and let
$\langle\varphi,\bar{\xi}_{t}\rangle=\int_{\mathbb{R}^{d}}\varphi(\mathbf{x})\bar{\xi}_{t}(d\mathbf{x})$.
Then
$\langle\varphi,\bar{\xi}_{t}\rangle=\langle\varphi,\mathbbm{\bar{x}}(t)\rangle$
and

$\displaystyle\langle\varphi,\operatorname{\mathbb{E}}\bar{\xi}_{t}\rangle=$
$\displaystyle\langle\varphi,\operatorname{\mathbb{E}}\bar{\xi}_{0}\rangle+\int_{0}^{t}\left\langle
B_{\varrho,s}\varphi,\operatorname{\mathbb{E}}\bar{\xi}_{s}\right\rangle+\left\langle\lambda(\cdot,\varrho(s,\cdot))\varphi(\cdot),\operatorname{\mathbb{E}}\bar{\xi}_{s}\right\rangle
ds,$

which means that

$\langle\varphi,\bar{\mu}_{t}\rangle=\langle\varphi,\bar{\mu}_{0}\rangle+\int_{0}^{t}\left\langle
B_{\varrho,s}\varphi,\bar{\mu}_{s}\right\rangle+\left\langle\lambda(\cdot,\varrho(s,\cdot))\varphi(\cdot),\bar{\mu}_{s}\right\rangle
ds.$ (1z)

This is the week version of Equation (1ma) and it is well known that it admits
a unique solution which is absolutely continuous with respect to Lebesgue
measure for $t>0$ even if $\bar{\mu}_{0}$ is not.

Let now $(\mathbf{X},M)$ be a process obtained as a solution to (1x) with
given $\varrho$ and $\varphi\in C^{2}_{b}(\mathbb{R}^{d})$. Then, by Itô
formula we have

$\displaystyle\varphi(\mathbf{X}(t))M(t)=$
$\displaystyle\varphi(\mathbf{X}(0))M(0)+\int_{0}^{t}\biggl{[}M(s)\nabla\varphi(\mathbf{X}(s))\cdot\mathbf{b}(\mathbf{X}(s),\nabla\varrho(s,\mathbf{X}(s)))$
$\displaystyle+\varphi(\mathbf{X}(s))\lambda\bigl{(}\mathbf{X}(s),\varrho(s,\mathbf{X}(s))\bigr{)}M(s)+\frac{\sigma^{2}}{2}\Delta\varphi(\mathbf{X}(s))\biggl{]}\,ds$
$\displaystyle-\int_{0}^{t}\nabla\varphi(\mathbf{X}(s))\cdot\mathbf{b}(\mathbf{X}(s),\nabla\varrho(s,\mathbf{X}(s)))dW(s)$

Note that for $\mu_{t}$ given by (1o) we have
$\langle\varphi,\mu_{t}\rangle=\int_{\mathbb{R}^{d}}\varphi(\mathbf{x})\mu_{t}(d\mathbf{x})=\operatorname{\mathbb{E}}[\varphi(\mathbf{X}(t))M(t)]$.
Taking expectation on both sides of the equation above we get

$\langle\varphi,\mu_{t}\rangle=\int_{0}^{t}\left\langle
B_{\varrho,s}\varphi(\cdot)+\lambda(\cdot,\varrho(s,\cdot))\varphi(\cdot),\mu_{s}\right\rangle
ds.$ (1aa)

which is exactly the same as (1z). ∎

###  5.2 Proof of convergence

###### Proof of Theorem 5.

In order to prove the point (i) we check that the sequence $\xi^{n_{0}}_{t}$
is tight on $D_{\mathcal{M}[0,t]}$ and then we check that the limit of any
subsequence has to coincide with $\bar{\mu}_{t}$. Similarly, we prove
tightness of $\varrho^{n_{0}}_{t}$ in $C_{C^{1}_{b}(\mathbb{R}^{d})}[0,T]$ and
check that the limit has to satisfy (1u).

Tightness of $\\{\xi^{n_{0}}\\}_{n_{0}\in\mathbb{N}}$. The process
$\xi^{n_{0}}$ has values in the space $\mathcal{M}$ of finite positive Radon
measures on $\mathbb{R}^{d}$. Note that the $\mathcal{M}$ with the vague
convergence topology can be metrizable, eg. with metric

$d_{\mathcal{M}}(\mu,\nu)=\sum_{k=1}^{\infty}\frac{1}{2^{k}}\min\\{1,\langle\varphi_{k},\mu-\nu\rangle\\}$

with some sequence $\varphi_{k}\in C_{c}(\mathbb{R}^{d})$ (cf. [36, Section
§31]) in such a way that $\mathcal{M}$ is complete. Moreover, set
$H\subset\mathcal{M}$ is vaguely relatively compact if and only if

$\sup_{\mu\in H}|\langle f,\mu\rangle|\quad\text{ for all }f\in
C_{c}(\mathbb{R}^{d}),$

where one can take $(\varphi_{k})_{k\in\mathbb{N}}$ instead of all $f\in
C_{c}(\mathbb{R}^{d})$. Therefore, Proposition 1.7 from [37, Chapter 4] holds
for processes with values in $(\mathcal{M},d_{\mathcal{M}})$. Now, thanks to
Aldous criterion (see, eg. [38, Chapter VI, Theorem 4.5]) for the relative
compactness of $\\{\xi^{n_{0}}_{t}\\}$ it suffices to check for all
$\varphi_{k}$ that for any $\varepsilon>0$ there exists $M>0$ such that

