# Coagulation dynamics under environmental noise:
Scaling limit to SPDE 111Keywords and phrases: scaling limits; coagulation
dynamics; stochastic PDE; environmental noise; interacting diffusions;
rainfall formation. 222AMS subject classification (2020): 60K35, 82C21, 60H15,
86A08.

Franco Flandoli and Ruojun Huang

###### Abstract

We prove that a system of locally interacting diffusions carrying discrete
masses, subject to an environmental noise and undergoing mass coagulation,
converges to a system of Stochastic Partial Differential Equations (SPDEs)
with Smoluchowski-type nonlinearity. Existence, uniqueness and regularity of
the SPDEs are also proven.

##  1 Introduction

Environmental noise in deterministic or stochastic interacting particle
systems is a space-dependent noise acting on all particles, opposite to the
more common independent noise for each particle. For particles in a fluid, the
environmental noise may be an idealized description of a turbulent fluid. An
example which motivates the model studied in the present work are the small
rain droplets of which clouds are made. Droplets move in the cloud due to the
force exerted on them by the surrounding turbulent air, and coagulate when
they become sufficiently close to each other. We introduce a particle system
modeling these two phenomena and investigate its scaling limit to a continuous
density model, which is a stochastic Smoluchowski system. Opposite to
independent noise for each particle which becomes a Laplacian in the scaling
limit, the environmental noise yields a stochastic transport term in the
continuous limit. Raindrop formation in turbulent fluids has been studied in
the Physics literature [44, 9, 2, 41]; our paper provides foundational results
on the particle system viewpoint and its continuum limit to a Stochastic
Partial Differential Equation (SPDE).

We model the individual rain droplets as diffusions on $\mathbb{R}^{d}$,
$d\geq 1$, with a small molecular diffusivity, and subject to a common
Stratonovich transport-type noise. Any pair of particles has a propensity to
coagulate into one, combining their masses, when their positions get locally
close to each other. Without the common noise, this is in the spirit of
classical Smoluchowski coagulation model which leads to his famous PDEs [42,
43], mathematically derived before in the kinetic limit from interacting
particle systems in [32, 23, 24]. We prove in this work that as the total
number of particles in our system tends to infinity, the empirical measures as
indexed by mass parameter, converge to a system of SPDEs with the same
Stratonovich transport-type noise. From Stratonovich-to-Itô correction, we
obtain an extra second-order divergence-form operator, the “eddy diffusion”.
This opens the door to the investigation of diffusion and coagulation
enhancement, along the lines of [11, 12, 13, 7, 20, 18, 17] and references
therein (also for other independent works on mixing or diffusion enhancement).
We stress that the philosophy underlying the emergence of enhanced diffusion
starting from a transport-type noise and through a suitable scaling limit was
first discovered by Galeati [20]. Numerical results are recently obtained for
a closely related coagulation model in [38].

Given a filtered probability space
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\},\mathbb{P})$, for each
$N\in\mathbb{N}$ consider an interacting particle system in $\mathbb{R}^{d}$,
$d\geq 1$, consisting initially of $N(0)=N$ particles. Each particle $i$ has a
position $x_{i}^{N}(t)\in\mathbb{R}^{d}$ and carries an integer mass
$m_{i}^{N}(t)$, which takes values in

$\displaystyle\mathbb{S}:=\\{1,2,...,M,\emptyset\\},$

where $M\in\mathbb{N}$ is the largest possible mass, and $\emptyset$ is a
fictitious element. The particles are subject to pairwise coagulation, at
which time some particles may cease to be active in the system. So long as
particle $i$ is still active (how the index set changes when a coagulation
event happens will be explained below), its position $x_{i}^{N}(t)$ obeys the
SDE:

$\displaystyle dx_{i}^{N}(t)=\sum_{k\in
K}\sigma_{k}\left(x_{i}^{N}(t)\right)\,\circ\,dW^{k}_{t}+\lambda\,d\beta_{i}(t),\quad
i\in\mathcal{N}(t)$ (1)

where $\circ$ denotes Stratonovich integration, scalar $\lambda>0$,
$\mathcal{N}(t)\subset\\{1,...,N\\}$ denotes the set of indices of active
particles at time $t$, whose cardinality $|\mathcal{N}(t)|=N(t)\leq N$, $K$ is
a finite set, $\\{W_{t}^{k}\\}_{k\in K}$ is a given finite collection of
independent standard Brownian motions in $\mathbb{R}$, and
$\\{\beta_{i}(t)\\}_{i=1}^{\infty}$ are given independent standard Brownian
motions in $\mathbb{R}^{d}$, $\\{\sigma_{k}(x)\\}_{k\in K}$ are given
divergence-free vector field of class
$C_{b}^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d})$, where $C_{b}^{\infty}(\cdot)$
denotes the space of smooth functions with derivatives of all orders uniformly
bounded. Following [3], we call

$\displaystyle\frac{d\mathscr{W}}{dt}(t,x):=\sum_{k\in
K}\sigma_{k}(x)\frac{dW^{k}_{t}}{dt}$

the environmental noise or common noise acting simultaneously on all
particles. We denote the $d\times d$ spatial covariance matrix of
$\mathscr{W}$ by

$\displaystyle Q(x,y):=\sum_{k\in K}\sigma_{k}(x)\otimes\sigma_{k}(y).$

As the set $K$ is finite, we do not assume $Q$ is spatially homogeneous as in
some prior studies using such transport noise.

From Stratonovich to Itô, (1) can be equivalently written as

$\displaystyle dx_{i}^{N}(t)=\sum_{k\in
K}\sigma_{k}\left(x_{i}^{N}(t)\right)dW^{k}_{t}+\frac{1}{2}\sum_{k\in
K}\left(\nabla\sigma_{k}\cdot\sigma_{k}\right)\left(x_{i}^{N}(t)\right)dt+\lambda\,d\beta_{i}(t),\quad
i\in\mathcal{N}(t)$

where component-wise,

$\displaystyle\left(\nabla\sigma_{k}\cdot\sigma_{k}\right)^{\alpha}(x):=\sum_{\beta=1}^{d}\sigma_{k}^{\beta}(x)\partial_{\beta}\sigma_{k}^{\alpha}(x),\quad\alpha=1,...,d.$
(2)

The initial masses $\\{m_{i}(0)\\}_{i=1}^{N}$ are chosen i.i.d. from
$\\{1,...,M\\}$ with probability $r_{m}$ to be mass $m$, so that
$\sum_{m=1}^{M}r_{m}=1$; and once the masses are determined, the distributions
of $x_{i}(0),i=1,2,..$ are independent with density
$p_{m_{i}(0)}(x),i=1,2,..$, for some given probability density functions
$p_{m}(x):\mathbb{R}^{d}\to\mathbb{R}_{+}$, $m=1,...,M$, all of which satisfy

###### Condition 1

  1. 1.

They are compactly supported in the Euclidean ball $\mathbb{B}(0,R)$ of finite
radius $R$ centered at the origin;

  2. 2.

They are uniformly bounded above by some finite constant $\Gamma$, i.e.
$\|p_{m}\|_{\infty}\leq\Gamma$;

  3. 3.

There exists some integer $n>d/4$ such that

$p_{m}(x)\in W^{2n,2}(\mathbb{R}^{d}),$

where $W^{k,p}(\mathbb{R}^{d})$ are standard Sobolev spaces.

Note that the initial conditions for different $N\in\mathbb{N}$ are naturally
coupled together, hence we do not stress the dependence of $x_{i}(0),m_{i}(0)$
on $N$. The probability space is endowed with the canonical filtration

$\mathcal{F}_{t}:=\sigma\left\\{\\{m_{i}(0)\\}_{i=1}^{\infty},\\{x_{i}(0)\\}_{i=1}^{\infty},\\{\beta_{i}(s)\\}_{i=1}^{\infty},\\{W^{k}_{s}\\}_{k\in
K}:s\in[0,t]\right\\},\quad t\geq 0.$

The interaction between particles is by means of coagulation of masses,
heuristically described below. It was studied, without the environmental
noise, by Hammond and Rezakhanlou in [23, 24] (however the limit there is a
system of PDEs instead of SPDEs, among other differences). Let
$\theta:\mathbb{R}^{d}\to\mathbb{R}_{+}$ be nonnegative, of class
$C^{\alpha}(\mathbb{R}^{d})$ (the space of Hölder continuous functions on
$\mathbb{R}^{d}$) for some $\alpha\in(0,1)$, compactly supported in
$\mathbb{B}(0,C_{0})$ for some finite constant $C_{0}$, with $\theta(0)=0$ and
$\int_{\mathbb{R}^{d}}\theta=1$, where we denote by $\mathbb{B}(x,r)$ the open
Euclidean ball of radius $r$ centered at $x\in\mathbb{R}^{d}$. For every
$\epsilon>0$, we denote

$\displaystyle\theta^{\epsilon}(x):=\epsilon^{-d}\theta(\epsilon^{-1}x).$

Throughout we impose the following relation between $N$ and $\epsilon$:

$\displaystyle\epsilon=\epsilon(N)\to 0$ $\displaystyle\text{as}\quad
N\to\infty,$ (3)
$\displaystyle\limsup_{N\to\infty}\frac{\epsilon^{1-d}}{N}<\infty,\quad d\geq
2;$ $\displaystyle\limsup_{N\to\infty}\frac{|\log\epsilon|}{N}<\infty,\quad
d=1.$

This regime includes local interaction (when $\epsilon^{-d}\asymp N$), see
discussion below, which is the scaling of interest for modelling raindrop
formations. A configuration of the stochastic system is a finite string of
positions and masses (whose length is at most $2N$)

$\displaystyle\eta=(x_{1},x_{2},...;m_{1},m_{2},..)\in(\mathbb{R}^{d})^{N}\times\mathbb{S}^{N}.$

On top of individual diffusions (1), we impose the following rule of mass
coagulation. Suppose the current configuration is $\eta$. For each pair of
particles $(i,j)$ active in the system (i.e. $m_{i},m_{j}\neq\emptyset$), with
positions $(x_{i},x_{j})$ and masses $(m_{i},m_{j})$, where $i\neq j$ each
ranges over the index set, with a rate

$\displaystyle\frac{1}{N}\theta^{\epsilon}(x_{i}-x_{j})$ (4)

we remove both particles from the system, and then add a new particle with
mass $(m_{i}+m_{j})$ at either $x_{i}$ or $x_{j}$ chosen with probability
$\frac{m_{i}}{m_{i}+m_{j}}$ for the former (and $\frac{m_{j}}{m_{i}+m_{j}}$
for the latter). However, we do this only if $m_{i}+m_{j}\leq M$, otherwise
after the pair is removed, no new particle is added to the system, or
equivalently we add a fictitious particle with mass $\emptyset$ at the origin.
We denote the new configuration obtained from $\eta$ this way by
$S_{i,j}^{1}\eta$, and respectively $S_{i,j}^{2}\eta$. For labelling purposes,
in the new configuration $S_{i,j}^{1}\eta$, we call particle $i$ the new mass-
combined particle found at position $x_{i}$, whereas there is no longer a
particle $j$ in the system; and analogously for $S_{i,j}^{2}\eta$. We will
only be tracking the empirical measures of particles with mass $\leq M$. The
interpretation is that, in terms of raindrop formations, when the mass of a
raindrop exceeds a certain threshold, it falls.

For $N$ diffusion particles evolving in a unit-order volume in
$\mathbb{R}^{d}$, the typical inter-particle distance is on the order of
$N^{-1/d}$. With $\epsilon$ representing the typical length scale over which
pairs of particles can interact, see (4), the relation (3) includes the most
relevant case $\epsilon\asymp N^{-1/d}$ for our modelling of raindrops
formation, namely when the interaction is so-called “local”. In such a regime,
each particle typically interacts with a bounded number of others at any given
time, which is an analog of the nearest-neighbor, or finite-range,
interactions widely studied in the discrete setting, e.g. on lattices [4, 28].
If $\epsilon^{-d}\ll N$, then the interaction is no longer local, but more
spread-out, in particular when $\epsilon$ is independent of $N$ it is the
“mean-field” case, and the regime between mean-field and local we can term
“moderate” following Oelschläger. In those regimes, each particle interacts
with a diverging (in $N$) number of others, see [15, 16, 14] for more
discussion on local versus moderate or mean-field interactions for diffusion
systems. In the latter cases techniques are much more developed, see e.g. [34,
35, 45, 33, 27, 15]. Also in those regimes, particle systems subject to
environmental noise, with even “singular” interactions, have been studied, see
[3, 19, 21] among others, where the interaction can occur in the drift of the
SDEs and not just in auxiliary variables (like mass). Localizing the range of
interaction for diffusions is a non-trivial task, in particular, in this work
(as well as in [14]) we have to utilize some techiniques developed in [23, 24]
in the spirit of the classical Itô-Tanaka trick, to handle the convergence of
the nonlinear terms, see Section 3. In those papers [23, 24], a system of
Smochulowski PDEs is rigorously derived from interacting particles in the so-
called “mean-free path” regime, which corresponds to diluted gases (not
covered by our result). Some of the other classical works on diffusions with
local interactions (that occur in the drift) are [47, 36, 37, 46].

Now we can formally give the infinitesimal generator of the system

$\displaystyle\mathcal{L}^{N}F(\eta):=\mathcal{L}^{N}_{D}F(\eta)+\mathcal{L}^{N}_{J}F(\eta)$

where the first, diffusion part of the generator is

$\displaystyle\mathcal{L}^{N}_{D}F(\eta)$ (5)
$\displaystyle:=\frac{\lambda^{2}}{2}\sum_{i\in\mathcal{N}(\eta)}\Delta_{x_{i}}F(\eta)+\frac{1}{2}\sum_{i,j\in\mathcal{N}(\eta)}\sum_{\alpha,\beta=1}^{d}\frac{\partial^{2}F}{\partial
x_{i}^{\alpha}\partial x_{j}^{\beta}}(\eta)\sum_{k\in
K}\sigma_{k}^{\alpha}(x_{i})\sigma_{k}^{\beta}(x_{j})+\frac{1}{2}\sum_{i\in\mathcal{N}(\eta)}\sum_{\alpha,\beta=1}^{d}\partial_{x_{i}^{\alpha}}F(\eta)\sum_{k\in
K}\sigma_{k}^{\beta}(x_{i})\partial_{\beta}\sigma_{k}^{\alpha}(x_{i})$
$\displaystyle=\frac{\lambda^{2}}{2}\sum_{i\in\mathcal{N}(\eta)}\Delta_{x_{i}}F(\eta)+\frac{1}{2}\sum_{i,j\in\mathcal{N}(\eta)}\sum_{\alpha,\beta=1}^{d}\frac{\partial^{2}F}{\partial
x_{i}^{\alpha}\partial
x_{j}^{\beta}}(\eta)Q^{\alpha\beta}(x_{i},x_{j})+\frac{1}{2}\sum_{i\in\mathcal{N}(\eta)}\sum_{\alpha,\beta=1}^{d}\partial_{x_{i}^{\alpha}}F(\eta)\partial_{x_{i}^{\beta}}\left(Q^{\alpha\beta}(x_{i},x_{i})\right)$

using $\operatorname{div}\sigma_{k}(x)=0$ in the last line, and
$\mathcal{N}(\eta)$ denotes the set of indices of active particles in
configuration $\eta$; and the second, jump (or coagulation) part of the
generator is

$\displaystyle\mathcal{L}^{N}_{J}F(\eta):=\frac{1}{N}\sum_{i\neq
j\in\mathcal{N}(\eta)}\theta^{\epsilon}(x_{i}-x_{j})\left[\frac{m_{i}}{m_{i}+m_{j}}F(S_{i,j}^{1}\eta)+\frac{m_{j}}{m_{i}+m_{j}}F(S_{i,j}^{2}\eta)-F(\eta)\right].$
(6)

We henceforth denote by $\eta(t)$ the random configuration at time $t$. Note
that if at time $t$, a coagulation event happens for some pair of particles,
the cardinality of active particles $N(t)$ decreases either by one or by two.

For each $N\in\mathbb{N}$ and $1\leq m\leq M$, we denote the empirical measure
of mass-$m$ particles

$\displaystyle\mu^{N,m}_{t}(dx):=\frac{1}{N}\sum_{i\in\mathcal{N}(t)}\delta_{x_{i}^{N}(t)}(dx)1_{\\{m_{i}^{N}(t)=m\\}},\quad
t\geq 0.$ (7)

For each $t$, these are nonnegative random measures on $\mathbb{R}^{d}$ with
total mass bounded above by $1$. Denote by $\mathcal{M}_{+,1}(\mathbb{R}^{d})$
the space of subprobability measures on $\mathbb{R}^{d}$ endowed with the weak
topology, and by
$\mathcal{D}([0,T];\mathcal{M}_{+,1}(\mathbb{R}^{d}))\equiv\mathcal{D}_{T}(\mathcal{M}_{+,1})$
the space of càdlàg functions taking values in
$\mathcal{M}_{+,1}(\mathbb{R}^{d})$, endowed with the Skorohod topology. Then,
$\\{\mu^{N,m}_{t}:t\in[0,T]\\}_{m\leq M}$ are
$\mathcal{D}_{T}(\mathcal{M}_{+,1})^{M}$-valued random variables.

Let us also introduce a finite system of SPDEs

$\displaystyle\begin{cases}du_{m}(t,x)&=\frac{1}{2}\lambda^{2}\Delta
u_{m}(t,x)dt+\sum_{n=1}^{m-1}u_{n}(t,x)u_{m-n}(t,x)dt-2\sum_{n=1}^{M}u_{n}(t,x)u_{m}(t,x)dt\\\
\\\ &+\frac{1}{2}\operatorname{div}\left(Q(x,x)\nabla
u_{m}(t,x)\right)dt+\sum_{k\in K}\sigma_{k}(x)\cdot\nabla
u_{m}(t,x)\;dW_{t}^{k},\quad(t,x)\in[0,T]\times\mathbb{R}^{d},\\\ \\\
u_{m}(0,x)&=r_{m}p_{m}(x),\quad m=1,...,M\end{cases}$ (8)

where by convention, when $m=1$, set $\sum_{n=1}^{m-1}[...]=0$. For our notion
of (analytically) weak solutions of the system (8), see Definition 24. The
SPDE can be equivalently written as

$\displaystyle\begin{cases}du_{m}(t,x)&=\frac{1}{2}\lambda^{2}\Delta
u_{m}(t,x)dt+\sum_{n=1}^{m-1}u_{n}(t,x)u_{m-n}(t,x)dt-2\sum_{n=1}^{M}u_{n}(t,x)u_{m}(t,x)dt\\\
\\\ &+\sum_{k\in K}\sigma_{k}(x)\cdot\nabla
u_{m}(t,x)\,\circ\,dW_{t}^{k},\quad(t,x)\in[0,T]\times\mathbb{R}^{d},\\\ \\\
u_{m}(0,x)&=r_{m}p_{m}(x),\quad m=1,...,M\end{cases}$

since by $\operatorname{div}\sigma_{k}(x)=0$, the Stratonovich-to-Itô
correction takes the form

$\displaystyle\frac{1}{2}\sum_{k\in
K}\sigma_{k}(x)\cdot\nabla\left(\sigma_{k}(x)\cdot\nabla
u_{m}(t,x)\right)dt=\frac{1}{2}\sum_{k\in
K}\sum_{\alpha,\beta=1}^{d}\sigma_{k}^{\alpha}(x)\partial_{\alpha}\left(\sigma_{k}^{\beta}(x)\partial_{\beta}u_{m}(t,x)\right)dt$
$\displaystyle=\frac{1}{2}\sum_{k\in
K}\sum_{\alpha,\beta=1}^{d}\partial_{\alpha}\left(\sigma_{k}^{\alpha}(x)\sigma_{k}^{\beta}(x)\partial_{\beta}u_{m}(t,x)\right)dt=\frac{1}{2}\operatorname{div}\left(Q(x,x)\nabla
u_{m}(t,x)\right)dt.$

The main result of the article is the following.