$\mathbb{P}\left(\langle\varphi_{k},\xi^{n_{0}}_{t}\rangle>M\right)<\varepsilon,\text{
for all }t\in[0,T]\text{ and }n_{0}\in\mathbb{N}$ (1ab)

and

$\lim_{\gamma\to
0}\limsup_{n_{0}\to\infty}\sup_{\tau\in\mathcal{T}_{T},\theta<\gamma}\mathbb{P}\left(|\langle\varphi_{k},\xi^{n_{0}}_{\tau+\theta}\rangle-\langle\varphi_{k},\xi^{n_{0}}_{\tau}\rangle|>\varepsilon\right)=0,$
(1ac)

where $\mathcal{T}_{T}$ is the set of all stopping times bounded by $T$. Note
that (1ab) follows by Markov’s inequality from Lemma 1. To prove (1ac), using
Itô’s Lemma we calculate

$\displaystyle\langle\varphi_{k},\xi^{n_{0}}_{t}\rangle=$
$\displaystyle\langle\varphi_{k},\tfrac{1}{n_{0}}\mathbbm{x}^{n_{0}}(t)\rangle=\langle\varphi_{k},\tfrac{1}{n_{0}}\mathbbm{x}^{n_{0}}(0)\rangle$
$\displaystyle+\frac{1}{n_{0}}\int_{0}^{t}\left(\sum_{i=1}^{n_{0}}\sum_{\mathbbm{j}\in\mathbb{J}}\mathbf{b}(\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s),\nabla\varrho^{n_{0}}(t,\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s)))\nabla\varphi_{k}(\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s))+\frac{\sigma^{2}}{2}\Delta\varphi_{k}(\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s))\right)ds$
$\displaystyle+\frac{1}{n_{0}}\int_{0}^{t}\sigma\sum_{i=1}^{n_{0}}\sum_{\mathbbm{j}\in\mathbb{J}}\nabla\varphi_{k}(\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s))dW_{i,\mathbbm{j}}(s)+\frac{1}{n_{0}}\sum_{i=1}^{n_{0}}\sum_{s\leq
t}\left(\langle\varphi_{k}(\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s))-\varphi_{k}(\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s^{-}))\rangle\right)$
$\displaystyle=$
$\displaystyle\langle\varphi_{k},\xi^{n_{0}}_{0}\rangle+\int_{0}^{t}\left\langle\mathbf{b}(\cdot,\nabla\varrho^{n_{0}}(s,\cdot))\nabla\varphi_{k}(\cdot)+\frac{\sigma^{2}}{2}\Delta\varphi_{k}(\cdot)+\lambda(\cdot,\varrho^{n_{0}}(s,\cdot))\varphi_{k}(\cdot),\xi^{n_{0}}_{s}\right\rangle
ds$ $\displaystyle M^{n_{0}}_{1,k}(t)\begin{cases}\\\\[8.61108pt] \end{cases}$
$\displaystyle+\frac{\sigma}{n_{0}}\sum_{i=1}^{n_{0}}\sum_{\mathbbm{j}\in\mathbb{J}}\int_{0}^{t}\nabla\varphi_{k}(\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s))dW_{i,\mathbbm{j}}(s)$
$\displaystyle M^{n_{0}}_{2,k}(t)\begin{cases}\\\ \\\ \\\ \\\ \\\ \\\
\end{cases}$ $\displaystyle\begin{aligned}
&-\frac{1}{n_{0}}\sum_{i=1}^{n_{0}}\sum_{\mathbbm{j}\in\mathbb{J}}\int_{0}^{t}\varphi_{k}(\mathbf{X}^{n_{0}}_{i,\mathbbm{j}})\mathds{1}_{[0,\lambda_{\text{b}}(X_{\mathbbm{j}}(s^{-}),\rho^{n_{0}}(s^{-},\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s^{-})))+\lambda_{\text{d}}(X_{\mathbbm{j}}(s^{-}),\rho^{n_{0}}(s^{-},\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s^{-})))]}(z)\mathcal{N}_{i,\mathbbm{j}}(ds,dz)\\\
&+\frac{1}{n_{0}}\sum_{i=1}^{n_{0}}\sum_{\mathbbm{j}\in\mathbb{J}}\int_{0}^{t}2\varphi_{k}(\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s^{-}))\mathds{1}_{[0,\lambda_{\text{b}}(X_{\mathbbm{j}}(s^{-}),\rho^{n_{0}}(s^{-},\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s^{-})))]}(z)\mathcal{N}_{i,\mathbbm{j}}(ds,dz)\\\
&-\int_{0}^{t}\left\langle\lambda(\cdot,\varrho^{n_{0}}(s,\cdot))\varphi_{k}(\cdot),\xi^{n_{0}}_{s}\right\rangle
ds,\end{aligned}$

where $M^{n_{0}}_{1,k}(t)$ and $M^{n_{0}}_{1,k}(t)$ are martingales.
Therefore,

$\displaystyle\langle\varphi_{k},\xi^{n_{0}}_{\tau+\theta}-\xi^{n_{0}}_{\tau}\rangle=$
$\displaystyle\int_{\tau}^{\tau+\theta}\left\langle\mathbf{b}(\cdot,\nabla\varrho^{n_{0}}(s,\cdot))\nabla\varphi_{k}(\cdot)+\frac{\sigma^{2}}{2}\Delta\varphi_{k}(\cdot)+\lambda(\cdot,\varrho^{n_{0}}(s,\cdot))\varphi_{k}(\cdot),\xi^{n_{0}}_{s}\right\rangle
ds$
$\displaystyle+M^{n_{0}}_{1,k}(\tau+\theta)-M^{n_{0}}_{1,k}(\tau)+M^{n_{0}}_{2,k}(\tau+\theta)-M^{n_{0}}_{2,k}(\tau)$