###### Theorem 2

Assume that $N$ and $\epsilon$ satisfy (3), and Condition 1 holds. Then, for
every finite $T$ and $d\geq 1$, the empirical measure of the particle system
$\\{\mu^{N,m}_{t}(dx):t\in[0,T]\\}_{m\leq M}$ defined in (7) converges in
probability as $N\to\infty$, in the space
$\mathcal{D}_{T}(\mathcal{M}_{+,1})^{M}$ to a limit
$\\{\overline{\mu}_{t}^{m}(dx):t\in[0,T]\\}_{m\leq M}$. The latter random
measure is absolutely continuous with respect to Lebesgue measure, with a
uniformly bounded density
$\\{u_{m}(t,x):t\in[0,T],x\in\mathbb{R}^{d}\\}_{m\leq M}$ that is the pathwise
unique weak solution to the SPDE (8).

Besides pathwise uniqueness (Corollary 26), we also obtain regularity results
for solutions of such nonlinear SPDEs (8), see Proposition 27, that may be of
independent interest.

Regarding some of our assumptions, we have the following remarks.

###### Remark 3

(a). We have assumed $\sigma_{k}(x)\in
C_{b}^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d})$ for $k\in K$ to facilitate the
nontrivial SPDE arguments in Section 8. While we did not try to optimize the
regularity and perhaps there exists a different proof that imposes less
regularity (see some remarks on this issue in Section 8), we believe that our
arguments herein are of independent interest.

(b). The reason we chose finitely many mass levels (instead of infinite, i.e.
$M=\infty$), is in order to have a finite system of SPDEs (8) in the limit,
whose pathwise uniqueness of solution can be proved, see Section 5. Note that
$M=\infty$ is assumed in [23, 24], with uniqueness for their PDE system proved
in [25] under appropriate assumptions.

(c). The assumption $\theta(0)=0$ seems to be necessary in order to rewrite
certain quantity by means of the empirical measure, see (LABEL:theta-redef). A
potential criticism is that it is unrealistic: $\theta(0)=0$ and smooth
implies $\theta(\cdot)$ is very small near the origin, hence two particles
which are “too close” to each other have also little rate to coagulate.
However, “physics” is saved since: (i) the region, call it
$\mathbb{B}(0,\iota\epsilon)$, where $\theta^{\epsilon}(\cdot)$ is very small
near the origin can be extremely small compared to
$\mathbb{B}(0,C_{0}\epsilon)$, and (ii) before getting
$(\iota\epsilon)$-close, two particles have to be $(C_{0}\epsilon)$-close for
a while, hence the loss of rate in $\mathbb{B}(0,\iota\epsilon)$ is
practically not influential.

(d). Our dynamics (1) is less general than the setting of [23, 24] in that the
coefficients $\sigma_{k}$ and $\lambda$ do not depend on the mass parameter
$m_{i}^{N}$. This is for technical reasons, with the need to couple with an
auxiliary free system in Sections 4 and 7, notably Proposition 11. How to
incorporate the additional mass dependence in (1) is an open problem.

The strategy of proof is summarised as follows. In Section 6, we show that the
sequence of probability laws $\\{\mathscr{P}^{N}\\}_{N}$ induced by the
$\mathcal{D}_{T}(\mathcal{M}_{+,1})^{M}$-valued random variables

$\displaystyle\big{\\{}\mu^{N,m}_{t}(dx):t\in[0,T]\big{\\}}_{m\leq M},\quad
N\in\mathbb{N}$

is tight hence weakly relatively compact. Fix any weak subsequential limit
$\overline{\mathscr{P}}$ of $\\{\mathscr{P}^{N_{j}}\\}_{j\in\mathbb{N}}$ along
a subsequence $N_{j}$. By Skorohod’s representation theorem, we can construct
on an auxiliary probability space
$(\widehat{\Omega},\widehat{\mathcal{F}},\mathbf{P})$ (that depends on the
subsequence) random variables $\\{\widehat{\mu}^{N_{j},m}\\}_{m\leq M}$,
$j\geq 1$, and $\\{\overline{\mu}^{m}\\}_{m\leq M}$, having the laws
$\mathscr{P}^{N_{j}}$, $j\geq 1$, and $\overline{\mathscr{P}}$, respectively,
such that $\mathbf{P}$-a.s.

$\\{\widehat{\mu}^{N_{j},m}\\}_{m\leq M}\to\\{\overline{\mu}^{m}\\}_{m\leq
M},\quad j\to\infty.$

In Sections 7-8, we show that any subsequential limit measure
$\\{\overline{\mu}^{m}_{t}(dx):t\in[0,T]\\}_{m\leq M}$ is supported on the
subset of measures that are absolutely continuous with respect to Lebesgue
measure, with density

$\displaystyle\left\\{u_{m}(t,x),t\in[0,T]\right\\}_{m\leq M}$

uniformly bounded by the deterministic constant $\Gamma$ in Condition 1. This
is achieved by considering an auxiliary (“free”) particle system without mass-
coagulation, that dominates our true system, and studying the regularity and
boundedness of solutions of its associated SPDE (62), see in particular
Theorem 38 and Lemma 39. In Sections 2-4, we show that on
$(\widehat{\Omega},\widehat{\mathcal{F}},\mathbf{P})$, there exist a finite
collection of independent Brownian motions $(\overline{W}_{t}^{k})_{k\in K}$
such that $\left(\\{u_{m}\\}_{m\leq M},(\overline{W}^{k})_{k\in K}\right)$ is
a weak solution to the finite system of SPDEs (8). Here we need to deal with
the difficulty posed by local interaction.

In Section 5 we show that (analytically) weak solutions to (8) are pathwise
unique. By a well-known theorem of Gyöngy-Krylov [22, Lemma 1.1, Theorem 2.4],
see also [12, Chapter 2], [13, Appendix C], having pathwise uniqueness, one
can strengthen the convergence in law along subsequences to convergence in
probability along the full sequence, of $\\{\mu^{N,m}\\}_{m\in M}$,
$N\in\mathbb{N}$, on the original probability space
$(\Omega,\mathcal{F},\mathbb{P})$. The details involve representing two copies
of the empirical measures by Skorohod’s representation, and showing that any
subsequential limit pair is concentrated on the diagonal. As this is similar
to the Skorohod argument we already elaborate in Section 2 as well as those in
the above references, we omit the proof of the Gyöngy-Krylov step.

##  2 Identity involving the empirical measure

We start by deriving an identity involving the empirical measure
$\\{\mu_{t}^{m}(dx)\\}_{m\leq M}$. Taking any $\phi\in
C_{c}^{\infty}(\mathbb{R}^{d})$, we consider the functional

$\displaystyle
F_{1}(\eta):=\frac{1}{N}\sum_{i\in\mathcal{N}(\eta)}\phi(x_{i})1_{\\{m_{i}=m\\}},$

for any fixed $1\leq m\leq M$. Applying Itô formula to the process

$\displaystyle
F_{1}(\eta(t))=\frac{1}{N}\sum_{i\in\mathcal{N}(t)}\phi(x_{i}^{N}(t))1_{\\{m_{i}^{N}(t)=m\\}}=\left\langle\phi(x),\mu^{N,m}_{t}(dx)\right\rangle,\quad
t\geq 0$

where the notation $\langle f,\nu\rangle$ denotes integrating a function $f$
against a measure $\nu$, in view of (5)-(6), we get that for every finite $T$,

$\displaystyle\left\langle\phi(x),\mu^{N,m}_{T}(dx)\right\rangle$
$\displaystyle=\left\langle\phi(x),\mu^{N,m}_{0}(dx)\right\rangle+\int_{0}^{T}dt\frac{\lambda^{2}}{2N}\sum_{i\in\mathcal{N}(t)}\Delta\phi(x_{i}^{N}(t))1_{\\{m_{i}^{N}(t)=m\\}}$
$\displaystyle+\int_{0}^{T}dt\frac{1}{2N}\sum_{i\in\mathcal{N}(t)}\sum_{\alpha,\beta=1}^{d}Q^{\alpha\beta}\left(x_{i}^{N}(t),x_{i}^{N}(t)\right)\left(\partial^{2}_{\alpha\beta}\phi\right)(x_{i}^{N}(t))1_{\\{m_{i}^{N}(t)=m\\}}$
$\displaystyle+\int_{0}^{T}dt\frac{1}{2N}\sum_{i\in\mathcal{N}(t)}\sum_{\alpha,\beta=1}^{d}\partial_{\alpha}\phi\left(x_{i}^{N}(t)\right)\partial_{\beta}\left(Q^{\alpha\beta}\left(x_{i}^{N}(t),x_{i}^{N}(t)\right)\right)1_{\\{m_{i}^{N}(t)=m\\}}$
$\displaystyle+\int_{0}^{T}dt\frac{1}{N}\sum_{i\neq
j\in\mathcal{N}(t)}\frac{1}{N}\theta^{\epsilon}(x_{i}^{N}(t)-x_{j}^{N}(t))\Big{[}\frac{m_{i}^{N}(t)}{m_{i}^{N}(t)+m_{j}^{N}(t)}\phi(x_{i}^{N}(t))1_{\\{m_{i}^{N}(t)+m_{j}^{N}(t)=m\\}}$
$\displaystyle\qquad+\frac{m_{j}^{N}(t)}{m_{i}^{N}(t)+m_{j}^{N}(t)}\phi(x_{j}^{N}(t))1_{\\{m_{i}^{N}(t)+m_{j}^{N}(t)=m\\}}-\phi(x_{i}^{N}(t))1_{\\{m_{i}^{N}(t)=m\\}}-\phi(x_{j}^{N}(t))1_{\\{m_{j}^{N}(t)=m\\}}\Big{]}$
$\displaystyle+M^{1,D,\phi}_{T}+M^{2,D,\phi}_{T}+M_{T}^{J,\phi}.$

Since for every $\alpha,\beta=1,...,d$,

$Q^{\alpha\beta}\left(x,x\right)\left(\partial^{2}_{\alpha\beta}\phi\right)(x)+\partial_{\alpha}\phi\left(x\right)\partial_{\beta}\left(Q^{\alpha\beta}\left(x,x\right)\right)=\partial_{\beta}\left(Q^{\alpha\beta}\left(x,x\right)\partial_{\alpha}\phi(x)\right),$

the above identity can be written as

$\displaystyle\left\langle\phi(x),\mu^{N,m}_{T}(dx)\right\rangle=$
$\displaystyle\left\langle\phi(x),\mu^{N,m}_{0}(dx)\right\rangle+\int_{0}^{T}\left\langle\mu_{t}^{N,m}(dx),\frac{\lambda^{2}}{2}\Delta\phi(x)\right\rangle
dt$ (9)
$\displaystyle+\int_{0}^{T}\left\langle\mu^{N,m}_{t}(dx),\frac{1}{2}\operatorname{div}\left(Q(x,x)\nabla\phi(x)\right)\right\rangle
dt$ $\displaystyle+\int_{0}^{T}dt\sum_{n=1}^{m-1}\frac{1}{N^{2}}\sum_{i\neq
j\in\mathcal{N}(t)}\theta^{\epsilon}(x_{i}^{N}(t)-x_{j}^{N}(t))\frac{n}{m}\phi(x_{i}^{N}(t))1_{\\{m_{i}^{N}(t)=n\\}}1_{\\{m_{j}^{N}(t)=m-n\\}}$
$\displaystyle+\int_{0}^{T}dt\sum_{n=1}^{m-1}\frac{1}{N^{2}}\sum_{i\neq
j\in\mathcal{N}(t)}\theta^{\epsilon}(x_{i}^{N}(t)-x_{j}^{N}(t))\frac{m-n}{m}\phi(x_{j}^{N}(t))1_{\\{m_{i}^{N}(t)=n\\}}1_{\\{m_{j}^{N}(t)=m-n\\}}$
$\displaystyle-\int_{0}^{T}dt\sum_{n=1}^{M}\frac{1}{N^{2}}\sum_{i\neq
j\in\mathcal{N}(t)}\theta^{\epsilon}(x_{i}^{N}(t)-x_{j}^{N}(t))\phi(x_{i}^{N}(t))1_{\\{m_{i}^{N}(t)=m\\}}1_{\\{m_{j}^{N}(t)=n\\}}$
$\displaystyle-\int_{0}^{T}dt\sum_{n=1}^{M}\frac{1}{N^{2}}\sum_{i\neq
j\in\mathcal{N}(t)}\theta^{\epsilon}(x_{i}^{N}(t)-x_{j}^{N}(t))\phi(x_{j}^{N}(t))1_{\\{m_{j}^{N}(t)=m\\}}1_{\\{m_{i}^{N}(t)=n\\}}$
$\displaystyle+M_{T}^{1,D,\phi}+M_{T}^{2,D,\phi}+M_{T}^{J,\phi}$

where $\\{M^{J,\phi}_{t}\\}_{t\geq 0}$ denotes the martingale associated with
jumps (which we do not write out explicitly), and there are two martingales
associated with diffusions

$\displaystyle M_{T}^{1,D,\phi}$
$\displaystyle:=\int_{0}^{T}\frac{\lambda}{N}\sum_{i\in\mathcal{N}(t)}\nabla\phi(x_{i}^{N}(t))1_{\\{m_{i}^{N}(t)=m\\}}\cdot
d\beta_{i}(t)$ $\displaystyle M_{T}^{2,D,\phi}$
$\displaystyle:=\int_{0}^{T}\sum_{k\in
K}\sigma_{k}(x_{i}^{N}(t))\cdot\frac{1}{N}\sum_{i\in\mathcal{N}(t)}\nabla\phi(x_{i}^{N}(t))1_{\\{m_{i}^{N}(t)=m\\}}dW_{t}^{k}$
$\displaystyle=\int_{0}^{T}\sum_{k\in
K}\left\langle\mu_{t}^{N,m}(dx),\sigma_{k}(x)\cdot\nabla\phi(x)\right\rangle
dW_{t}^{k}.$

Firstly, by Itô isometry and the independence among $\beta_{i}(t)$, we have
that

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\left|M_{t}^{1,D,\phi}\right|^{2}\right]$
$\displaystyle\leq
4\mathbb{E}\left[\int_{0}^{T}dt\frac{\lambda^{2}}{N^{2}}\sum_{i\in\mathcal{N}(t)}|\nabla\phi(x_{i}^{N}(t))|^{2}1_{\\{m_{i}^{N}(t)=m\\}}dt\right]\leq\frac{C\left(\|\phi\|_{C^{1}},T\right)}{N}.$
(10)

The second diffusion martingale $M^{2,D,\phi}$ is not negligible, and is
responsible for the stochastic term in the limit SPDE. Secondly, we bound the
jump martingale by (cf. [5, Proposition 8.7])

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\left|M_{t}^{J,\phi}\right|^{2}\right]$
$\displaystyle\leq 4\mathbb{E}\int_{0}^{T}dt\frac{1}{N^{2}}\sum_{i\neq
j\in\mathcal{N}(t)}\frac{1}{N}\theta^{\epsilon}(x_{i}^{N}(t)-x_{j}^{N}(t))\Big{[}\frac{m_{i}^{N}(t)}{m}\phi(x_{i}^{N}(t))1_{\\{m_{i}^{N}(t)+m_{j}^{N}(t)=m\\}}+\frac{m_{j}^{N}(t)}{m}\phi(x_{j}^{N}(t))1_{\\{m_{i}^{N}(t)+m_{j}^{N}(t)=m\\}}$
$\displaystyle\quad-\phi(x_{i}^{N}(t))1_{\\{m_{i}^{N}(t)=m\\}}-\phi(x_{j}^{N}(t))1_{\\{m_{j}^{N}(t)=m\\}}\Big{]}^{2}$
$\displaystyle\leq
64\|\phi\|^{2}_{\infty}\mathbb{E}\Big{[}\int_{0}^{T}dt\frac{1}{N^{3}}\sum_{i\neq
j\in\mathcal{N}(t)}\theta^{\epsilon}(x_{i}^{N}(t)-x_{j}^{N}(t))\Big{]}\leq\frac{C(\|\phi\|_{\infty})}{N},$
(11)

by Lemma 4 below.

###### Lemma 4

We have that

$\displaystyle\mathbb{E}\int_{0}^{T}dt\sum_{i\neq
j\in\mathcal{N}(t)}\theta^{\epsilon}(x_{i}^{N}(t)-x_{j}^{N}(t))\leq N^{2}.$

Proof. We apply the generator to the process of cardinality $N(t)$ of active
particles. The diffusion part of the generator does not affect $N(t)$, whereas
the coagulation part decreases cardinality by either one or two (depending on
if the combined mass exceeds $M$ or not). Hence, by Itô formula and taking
expectation, we have that

$\displaystyle\mathbb{E}N(T)\leq\mathbb{E}N(0)-\mathbb{E}\int_{0}^{T}dt\frac{1}{N}\sum_{i\neq
j\in\mathcal{N}(t)}\theta^{\epsilon}(x_{i}^{N}(t)-x_{j}^{N}(t)).$

Since $N(0)=N$, this completes the proof.

We observe that the middle four nonlinear terms of (9) can also be written by
means of empirical measure, since for every $m,n,t$,

$\displaystyle\frac{1}{N^{2}}\sum_{i\neq
j\in\mathcal{N}(t)}\theta^{\epsilon}(x_{i}^{N}(t)-x_{j}^{N}(t))\phi(x_{i}^{N}(t))\psi(x_{j}^{N}(t))1_{\\{m_{i}^{N}(t)=m\\}}1_{\\{m_{j}^{N}(t)=n\\}}$
(12)
$\displaystyle=\left\langle\theta^{\epsilon}(x-y)\phi(x)\psi(y),\mu^{N,m}_{t}(dx)\mu^{N,n}_{t}(dy)\right\rangle,$

where due to $\theta^{\epsilon}(0)=0$ we can include the terms with repeated
indices $i=j$ to the LHS of (LABEL:theta-redef).

By (9),(10), (11),(LABEL:theta-redef) for every $\phi\in
C_{c}^{\infty}(\mathbb{R}^{d})$ and $1\leq m\leq M$, we have that

$\displaystyle\limsup_{N\to\infty}\mathbb{E}\Big{|}\left\langle\phi(x),\mu^{N,m}_{T}(dx)\right\rangle$
$\displaystyle-\left\langle\phi(x),\mu^{N,m}_{0}(dx)\right\rangle-\int_{0}^{T}dt\left\langle\mu_{t}^{N,m}(dx),\frac{\lambda^{2}}{2}\Delta\phi(x)\right\rangle$
(13)
$\displaystyle-\int_{0}^{T}dt\left\langle\mu^{N,m}_{t}(dx),\frac{1}{2}\operatorname{div}\left(Q(x,x)\nabla\phi(x)\right)\right\rangle$
$\displaystyle-\int_{0}^{T}dt\sum_{n=1}^{m-1}\frac{n}{m}\left\langle\theta^{\epsilon}(x-y)\phi(x),\mu^{N,m-n}_{t}(dy)\mu^{N,n}_{t}(dx)\right\rangle$
$\displaystyle-\int_{0}^{T}dt\sum_{n=1}^{m-1}\frac{m-n}{m}\left\langle\theta^{\epsilon}(x-y)\phi(y),\mu^{N,m-n}_{t}(dy)\mu^{N,n}_{t}(dx)\right\rangle$
$\displaystyle+\int_{0}^{T}dt\sum_{n=1}^{M}\left\langle\theta^{\epsilon}(x-y)\phi(x),\mu^{N,n}_{t}(dy)\mu^{N,m}_{t}(dx)\right\rangle$
$\displaystyle+\int_{0}^{T}dt\sum_{n=1}^{M}\left\langle\theta^{\epsilon}(x-y)\phi(y),\mu^{N,m}_{t}(dy)\mu^{N,n}_{t}(dx)\right\rangle$
$\displaystyle-\int_{0}^{T}\sum_{k\in
K}\left\langle\mu^{N,m}_{t}(dx),\sigma_{k}(x)\cdot\nabla\phi(x)\right\rangle
dW_{t}^{k}\Big{|}=0.$

In Section 6, we show that the laws $\\{\mathscr{P}_{*}^{N}\\}_{N}$ of the
sequence of $\mathcal{D}_{T}(\mathcal{M}_{+,1})^{M}\times
C([0,T];\mathbb{R})^{|K|}$-valued random variables

$\big{\\{}\big{\\{}\mu^{N,m}_{t}:t\in[0,T]\big{\\}}_{m\leq
M},\big{\\{}W_{t}^{k}:t\in[0,T]\big{\\}}_{k\in K}\big{\\}},\quad
N\in\mathbb{N}$

are tight hence relatively compact (cf. Remark 31), where the space
$C([0,T];\mathbb{R})$ is endowed with the uniform topology. Fix any weak
subsequential limit $\overline{\mathscr{P}}_{*}$ of
$\\{\mathscr{P}_{*}^{N_{j}}\\}_{j\in\mathbb{N}}$ along a subsequence $N_{j}$.
By Skorohod’s representation theorem, we can construct on an auxiliary
probability space $(\widehat{\Omega},\widehat{\mathcal{F}},\mathbf{P})$ (that
depends on the subsequence) random variables
$\big{\\{}\\{\widehat{\mu}^{N_{j},m}\\}_{m\leq
M},\\{\widehat{W}^{N_{j},k}\\}_{k\in K}\big{\\}}$, $j\geq 1$, and
$\big{\\{}\\{\overline{\mu}^{m}\\}_{m\leq M},\\{\overline{W}^{k}\\}_{k\in
K}\big{\\}}$, having the laws $\mathscr{P}_{*}^{N_{j}}$, $j\geq 1$, and
$\overline{\mathscr{P}}_{*}$, respectively, such that $\mathbf{P}$-a.s.

$\big{\\{}\\{\widehat{\mu}^{N_{j},m}\\}_{m\leq
M},\\{\widehat{W}^{N_{j},k}\\}_{k\in
K}\big{\\}}\to\big{\\{}\\{\overline{\mu}^{m}\\}_{m\leq
M},\\{\overline{W}^{k}\\}_{k\in K}\big{\\}},\quad j\to\infty.$

Further, the limit measure has a uniformly bounded density
$\\{u_{m}(t,x):t\in[0,T]\\}_{m\leq M}$, i.e. for every $m$

$\overline{\mu}^{m}_{t}(dx)=u_{m}(t,x)dx,\quad\|u_{m}\|_{\infty}\leq\Gamma,$

as shown in Sections 7 and 8, and we also have that
$u_{m}(0,x)=r_{m}p_{m}(x)$.

By (13) and the representation, on
$(\widehat{\Omega},\widehat{\mathcal{F}},\mathbf{P})$ we have that for every
$\phi\in C_{c}^{\infty}(\mathbb{R}^{d})$ and $1\leq m\leq M$,

$\displaystyle\lim_{j\to\infty}\mathbf{E}\Big{|}\left\langle\phi(x),\widehat{\mu}^{N_{j},m}_{T}(dx)\right\rangle$
$\displaystyle-\left\langle\phi(x),\widehat{\mu}^{N_{j},m}_{0}(dx)\right\rangle-\int_{0}^{T}dt\left\langle\widehat{\mu}_{t}^{N_{j},m}(dx),\frac{\lambda^{2}}{2}\Delta\phi(x)\right\rangle$
(14)
$\displaystyle-\int_{0}^{T}dt\left\langle\widehat{\mu}^{N_{j},m}_{t}(dx),\frac{1}{2}\operatorname{div}\left(Q(x,x)\nabla\phi(x)\right)\right\rangle$
$\displaystyle-\int_{0}^{T}dt\sum_{n=1}^{m-1}\frac{n}{m}\left\langle\theta^{\epsilon}(x-y)\phi(x),\widehat{\mu}^{N_{j},m-n}_{t}(dy)\widehat{\mu}^{N_{j},n}_{t}(dx)\right\rangle$
$\displaystyle-\int_{0}^{T}dt\sum_{n=1}^{m-1}\frac{m-n}{m}\left\langle\theta^{\epsilon}(x-y)\phi(y),\widehat{\mu}^{N_{j},m-n}_{t}(dy)\widehat{\mu}^{N_{j},n}_{t}(dx)\right\rangle$
$\displaystyle+\int_{0}^{T}dt\sum_{n=1}^{M}\left\langle\theta^{\epsilon}(x-y)\phi(x),\widehat{\mu}^{N_{j},n}_{t}(dy)\widehat{\mu}^{N_{j},m}_{t}(dx)\right\rangle$
$\displaystyle+\int_{0}^{T}dt\sum_{n=1}^{M}\left\langle\theta^{\epsilon}(x-y)\phi(y),\widehat{\mu}^{N_{j},m}_{t}(dy)\widehat{\mu}^{N_{j},n}_{t}(dx)\right\rangle$
$\displaystyle-\int_{0}^{T}\sum_{k\in
K}\left\langle\widehat{\mu}^{N_{j},m}_{t}(dx),\sigma_{k}(x)\cdot\nabla\phi(x)\right\rangle
d\widehat{W}_{t}^{N_{j},k}\Big{|}=0.$

Since $\widehat{\mu}^{N_{j},m}\to\overline{\mu}^{m}$ in
$\mathcal{D}_{T}(\mathcal{M}_{+,1})$ under Skorohod topology,
$\mathbf{P}$-a.s. for every $1\leq m\leq M$, we have that $\mathbf{P}$-a.s.
for every $\phi\in C_{c}^{\infty}(\mathbb{R}^{d})$

$\displaystyle\sup_{t\in[0,T]}\left\langle\widehat{\mu}^{N_{j},m}_{t}(dx)-\overline{\mu}^{m}_{t}(dx),\phi(x)\right\rangle\to
0$ (15)
$\displaystyle\sup_{t\in[0,T]}\left\langle\widehat{\mu}_{t}^{N_{j},m}(dx)-\overline{\mu}_{t}^{m}(dx),\frac{\lambda^{2}}{2}\Delta\phi(x)\right\rangle\to
0$
$\displaystyle\sup_{t\in[0,T]}\left\langle\widehat{\mu}_{t}^{N_{j},m}(dx)-\overline{\mu}_{t}^{m}(dx),\frac{1}{2}\operatorname{div}\left(Q(x,x)\nabla\phi(x)\right)\right\rangle\to
0$

(cf. [8, Ch. 3, Proposition 5.3]). The convergences also hold in
$L^{1}(\mathbf{P})$ by dominated convergence, since the variables in (15) are
all uniformly bounded. The middle four nonlinear terms in (14) also converge
in $L^{1}(\mathbf{P})$ by Lemma 6 below.

We now argue that the last martingale term in (14) also converges in
$L^{1}(\mathbf{P})$, i.e. for every $1\leq m\leq M$,

$\displaystyle\int_{0}^{T}\sum_{k\in
K}\left\langle\widehat{\mu}^{N_{j},m}_{t}(dx),\sigma_{k}(x)\cdot\nabla\phi(x)\right\rangle
d\widehat{W}_{t}^{N_{j},k}-\int_{0}^{T}\sum_{k\in
K}\left\langle\overline{\mu}^{m}_{t}(dx),\sigma_{k}(x)\cdot\nabla\phi(x)\right\rangle
d\overline{W}_{t}^{k}\to 0,\quad j\to\infty.$ (16)

Indeed, by Burkholder-Davis-Gundy inequality, we first have that for every
$j\in\mathbb{N}$,

$\displaystyle\mathbf{E}\left|\int_{0}^{T}\sum_{k\in
K}\left\langle\widehat{\mu}_{t}^{N_{j},m}(dx)-\overline{\mu}_{t}^{m}(dx),\nabla\phi(x)\cdot\sigma_{k}(x)\right\rangle
d\widehat{W}_{t}^{N_{j},k}\right|$ $\displaystyle\leq
C_{1}\mathbf{E}\left[\left(\int_{0}^{T}\sum_{k\in
K}\left|\left\langle\widehat{\mu}_{t}^{N_{j},m}(dx)-\overline{\mu}_{t}^{m}(dx),\nabla\phi(x)\cdot\sigma_{k}(x)\right\rangle\right|^{2}dt\right)^{1/2}\right]$
$\displaystyle\leq C_{1}\sqrt{T}\mathbf{E}\sum_{k\in
K}\sup_{t\in[0,T]}\left|\left\langle\widehat{\mu}_{t}^{N_{j},m}(dx)-\overline{\mu}_{t}^{m}(dx),\nabla\phi(x)\cdot\sigma_{k}(x)\right\rangle\right|.$
(17)

Since the variable inside the expectation in the last line is uniformly
bounded (by $2\sum_{k\in K}\|\phi\|_{C^{1}}\|\sigma_{k}\|_{\infty}$), and
converges to zero $\mathbf{P}$-a.s. as $j\to\infty$, we have that (2)
converges to zero.

Secondly, since $\widehat{W}_{t}^{N_{j},k}-\overline{W}_{t}^{k}$ is a
$\mathbf{P}$-martingale on $t\in[0,T]$ for every $k\in K$, we denote by
$\big{[}\widehat{W}^{N_{j},k}-\overline{W}^{k}\big{]}_{t}$ its quadratic
variation. By Burkholder-Davis-Gundy inequality,

$\displaystyle\mathbf{E}\left|\int_{0}^{T}\sum_{k\in
K}\left\langle\overline{\mu}_{t}^{m}(dx),\nabla\phi(x)\cdot\sigma_{k}(x)\right\rangle
d\left(\widehat{W}_{t}^{N_{j},k}-\overline{W}_{t}^{k}\right)\right|$
$\displaystyle\leq\sum_{k\in
K}\mathbf{E}\left|\int_{0}^{T}\left\langle\overline{\mu}_{t}^{m}(dx),\nabla\phi(x)\cdot\sigma_{k}(x)\right\rangle
d\left(\widehat{W}_{t}^{N_{j},k}-\overline{W}_{t}^{k}\right)\right|$
$\displaystyle\leq C_{1}^{\prime}\sum_{k\in
K}\mathbf{E}\left[\left(\int_{0}^{T}\left|\left\langle\overline{\mu}^{m}_{t},\nabla\phi(x)\cdot\sigma_{k}(x)\right\rangle\right|^{2}d\big{[}\widehat{W}^{N_{j},k}-\overline{W}^{k}\big{]}_{t}\right)^{1/2}\right]$
$\displaystyle\leq C^{\prime}_{1}\|\phi\|_{C^{1}}\sum_{k\in
K}\|\sigma_{k}\|_{\infty}\mathbf{E}\left[\big{[}\widehat{W}^{N_{j},k}-\overline{W}^{k}\big{]}_{T}^{1/2}\right].$
(18)

Now by the definition of quadratic variation,

$\displaystyle\mathbf{E}\big{[}\widehat{W}^{N_{j},k}-\overline{W}^{k}\big{]}_{T}=\mathbf{E}\left[\big{|}\widehat{W}^{N_{j},k}_{T}-\overline{W}^{k}_{T}\big{|}^{2}\right].$
(19)

Since $\widehat{W}^{N_{j},k}-\overline{W}^{k}\to 0$ in the uniform topoplogy
of $C([0,T];\mathbb{R})$, $\mathbf{P}$-a.s. as $j\to\infty$, we have that
$\widehat{W}^{N_{j},k}_{T}\to\overline{W}^{k}_{T}$, $\mathbf{P}$-a.s.; and
besides, for all $j\in\mathbb{N}$ and $p>2$,

$\mathbf{E}\big{[}|\widehat{W}_{T}^{N_{j},k}|^{p}\big{]}=\mathbf{E}\big{[}|\overline{W}^{k}_{T}|^{p}\big{]}<\infty.$

By Vitali convergence theorem, RHS of (19) converges to zero as $j\to\infty$,
and as a consequence (2) also converges to zero.

Combining (2), (2) yields our claim (16) by the triangle inquality. Then,
combining (14), (15), (16) and Lemma 6, we conclude that for every $\phi\in
C_{c}^{\infty}(\mathbb{R}^{d})$ and $1\leq m\leq M$, it holds that
$\mathbf{P}$-a.s.

$\displaystyle\left\langle\phi(x),u_{m}(T,x)\right\rangle$
$\displaystyle-\left\langle\phi(x),r_{m}p_{m}(x)\right\rangle-\int_{0}^{T}dt\left\langle
u_{m}(t,x),\frac{\lambda^{2}}{2}\Delta\phi(x)\right\rangle$
$\displaystyle-\int_{0}^{T}dt\left\langle
u_{m}(t,x),\frac{1}{2}\operatorname{div}\left(Q(x,x)\nabla\phi(x)\right)\right\rangle$
$\displaystyle-\int_{0}^{T}dt\sum_{n=1}^{m-1}\left\langle
u_{m-n}(t,x)u_{n}(t,x),\phi(x)\right\rangle$
$\displaystyle+2\int_{0}^{T}dt\sum_{n=1}^{M}\left\langle
u_{n}(t,x)u_{m}(t,x),\phi(x)\right\rangle$
$\displaystyle-\int_{0}^{T}\sum_{k\in
K}\left\langle\overline{\mu}^{m}_{t}(dx),\sigma_{k}(x)\cdot\nabla\phi(x)\right\rangle
d\overline{W}_{t}^{k}=0.$

(We used $n/m+(m-n)/m=1$ for every $1\leq n\leq m-1$ in (14).) By the
separability of the space $C_{c}^{\infty}(\mathbb{R}^{d})$, we can combine
countably many null sets such that $\mathbf{P}$-a.s. for all $\phi\in
C_{c}^{\infty}(\mathbb{R}^{d})$, the preceding identity holds, which means
that $\big{(}\\{u_{m}\\}_{m\leq M},\\{\overline{W}^{k}\\}_{k\in K}\big{)}$ is
a (both analytically and probabilistically) weak solution to the SPDE (8) on
$(\widehat{\Omega},\widehat{\mathcal{F}},\mathbf{P})$ endowed with the
filtration

$\displaystyle\mathcal{G}_{t}:=\sigma\big{\\{}\\{u_{m}(s,\cdot)\\}_{m\leq
M},\\{\overline{W}^{k}_{s}\\}_{k\in K}:s\in[0,t]\big{\\}},\quad t\geq 0.$

To treat the nonlinear terms in (14), in Sections 3-4 we use Itô-Tanaka trick
to show that

###### Proposition 5

On the original probability space $(\Omega,\mathcal{F},\mathbb{P})$, for every
$T$ finite, $1\leq m,n\leq M$ and $\phi,\psi\in
C_{c}^{\infty}(\mathbb{R}^{d})$, we have that

$\displaystyle\lim_{|z|\to 0}\limsup_{N\to\infty}\mathbb{E}$
$\displaystyle\sup_{t\in[0,T]}\Big{|}\int_{0}^{t}ds\frac{1}{N^{2}}\sum_{i\neq
j\in\mathcal{N}(s)}\phi\left(x_{i}^{N}(s)\right)\psi\left(x_{j}^{N}(s)\right)1_{\\{m_{i}^{N}(s)=m,m_{j}^{N}(s)=n\\}}$
(20)
$\displaystyle\cdot\left[\theta^{\epsilon}\left(x_{i}^{N}(s)-x_{j}^{N}(s)+z\right)-\theta^{\epsilon}\left(x_{i}^{N}(s)-x_{j}^{N}(s)\right)\right]\Big{|}=0.$

Given (20), we can show the convergence in $L^{1}(\mathbf{P})$ of each of the
nonlinear terms in (14) as in the following lemma.