The integral over $ds$ can be estimated by a constant times
$\theta\sup_{s\in[0,T]}\langle 1,\xi^{n_{0}}_{s}\rangle$, so, since
$\theta\leq\gamma$, by Lemma 1 and Markov inequality, probability that it is
greater then $\varepsilon$ goes to zero as $\gamma\to 0$. By Itô’s Lemma,

$(M^{n_{0}}_{1,k}(t))^{2}=\frac{\sigma^{2}}{n_{0}^{2}}\sum_{i=1}^{n_{0}}\sum_{\mathbbm{j}\in\mathbb{J}}\int_{0}^{t}\left(\nabla\varphi_{k}(\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s))\right)^{2}ds+\tilde{M}^{n_{0}}_{1,k}(t),$

where $\tilde{M}^{n_{0}}_{1,k}(t)$ is a martingale, so

$\operatorname{\mathbb{E}}\left[\left(M^{n_{0}}_{1,k}(\tau+\theta)-M^{n_{0}}_{1,k}(\tau)\right)^{2}\right]=\operatorname{\mathbb{E}}\left[\left(M^{n_{0}}_{1,k}(\tau+\theta)\right)^{2}\right]-\operatorname{\mathbb{E}}\left[\left(M^{n_{0}}_{1,k}(\tau)\right)^{2}\right]=\\\
\operatorname{\mathbb{E}}\left[\frac{\sigma^{2}}{n_{0}^{2}}\sum_{i=1}^{n_{0}}\sum_{\mathbbm{j}\in\mathbb{J}}\int_{\tau}^{\tau+\theta}\left(\nabla\varphi_{k}(\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s))\right)^{2}ds\right]\leq\theta\frac{\sigma^{2}}{n_{0}}\|\nabla\varphi_{k}\|_{\infty}\operatorname{\mathbb{E}}\left[\sup_{s\in[0,T]}\frac{\langle
1,\xi^{n_{0}}\rangle}{n_{0}}\right].$ (1ad)

Similarly, using Itô’s formula again we get

$\displaystyle(M^{n_{0}}_{2,k}(t))^{2}=$
$\displaystyle-\int_{0}^{t}2M^{n_{0}}_{2,k}(s)\left\langle\lambda(\cdot,\varrho^{n_{0}}(s,\cdot))\varphi_{k}(\cdot),\xi^{n_{0}}_{s}\right\rangle
ds$
$\displaystyle-\sum_{i=1}^{n_{0}}\sum_{\mathbbm{j}\in\mathbb{J}}\int_{0}^{t}\left(\frac{1}{n_{0}^{2}}\varphi_{k}(\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s^{-}))^{2}-\frac{2}{n_{0}}M^{n_{0}}_{2,k}(s)\varphi_{k}(\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s^{-}))\right)$
$\displaystyle\qquad\qquad\qquad\times\mathds{1}_{[0,\lambda_{\text{b}}(\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s^{-}),\varrho^{n_{0}}(s^{-},\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s^{-})))+\lambda_{\text{d}}(\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s^{-}),\varrho^{n_{0}}(s^{-},\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s^{-})))]}(z)\mathcal{N}_{i,\mathbbm{j}}(ds,dz)$
$\displaystyle+\sum_{i=1}^{n_{0}}\sum_{\mathbbm{j}\in\mathbb{J}}\int_{0}^{t}\left(\frac{4}{n_{0}^{2}}\varphi_{k}\left(\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s^{-})\right)^{2}+\frac{4}{n_{0}}M^{n_{0}}_{2,k}(s)\varphi_{k}\left(\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s^{-})\right)\right)$
$\displaystyle\qquad\qquad\qquad\times\mathds{1}_{[0,\lambda_{\text{b}}(\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s^{-}),\varrho^{n_{0}}(s^{-},\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(s^{-})))]}(z)\mathcal{N}_{i,\mathbbm{j}}(ds,dz)$
$\displaystyle=$
$\displaystyle\frac{1}{n_{0}}\int_{0}^{t}\left\langle[3\lambda_{\text{b}}(\cdot,\varrho^{n_{0}}(s^{-},\cdot))-\lambda_{\text{d}}(\cdot,\varrho^{n_{0}}(s^{-},\cdot))]\varphi_{k}(\cdot)^{2},\xi^{n_{0}}_{s}\right\rangle
ds+\tilde{M}^{n_{0}}_{2,k}(t).$

where $\tilde{M}^{n_{0}}_{2,k}(t)$ is a martingale, thus again

$\operatorname{\mathbb{E}}\left[\left(M^{n_{0}}_{2,k}(\tau+\theta)-M^{n_{0}}_{2,k}(\tau)\right)^{2}\right]\leq\theta\frac{3\bar{\lambda}}{n_{0}}\|\varphi_{k}^{2}\|_{\infty}\operatorname{\mathbb{E}}\left[\sup_{s\in[0,T]}\frac{\langle
1,\xi^{n_{0}}\rangle}{n_{0}}\right],$ (1ae)

which completes the proof of (1ac).