###### Lemma 6

Granted Proposition 5. On
$(\widehat{\Omega},\widehat{\mathcal{F}},\mathbf{P})$, for every $T$ finite,
$1\leq m,n\leq M$ and $\phi,\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$, we have
that

$\displaystyle\lim_{j\to\infty}\mathbf{E}\sup_{t\in[0,T]}\Big{|}\int_{0}^{t}\left\langle\theta^{\epsilon}(x-y)\phi(x)\psi(y),\widehat{\mu}_{s}^{N_{j},m}(dx)\widehat{\mu}^{N_{j},n}_{s}(dy)\right\rangle
ds-\int_{0}^{t}ds\int_{\mathbb{R}^{d}}dw\phi(w)\psi(w)u_{m}(s,w)u_{n}(s,w)\Big{|}=0.$
(21)

Proof. First notice that the quantity inside the absolute value on the LHS of
(20) is a function of the empirical measure (using $\theta^{\epsilon}(0)=0$)

$\int_{0}^{t}\left\langle\big{[}\theta^{\epsilon}(x-y+z)-\theta^{\epsilon}(x-y)\big{]}\phi(x)\psi(y),\mu_{s}^{N,m}(dx)\mu^{N,n}_{s}(dy)\right\rangle
ds.$

Thus, under Skorohod’s representation, the same limit (20) holds on the
auxiliary probability space
$(\widehat{\Omega},\widehat{\mathcal{F}},\mathbf{P})$ along the subsequence
$N_{j}$, namely

$\displaystyle\lim_{|z|\to
0}\limsup_{j\to\infty}\mathbf{E}\sup_{t\in[0,T]}\Big{|}\int_{0}^{t}\left\langle\big{[}\theta^{\epsilon}(x-y+z)-\theta^{\epsilon}(x-y)\big{]}\phi(x)\psi(y),\widehat{\mu}_{s}^{N_{j},m}(dx)\widehat{\mu}^{N_{j},n}_{s}(dy)\right\rangle
ds\Big{|}=0.$ (22)

The subsequent argument is similar to [23, pages 42-43]. We introduce an
auxiliary mollifier $\chi^{\delta}(x)=\delta^{-d}\chi(\delta^{-1}x)$ for some
$C^{\infty}_{b}(\mathbb{R}^{d})$ function
$\chi:\mathbb{R}^{d}\to\mathbb{R}_{+}$, nonnegative, compactly supported in
$\mathbb{B}(0,1)$, with $\int\chi=1$. By (22), we can write

$\displaystyle\sup_{t\in[0,T]}\Big{|}\int_{0}^{t}\left\langle\theta^{\epsilon}(x-y)\phi(x)\psi(y),\widehat{\mu}_{s}^{N_{j},m}(dx)\widehat{\mu}^{N_{j},n}_{s}(dy)\right\rangle
ds$ (23)
$\displaystyle-\iint_{\mathbb{R}^{2d}}dz_{1}dz_{2}\chi^{\delta}(z_{1})\chi^{\delta}(z_{2})\int_{0}^{t}\left\langle\theta^{\epsilon}(x-y+z_{2}-z_{1})\phi(x)\psi(y),\widehat{\mu}_{s}^{N_{j},m}(dx)\widehat{\mu}^{N_{j},n}_{s}(dy)\right\rangle
ds\Big{|}$ $\displaystyle=:\text{error}(N,\delta)$
$\displaystyle\quad\text{where }\quad\lim_{\delta\to
0}\limsup_{N\to\infty}\mathbf{E}|\text{error}(N,\delta)|=0.$

Shifting the arguments of $\phi$ and $\psi$ by $z_{1}$ and $z_{2}$
respectively, with $|z_{1}|,|z_{2}|\leq\delta$, we have that

$\displaystyle\Big{|}\iint_{\mathbb{R}^{2d}}dz_{1}dz_{2}\chi^{\delta}(z_{1})\chi^{\delta}(z_{2})\int_{0}^{t}\left\langle\theta^{\epsilon}(x-y+z_{2}-z_{1})\phi(x)\psi(y),\widehat{\mu}_{s}^{N_{j},m}(dx)\widehat{\mu}^{N_{j},n}_{s}(dy)\right\rangle
ds$ (24) $\displaystyle-$
$\displaystyle\iint_{\mathbb{R}^{2d}}dz_{1}dz_{2}\chi^{\delta}(z_{1})\chi^{\delta}(z_{2})\int_{0}^{t}\left\langle\theta^{\epsilon}(x-y+z_{2}-z_{1})\phi(x-z_{1})\psi(y-z_{2}),\widehat{\mu}_{s}^{N_{j},m}(dx)\widehat{\mu}^{N_{j},n}_{s}(dy)\right\rangle
ds\Big{|}$ $\displaystyle\leq
C(\phi,\psi)\delta\iint_{\mathbb{R}^{2d}}dz_{1}dz_{2}\chi^{\delta}(z_{1})\chi^{\delta}(z_{2})\int_{0}^{t}\left\langle\theta^{\epsilon}(x-y+z_{2}-z_{1}),\widehat{\mu}_{s}^{N_{j},m}(dx)\widehat{\mu}^{N_{j},n}_{s}(dy)\right\rangle
ds.$

By a change of variables, the second term in (LABEL:shift)

$\displaystyle\iint_{\mathbb{R}^{2d}}dz_{1}dz_{2}\chi^{\delta}(z_{1})\chi^{\delta}(z_{2})\int_{0}^{t}\left\langle\theta^{\epsilon}(x-y+z_{2}-z_{1})\phi(x-z_{1})\psi(y-z_{2}),\widehat{\mu}_{s}^{N_{j},m}(dx)\widehat{\mu}^{N_{j},n}_{s}(dy)\right\rangle
ds$
$\displaystyle=\iint_{\mathbb{R}^{2d}}dw_{1}dw_{2}\theta^{\epsilon}(w_{1}-w_{2})\phi(w_{1})\psi(w_{2})\int_{0}^{t}\left\langle\chi^{\delta}(x-w_{1}),\widehat{\mu}_{s}^{N_{j},m}(dx)\right\rangle\left\langle\chi^{\delta}(y-w_{2})\widehat{\mu}^{N_{j},n}_{s}(dy)\right\rangle
ds.$

We now shift $w_{2}$ to $w_{1}$ in some arguments, using that
$|w_{1}-w_{2}|\leq 2C_{0}\epsilon$ being in the support of
$\theta^{\epsilon}$, we get for every $t\in[0,T]$,

$\displaystyle\Big{|}\int_{0}^{t}ds\iint_{\mathbb{R}^{2d}}dw_{1}dw_{2}\theta^{\epsilon}\left(w_{1}-w_{2}\right)\phi(w_{1})\psi(w_{2})\left\langle\chi^{\delta}(\cdot-
w_{1}),\widehat{\mu}_{s}^{N_{j},m}\right\rangle\left\langle\chi^{\delta}(\cdot-
w_{2}),\widehat{\mu}_{s}^{N_{j},n}\right\rangle$
$\displaystyle-\int_{0}^{t}ds\iint_{\mathbb{R}^{2d}}dw_{1}dw_{2}\theta^{\epsilon}\left(w_{1}-w_{2}\right)\phi(w_{1})\psi(w_{1})\left\langle\chi^{\delta}(\cdot-
w_{1}),\widehat{\mu}_{s}^{N_{j},m}\right\rangle\left\langle\chi^{\delta}(\cdot-
w_{1}),\widehat{\mu}_{s}^{N_{j},n}\right\rangle\Big{|}$ $\displaystyle\leq
C(\phi,\psi,C_{0},T)\epsilon\delta^{-2d-1}.$

Since $\int\theta^{\epsilon}(w_{1}-w_{2})dw_{2}=1$, the second term above
equals

$\displaystyle\int_{0}^{t}ds\iint_{\mathbb{R}^{2d}}dw_{1}dw_{2}\theta^{\epsilon}\left(w_{1}-w_{2}\right)\phi(w_{1})\psi(w_{1})\left\langle\chi^{\delta}(\cdot-
w_{1}),\widehat{\mu}_{s}^{N_{j},m}\right\rangle\left\langle\chi^{\delta}(\cdot-
w_{1}),\widehat{\mu}_{s}^{N_{j},n}\right\rangle$
$\displaystyle=\int_{0}^{t}ds\int_{\mathbb{R}^{d}}dw_{1}\phi(w_{1})\psi(w_{1})\left\langle\chi^{\delta}(\cdot-
w_{1}),\widehat{\mu}_{s}^{N_{j},m}\right\rangle\left\langle\chi^{\delta}(\cdot-
w_{1}),\widehat{\mu}_{s}^{N_{j},n}\right\rangle.$

In Section 7, it is shown that
$\\{\widehat{\mu}_{t}^{N_{j},m}(dx):t\in[0,T]\\}_{m\leq
M}\to\\{u_{m}(t,x)dx:t\in[0,T]\\}_{m\leq M}$, $\mathbf{P}$-a.s., we get that
for fixed $\delta>0$, as $N_{j}\to\infty$ (hence $\epsilon=\epsilon(N_{j})\to
0$), $\mathbf{P}$-a.s.

$\displaystyle\sup_{t\in[0,T]}\Big{|}\int_{0}^{t}ds\int_{\mathbb{R}^{d}}dw_{1}\phi(w_{1})\psi(w_{1})\left\langle\chi^{\delta}(\cdot-
w_{1}),\widehat{\mu}_{s}^{N_{j},m}\right\rangle\left\langle\chi^{\delta}(\cdot-
w_{1}),\widehat{\mu}_{s}^{N_{j},n}\right\rangle$
$\displaystyle-\int_{0}^{t}ds\int_{\mathbb{R}^{d}}dw_{1}\phi(w_{1})\psi(w_{1})\left\langle\chi^{\delta}(\cdot-
w_{1}),u_{m}(s,\cdot)\right\rangle\left\langle\chi^{\delta}(\cdot-
w_{1}),u_{n}(s,\cdot)\right\rangle\Big{|}\to 0.$

The convergence also holds in $L^{1}(\mathbf{P})$ by dominated convergence
(note that at this step $\delta$ is fixed). Then, since $\int\chi^{\delta}=1$
for any $\delta>0$ and $\\{u_{m}\\}_{m\leq M}$ is bounded above uniformly by
$\Gamma$, we have that

$\left\langle\chi^{\delta}(\cdot-
w_{1}),u_{m}(s,\cdot)\right\rangle\leq\Gamma.$

By dominated convergence theorem, as $\delta\to 0$ we get that
$\mathbf{P}$-a.s.

$\displaystyle\sup_{t\in[0,T]}\Big{|}\int_{0}^{t}ds\int_{\mathbb{R}^{d}}dw_{1}\phi(w_{1})\psi(w_{1})\left\langle\chi^{\delta}(\cdot-
w_{1}),u_{m}(s,\cdot)\right\rangle\left\langle\chi^{\delta}(\cdot-
w_{1}),u_{n}(s,\cdot)\right\rangle$
$\displaystyle-\int_{0}^{t}ds\int_{\mathbb{R}^{d}}dw_{1}\phi(w_{1})\psi(w_{1})u_{m}(s,w_{1})u_{n}(s,w_{1})\Big{|}\to
0.$

The convergence also holds in $L^{1}(\mathbf{P})$. There is also the minor
term on the RHS of (LABEL:shift)

$C(\phi,\psi)\delta\sup_{t\in[0,T]}\Big{|}\int_{0}^{t}\iint_{\mathbb{R}^{2d}}dz_{1}dz_{2}\chi^{\delta}(z_{1})\chi^{\delta}(z_{2})\left\langle\theta^{\epsilon}(x-y+z_{2}-z_{1}),\widehat{\mu}_{s}^{N_{j},m}(dx)\widehat{\mu}^{N_{j},n}_{s}(dy)\right\rangle
ds\Big{|}$

that can be shown in a similar way to vanish in $L^{1}(\mathbf{P})$ as
$j\to\infty$ followed by $\delta\to 0$.

By (23), (LABEL:shift) and the previous chain of limits, we get (21).

##  3 Itô-Tanaka procedure

Our goal in this and next sections is to prove Proposition 5, which we argue
on the original probability space $(\Omega,\mathcal{F},\mathbb{P})$.

The Itô-Tanaka trick, well-known in the setting of SDEs, is a way to
substitute a less regular function (here $\theta^{\epsilon}(\cdot)$) by a more
regular one, via the application of Itô formula. Fix $1\leq m,n\leq N$ and
consider the (time-dependent) functional

$\displaystyle F_{2}(t,\eta):=\frac{1}{N^{2}}\sum_{i\neq
j\in\mathcal{N}(\eta)}v^{\epsilon,z}\left(t,x_{i},x_{j}\right)\phi(x_{i})\psi(x_{j})1_{\\{m_{i}=m,m_{j}=n\\}}$

where
$v^{\epsilon,z}(t,x,y):[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}$
also depends on $z\in\mathbb{R}^{d}$. In fact, it is of the form of a
difference

$\displaystyle
v^{\epsilon,z}(t,x,y)=r^{\epsilon,z}(t,x,y)-r^{\epsilon,0}(t,x,y)$

where $r^{\epsilon,z}(t,x,y)$ is a family (indexed by $z$) of nonnegative
functions in the parabolic Hölder space
$C^{1+\alpha/2,2+\alpha}([0,T]\times\mathbb{R}^{2d})$ (cf. [29]), for some
$\alpha\in(0,1)$, defined in (35). Applying Itô formula to the process

$\displaystyle F_{2}(t,\eta(t)):=\frac{1}{N^{2}}\sum_{i\neq
j\in\mathcal{N}(t)}v^{\epsilon,z}\left(t,x_{i}^{N}(t),x_{j}^{N}(t)\right)\phi(x_{i}^{N}(t))\psi(x_{j}^{N}(t))1_{\\{m_{i}^{N}(t)=m,m_{j}^{N}(t)=n\\}},\quad
t\geq 0$

and integrating on $[0,T^{\prime}]$, for any $T^{\prime}\leq T$, we get the
following terms from the action of the diffusion generator
$\mathcal{L}^{N}_{D}$ (5):

$\displaystyle H_{1}:=\int_{0}^{T^{\prime}}dt\frac{1}{N^{2}}\sum_{i\neq
j\in\mathcal{N}(t)}\left(\left(\partial_{t}+\frac{\lambda^{2}}{2}\Delta_{x}+\frac{\lambda^{2}}{2}\Delta_{y}\right)v^{\epsilon,z}\right)\left(t,x_{i}^{N}(t),x_{j}^{N}(t)\right)\phi(x_{i}^{N}(t))\psi(x_{j}^{N}(t))1_{\\{m_{i}^{N}(t)=m,m_{j}^{N}(t)=n\\}}\;;$
(25) $\displaystyle H_{2}$
$\displaystyle:=\int_{0}^{T^{\prime}}dt\frac{\lambda^{2}}{N^{2}}\sum_{i\neq
j\in\mathcal{N}(t)}\Big{[}\frac{1}{2}v^{\epsilon,z}\left(t,x_{i}^{N}(t),x_{j}^{N}(t)\right)\left(\Delta\phi(x_{i}^{N}(t))\psi(x_{j}^{N}(t))+\phi(x_{i}^{N}(t))\Delta\psi(x_{j}^{N}(t))\right)$
(26)
$\displaystyle+\left(\nabla_{x}v^{\epsilon,z}\right)\left(t,x_{i}^{N}(t),x_{j}^{N}(t)\right)\cdot\nabla\phi(x_{i}^{N}(t))\psi(x_{j}^{N}(t))$
$\displaystyle+\left(\nabla_{y}v^{\epsilon,z}\right)\left(t,x_{i}^{N}(t),x_{j}^{N}(t)\right)\cdot\nabla\psi(x_{j}^{N}(t))\phi(x_{i}^{N}(t))\Big{]}1_{\\{m_{i}^{N}(t)=m,m_{j}^{N}(t)=n\\}}\;;$

where $\Delta_{x}$ denotes Laplacian with respect to the first $d$ spatial
coordinates, and $\Delta_{y}$ with respect to the last $d$ spatial
coordinates; the same interpretation applies for gradients
$\nabla_{x},\nabla_{y}$;

$\displaystyle H_{3}:=$
$\displaystyle\int_{0}^{T^{\prime}}dt\frac{1}{N^{2}}\sum_{i\neq
j\in\mathcal{N}(t)}\frac{1}{2}\sum_{\alpha,\beta=1}^{d}\Big{[}Q^{\alpha\beta}(x_{i}^{N}(t),x_{i}^{N}(t))\left(\partial^{2}_{x^{\alpha}x^{\beta}}v^{\epsilon,z}\right)\left(t,x_{i}^{N}(t),x_{j}^{N}(t)\right)$
(27)
$\displaystyle+Q^{\alpha\beta}(x_{j}^{N}(t),x_{j}^{N}(t))\left(\partial^{2}_{y^{\alpha}y^{\beta}}v^{\epsilon,z}\right)\left(t,x_{i}^{N}(t),x_{j}^{N}(t)\right)$
$\displaystyle+Q^{\alpha\beta}(x_{i}^{N}(t),x_{j}^{N}(t))\left(\partial^{2}_{x^{\alpha}y^{\beta}}v^{\epsilon,z}\right)\left(t,x_{i}^{N}(t),x_{j}^{N}(t)\right)$
$\displaystyle+Q^{\alpha\beta}(x_{j}^{N}(t),x_{i}^{N}(t))\left(\partial^{2}_{y^{\alpha}x^{\beta}}v^{\epsilon,z}\right)\left(t,x_{i}^{N}(t),x_{j}^{N}(t)\right)\Big{]}\phi(x_{i}^{N}(t))\psi(x_{j}^{N}(t))1_{\\{m_{i}^{N}(t)=m,m_{j}^{N}(t)=n\\}}\;;$
$\displaystyle H_{4}:=$
$\displaystyle\int_{0}^{T^{\prime}}dt\frac{1}{N^{2}}\sum_{i\neq
j\in\mathcal{N}(t)}\frac{1}{2}\sum_{\alpha,\beta=1}^{d}\Big{[}Q^{\alpha\beta}(x_{i}^{N}(t),x_{i}^{N}(t))v^{\epsilon,z}\left(t,x_{i}^{N}(t),x_{j}^{N}(t)\right)\partial^{2}_{\alpha\beta}\phi(x_{i}^{N}(t))\psi(x_{j}^{N}(t))$
(28)
$\displaystyle+Q^{\alpha\beta}(x_{j}^{N}(t),x_{j}^{N}(t))v^{\epsilon,z}\left(t,x_{i}^{N}(t),x_{j}^{N}(t)\right)\phi(x_{i}^{N}(t))\partial^{2}_{\alpha\beta}\psi(x_{j}^{N}(t))$
$\displaystyle+2Q^{\alpha\beta}(x_{i}^{N}(t),x_{i}^{N}(t))\left(\partial_{x^{\alpha}}v^{\epsilon,z}\right)\left(t,x_{i}^{N}(t),x_{j}^{N}(t)\right)\partial_{\beta}\phi(x_{i}^{N}(t))\psi(x_{j}^{N}(t))$
$\displaystyle+2Q^{\alpha\beta}(x_{j}^{N}(t),x_{j}^{N}(t))\left(\partial_{y^{\alpha}}v^{\epsilon,z}\right)\left(t,x_{i}^{N}(t),x_{j}^{N}(t)\right)\partial_{\beta}\psi(x_{j}^{N}(t))\phi(x_{i}^{N}(t))$
$\displaystyle+Q^{\alpha\beta}(x_{i}^{N}(t),x_{j}^{N}(t))v^{\epsilon,z}\left(t,x_{i}^{N}(t),x_{j}^{N}(t)\right)\partial_{\alpha}\phi(x_{i}^{N}(t))\partial_{\beta}\psi(x_{j}^{N}(t))$
$\displaystyle+Q^{\alpha\beta}(x_{i}^{N}(t),x_{j}^{N}(t))\left(\partial_{y^{\beta}}v^{\epsilon,z}\right)\left(t,x_{i}^{N}(t),x_{j}^{N}(t)\right)\partial_{\alpha}\phi(x_{i}^{N}(t))\psi(x_{j}^{N}(t))$
$\displaystyle+Q^{\alpha\beta}(x_{i}^{N}(t),x_{j}^{N}(t))\left(\partial_{x^{\alpha}}v^{\epsilon,z}\right)\left(t,x_{i}^{N}(t),x_{j}^{N}(t)\right)\phi(x_{i}^{N}(t))\partial_{\beta}\psi(x_{j}^{N}(t))$
$\displaystyle+Q^{\alpha\beta}(x_{j}^{N}(t),x_{i}^{N}(t))v^{\epsilon,z}\left(t,x_{i}^{N}(t),x_{j}^{N}(t)\right)\partial_{\beta}\phi(x_{i}^{N}(t))\partial_{\alpha}\psi(x_{j}^{N}(t))$
$\displaystyle+Q^{\alpha\beta}(x_{j}^{N}(t),x_{i}^{N}(t))\left(\partial_{x^{\beta}}v^{\epsilon,z}\right)\left(t,x_{i}^{N}(t),x_{j}^{N}(t)\right)\phi(x_{i}^{N}(t))\partial_{\alpha}\psi(x_{j}^{N}(t))$
$\displaystyle+Q^{\alpha\beta}(x_{j}^{N}(t),x_{i}^{N}(t))\left(\partial_{y^{\alpha}}v^{\epsilon,z}\right)\left(t,x_{i}^{N}(t),x_{j}^{N}(t)\right)\partial_{\beta}\phi(x_{i}^{N}(t))\psi(x_{j}^{N}(t))\Big{]}1_{\\{m_{i}^{N}(t)=m,m_{j}^{N}(t)=n\\}}\;;$
$\displaystyle H_{5}:=$ (29)
$\displaystyle\int_{0}^{T^{\prime}}dt\frac{1}{2N^{2}}\sum_{i\neq
j\in\mathcal{N}(t)}\sum_{\alpha,\beta=1}^{d}\partial_{\alpha}\left(v^{\epsilon,z}\left(t,\cdot,x_{j}^{N}(t)\right)\phi(\cdot)\right)\left(x_{i}^{N}(t)\right)\partial_{\beta}\left(Q^{\alpha\beta}\left(x_{i}^{N}(t),x_{i}^{N}(t)\right)\right)\psi(x_{j}^{N}(t))1_{\\{m_{i}^{N}(t)=m,m_{j}^{N}(t)=n\\}}$
$\displaystyle+\int_{0}^{T^{\prime}}dt\frac{1}{2N^{2}}\sum_{i\neq
j\in\mathcal{N}(t)}\sum_{\alpha,\beta=1}^{d}\partial_{\alpha}\left(v^{\epsilon,z}\left(t,x_{i}^{N}(t),\cdot\right)\psi(\cdot)\right)\left(x_{j}^{N}(t)\right)\partial_{\beta}\left(Q^{\alpha\beta}\left(x_{j}^{N}(t),x_{j}^{N}(t)\right)\right)\phi(x_{i}^{N}(t))1_{\\{m_{i}^{N}(t)=m,m_{j}^{N}(t)=n\\}}\;;$

we note here that only $H_{1}$ and $H_{3}$ involve second partial derivatives
of $v^{\epsilon,z}$.

From the action of the coagulation generator $\mathcal{L}^{N}_{J}$ (6) we get:

$\displaystyle H_{6}:=$
$\displaystyle\int_{0}^{T^{\prime}}dt\frac{1}{N^{2}}\sum_{i\in\mathcal{N}(t)}\sum_{k\in\mathcal{N}(t),k\neq
i}\frac{1}{N}\theta^{\epsilon}(x_{i}^{N}(t)-x_{k}^{N}(t))$ (30)
$\displaystyle\cdot\Big{[}\sum_{j\in\mathcal{N}(t),j\neq
i,k}\frac{m_{i}^{N}(t)}{m_{i}^{N}(t)+m_{k}^{N}(t)}v^{\epsilon,z}(t,x_{i}^{N}(t),x_{j}^{N}(t))\phi(x_{i}^{N}(t))1_{\\{m_{i}^{N}(t)+m_{k}^{N}(t)=m\\}}\psi(x_{j}^{N}(t))1_{\\{m_{j}^{N}(t)=n\\}}$
$\displaystyle+\sum_{j\in\mathcal{N}(t),j\neq
i,k}\frac{m_{k}^{N}(t)}{m_{i}^{N}(t)+m_{k}^{N}(t)}v^{\epsilon,z}(t,x_{k}^{N}(t),x_{j}^{N}(t))\phi(x_{k}^{N}(t))1_{\\{m_{i}^{N}(t)+m_{k}^{N}(t)=m\\}}\psi(x_{j}^{N}(t))1_{\\{m_{j}^{N}(t)=n\\}}$
$\displaystyle-\sum_{j\in\mathcal{N}(t),j\neq
i,k}v^{\epsilon,z}(t,x_{i}^{N}(t),x_{j}^{N}(t))\phi(x_{i}^{N}(t))1_{\\{m_{i}^{N}(t)=m\\}}\psi(x_{j}^{N}(t))1_{\\{m_{j}^{N}(t)=n\\}}$
$\displaystyle-\sum_{j\in\mathcal{N}(t),j\neq
i,k}v^{\epsilon,z}(t,x_{k}^{N}(t),x_{j}^{N}(t))\phi(x_{k}^{N}(t))1_{\\{m_{i}^{N}(t)=m\\}}\psi(x_{j}^{N}(t))1_{\\{m_{j}^{N}(t)=n\\}}\Big{]}$
$\displaystyle+$
$\displaystyle\int_{0}^{T^{\prime}}dt\frac{1}{N^{2}}\sum_{i\in\mathcal{N}(t)}\sum_{k\in\mathcal{N}(t),k\neq
i}\frac{1}{N}\theta^{\epsilon}(x_{i}^{N}(t)-x_{k}^{N}(t))$
$\displaystyle\cdot\Big{[}\sum_{j\in\mathcal{N}(t),j\neq
i,k}\frac{m_{i}^{N}(t)}{m_{i}^{N}(t)+m_{k}^{N}(t)}v^{\epsilon,z}(t,x_{j}^{N}(t),x_{i}^{N}(t))\phi(x_{j}^{N}(t))1_{\\{m_{j}^{N}(t)=m\\}}\psi(x_{i}^{N}(t))1_{\\{m_{i}^{N}(t)+m_{k}^{N}(t)=n\\}}$
$\displaystyle+\sum_{j\in\mathcal{N}(t),j\neq
i,k}\frac{m_{k}^{N}(t)}{m_{i}^{N}(t)+m_{k}^{N}(t)}v^{\epsilon,z}(t,x_{j}^{N}(t),x_{k}^{N}(t))\phi(x_{j}^{N}(t))1_{\\{m_{j}^{N}(t)=m\\}}\psi(x_{k}^{N}(t))1_{\\{m_{i}^{N}(t)+m_{k}^{N}(t)=n\\}}$
$\displaystyle-\sum_{j\in\mathcal{N}(t),j\neq
i,k}v^{\epsilon,z}(t,x_{j}^{N}(t),x_{i}^{N}(t))\phi(x_{j}^{N}(t))1_{\\{m_{j}^{N}(t)=m\\}}\psi(x_{i}^{N}(t))1_{\\{m_{i}^{N}(t)=n\\}}$
$\displaystyle-\sum_{j\in\mathcal{N}(t),j\neq
i,k}v^{\epsilon,z}(t,x_{j}^{N}(t),x_{k}^{N}(t))\phi(x_{j}^{N}(t))1_{\\{m_{i}^{N}(t)=m\\}}\psi(x_{k}^{N}(t))1_{\\{m_{k}^{N}(t)=n\\}}\Big{]}\;;$

$\displaystyle H_{7}:=$
$\displaystyle-\int_{0}^{T^{\prime}}dt\frac{1}{N^{3}}\sum_{i\in\mathcal{N}(t)}\sum_{k\in\mathcal{N}(t),k\neq
i}\theta^{\epsilon}(x_{i}^{N}(t)-x_{k}^{N}(t))v^{\epsilon,z}\left(t,x_{i}^{N}(t),x_{k}^{N}(t)\right)\phi(x_{i}^{N}(t))\psi(x_{k}^{N}(t))1_{\\{m_{i}^{N}(t)=m,m_{k}^{N}(t)=n\\}}.$
(31)

###### Remark 7

This negative term $H_{7}$ arises because of the specific coagulation rule,
namely if particles $(i,k)$ coagulate, then they are both removed from the
system. There may be a new particle added, but it is of a different mass hence
has to be reconsidered (in $H_{6}$). Unlike [23, 24], under our scaling (3)
$H_{7}$ turns out to be negligible, see Lemma 17.

Regarding the martingale terms, of which $M_{1},M_{2}$ come from diffusion

$\displaystyle M_{1}:=$
$\displaystyle\int_{0}^{T^{\prime}}\frac{\lambda}{N^{2}}\sum_{i\in\mathcal{N}(t)}\sum_{j\in\mathcal{N}(t),j\neq
i}\nabla\left[v^{\epsilon,z}\left(t,\cdot,x_{j}^{N}(t)\right)\phi(\cdot)\right]\left(x_{i}^{N}(t)\right)\psi(x_{j}^{N}(t))1_{\\{m_{i}^{N}(t)=m,m_{j}^{N}(t)=n\\}}\cdot
d\beta_{i}(t)$
$\displaystyle+\int_{0}^{T^{\prime}}\frac{\lambda}{N^{2}}\sum_{j\in\mathcal{N}(t)}\sum_{i\in\mathcal{N}(t),i\neq
j}\nabla\left[v^{\epsilon,z}\left(t,x_{i}^{N}(t),\cdot\right)\psi(\cdot)\right]\left(x_{j}^{N}(t)\right)\phi(x_{i}^{N}(t))1_{\\{m_{i}^{N}(t)=m,m_{j}^{N}(t)=n\\}}\cdot
d\beta_{j}(t)$

whose quadratic variation is

$\displaystyle B_{1}$
$\displaystyle=\int_{0}^{T^{\prime}}\frac{\lambda^{2}}{N^{4}}\sum_{i\in\mathcal{N}(t)}\left(\sum_{j\in\mathcal{N}(t),j\neq
i}\nabla\left[v^{\epsilon,z}\left(t,\cdot,x_{j}^{N}(t)\right)\phi(\cdot)\right]\left(x_{i}^{N}(t)\right)\psi(x_{j}^{N}(t))1_{\\{m_{i}^{N}(t)=m,m_{j}^{N}(t)=n\\}}\right)^{2}dt$
(32)
$\displaystyle+\int_{0}^{T^{\prime}}\frac{\lambda^{2}}{N^{4}}\sum_{j=1}^{N(t)}\left(\sum_{i\in\mathcal{N}(t),i\neq
j}\nabla\left[v^{\epsilon,z}\left(t,x_{i}^{N}(t),\cdot\right)\psi(\cdot)\right]\left(x_{j}^{N}(t)\right)\phi(x_{i}^{N}(t))1_{\\{m_{i}^{N}(t)=m,m_{j}^{N}(t)=n\\}}\right)^{2}dt\;;$

and

$\displaystyle M_{2}$ $\displaystyle=\int_{0}^{T^{\prime}}\sum_{k\in
K}\sigma_{k}(x_{i}^{N}(t))\cdot\frac{1}{N^{2}}\sum_{i\in\mathcal{N}(t)}\sum_{j\in\mathcal{N}(t),j\neq
i}\nabla\left[v^{\epsilon,z}\left(t,\cdot,x_{j}^{N}(t)\right)\phi(\cdot)\right]\left(x_{i}^{N}(t)\right)\psi(x_{j}^{N}(t))1_{\\{m_{i}^{N}(t)=m,m_{j}^{N}(t)=n\\}}dW_{k}(t)$
$\displaystyle+\int_{0}^{T^{\prime}}\sum_{k\in
K}\sigma_{k}(x_{j}^{N}(t))\cdot\frac{1}{N^{2}}\sum_{j\in\mathcal{N}(t)}\sum_{i\in\mathcal{N}(t),i\neq
j}\nabla\left[v^{\epsilon,z}\left(t,x_{i}^{N}(t),\cdot\right)\psi(\cdot)\right]\left(x_{j}^{N}(t)\right)\phi(x_{i}^{N}(t))1_{\\{m_{i}^{N}(t)=m,m_{j}^{N}(t)=n\\}}dW_{k}(t)$

whose quadratic variation is

$\displaystyle B_{2}$
$\displaystyle:=\int_{0}^{T^{\prime}}\frac{1}{N^{4}}\sum_{k\in
K}\left(\sigma_{k}(x_{i}^{N}(t))\cdot\sum_{i\neq
j\in\mathcal{N}(t)}\nabla\left[v^{\epsilon,z}\left(t,\cdot,x_{j}^{N}(t)\right)\phi(\cdot)\right]\left(x_{i}^{N}(t)\right)\psi(x_{j}^{N}(t))1_{\\{m_{i}^{N}(t)=m,m_{j}^{N}(t)=n\\}}\right)^{2}dt$
(33) $\displaystyle+\int_{0}^{T^{\prime}}\frac{1}{N^{4}}\sum_{k\in
K}\left(\sigma_{k}(x_{j}^{N}(t))\cdot\sum_{i\neq
j\in\mathcal{N}(t)}\nabla\left[v^{\epsilon,z}\left(t,x_{i}^{N}(t),\cdot\right)\psi(\cdot)\right]\left(x_{j}^{N}(t)\right)\phi(x_{i}^{N}(t))1_{\\{m_{i}^{N}(t)=m,m_{j}^{N}(t)=n\\}}\right)^{2}dt\;;$

and the jump part of the martingale $M_{3}$ (which we do not write explicitly)
has its second moment bounded above by

$\displaystyle B_{3}:=$
$\displaystyle\int_{0}^{T^{\prime}}dt\frac{4}{N^{4}}\sum_{i\in\mathcal{N}(t)}\sum_{k\in\mathcal{N}(t),k\neq
i}\frac{1}{N}\theta^{\epsilon}(x_{i}^{N}(t)-x_{k}^{N}(t))$ (34)
$\displaystyle\cdot\Big{[}\sum_{j\neq
i,k}\frac{m_{i}^{N}(t)}{m_{i}^{N}(t)+m_{k}^{N}(t)}v^{\epsilon,z}(t,x_{i}^{N}(t),x_{j}^{N}(t))\phi(x_{i}^{N}(t))1_{\\{m_{i}^{N}(t)+m_{k}^{N}(t)=m\\}}\psi(x_{j}^{N}(t))1_{\\{m_{j}^{N}(t)=n\\}}$
$\displaystyle+\sum_{j\neq
i,k}\frac{m_{k}^{N}(t)}{m_{i}^{N}(t)+m_{k}^{N}(t)}v^{\epsilon,z}(t,x_{k}^{N}(t),x_{j}^{N}(t))\phi(x_{k}^{N}(t))1_{\\{m_{i}^{N}(t)+m_{k}^{N}(t)=m\\}}\psi(x_{j}^{N}(t))1_{\\{m_{j}^{N}(t)=n\\}}$
$\displaystyle-\sum_{j\neq
i,k}v^{\epsilon,z}(t,x_{i}^{N}(t),x_{j}^{N}(t))\phi(x_{i}^{N}(t))1_{\\{m_{i}^{N}(t)=m\\}}\psi(x_{j}^{N}(t))1_{\\{m_{j}^{N}(t)=n\\}}$
$\displaystyle-\sum_{j\neq
i,k}v^{\epsilon,z}(t,x_{k}^{N}(t),x_{j}^{N}(t))\phi(x_{k}^{N}(t))1_{\\{m_{i}^{N}(t)=m\\}}\psi(x_{j}^{N}(t))1_{\\{m_{j}^{N}(t)=n\\}}$
$\displaystyle-v^{\epsilon,z}\left(t,x_{i}^{N}(t),x_{k}^{N}(t)\right)\phi(x_{i}^{N}(t))\psi(x_{k}^{N}(t))1_{\\{m_{i}^{N}(t)=m,m_{k}^{N}(t)=n\\}}\Big{]}^{2}$
$\displaystyle+\int_{0}^{T^{\prime}}\frac{4}{N^{4}}\sum_{i\in\mathcal{N}(t)}\sum_{k\in\mathcal{N}(t),k\neq
i}\frac{1}{N}\theta^{\epsilon}(x_{i}^{N}(t)-x_{k}^{N}(t))$
$\displaystyle\cdot\Big{[}\sum_{j\neq
i,k}\frac{m_{i}^{N}(t)}{m_{i}^{N}(t)+m_{k}^{N}(t)}v^{\epsilon,z}(t,x_{j}^{N}(t),x_{i}^{N}(t))\phi(x_{j}^{N}(t))1_{\\{m_{i}^{N}(t)=m\\}}\psi(x_{i}^{N}(t))1_{\\{m_{i}^{N}(t)+m_{k}^{N}(t)=n\\}}$
$\displaystyle+\sum_{j\neq
i,k}\frac{m_{k}^{N}(t)}{m_{i}^{N}(t)+m_{k}^{N}(t)}v^{\epsilon,z}(t,x_{j}^{N}(t),x_{k}^{N}(t))\phi(x_{j}^{N}(t))1_{\\{m_{j}^{N}(t)=m\\}}\psi(x_{k}^{N}(t))1_{\\{m_{i}^{N}(t)+m_{k}^{N}(t)=n\\}}$
$\displaystyle-\sum_{j\neq
i,k}v^{\epsilon,z}(t,x_{j}^{N}(t),x_{i}^{N}(t))\phi(x_{j}^{N}(t))1_{\\{m_{j}^{N}(t)=m\\}}\psi(x_{i}^{N}(t))1_{\\{m_{i}^{N}(t)=n\\}}$
$\displaystyle-\sum_{j\neq
i,k}v^{\epsilon,z}(t,x_{j}^{N}(t),x_{k}^{N}(t))\phi(x_{j}^{N}(t))1_{\\{m_{i}^{N}(t)=m\\}}\psi(x_{k}^{N}(t))1_{\\{m_{k}^{N}(t)=n\\}}$
$\displaystyle-v^{\epsilon,z}\left(t,x_{i}^{N}(t),x_{k}^{N}(t)\right)\phi(x_{i}^{N}(t))\psi(x_{k}^{N}(t))1_{\\{m_{i}^{N}(t)=m,m_{k}^{N}(t)=n\\}}\Big{]}^{2}dt.$