Tightness of $\\{\varrho^{n_{0}}\\}_{n_{0}\in\mathbb{N}}$. Although it would
be sufficient to use the topology of locally uniform convergence, we need a
stronger convergence in the next proof, so let us consider
$C^{1}_{b}(\mathbb{R}^{d})$ with the topology of locally uniform convergence
of function and its derivative, that is we us a norm

$\|f\|_{C^{2}_{b}}=\sum_{R=1}^{\infty}\frac{1}{2^{R}}\left(\sup_{x\in
B(0,R)}|f(x)|+\sup_{x\in B(0,R)}\|\nabla f(x)\|\right).$ (1af)

Note that a set $K_{M}=\\{f\in C^{1}_{b}(\mathbb{R}^{d}):\|f\|_{\infty}\leq
M,\|\nabla f\|_{\infty}\leq M,\|\text{Hess}f\|_{\infty}\leq M\\},$ where
$\|\text{Hess}f\|_{\infty}=\sup_{\mathbf{x}\in\mathbb{R}^{d}}\max_{1\leq
i,j\leq d}|\partial_{i}\partial_{j}f(\mathbf{x})|$, is relatively compact in
this norm. Using the version of Ascoli Theorem (see eg. [39, Theorem 47.1]) we
know that a family
$\mathcal{K}_{M}\subset\mathcal{C}_{T}=C_{C^{1}_{b}(\mathbb{R}^{d})}[0,T]$ of
functions $g$, which are equicontinuous in $t$ and such that
$\\{g(t):g\in\mathcal{K}_{M}\\}\subset K_{M}$ for each $t\in[0,T]$, is
relatively compact. Now, in order to prove tightness of $\varrho^{n_{0}}_{t}$,
we need to check that for any $\varepsilon>0$ there exists $M>0$ such that
$\mathbb{P}(\varrho^{n_{0}}\in\mathcal{K}_{M})>1-\varepsilon$. To that end,
recall that
$\varrho^{n_{0}}_{t}=S_{t}\rho_{0}+\alpha\int_{0}^{t}S_{t-s}\kappa*\xi^{n_{0}}_{s}(x)ds$
where the first summand is continuous in $\mathcal{C}_{T}$ and the latter is
Lipschitz with probability $1-\varepsilon$, because

$\left|\alpha\int_{t}^{t+\theta}S_{t+\theta-s}[\kappa*\xi^{n_{0}}_{s}](\mathbf{x})\,ds\right|\leq\theta\alpha\|\kappa\|_{\infty}\sup_{s\in[0,T]}\langle
1,\xi^{n_{0}}_{s}\rangle$

and

$\left|\alpha\nabla\int_{t}^{t+\theta}S_{t+\theta-s}[\kappa*\xi^{n_{0}}_{s}](\mathbf{x})\,ds\right|\leq\theta\alpha\|\nabla\kappa\|_{\infty}\sup_{s\in[0,T]}\langle
1,\xi^{n_{0}}_{s}\rangle.$

Moreover, we have

$|\varrho^{n_{0}}_{t}(x)|\leq|S_{t}\rho_{0}(x)|+\sup_{s\in[0,T]}\|\kappa*\xi^{n_{0}}_{s}\|_{\infty}\leq\|\rho_{0}\|_{\infty}+t\|\kappa\|_{\infty}\sup_{s\in[0,T]}\langle
1,\xi^{n_{0}}_{s}\rangle,$

similarly

$|\nabla\varrho^{n_{0}}_{t}(x)|\leq\|\nabla\rho_{0}\|_{\infty}+t\,\|\nabla\kappa\|_{\infty}\sup_{s\in[0,T]}\langle
1,\xi^{n_{0}}_{s}\rangle,$ (1ag)

and further

$|\partial_{i}\partial_{j}\varrho^{n_{0}}_{t}(x)|\leq\|\partial_{i}\partial_{j}\rho_{0}\|_{\infty}+t\,\|\partial_{i}\partial_{j}\kappa\|_{\infty}\sup_{s\in[0,T]}\langle
1,\xi^{n_{0}}_{s}\rangle.$ (1ah)

These estimates with Lemma 1 and Markov’s inequality complete the proof of
tightness.

Identification of the limit. By the similar estimates as in (1ad) and (1ae) we
get that the limit has to satisfy (1z) for any $\varphi\in
C^{2}_{c}(\mathbb{R}^{d})$ and (1u) which admit a unique solution. ∎

###  5.3 Proof of Theorem 6

Fix $T>0$ and $\varepsilon>0$. We use here the coupling of
${\mathbbm{x}}^{n_{0}}_{1}(t)$ and $\bar{\mathbbm{x}}(t)$ obtained by using
the same processes $W_{1,\mathbbm{j}}$ and $\mathcal{N}_{1,\mathbbm{j}}$, and
the fact from Theorem 5 that $\varrho^{n_{0}}$ converges in probability to
$\varrho$.

Let $\bar{\sigma}_{\mathbbm{j}}$ and $\bar{\tau}_{\mathbbm{j}}$ denote times
of birth and death, respectively, of the $\mathbbm{j}$-th particle of
$\bar{\mathbbm{x}}(t)$ and let us construct such a process
$\mathbbm{\tilde{x}}^{n_{0}}_{1}$ that its $\mathbbm{j}$-th particle lives
from $\bar{\sigma}_{\mathbbm{j}}$ to $\bar{\tau}_{\mathbbm{j}}$ and moves
during this time according to the equation

$\mathbf{\tilde{X}}^{n_{0}}_{1,\mathbbm{j}}(t)=\mathbf{\tilde{X}}^{n_{0}}_{1,\mathbbm{j}}\left(\bar{\sigma}_{\mathbbm{j}}\right)+\int_{\bar{\sigma}_{\mathbbm{j}}}^{t}\mathbf{b}(\mathbf{\tilde{X}}^{n_{0}}_{1,\mathbbm{j}}(s),\nabla\varrho^{n_{0}}(s,\mathbf{\tilde{X}}^{n_{0}}_{i,\mathbbm{j}}(s)))\,ds+\sigma\left(W_{1,\mathbbm{j}}(t)-W_{1,\mathbbm{j}}(\bar{\sigma}_{\mathbbm{j}})\right),$

for $t\in\left[\bar{\sigma}_{\mathbbm{j}},\bar{\tau}_{\mathbbm{j}}\right)$. It
means that the particles of $\mathbbm{\tilde{x}}^{n_{0}}_{1}$ die and are born
in the same times as particles of $\mathbbm{\bar{x}}$ but their dynamics is
the same as the dynamics of $\mathbbm{x}^{n_{0}}_{1}$. The idea is to prove,
that for $n_{0}$ large enough with high probability
$\mathbbm{\tilde{x}}^{n_{0}}_{1}$ is close to $\mathbbm{\bar{x}}$ and equal to
$\mathbbm{x}^{n_{0}}_{1}$.