We also have the initial and terminal conditions

$\displaystyle H_{8}:=$ $\displaystyle\frac{1}{N^{2}}\sum_{i\neq
j\in\mathcal{N}(0)}v^{\epsilon,z}\left(0,x_{i}(0),x_{j}(0)\right)\phi(x_{i}(0))\psi(x_{j}(0))1_{\\{m_{i}(0)=m,m_{j}(0)=n\\}};$
$\displaystyle H_{9}:=$ $\displaystyle\frac{1}{N^{2}}\sum_{i\neq
j\in\mathcal{N}(T^{\prime})}v^{\epsilon,z}\left(T^{\prime},x^{N}_{i}(T^{\prime}),x^{N}_{j}(T^{\prime})\right)\phi(x^{N}_{i}(T^{\prime}))\psi(x^{N}_{j}(T^{\prime}))1_{\\{m^{N}_{i}(T^{\prime})=m,m^{N}_{j}(T^{\prime})=n\\}}.$

We set up the following family (indexed by $z\in\mathbb{R}^{d}$) of auxiliary
PDE terminal value problems, whose unique nonnegative solution is called
$r^{\epsilon,z}(t,x,y):[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}_{+}$:

$\displaystyle\begin{cases}\left[\partial_{t}+\frac{\lambda^{2}}{2}\big{(}\Delta_{x}+\Delta_{y}\big{)}+\frac{1}{2}\sum_{\alpha,\beta=1}^{d}\big{(}Q^{\alpha\beta}(x,x)\partial^{2}_{x^{\alpha}x^{\beta}}+Q^{\alpha\beta}(y,y)\partial^{2}_{y^{\alpha}y^{\beta}}+Q^{\alpha\beta}(x,y)\partial^{2}_{x^{\alpha}y^{\beta}}+Q^{\alpha\beta}(y,x)\partial^{2}_{y^{\alpha}x^{\beta}}\big{)}\right]r^{\epsilon,z}\left(t,x,y\right)\\\
=-\theta^{\epsilon}(x-y+z),\quad(t,x,y)\in[0,T]\times\mathbb{R}^{2d}\\\ \\\
r^{\epsilon,z}(T,x,y)=0.\end{cases}$ (35)

To be more transparant, if we denote $\mathbf{x}:=(x,y)\in\mathbb{R}^{2d}$,
$\Delta_{\mathbf{x}}:=\Delta_{x}+\Delta_{y}$,

$D^{2}_{\mathbf{x}}:=\left(\begin{matrix}D^{2}_{xx},D^{2}_{xy}\\\
D^{2}_{yx},D^{2}_{yy}\end{matrix}\right)$

and the $(2d)\times(2d)$ non-negative definite matrix

$\displaystyle\widehat{Q}(\mathbf{x}):=\left(\begin{matrix}Q(x,x),Q(x,y)\\\
Q(y,x),Q(y,y)\end{matrix}\right),\quad\mathbf{x}=(x,y)$

then, (35) can be rewritten as

$\displaystyle\begin{cases}\left[\partial_{t}+\frac{\lambda^{2}}{2}\Delta_{\mathbf{x}}+\frac{1}{2}\text{tr}\big{(}\widehat{Q}(\mathbf{x})D^{2}_{\mathbf{x}}\big{)}\right]r^{\epsilon,z}(t,\mathbf{x})=-\theta^{\epsilon}(x-y+z)\\\
r^{\epsilon,z}(T,\mathbf{x})=0,\quad\mathbf{x}=(x,y),t\in[0,T].\end{cases}$
(36)

To see that $\widehat{Q}(\mathbf{x})$ is non-negative, take any
$\xi=(\xi_{1},\xi_{2})\in\mathbb{R}^{2d}$, we have that

$\xi\widehat{Q}(\mathbf{x})\xi^{T}=\sum_{k\in
K}|\xi_{1}\cdot\sigma_{k}(x)+\xi_{2}\cdot\sigma_{k}(y)|^{2}\geq 0.$

Since $\theta^{\epsilon}\in C^{\alpha}(\mathbb{R}^{d})$ for some
$\alpha\in(0,1)$ and $\widehat{Q}$ is smooth of class $C_{b}^{\infty}$, the
solution $r^{\epsilon,z}(t,x,y)\in
C^{1+\alpha/2,2+\alpha}([0,T]\times\mathbb{R}^{2d})$, for every
$\epsilon\in(0,1)$ and $z\in\mathbb{R}^{d}$ (cf. [29, Theorem 8.10.1]). Let us
denote the non-divergence form operator with $C^{\infty}_{b}(\mathbb{R}^{2d})$
coefficients

$\displaystyle\mathcal{A}_{\mathbf{x}}:=\frac{\lambda^{2}}{2}\Delta_{\mathbf{x}}+\frac{1}{2}\text{tr}\left(\widehat{Q}(\mathbf{x})D^{2}_{\mathbf{x}}\right).$
(37)

By the parabolic Maximum Principle, since $\theta^{\epsilon}\geq 0$, the
unique solution $r^{\epsilon,z}(t,x,y)$ of (36) is nonnegative. Since

$\displaystyle
v^{\epsilon,z}(t,x,y)=r^{\epsilon,z}(t,x,y)-r^{\epsilon,0}(t,x,y),$

by linearity it also follows that

$\displaystyle\begin{cases}\left[\partial_{t}+\frac{\lambda^{2}}{2}\Delta_{\mathbf{x}}+\frac{1}{2}\text{tr}\big{(}\widehat{Q}(\mathbf{x})D^{2}_{\mathbf{x}}\big{)}\right]v^{\epsilon,z}(t,\mathbf{x})=-\theta^{\epsilon}(x-y+z)+\theta^{\epsilon}(x-y)\\\
v^{\epsilon,z}(T,\mathbf{x})=0,\quad\mathbf{x}=(x,y),t\in[0,T].\end{cases}$
(38)

From (25), (27), (38), we have that (recall $T^{\prime}\leq T$)

$\displaystyle H_{1}+H_{3}$
$\displaystyle=\int_{0}^{T^{\prime}}dt\frac{1}{N^{2}}\sum_{i\neq
j\in\mathcal{N}(t)}\phi(x_{i}^{N}(t))\psi(x_{j}^{N}(t))1_{\\{m_{i}^{N}(t)=m,m_{j}^{N}(t)=n\\}}$
(39)
$\displaystyle\qquad\cdot\left[\partial_{t}+\frac{\lambda^{2}}{2}\Delta_{\mathbf{x}}+\frac{1}{2}\text{tr}\big{(}\widehat{Q}\big{(}x_{i}^{N}(t),x_{j}^{N}(t)\big{)}D^{2}_{\mathbf{x}}\big{)}\right]v^{\epsilon,z}\left(t,x_{i}^{N}(t),x_{j}^{N}(t)\right)$
$\displaystyle=-\int_{0}^{T^{\prime}}dt\frac{1}{N^{2}}\sum_{i\neq
j\in\mathcal{N}(t)}\phi(x_{i}^{N}(t))\psi(x_{j}^{N}(t))1_{\\{m_{i}^{N}(t)=m,m_{j}^{N}(t)=n\\}}$
$\displaystyle\qquad\cdot\left[\theta^{\epsilon}\left(x_{i}^{N}(t)-x_{j}^{N}(t)+z\right)-\theta^{\epsilon}\left(x_{i}^{N}(t)-x_{j}^{N}(t)\right)\right].$

In view of the identity from Itô formula

$\displaystyle H_{1}+...+H_{9}+M_{1}+M_{2}+M_{3}=0,$ (40)

we can accomplish (20) if we can show that the rest of the terms in (40),
namely those apart from $H_{1},H_{3}$, are all negligible in the sense that

$\displaystyle\lim_{|z|\to
0}\limsup_{N\to\infty}\mathbb{E}\sup_{T^{\prime}\in[0,T]}|H_{i}|=0,\quad
i\in\\{2,4,5,6,7,8,9\\}$ (41) $\displaystyle\lim_{|z|\to
0}\limsup_{N\to\infty}\mathbb{E}\sup_{T^{\prime}\in[0,T]}|M_{i}|=0,\quad
i\in\\{1,2,3\\}.$

These terms only contain up to first partial derivatives of $v^{\epsilon,z}$,
hence their regularity is strictly better than $\theta^{\epsilon}$.

The following key proposition provides uniform bounds on $r^{\epsilon}$ and
its gradient. The proof is similar to [14, Proposition 5], due to which we
omit some details. Recall that $C_{0}$ is the maximal radius of the compact
support of $\theta$.

###### Proposition 8

Let $r^{\epsilon,z}(t,x,y)$ be the unique solution of the PDE (36). Then,
there exists some finite constant $C(d,T,C_{0},\\\ \\{\sigma_{k}\\}_{k\in
K},\lambda)$ such that for any $x,y,z\in\mathbb{R}^{d}$, $t\in[0,T]$ and
$\epsilon\leq\epsilon_{0}$ for some small $\epsilon_{0}(C_{0})$, we have that

$\displaystyle
r^{\epsilon,z}(t,x,y)\leq\begin{cases}Ce^{-C|y-x-z|^{2}}1_{\\{|y-x-z|\geq
4\\}}+C\left(|y-x-z|\vee\epsilon\right)^{2-d}1_{\\{|y-x-z|<4\\}},\quad d\geq
3\\\ \\\ Ce^{-C|y-x-z|^{2}}1_{\\{|y-x-z|\geq
4\\}}+C\left|\log\left(|y-x-z|\vee\epsilon\right)\right|1_{\\{|y-x-z|<4\\}},\quad
d=2\\\ \\\ Ce^{-C|y-x-z|^{2}}1_{\\{|y-x-z|\geq
4\\}}+C1_{\\{|y-x-z|<4\\}},\quad d=1.\end{cases}$ (42)
$\displaystyle\left|\nabla_{x}r^{\epsilon,z}(t,x,y)\right|$
$\displaystyle\leq\begin{cases}Ce^{-C|y-x-z|^{2}}1_{\\{|y-x-z|\geq
4\\}}+C\left(|y-x-z|\vee\epsilon\right)^{1-d}1_{\\{|y-x-z|<4\\}},\quad d\geq
2\\\ \\\ Ce^{-C|y-x-z|^{2}}1_{\\{|y-x-z|\geq
4\\}}+C\left|\log(|y-x-z|\vee\epsilon)\right|1_{\\{|y-x-z|<4\\}},\quad
d=1.\end{cases}$ (43)

The same also holds for $\nabla_{y}$ in place of $\nabla_{x}$.

###### Remark 9

In particular, we have the following useful crude bounds: there exists
$C=C(d,T,C_{0},\\{\sigma_{k}\\}_{k\in K},\lambda)$, such that for any
$x,y,z\in\mathbb{R}^{d}$, $t\in[0,T]$ and $\epsilon\in(0,1)$ small

$\displaystyle r^{\epsilon,z}(t,x,y)$
$\displaystyle\leq\begin{cases}C\epsilon^{2-d},\quad d\geq 3\\\
C|\log\epsilon|,\quad d=2\\\ C,\quad d=1\end{cases}$ (44)
$\displaystyle|\nabla_{x}r^{\epsilon,z}(t,x,y)|$
$\displaystyle\leq\begin{cases}C\epsilon^{1-d},\quad d\geq 2\\\
C|\log\epsilon|,\quad d=1.\end{cases}$ (45)

###### Remark 10

Let us explain the first inequality in (44) by a simple heuristic argument.
First, notice that $\theta_{\epsilon}\left(x\right)\leq
c\epsilon^{-d}1_{\mathbb{B}\left(0,\epsilon C_{0}\right)}\left(x\right)$ for
some constant $c>0$; take $C_{0}=1$ below for notational simplicity. Due to
the Gaussian upper bound described at the beginning of the proof, the problem
reduces to prove that

$\epsilon^{-d}\int_{0}^{t}\mathbb{P}\left(B_{s}\in
S\left(y-x-z,\epsilon\right)\right)ds\leq C\epsilon^{-d+2}$

for some constant $C>0$, where $\left(B_{t}\right)$ is a standard Brownian
motion in $\mathbb{R}^{2d}$ and $S\left(u,\epsilon\right)$ is the strip in
$\mathbb{R}^{2d}$ defined as

$S\left(u,\epsilon\right):=\left\\{\left(x^{\prime},y^{\prime}\right)\in\mathbb{R}^{2d}:\left|y^{\prime}-x^{\prime}-u\right|\leq\epsilon\right\\},\quad
u\in\mathbb{R}^{d}.$

When $\left|y-x-z\right|$ is not infinitesimal in $\epsilon$, say
$\left|y-x-z\right|\geq 4$ as in the proof above, then the strip
$S\left(y-x-z,\epsilon\right)$ is far from the origin (where
$\left(B_{t}\right)$ starts deterministically), hence, thanks also to the
exponential decay of the Brownian density, the problem reduces to estimate
(where $Leb$ denotes Lebesgue measure in $\mathbb{R}^{2d}$)

$\epsilon^{-d}\int_{0}^{t}Leb\left(S\left(y-x-z,\epsilon\right)\cap\mathbb{B}\left(0,1\right)\right)ds.$

It is easily seen that
$Leb\left(S\left(y-x-z,\epsilon\right)\cap\mathbb{B}\left(0,1\right)\right)\leq\epsilon^{d}$
(up to constants) and thus the bound above holds (see also the first
inequality in (42)). If $\left|y-x-z\right|$ is very small, on the contrary,
let us examine the worst case,

$\epsilon^{-d}\int_{0}^{t}\mathbb{P}\left(B_{s}\in
S\left(0,\epsilon\right)\right)ds.$

The problem is that $\mathbb{P}\left(B_{s}\in S\left(0,\epsilon\right)\right)$
is not infinitesimal with $\epsilon$ for $s=0$. The very crude but convincing
argument is that the Brownian motion $B_{s}$ remains in the strip
$S\left(0,\epsilon\right)$ for a time of order $\epsilon^{2}$, contributing to
the integral with the term

$\int_{0}^{\epsilon^{2}}\mathbb{P}\left(B_{s}\in
S\left(0,\epsilon\right)\right)ds\sim\epsilon^{2}.$

After this time the estimate is better. Thus there is an addend
$\epsilon^{-d}\epsilon^{2}$ in the estimate.

Proof. We first show (42), starting with the representation of solution of
(36)

$\displaystyle r^{\epsilon,z}(T-t,x,y)=$
$\displaystyle\int_{[0,t]\times\mathbb{R}^{2d}}q_{t-s}(x,y;x^{\prime},y^{\prime})\theta^{\epsilon}(x^{\prime}-y^{\prime}+z)dsdx^{\prime}dy^{\prime}$

where $q_{t}(x,y;x^{\prime},y^{\prime})$ is the $(2d)$-dimensional heat kernel
(i.e. fundamental solution) associated with the operator
$\mathcal{A}_{\mathbf{x}}$ (37). Since the latter operator is uniformly
elliptic, we have Gaussian upper (and lower) bounds for the heat kernel (cf.
[26, Theorem 1])

$\displaystyle
q_{t}(x,y;x^{\prime},y^{\prime})\leq\frac{C}{t^{\frac{2d}{2}}}e^{-\frac{|x-x^{\prime}|^{2}+|y-y^{\prime}|^{2}}{Ct}}$

for any $t>0,x,y,x^{\prime},y^{\prime}\in\mathbb{R}^{d}$, where
$C=C\left(d,\\{\sigma_{k}\\}_{k\in K},\lambda\right)$ finite. One can show,
for details cf. the proof of [14, Proposition 5, pp. 615], that for some
$C=C(d,T)$ and any $t\in[0,T]$, $x,y,x^{\prime},y^{\prime}\in\mathbb{R}^{d}$
and $d>1$ (hence $2d>2$), we have that

$\displaystyle\int_{0}^{t}q_{s}(x,y;x^{\prime},y^{\prime})ds\leq\int_{0}^{t}\frac{C}{s^{\frac{2d}{2}}}e^{-\frac{|x-x^{\prime}|^{2}+|y-y^{\prime}|^{2}}{Cs}}ds$
$\displaystyle\leq
C\left(|x-x^{\prime}|^{2}+|y-y^{\prime}|^{2}\right)^{\frac{2-2d}{2}}e^{-\frac{|x-x^{\prime}|^{2}+|y-y^{\prime}|^{2}}{C}}+C1_{\\{|x-x^{\prime}|^{2}+|y-y^{\prime}|^{2}\leq
1\\}}.$

Now let us integrate in $x^{\prime},y^{\prime}$ of the preceding expression
against $\theta^{\epsilon}(x^{\prime}-y^{\prime}+z)$, up to constant $C$ we
have that

$\displaystyle\iint_{\mathbb{R}^{2d}}\left(|x-x^{\prime}|^{2}+|y-y^{\prime}|^{2}\right)^{\frac{2-2d}{2}}e^{-\frac{|x-x^{\prime}|^{2}+|y-y^{\prime}|^{2}}{C}}\theta^{\epsilon}(x^{\prime}-y^{\prime}+z)dx^{\prime}dy^{\prime}$
$\displaystyle\quad+\iint_{\mathbb{R}^{2d}}1_{\\{|x-x^{\prime}|^{2}+|y-y^{\prime}|^{2}\leq
1\\}}\theta^{\epsilon}(x^{\prime}-y^{\prime}+z)dx^{\prime}dy^{\prime}$
$\displaystyle\overset{\zeta=x^{\prime}-y^{\prime}+z,\;\gamma=x-x^{\prime}}{=}\iint_{\mathbb{R}^{2d}}\left(|\gamma|^{2}+|y-x-z+\gamma+\zeta|^{2}\right)^{\frac{2-2d}{2}}e^{-\frac{|\gamma|^{2}+|y-x-z+\gamma+\zeta|^{2}}{C}}\theta^{\epsilon}(\zeta)d\gamma
d\zeta$ (46)
$\displaystyle\quad+\iint_{\mathbb{R}^{2d}}1_{\\{|\gamma|^{2}+|y-x-z+\gamma+\zeta|^{2}\leq
1\\}}\theta^{\epsilon}(\zeta)d\gamma d\zeta.$ (47)

We first argue about the exponential decay when $|y-x-z|\geq 4$. In this case,
if we look at (47), since the support of $\zeta$ is in $|\zeta|\leq
C_{0}\epsilon$, and the support of $\gamma$ is in $|\gamma|\leq 1$ (otherwise
the indicator in (47) is $0$), we see that when $|y-x-z|\geq 4$, then
necessarily $|y-x-z+\gamma+\zeta|>1$ for $\epsilon$ small enough, rendering
the indicator again $0$. Thus, we can focus solely on the first term (46) and
we argue exponential decay in $|y-x-z|$ when $|y-x-z|\geq 4$.