######  Lemma 2.

Fix $T>0$. For any $\eta>0$ we can find a set $\Omega_{\eta}$ such that
$\mathbb{P}(\Omega_{\eta})>1-\eta$ and

  1. (i)

there exists $\bar{N}>0$ such that $\sup_{t\in[0,T]}\langle
1,\bar{\mathbbm{x}}(t)\rangle\leq\bar{N}$ i.e. there is at most $\bar{N}$
particles of $\bar{\mathbbm{x}}(t)$ alive to time $T$

  2. (ii)

there exists $R>0$ such that all particles of $\bar{\mathbbm{x}}(t)$ live in
the ball of radius $R$, i.e.

$\sup_{t\in[0,T]}\max_{\mathbbm{j}\in\mathbb{J}}\|\mathbf{\bar{X}}_{\mathbbm{j}}\|\leq
R$

(there is at most $\bar{N}$ particles in the maximum).

for $\omega\in\Omega_{\eta}$.

###### Proof.

The first point is a simple consequence of Lemma 1. Once we have finite nuber
of particles, their positions are described by a finite number of Itô
equations with bounded drift $\mathbf{b}$, so (ii) obviously follows. ∎

Let us denote by $\mathbb{J}_{\bar{N}}$ the (finite) set of all indices of the
length at most $\bar{N}$. Apparently, if there were not more than $\bar{N}$
particles of $\bar{\mathbbm{x}}(t)$ up to time $T$, then their indices are in
$\mathbb{J}_{\bar{N}}$.

######  Lemma 3.

For any $\eta>0$ there exists $\delta>0$ such that if

$\sup_{t\in[0,T]}\left(\sup_{\mathbf{x}\in
B(0,R)}\|\varrho^{n_{0}}(\mathbf{x})-\varrho(\mathbf{x})\|+\sup_{\mathbf{x}\in
B(0,R)}\|\nabla\varrho^{n_{0}}(\mathbf{x})-\nabla\varrho(\mathbf{x})\|\right)<\delta,$
(1ai)

where $R$ is from Lemma 2, then the probability that process
$\mathbbm{\tilde{x}}^{n_{0}}_{1}$ is different than $\mathbbm{x}^{n_{0}}_{1}$
is less than $3\eta$.

###### Proof.

Throughout the proof we assume we are in $\Omega_{\eta}$ from Lemma 2,
$\bar{N}$ and $R$ are as in Lemma 2 and every particle of $\mathbbm{\bar{x}}$
alive during $[0,T]$ has index $\mathbbm{j}\in\mathbb{J}_{\bar{N}}$. The
movement of particles of $\mathbbm{\tilde{x}}^{n_{0}}_{1}$ and
$\mathbbm{x}^{n_{0}}_{1}$ is given by the same equation, so the processes are
different if and only if any time of birth or death is different. Recall that
$\sigma^{n_{0}}_{1,\mathbbm{j}}$ and $\tau^{n_{0}}_{1,\mathbbm{j}}$ are given
by (1g) and (1h), and $\bar{\sigma}_{\mathbbm{j}}$ and
$\bar{\tau}_{\mathbbm{j}}$ analogously with
$\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}$ and $\varrho^{n_{0}}$ replaced by
$\mathbf{\bar{X}}_{\mathbbm{j}}$ and $\varrho$. Therefore, if for every
$\mathbbm{j}\in\mathbb{J}_{\bar{N}}$ there are no points of
$\mathcal{N}_{1,\mathbbm{j}}$ in between
$\lambda\bigl{(}\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(t),\varrho^{n_{0}}(t,\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(t))\bigr{)}$
and
$\lambda\bigl{(}\mathbf{\bar{X}}_{\mathbbm{j}}(t),\varrho(t,\mathbf{\bar{X}}_{\mathbbm{j}}(t))\bigr{)}$
nor between
$\lambda_{\text{b}}\bigl{(}\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(t),\varrho^{n_{0}}(t,\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(t))\bigr{)}$
and
$\lambda_{\text{b}}\bigl{(}\mathbf{\bar{X}}_{\mathbbm{j}}(t),\varrho(t,\mathbf{\bar{X}}_{\mathbbm{j}}(t))\bigr{)}$
for $t\in\left[\bar{\sigma}_{\mathbbm{j}},\bar{\tau}_{\mathbbm{j}}\right)$,
then for all $\mathbbm{j}\in\mathbb{J}_{\bar{N}}$ we have
$\sigma^{n_{0}}_{1,\mathbbm{j}}=\bar{\sigma}_{\mathbbm{j}}$,
$\tau^{n_{0}}_{1,\mathbbm{j}0}=\bar{\tau}_{1,\mathbbm{j}0}$ and
$\tau^{n_{0}}_{1,\mathbbm{j}1}=\bar{\tau}_{1,\mathbbm{j}1}$.

Note that for
$t\in\left[\bar{\sigma}_{\mathbbm{j}},\bar{\tau}_{\mathbbm{j}}\right)$ we have

$\displaystyle\mathbf{\tilde{X}}^{n_{0}}_{1,\mathbbm{j}}(t)-\mathbf{\bar{X}}_{\mathbbm{j}}(t)=$
$\displaystyle\mathbf{\tilde{X}}^{n_{0}}_{1,\mathbbm{j}}\left(\bar{\sigma}_{\mathbbm{j}}\right)-\mathbf{\bar{X}}_{\mathbbm{j}}\left(\bar{\sigma}_{\mathbbm{j}}\right)$
$\displaystyle+\int_{\bar{\sigma}_{\mathbbm{j}}}^{t}\left[\mathbf{b}(\mathbf{\tilde{X}}^{n_{0}}_{1,\mathbbm{j}}(s),\nabla\varrho^{n_{0}}(s,\mathbf{\tilde{X}}^{n_{0}}_{i,\mathbbm{j}}(s)))-\mathbf{b}(\mathbf{\bar{X}}_{\mathbbm{j}}(s),\nabla\varrho(s,\mathbf{\bar{X}}_{\mathbbm{j}}(s)))\right]ds$

where

$\left|\mathbf{b}(\mathbf{\tilde{X}}^{n_{0}}_{1,\mathbbm{j}}(s),\nabla\varrho^{n_{0}}(s,\mathbf{\tilde{X}}^{n_{0}}_{i,\mathbbm{j}}(s)))-\mathbf{b}(\mathbf{\bar{X}}_{\mathbbm{j}}(s),\nabla\varrho(s,\mathbf{\bar{X}}_{\mathbbm{j}}(s)))\right|\leq\\\
L_{\mathbf{b}}\biggl{(}|\mathbf{\tilde{X}}^{n_{0}}_{i,\mathbbm{j}}(s)-\mathbf{\bar{X}}_{\mathbbm{j}}(s))|+\left|\nabla\varrho^{n_{0}}(s,\mathbf{\tilde{X}}^{n_{0}}_{i,\mathbbm{j}}(s)))-\nabla\varrho^{n_{0}}(s,\mathbf{\bar{X}}_{\mathbbm{j}}(s)))\right|+\\\
\left|\nabla\varrho^{n_{0}}(s,\mathbf{\bar{X}}_{\mathbbm{j}}(s)))-\nabla\varrho(s,\mathbf{\bar{X}}_{\mathbbm{j}}(s)))\right|\biggr{)}.$

By (1ag) and (1ah), functions $\varrho^{n_{0}}$ and $\nabla\varrho^{n_{0}}$
are Lipschitz with some constant $L_{\varrho,\eta}$ with probability greater
then $1-\eta$, so if we denote
$\Delta_{1}(t)=\max_{\mathbbm{j}\in\mathbb{J}_{\bar{N}}}\left|\mathbf{\tilde{X}}^{n_{0}}_{1,\mathbbm{j}}(t)-\mathbf{\bar{X}}_{\mathbbm{j}}(t)\right|$
then by (1ai)

$\displaystyle\Delta_{1}(t)\leq$ $\displaystyle
L_{\mathbf{b}}\int_{0}^{t}\Bigr{(}(1+L_{\varrho,\eta})\Delta_{1}(s)+\sup_{t\in[0,T]}\sup_{\mathbf{x}\in
B(0,R)}\left\|\nabla\varrho^{n_{0}}_{t}(\mathbf{x})-\nabla\varrho_{t}(\mathbf{x})\right\|\Bigr{)}ds$
$\displaystyle\leq$ $\displaystyle
L_{\mathbf{b}}\int_{0}^{t}\Bigl{(}(1+L_{\varrho,\eta})\Delta_{1}(s)+\delta\Bigr{)}ds.$

By Gronwall’s inequality we have
$\Delta_{1}(t)\leq\delta\,t\,L_{\mathbf{F}}e^{L_{\mathbf{F}}(1+L_{\varrho,\eta})t}$
so

$\sup_{t\in[0,T]}\Delta_{1}(t)\leq c^{(1)}_{T,\eta}\delta.$ (1aj)

and thus

$\left|\lambda\bigl{(}\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(t),\varrho^{n_{0}}(t,\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(t))\bigr{)}-\lambda\bigl{(}\mathbf{\bar{X}}_{\mathbbm{j}}(t),\varrho(t,\mathbf{\bar{X}}_{\mathbbm{j}}(t))\bigr{)}\right|\leq
L_{\lambda}((1+L_{\varrho,\eta})\Delta_{1}(t)+\delta)\leq
c^{(2)}_{T,\eta}\delta,$ (1ak)

where $c^{(i)}_{T,\eta}$, $i=1,2$ depend only on $T$ and $L_{\varrho,\eta}$.
Similar estimate holds for $\lambda_{\text{b}}$. Let

$A_{\mathbbm{j}}=\biggl{\\{}(t,z)\subset[0,T]\times[0,\bar{\lambda}]:t\in\left[\bar{\sigma}_{\mathbbm{j}},\bar{\tau}_{\mathbbm{j}}\right),z\in[\hat{\lambda}^{n_{0}}_{\text{b}}(t),\check{\lambda}^{n_{0}}_{\text{b}}(t)]\text{
or }z\in[\hat{\lambda}^{n_{0}}(t),\check{\lambda}^{n_{0}}(t)]\biggr{\\}},$