We separate the integral in $\gamma$ of (46) according to
$|\gamma|\leq|y-x-z|/2$ and $|\gamma|>|y-x-z|/2$. In the former case,
$|y-x-z+\gamma+\zeta|\geq|y-x-z|/4\geq 1$ for $\epsilon$ small enough, thus we
can bound part of the integral by (noting $2-2d<0$, and
$\int\theta^{\epsilon}(\zeta)d\zeta=1$)

$\displaystyle
C\int_{|\gamma|\leq|y-x-z|/2}e^{-\frac{|y-x-z|^{2}}{C}}d\gamma\leq
C|y-x-z|^{d}e^{-\frac{|y-x-z|^{2}}{C}}\leq
C^{\prime}e^{-\frac{|y-x-z|^{2}}{C^{\prime}}}$

for some $C^{\prime}>C$. In the latter case that $|\gamma|>|y-x-z|/2\geq 2$,
we can bound the other part of the integral by

$\displaystyle
C\int_{|\gamma|>|y-x-z|/2}e^{-\frac{|\gamma|^{2}}{C}}d\gamma\leq
C|y-x-z|^{d-2}e^{-\frac{|y-x-z|^{2}}{C}}\leq
C^{\prime}e^{-\frac{|y-x-z|^{2}}{C^{\prime}}}.$

That is, we have the claimed exponential decay when $|y-x-z|\geq 4$.

Now we turn to the case when $|y-x-z|<4$. Here the term (47) is merely bounded
(due to $\int\theta^{\epsilon}=1$ and that the indicator is over a compact
ball), so the total bound cannot be smaller than a constant bound. We now
focus on the first integral (46). We separate two cases: $|y-x-z|\geq
4C_{0}\epsilon$ or $|y-x-z|<4C_{0}\epsilon$.

If $|y-x-z|\geq 4C_{0}\epsilon$ and $|\gamma|\leq|y-x-z|/4$, then
$|y-x-z+\gamma+\zeta|\geq|y-x-z|/4$ since $|\zeta|\leq C_{0}\epsilon$, we can
bound part of the integral (46) by (ignore the exponential)

$\displaystyle C\int_{|\gamma|\leq|y-x-z|/4}|y-x-z|^{2-2d}d\gamma\leq
C|y-x-z|^{2-d}.$

If $|y-x-z|\geq 4C_{0}\epsilon$ and $|\gamma|>|y-x-z|/4$, then we bound the
other part of the integral by

$\displaystyle
C\int_{|\gamma|>|y-x-z|/4}|\gamma|^{2-2d}e^{-|\gamma|^{2}/C}d\gamma\leq
C\int_{1\geq|\gamma|>|y-x-z|/4}|\gamma|^{2-2d}d\gamma+\int_{|\gamma|>1}e^{-|\gamma|^{2}/C}d\gamma$
$\displaystyle\leq\begin{cases}C|y-x-z|^{2-d}+C,\quad d\geq 3\\\
C|\log|y-x-z||+C,\quad d=2\end{cases}\leq C\begin{cases}|y-x-z|^{2-d},\quad
d\geq 3\\\ |\log|y-x-z||,\quad d=2.\end{cases}$

If on the other hand $|y-x-z|<4C_{0}\epsilon$, then we have that
$|y-x-z+\zeta|\leq 5C_{0}\epsilon$. We note that for any
$\gamma\in\mathbb{R}^{d}$,

$|\gamma|\vee|y-x-z+\gamma+\zeta|\geq\frac{|x+z-y-\zeta|}{2},$

hence

$|\gamma|^{2}+|y-x-z+\gamma+\zeta|^{2}\geq\frac{|x+z-y-\zeta|^{2}}{4}.$

We separate the integral (46) according to
$\gamma\in\mathbb{B}\left(\frac{x+z-y-\zeta}{2},|x+z-y-\zeta|\right)$ or
otherwise. In the former case, we can bound part of the integral

$\displaystyle
C\int_{\mathbb{R}^{d}}\int_{\mathbb{B}\left(\frac{x+z-y-\zeta}{2},|x+z-y-\zeta|\right)}|x+z-y-\zeta|^{2-2d}\theta^{\epsilon}(\zeta)d\gamma
d\zeta\leq C\int|x+z-y-\zeta|^{2-d}\theta^{\epsilon}(\zeta)d\zeta$
$\displaystyle\leq C\epsilon^{-d}\int_{|\zeta|\leq
C_{0}\epsilon}|x+z-y-\zeta|^{2-d}d\zeta\leq C\epsilon^{-d}\int_{|w|\leq
5C_{0}\epsilon}|w|^{2-d}dw\leq C\epsilon^{2-d}$

where we have bounded
$\|\theta^{\epsilon}\|_{\infty}\leq\epsilon^{-d}\|\theta\|_{\infty}$ and used
$|y-x-z+\zeta|\leq 5C_{0}\epsilon$. The second part of (46) we bound by

$\displaystyle
C\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}\backslash\mathbb{B}\left(\frac{x+z-y-\zeta}{2},|x+z-y-\zeta|\right)}|\gamma|^{2-2d}e^{-|\gamma|^{2}/C}\theta^{\epsilon}(\zeta)d\gamma
d\zeta\leq
C\int\int_{|\gamma|>\frac{x+z-y-\zeta}{2}}|\gamma|^{2-2d}e^{-|\gamma|^{2}/C}\theta^{\epsilon}(\zeta)d\gamma
d\zeta$ $\displaystyle\leq C\begin{cases}\int_{|\zeta|\leq
C_{0}\epsilon}|x+z-y-\zeta|^{2-d}\theta^{\epsilon}(\zeta)d\zeta,\quad d\geq
3\\\ \int_{|\zeta|\leq
C_{0}\epsilon}|\log|x+z-y-\zeta||\theta^{\epsilon}(\zeta)d\zeta,\quad
d=2\end{cases}$ $\displaystyle\leq C\epsilon^{-d}\begin{cases}\int_{|w|\leq
5C_{0}\epsilon}|w|^{2-d}d\zeta,\quad d\geq 3\\\ \int_{|w|\leq
5C_{0}\epsilon}|\log|w||d\zeta,\quad d=2\end{cases}\leq
C\begin{cases}\epsilon^{2-d},\quad d\geq 3\\\ |\log\epsilon|,\quad
d=2.\end{cases}$

To summarise, we showed here that when $|y-x-z|<4$, then a bound of the form
$C|y-x-z|^{2-d}\wedge C\epsilon^{2-d}$ holds for $d\geq 3$ and
$C|\log|y-x-z||\wedge C|\log\epsilon|$ for $d=2$. This, together with the
exponential decay when $|y-x-z|\geq 4$, completes the proof of the first two
items of (42).

The $d=1$ case of (42) requires some changes. Since $2d=2$, we have that (cf.
[14, Proposition 5, pp. 615])

$\displaystyle\int_{0}^{t}q_{s}(x,y;x^{\prime},y^{\prime})ds\leq
Ce^{-\frac{|x-x^{\prime}|^{2}+|y-y^{\prime}|^{2}}{C}}-C\log\left(|x-x^{\prime}|^{2}+|y-y^{\prime}|^{2}\right)1_{\\{|x-x^{\prime}|^{2}+|y-y^{\prime}|^{2}\leq
1\\}}$

and thus,

$\displaystyle\iint_{\mathbb{R}^{2}}\int_{0}^{t}q_{s}(x,y;x^{\prime},y^{\prime})dsdx^{\prime}dy^{\prime}$
$\displaystyle\leq
C\iint_{\mathbb{R}^{2}}e^{-\frac{|\gamma|^{2}+|y-x-z+\gamma+\zeta|^{2}}{C}}\theta^{\epsilon}(\zeta)d\gamma
d\zeta$
$\displaystyle\quad-C\iint_{\mathbb{R}^{2d}}\log\left(|\gamma|^{2}+|y-x-z+\gamma+\zeta|^{2}\right)1_{\\{|\gamma|^{2}+|y-x-z+\gamma+\zeta|^{2}\leq
1\\}}\theta^{\epsilon}(\zeta)d\gamma d\zeta.$

Proceeding similarly as the $d\geq 2$ case yields the thesis, where the
constant bound is essentially due to the integrability of $-\log r$ function
near $r=0$. The gradient bounds (43) can also be proved analogously; we only
comment that we start with

$\displaystyle\nabla_{x}r^{\epsilon,z}(T-t,x,y)=$
$\displaystyle\int_{[0,t]\times\mathbb{R}^{2d}}\nabla_{x}q_{t-s}(x,y;x^{\prime},y^{\prime})\theta^{\epsilon}(x^{\prime}-y^{\prime}+z)dsdx^{\prime}dy^{\prime}$

and the fact that due to $\mathcal{A}_{\mathbf{x}}$ uniformly elliptic, the
gradient of its $(2d)$-dimensional heat kernel satisfies

$\displaystyle|\nabla_{x}q_{t}(x,y;x^{\prime},y^{\prime})|\leq\frac{C}{t^{\frac{1+2d}{2}}}e^{-\frac{|x-x^{\prime}|^{2}+|y-y^{\prime}|^{2}}{Ct}}$

for any $t>0$, $x,y,x^{\prime},y^{\prime}\in\mathbb{R}^{d}$ (cf. [26, Theorem
1]). Correspondingly, we have that for any $d\geq 1$ (cf. [14, Proposition 5,
pp. 615]),

$\displaystyle\int_{0}^{t}|\nabla_{x}q_{s}(x,y;x^{\prime},y^{\prime})|ds\leq\int_{0}^{t}\frac{C}{s^{\frac{1+2d}{2}}}e^{-\frac{|x-x^{\prime}|^{2}+|y-y^{\prime}|^{2}}{Cs}}ds$
$\displaystyle\leq
C\left(|x-x^{\prime}|^{2}+|y-y^{\prime}|^{2}\right)^{\frac{1-2d}{2}}e^{-\frac{|x-x^{\prime}|^{2}+|y-y^{\prime}|^{2}}{C}}+C1_{\\{|x-x^{\prime}|^{2}+|y-y^{\prime}|^{2}\leq
1\\}}.$

The rest of the proof is analogous to that for the uniform bounds given above.

##  4 Bounding various terms

Building on the Itô-Tanaka procedure of Section 3, still working on the
original probability space $(\Omega,\mathcal{F},\mathbb{P})$ we complete the
proof of Proposition 5 by showing the negligibility of all the minor terms in
(40).

To prepare, for each $N\in\mathbb{N}$ let us denote by $[0,\tau_{i}^{N})$ the
lifespan of particle $i=1,...,N$ in our system, namely at the stopping time
$\tau_{i}^{N}\in(0,\infty]$ particle $i$ is removed from the system due to the
coagulation rule. Let us also consider on the same probability space
$(\Omega,\mathcal{F},\mathbb{P})$ an auxiliary “free” particle system
$(x_{i}^{f}(t))_{i=1}^{\infty}$, that obeys the same dynamics as (1), with
$x_{i}^{f}(0)=x_{i}(0)$, $i\in\mathbb{N}$, and the same driving Brownian
motions $\\{W_{t}^{k}\\}_{k\in K}$, $\\{\beta_{i}(t)\\}_{i=1}^{\infty}$, but
without coagulation of masses. In other words,

$\displaystyle dx_{i}^{f}(t)=\sum_{k\in
K}\sigma_{k}\left(x_{i}^{f}(t)\right)\,\circ\,dW^{k}_{t}+\lambda\,d\beta_{i}(t),\quad
i\in\mathbb{N}$ (48)

with lifespan $[0,\infty)$. This auxiliary particle system dominates our true
system in the following sense: $\mathbb{P}$-a.s.

$\displaystyle x_{i}^{f}(t)=x_{i}^{N}(t),\quad
t\in[0,\tau_{i}^{N}),\;i=1,...,N$ (49)

due to the fact that the coefficients of the SDE (1) do not depend on the mass
parameter.

We start with a lemma proving that any fixed number of particles in the
auxiliary system have a joint density that is uniformly bounded and decays
exponentially at infinity. Recall that $\Gamma$ is the uniform upper bound on
the initial density of an individual particle, and $R$ the maximal radius of
its compact support.

###### Proposition 11

Let $\ell\in\mathbb{N},d\geq 1$ be fixed, and denote

$\displaystyle\overline{X}_{t}^{f}$
$\displaystyle:=\left(x_{1}^{f}(t),...,x_{\ell}^{f}(t)\right)\in\mathbb{R}^{\ell
d}$ $\displaystyle\overline{x}$
$\displaystyle:=\left(x_{1},...,x_{\ell}\right)\in\mathbb{R}^{\ell d}$

where $x_{i}^{f}(t)$ are defined above. Then $\overline{X}_{t}^{f}$ has a
probability density $\mathbf{p}_{t}(\overline{x})$ on $\mathbb{R}^{\ell d}$
that satisfies for any $t\in[0,T]$,

$\displaystyle\mathbf{p}_{t}(\overline{x})\leq c^{-1}e^{-c|\overline{x}|}$
(50)

where $c=c(d,\lambda,\ell,T,R,\Gamma,\\{\sigma_{k}\\}_{k\in K},)$ is a
positive constant.

Proof. Let us denote

$\displaystyle\overline{B}_{t}$
$\displaystyle:=(\beta_{1}(t),...,\beta_{\ell}(t))\in\mathbb{R}^{\ell d}$
$\displaystyle\Sigma_{k}(\overline{x})$
$\displaystyle:=(\sigma_{k}(x_{1}),...,\sigma_{k}(x_{\ell}))\in\mathbb{R}^{\ell
d}$ $\displaystyle\left(\nabla\Sigma_{k}\cdot\Sigma_{k}\right)(\overline{x})$
$\displaystyle:=\left(\left(\nabla\sigma_{k}\cdot\sigma_{k}\right)(x_{1}),...,\left(\nabla\sigma_{k}\cdot\sigma_{k}\right)(x_{\ell})\right)\in\mathbb{R}^{\ell
d}$

where $\nabla\sigma_{k}\cdot\sigma_{k}$ is defined in (2). Then, by (48) the
$\ell$-tuple $\overline{X}_{t}^{f}$ satisfies the SDE in $\mathbb{R}^{\ell
d}$:

$\displaystyle d\overline{X}_{t}^{f}$ $\displaystyle=\sum_{k\in
K}\Sigma_{k}(\overline{X}_{t}^{f})\,\circ\,dW^{k}_{t}+\lambda\,d\overline{B}_{t}$
$\displaystyle=\sum_{k\in
K}\Sigma_{k}(\overline{X}_{t}^{f})\,dW^{k}_{t}+\frac{1}{2}\sum_{k\in
K}\left(\nabla\Sigma_{k}\cdot\Sigma_{k}\right)(\overline{X}_{t}^{f})dt+\lambda\,d\overline{B}_{t}.$

This is an Itô diffusion associated with the operator with coefficients of
class $C_{b}^{\infty}(\mathbb{R}^{\ell d})$ that acts on functions on
$\mathbb{R}^{\ell d}$,

$\displaystyle\mathscr{L}=\frac{\lambda^{2}}{2}\Delta+\frac{1}{2}\sum_{k\in
K}\text{tr}\left(\Sigma_{k}^{T}\Sigma_{k}(\overline{x})D^{2}\right)+\frac{1}{2}\sum_{k\in
K}\left(\nabla\Sigma_{k}\cdot\Sigma_{k}\right)(\overline{x})\cdot D$

thus admits a transition density (i.e. heat kernel)
$q^{(\ell)}_{t}(\overline{x},\overline{y})$ that is bounded above (and below)
by Gaussian kernel (cf. [26, Theorem 1]), i.e.

$q^{(\ell)}_{t}(\overline{x},\overline{y})\leq
CQ_{t}(\overline{x}-\overline{y}),\quad\forall
t>0,\;\overline{x},\overline{y}\in\mathbb{R}^{\ell d},$

where $Q_{t}(\cdot)$ is the density of a centered Gaussian vector
$\overline{Y}^{0}_{t}$ in $\mathbb{R}^{\ell d}$ with covariance matrix $CtI$,
for some $C=C\left(d,\ell,\\{\sigma_{k}\\}_{k\in K},\lambda\right)$ finite.
Further, the initial condition $\overline{X}_{0}^{f}$ has a density
$\mathbf{p}_{0}(\overline{x})$ bounded above by $\Gamma^{\ell}$, and compactly
supported in the ball $\mathbb{B}(0,\sqrt{\ell}R)$, due to Condition 1. Then,
for any $t\in[0,T]$, the probability density of $\overline{X}_{t}^{f}$

$\displaystyle\mathbf{p}_{t}(\overline{x})=\left(q_{t}^{(\ell)}*\mathbf{p}_{0}\right)(\overline{x})\leq\int
Q_{t}^{(\ell)}(\overline{x}-\overline{y})\mathbf{p}_{0}(\overline{y})d\overline{y}$
$\displaystyle=\mathbb{E}\left[\mathbf{p}_{0}\left(\overline{x}+\overline{Y}_{t}^{0}\right)\right]\leq\Gamma^{\ell}\mathbb{P}\left(\overline{x}+\overline{Y}_{t}^{0}\in\mathbb{B}\left(0,\sqrt{\ell}R\right)\right)$
$\displaystyle\leq\Gamma^{\ell}\mathbb{P}\left(|\overline{Y}_{t}^{0}|\geq|\overline{x}|-\sqrt{\ell}R\right)\leq
c^{-1}e^{-c\left(|x|-\sqrt{\ell}R\right)_{+}},$

where we used the Gaussian tail of $\overline{Y}^{0}_{t}$ in the last line,
and $c=c(d,\lambda,\ell,T,R,\Gamma,\\{\sigma_{k}\\}_{k\in K})$ is a positive
constant. Adjusting the value of $c$ yields our thesis.