where $\hat{\lambda}^{n_{0}}(t)=$
$\min\left\\{\lambda\bigl{(}\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(t),\varrho^{n_{0}}(t,\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(t))\bigr{)},\lambda\bigl{(}\mathbf{\bar{X}}_{\mathbbm{j}}(t),\varrho(t,\mathbf{\bar{X}}_{\mathbbm{j}}(t))\bigr{)}\right\\}$
and
$\check{\lambda}^{n_{0}}(t)=$$\max\\{\lambda\bigl{(}\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(t),\varrho^{n_{0}}(t,\mathbf{X}^{n_{0}}_{i,\mathbbm{j}}(t))\bigr{)},\lambda\bigl{(}\mathbf{\bar{X}}_{\mathbbm{j}}(t),\varrho(t,\mathbf{\bar{X}}_{\mathbbm{j}}(t))\bigr{)}\\}$
and analogously for $\lambda_{\text{b}}$. By (1ak) the area of
$A_{\mathbbm{j}}$ is less than $T\,c_{T}\,\delta$ for any
$\mathbbm{j}\in\mathbb{J}_{\bar{N}}$. Therefore, taking $\delta$ sufficiently
small we have

$\mathbb{P}\left(\mathcal{N}_{1,\mathbbm{j}}(A_{\mathbbm{j}})>0\text{ for any
}\mathbbm{j}\in\mathbb{J}_{\bar{N}}\right)<\eta.$

∎

Now we are ready to prove Theorem 6.

###### Proof of Theorem 6.

Fix $\epsilon>0$. We have to prove that
$\mathbb{P}\left(\sup_{t\in[0,T]}d_{\mathbb{X}}\left(\mathbbm{x}^{n_{0}}_{1}(t),\mathbbm{\bar{x}}(t)\right)>\varepsilon\right)$
tends to 0 as ${n_{0}\to\infty}$. To that end fix $\eta>0$, and take $R>0$
from Lemma 2 for this $\eta$. Take $\delta>0$ small enough for Lemma 3 to be
satisfied and such that $c^{(1)}_{T,\eta}\delta\leq\varepsilon$ in (1aj).
Since, by Theorem 5, $\varrho^{n_{0}}$ converges to $\varrho$ in probability
on $\mathcal{C}_{T}$ in the norm
$\|f(\cdot,\cdot)\|_{\mathcal{C}_{T}}=\sup_{t\in[0,T]}\|f(t,\cdot)\|_{C^{1}_{b}}$,
for sufficiently big $n_{0}$ we have (1ai) with probability $1-\eta$. Using
Lemma 3 we know that with probability at least $1-3\eta$ we have
$\mathbbm{\tilde{x}}^{n_{0}}_{1}=\mathbbm{x}^{n_{0}}_{1}$ and
$\sup_{t\in[0,T]}d_{\mathbb{X}}\left(\mathbbm{x}^{n_{0}}_{1}(t),\mathbbm{\bar{x}}(t)\right)>\varepsilon$
by (1aj). ∎

## References

  * [1] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys. 15 (1953) 311–338.  doi:10.1007/bf02476407.
  * [2] E. F. Keller, L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol. 26 (3) (1970) 399–415.  doi:10.1016/0022-5193(70)90092-5.
  * [3] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein. 105 (3) (2003) 103–165.
  * [4] T. Hillen, K. J. Painter, A user’s guide to PDE models for chemotaxis, J. Math. Biol. 58 (1-2) (2009) 183–217.  doi:10.1007/s00285-008-0201-3.
  * [5] K. Oelschläger, A martingale approach to the law of large numbers for weakly interacting stochastic processes., Ann. Probab. 12 (1984) 458–479.
  * [6] A.-S. Sznitman, Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated, J. Funct. Anal. 56 (3) (1984) 311–336.  doi:10.1016/0022-1236(84)90080-6.
  * [7] D. Morale, V. Capasso, K. Oelschläger, An interacting particle system modelling aggregation behavior: from individuals to populations, J. Math. Biol. 50 (2005) 49–66.
  * [8] R. Rudnicki, R. Wieczorek, Phytoplankton dynamics: from the behaviour of cells to a transport equation, Math. Model. Nat. Phenom. 1 (1) (2006) 83–100.
  * [9] N. Fournier, B. Jourdain, Stochastic particle approximation of the Keller-Segel equation and two-dimensional generalization of Bessel processes, Ann. Appl. Probab. 27 (5) (2017) 2807–2861.  doi:10.1214/16-{AAP}1267.
  * [10] R. Wieczorek, A stochastic particles model of fragmentation process with shattering, Electron. J. Probab. 20 (2015) no. 86, 1–17.  doi:10.1214/{EJP}.v20-4060.
  * [11] A. Budhiraja, W.-T. Fan, Uniform in time interacting particle approximations for nonlinear equations of Patlak-Keller-Segel type, Electron. J. Probab. 22 (2017) Paper No. 8, 37.  doi:10.1214/17-{EJP}25.
  * [12] V. Capasso, R. Wieczorek, A hybrid stochastic model of retinal angiogenesis, Math. Methods Appl. Sci. 43 (18) (2020) 10578–10592.  doi:10.1002/mma.6725.
  * [13] S. Boi, V. Capasso, D. Morale, Modeling the aggregative behavior of ants of the species Polyergus rufescens, Vol. 1, 2000, pp. 163–176, spatial heterogeneity in ecological models (Alcalá de Henares, 1998).  doi:10.1016/{S}0362-546{X}(99)00399-5.
  * [14] A. Stevens, The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems, SIAM J. Appl. Math. 61 (1) (2000) 183–212.  doi:10.1137/{S}0036139998342065.
  * [15] K. Oelschläger, On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes, Probab. Theory Related Fields 82 (4) (1989) 565–586.  doi:10.1007/{BF}00341284.
  * [16] H. J. Hwang, K. Kang, A. Stevens, Drift-diffusion limits of kinetic models for chemotaxis: a generalization, Discrete Contin. Dyn. Syst. Ser. B 5 (2) (2005) 319–334.  doi:10.3934/dcdsb.2005.5.319.
  * [17] F. Bubba, T. Lorenzi, F. R. Macfarlane, From a discrete model of chemotaxis with volume-filling to a generalized Patlak-Keller-Segel model, Proc. A. 476 (2237) (2020) 20190871, 19.  doi:10.1098/rspa.2019.0871.
  * [18] S. Méléard, Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models, in: Probabilistic models for nonlinear partial differential equations (Montecatini Terme, 1995), Vol. 1627 of Lecture Notes in Math., Springer, Berlin, 1996, pp. 42–95.  doi:10.1007/{BF}b0093177.
  * [19] F. Bolley, A. Guillin, C. Villani, Quantitative concentration inequalities for empirical measures on non-compact spaces, Probab. Theory Related Fields 137 (3-4) (2007) 541–593.  doi:10.1007/s00440-006-0004-7.
  * [20] P. M. Kotelenez, T. G. Kurtz, Macroscopic limits for stochastic partial differential equations of McKean-Vlasov type, Probab. Theory Related Fields 146 (1-2) (2010) 189–222.  doi:10.1007/s00440-008-0188-0.
  * [21] L. Li, J.-G. Liu, P. Yu, On the mean field limit for Brownian particles with Coulomb interaction in 3D, J. Math. Phys. 60 (11) (2019) 111501, 34.  doi:10.1063/1.5114854.
  * [22] V. Capasso, D. Morale, A multiscale approach leading to hybrid mathematical models for angiogenesis: the role of randomness, in: Mathematical methods and models in biomedicine, Lect. Notes Math. Model. Life Sci., Springer, New York, 2013, pp. 87–115.
  * [23] V. Capasso, F. Flandoli, On the mean field approximation of a stochastic model of tumor-induced angiogenesis, European Journal of Applied Mathematics 30 (4) (2019) 619–658.  doi:10.1017/{S}0956792518000347.
  * [24] D. Woodward, R. Tyson, M. Myerscough, J. Murray, E. Budrene, H. Berg, Spatio-temporal patterns generated by Salmonella typhimurium, Biophysical Journal 68 (5) (1995) 2181–2189.  doi:https://doi.org/10.1016/{S}0006-3495(95)80400-5.
  * [25] X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: effects of motility and chemotaxis and dynamics, SIAM J. Math. Anal. 31 (3) (2000) 535–560.  doi:10.1137/{S}0036141098339897.
  * [26] M. Wang, R. M. Ford, R. W. Harvey, Coupled Effect of Chemotaxis and Growth on Microbial Distributions in Organic-Amended Aquifer Sediments: Observations from Laboratory and Field Studies, Environmental Science & Technology 42 (10) (2008) 3556–3562.  doi:10.1021/es702392h.
  * [27] V. Bansaye, F. Simatos, On the scaling limits of Galton-Watson processes in varying environments, Electron. J. Probab. 20 (2015) no. 75, 36.  doi:10.1214/{EJP}.v20-3812.
  * [28] V. Capasso, D. Morale, G. Facchetti, The role of stochasticity in a model of retinal angiogenesis, IMA J. Appl. Math. 77 (6) (2012) 729–747.
  * [29] S. R. McDougall, M. G. Watson, A. H. Devlin, C. A. Mitchell, M. A. J. Chaplain, A hybrid discrete-continuum mathematical model of pattern prediction in the developing retinal vasculature, Bull. Math. Biol. 74 (10) (2012) 2272–2314.  doi:10.1007/s11538-012-9754-9.
  * [30] N. L. Garcia, T. G. Kurtz, Spatial birth and death processes as solutions of stochastic equations, ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006) 281–303.
  * [31] M. L. Bujorianu, J. Lygeros, Toward a general theory of stochastic hybrid systems, in: Stochastic hybrid systems, Vol. 337 of Lect. Notes Control Inf. Sci., Springer, Berlin, 2006, pp. 3–30.  doi:10.1007/11587392_1.
  * [32] E. Buckwar, M. G. Riedler, An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution, J. Math. Biol. 63 (6) (2011) 1051–1093.
  * [33] L. L. Bonilla, V. Capasso, M. Alvaro, M. Carretero, Hybrid modeling of tumor-induced angiogenesis, Phys. Rev. E 90 (2014) 062716.  doi:10.1103/{P}hys{R}ev{E}.90.062716.
  * [34] A.-S. Sznitman, Topics in propagation of chaos, in: École d’Été de Probabilités de Saint-Flour XIX-1989, Vol. 1464 of Lecture Notes in Math., Springer, Berlin, 1991, pp. 165–251.  doi:10.1007/{BF}b0085169.
  * [35] P. Cattiaux, A. Guillin, F. Malrieu, Probabilistic approach for granular media equations in the non-uniformly convex case, Probab. Theory Related Fields 140 (1-2) (2008) 19–40.  doi:10.1007/s00440-007-0056-3.
  * [36] H. Bauer, Measure and integration theory, Vol. 26 of De Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 2001, translated from the German by Robert B. Burckel.  doi:10.1515/9783110866209.
  * [37] C. Kipnis, C. Landim, Scaling limits of interacting particle systems, Vol. 320 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1999\.  doi:10.1007/978-3-662-03752-2.
  * [38] J. Jacod, A. N. Shiryaev, Limit theorems for stochastic processes, Vol. 288 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1987.  doi:10.1007/978-3-662-02514-7.
  * [39] J. R. Munkres, Topology, Prentice Hall, Inc., Upper Saddle River, NJ, 2000, second edition of [ MR0464128].