###### Lemma 12

For any finite $T,d\geq 1$, we have that

$\displaystyle\lim_{|z|\to
0}\limsup_{N\to\infty}\mathbb{E}\int_{0}^{T}dt\frac{1}{N^{2}}\sum_{i\neq
j\in\mathcal{N}(t)}\left|v^{\epsilon,z}(t,x_{i}^{N}(t),x_{j}^{N}(t))\right|=0.$

Proof. Due to (49), it is enough to prove the thesis with the auxiliary system
replacing the true system, with the former having infinite lifespan, i.e. to
prove that

$\displaystyle\lim_{|z|\to
0}\limsup_{N\to\infty}\mathbb{E}\int_{0}^{T}dt\frac{1}{N^{2}}\sum_{i\neq
j=1}^{N}\left|v^{\epsilon,z}(t,x_{i}^{f}(t),x_{j}^{f}(t))\right|=0.$

(The same applies to all the subsequent lemmas in this section.) Consider any
fixed pair $(i,j)$, with $1\leq i\neq j\leq N$, and we go on to bound

$\displaystyle\mathbb{E}\int_{0}^{T}\left|v^{\epsilon,z}(t,x_{i}^{f}(t),x_{j}^{f}(t))\right|dt.$

By Proposition 11, $(x_{i}^{f}(t),x_{j}^{f}(t))$ has a joint density
$\mathbf{p}_{t}(x,y)$ in $\mathbb{R}^{2d}$ satisfying the bound (50) with
$\ell=2$, implying that

$\displaystyle\mathbb{E}\int_{0}^{T}\left|v^{\epsilon,z}(t,x_{i}^{f}(t),x_{j}^{f}(t))\right|dt$
$\displaystyle=\iint_{\mathbb{R}^{2d}}\int_{0}^{T}\left|v^{\epsilon,z}(t,x,y)\right|\mathbf{p}_{t}(x,y)dtdxdy$
$\displaystyle\leq
C\iint_{\mathbb{R}^{2d}}\sup_{t\in[0,T]}\left|v^{\epsilon,z}(t,x,y)\right|e^{-\sqrt{|x|^{2}+|y|^{2}}/C}dxdy\,,$
(51)

and thus the problem is reduced to prove that

$\lim_{|z|\to 0}\limsup_{\epsilon\to
0}\iint_{\mathbb{R}^{2d}}\sup_{t\in[0,T]}\left|v^{\epsilon,z}(t,x,y)\right|e^{-\sqrt{|x|^{2}+|y|^{2}}/C}dxdy=0.$

By Fatou’s lemma, we have that

$\limsup_{\epsilon\to
0}\iint_{\mathbb{R}^{2d}}\sup_{t\in[0,T]}\left|v^{\epsilon,z}(t,x,y)\right|e^{-\sqrt{|x|^{2}+|y|^{2}}/C}dxdy\leq\iint_{\mathbb{R}^{2d}}\limsup_{\epsilon\to
0}\sup_{t\in[0,T]}\left|v^{\epsilon,z}(t,x,y)\right|e^{-\sqrt{|x|^{2}+|y|^{2}}/C}dxdy$

and thus it is sufficient to prove that

$\lim_{|z|\to 0}\iint_{\mathbb{R}^{2d}}\limsup_{\epsilon\to
0}\sup_{t\in[0,T]}\left|v^{\epsilon,z}(t,x,y)\right|e^{-\sqrt{|x|^{2}+|y|^{2}}/C}dxdy=0.$

By a change of variables, for any $t\in[0,T]$,

$\displaystyle v^{\epsilon,z}(T-t,x,y)$
$\displaystyle=r^{\epsilon,z}(T-t,x,y)-r^{\epsilon,0}(T-t,x,y)$ (52)
$\displaystyle=\int_{[0,t]\times\mathbb{R}^{2d}}q_{t-s}(x,y;x^{\prime},y^{\prime})\left[\theta^{\epsilon}(x^{\prime}-y^{\prime}+z)-\theta^{\epsilon}(x^{\prime}-y^{\prime})\right]dsdx^{\prime}dy^{\prime}$
$\displaystyle=\int_{[0,t]\times\mathbb{R}^{2d}}\left[q_{t-s}(x,y;x^{\prime},y^{\prime}+z)-q_{t-s}(x,y;x^{\prime},y^{\prime})\right]\theta^{\epsilon}(x^{\prime}-y^{\prime})dsdx^{\prime}dy^{\prime}$

where $q_{t}(x,y;x^{\prime},y^{\prime})$ is the $(2d)$-dimensional heat kernel
(i.e. fundamental solution) associated with operator
$\mathcal{A}_{\mathbf{x}}$ introduced in the proof of Proposition 11. Hence,
as $\epsilon\to 0$,

$\displaystyle
v^{\epsilon,z}(T-t,x,y)\to\int_{[0,t]\times\mathbb{R}^{d}}\left[q_{t-s}(x,y;x^{\prime},x^{\prime}+z)-q_{t-s}(x,y;x^{\prime},x^{\prime})\right]dsdx^{\prime}.$

Thus, the problem is reduced to prove that

$\displaystyle\lim_{|z|\to
0}\iint_{\mathbb{R}^{2d}}\sup_{t\in[0,T]}\left|\int_{[0,t]\times\mathbb{R}^{d}}\left[q_{t-s}(x,y;x^{\prime},x^{\prime}+z)-q_{t-s}(x,y;x^{\prime},x^{\prime})\right]dsdx^{\prime}\right|e^{-\sqrt{|x|^{2}+|y|^{2}}/C}dxdy=0.$
(53)

Since the integrand converges to zero pointwise (see below), to apply Vitali
convergence theorem, it suffices to check the uniform integrability in the
parameter $|z|\leq 1$, with respect to the finite measure
$e^{-\sqrt{|x|^{2}+|y|^{2}}/C}dxdy$, of

$A_{x,y,z}:=\sup_{t\in[0,T]}\int_{\mathbb{R}^{d}}\int_{0}^{t}q_{t-s}(x,y;x^{\prime},x^{\prime}+z)dsdx^{\prime}\,;$

namely, to prove that for some $\delta>1$, we have that

$\sup_{|z|\leq
1}\iint_{\mathbb{R}^{2d}}|A_{x,y,z}|^{\delta}e^{-\sqrt{|x|^{2}+|y|^{2}}/C}dxdy<\infty.$

To this end, in the proof of Proposition 8, we already mentioned that for
$d\geq 2$,

$\displaystyle\int_{0}^{t}q_{t-s}(x,y;x^{\prime},x^{\prime}+z)ds\leq
C\left(|x-x^{\prime}|^{2}+|y-z-x^{\prime}|^{2}\right)^{\frac{2-2d}{2}}e^{-\frac{|x-x^{\prime}|^{2}+|y-z-x^{\prime}|^{2}}{C}}+C1_{\\{|x-x^{\prime}|^{2}+|y-z-x^{\prime}|^{2}\leq
1\\}}$ (54)

(and the $d=1$ case is similar, involving the logarithm). Let us integrate
this expression in the variable $x^{\prime}$ over $\mathbb{R}^{d}$. Note that,
for any $x^{\prime}\in\mathbb{R}^{d}$, it holds that

$|x-x^{\prime}|\vee|y-z-x^{\prime}|\geq\frac{|x-(y-z)|}{2}.$

Thus, for any $x^{\prime}$,

$|x-x^{\prime}|^{2}+|y-z-x^{\prime}|^{2}\geq\frac{|x-y+z|^{2}}{4}.$

We separate the integral in $x^{\prime}$ into one in the ball
$x^{\prime}\in\mathbb{B}\left(\frac{x+(y-z)}{2},|x-y+z|\right)$ and the other
outside the ball, and then it is easy to show that the first part is upper
bounded by $C|x-y+z|^{2-d}$, and the second part by $C|x-y+z|^{2-d}1_{\\{d\geq
3\\}}+C|\log|x-y+z||1_{\\{d=2\\}}$. In the case $d=1$, a constant bound can be
shown to hold.

Now we need to check, for some $\delta>1$, the uniform finiteness in $|z|\leq
1$ of

$\displaystyle\iint_{\mathbb{R}^{2d}}|A_{x,y,z}|^{\delta}e^{-\sqrt{|x|^{2}+|y|^{2}}/C}dxdy\leq\begin{cases}C\iint_{\mathbb{R}^{2d}}|x-y+z|^{(2-d)\delta}e^{-\sqrt{|x|^{2}+|y|^{2}}/C}dxdy,\quad
d\geq 3\\\
C\iint_{\mathbb{R}^{2d}}|\log|x-y+z||^{\delta}e^{-\sqrt{|x|^{2}+|y|^{2}}/C}dxdy,\quad
d=2\\\ C\iint_{\mathbb{R}^{2d}}e^{-\sqrt{|x|^{2}+|y|^{2}}/C}dxdy,\quad
d=1.\end{cases}$

Taking $d\geq 3$ for instance (the other cases being similar),

$\displaystyle\iint_{\mathbb{R}^{2d}}|x-y+z|^{(2-d)\delta}e^{-\sqrt{|x|^{2}+|y|^{2}}/C}dxdy\overset{u=\frac{x-y}{\sqrt{2}},\;v=\frac{x-y}{\sqrt{2}}}{=}\iint_{\mathbb{R}^{2d}}|\sqrt{2}u+z|^{(2-d)\delta}e^{-\sqrt{|u|^{2}+|v|^{2}}/C}dudv$
$\displaystyle\leq\iint|\sqrt{2}u+z|^{(2-d)\delta}e^{-(|u|+|v|)/(2C)}dudv\leq
C\int|\sqrt{2}u+z|^{(2-d)\delta}e^{-|u|/(2C)}du$
$\displaystyle\overset{|z|\leq 1}{\leq}C\int_{|u|\leq
2}|\sqrt{2}u+z|^{(2-d)\delta}du+C\int_{|u|>2}e^{-|u|/(2C)}du$
$\displaystyle\overset{|z|\leq 1}{\leq}C\int_{|w|\leq
4}|w|^{(2-d)\delta}dw+C\leq C^{\prime},$

provided we take $\delta\leq d/(d-2)$, where the constant $C^{\prime}$ is
independent of $|z|\leq 1$.

Note also that $z\mapsto A_{x,y,z}$ is continuous at $z=0$ by dominated
convergence, since LHS of (54) is continuous in $z$, a classical fact for
Green functions, and RHS of (54) is dominated by
$C|x-y|^{2-2d}e^{-|x-x^{\prime}|^{2}/C}+C1_{\\{|x-x^{\prime}|\leq 1\\}}$, for
all $|z|\leq|x-y|/4$ say (and $x\neq y$), which is integrable in
$x^{\prime}\in\mathbb{R}^{d}$.

With the integrand of (53) converging pointwise to $0$ as $|z|\to 0$ (i.e. for
a.e. fixed $(x,y)$), and uniformly integrable with respect to the finite
measure $e^{-\sqrt{|x|^{2}+|y|^{2}}/C}dxdy$, we have by Vitali convergence
theorem that

$\lim_{|z|\to 0}\limsup_{\epsilon\to
0}\iint_{\mathbb{R}^{2d}}\sup_{t\in[0,T]}\left|v^{\epsilon,z}(T-t,x,y)\right|e^{-\sqrt{|x|^{2}+|y|^{2}}/C}dxdy=0.$

Since the cardinality in the double sum in the thesis is (at most) $N^{2}$,
this completes the proof.

###### Lemma 13

For any finite $T,d\geq 1$, we have that

$\displaystyle\lim_{|z|\to
0}\limsup_{N\to\infty}\mathbb{E}\int_{0}^{T}dt\frac{1}{N^{2}}\sum_{i\neq
j\in\mathcal{N}(t)}\left|\nabla_{x}v^{\epsilon,z}(t,x_{i}^{N}(t),x_{j}^{N}(t))\right|=0.$

The same statement also holds for $\nabla_{y}$ instead of $\nabla_{x}$.

Proof. The proof is analogous to Lemma 12, using the estimate for
$|\nabla_{x}r^{\epsilon,z}(t,x,y)|$ (45) instead of (44). First we switch to
the free system, and it suffices to prove that

$\lim_{|z|\to
0}\limsup_{N\to\infty}\mathbb{E}\int_{0}^{T}dt\frac{1}{N^{2}}\sum_{i\neq
j=1}^{N}\left|\nabla_{x}v^{\epsilon,z}(t,x_{i}^{f}(t),x_{j}^{f}(t))\right|=0.$

Considering any fixed pair of particles $(i,j)$, $1\leq i\neq j\leq N$, we
have that

$\displaystyle\mathbb{E}\int_{0}^{T}\left|\nabla_{x}v^{\epsilon,z}\left(t,x_{i}^{f}(t),x_{j}^{f}(t)\right)\right|dt$
$\displaystyle=\iint_{\mathbb{R}^{2d}}\int_{0}^{T}\left|\nabla_{x}v^{\epsilon,z}(t,x,y)\right|\mathbf{p}_{t}(x,y)dtdxdy$
$\displaystyle\leq\iint_{\mathbb{R}^{2d}}\sup_{t\in[0,T]}\left|\nabla_{x}v^{\epsilon,z}(t,x,y)\right|e^{-\sqrt{|x|^{2}+|y|^{2}}/C}dxdy$
(55)

by (50). Further, by a change of variables, for any $t\in[0,T]$,

$\displaystyle\nabla_{x}v^{\epsilon,z}(T-t,x,y)$
$\displaystyle=\nabla_{x}r^{\epsilon,z}(T-t,x,y)-\nabla_{x}r^{\epsilon,0}(T-t,x,y)$
$\displaystyle=\int_{[0,t]\times\mathbb{R}^{2d}}\nabla_{x}q_{t-s}(x,y;x^{\prime},y^{\prime})\left[\theta^{\epsilon}(x^{\prime}-y^{\prime}+z)-\theta^{\epsilon}(x^{\prime}-y^{\prime})\right]dsdx^{\prime}dy^{\prime}$
$\displaystyle=\int_{[0,t]\times\mathbb{R}^{2d}}\left[\nabla_{x}q_{t-s}(x,y;x^{\prime},y^{\prime}+z)-\nabla_{x}q_{t-s}(x,y;x^{\prime},y^{\prime})\right]\theta^{\epsilon}(x^{\prime}-y^{\prime})dsdx^{\prime}dy^{\prime}.$

Hence, as $\epsilon\to 0$,

$\displaystyle\nabla_{x}v^{\epsilon,z}(T-t,x,y)\to\int_{[0,t]\times\mathbb{R}^{d}}\left[\nabla_{x}q_{t-s}(x,y;x^{\prime},x^{\prime}+z)-\nabla_{x}q_{t-s}(x,y;x^{\prime},x^{\prime})\right]e^{-\sqrt{|x|^{2}+|y|^{2}}/C}dsdx^{\prime}\,,$

and by Fatou’s lemma it is sufficient to prove that

$\displaystyle\lim_{|z|\to
0}\iint_{\mathbb{R}^{2d}}\sup_{t\in[0,T]}\left|\int_{[0,t]\times\mathbb{R}^{d}}\left[\nabla_{x}q_{t-s}(x,y;x^{\prime},x^{\prime}+z)-\nabla_{x}q_{t-s}(x,y;x^{\prime},x^{\prime})\right]dx^{\prime}ds\right|e^{-\sqrt{|x|^{2}+|y|^{2}}/C}dxdy=0.$

To check that the integrand above is uniformly integrable in $|z|\leq 1$, with
respect to the finite measure $e^{-\sqrt{|x|^{2}+|y|^{2}}/C}dxdy$, we note
that for $d\geq 1$ (already mentioned in the proof of Proposition 8)

$\int_{0}^{t}|\nabla_{x}q_{t-s}(x,y;x^{\prime},x^{\prime}+z)|ds\leq
C\left(|x-x^{\prime}|^{2}+|y-z-x^{\prime}|^{2}\right)^{\frac{1-2d}{2}}e^{-\frac{|x-x^{\prime}|^{2}+|y-z-x^{\prime}|^{2}}{C}}+C1_{\\{|x-x^{\prime}|^{2}+|y-z-x^{\prime}|^{2}\leq
1\\}}.$

By the same reasoning as in the proof of Lemma 12, as we integrate the
preceding expression in $x^{\prime}$ over $\mathbb{R}^{d}$, we get an upper
bound $C|x-y+z|^{1-d}1_{\\{d\geq 2\\}}+C|\log|x-y+z||1_{\\{d=1\\}}$. Then, for
$d\geq 2$ we check for some $\delta>1$, the uniform finiteness in the
parameter $|z|\leq 1$ of

$\iint_{\mathbb{R}^{2d}}|x-y+z|^{(1-d)\delta}e^{-\sqrt{|x|^{2}+|y|^{2}}/C}dxdy,$

which is true provided we take $\delta<d/(d-1)$. By Vitali convergence
theorem, we have that

$\lim_{|z|\to 0}\limsup_{\epsilon\to
0}\iint_{\mathbb{R}^{2d}}\sup_{t\in[0,T]}\left|\nabla_{x}v^{\epsilon,z}(T-t,x,y)\right|e^{-\sqrt{|x|^{2}+|y|^{2}}/C}dxdy=0$

which leads to our thesis, as in Lemma 12.

###### Remark 14

By Lemmas 12 and 13, the terms $H_{2},H_{4},H_{5},H_{8},H_{9}$ are negligible
in the sense of (41), once we bound test functions $\phi,\psi$ and their
derivatives, as well as the $C_{b}^{\infty}$ functions $Q^{\alpha\beta}$ and
its partial derivatives above by constants.

###### Lemma 15

For any finite $T,d\geq 1$, we have that

$\displaystyle\lim_{|z|\to
0}\limsup_{N\to\infty}\mathbb{E}\int_{0}^{T}dt\frac{1}{N^{3}}\sum_{i\neq
k\in\mathcal{N}(t)}\theta^{\epsilon}(x_{i}^{N}(t)-x_{k}^{N}(t))\sum_{j\in\mathcal{N}(t),j\neq
i,k}\left|v^{\epsilon,z}(t,x_{i}^{N}(t),x_{j}^{N}(t))\right|=0.$

Proof. First switch to the free system. Consider any triple of particles
$(i,j,k)$ with indices all distinct. To bound

$\displaystyle\mathbb{E}\int_{0}^{T}\theta^{\epsilon}(x_{i}^{f}(t)-x_{k}^{f}(t))\left|v^{\epsilon,z}(t,x_{i}^{f}(t),x_{j}^{f}(t))\right|dt$

by Proposition 11, $(x_{i}^{f}(t),x_{j}^{f}(t),x_{k}^{f}(t))$ has a joint
density $\mathbf{p}_{t}(x_{1},x_{2},x_{3})$ in $\mathbb{R}^{3d}$ satisfying
the bound (50) with $\ell=3$. Thus, we can write

$\displaystyle\mathbb{E}\int_{0}^{T}\theta^{\epsilon}(x_{i}^{f}(t)-x_{k}^{f}(t))\left|v^{\epsilon,z}(t,x_{i}^{f}(t),x_{j}^{f}(t))\right|dt$
$\displaystyle=\iiint_{\mathbb{R}^{3d}}\int_{0}^{T}\theta^{\epsilon}(x_{1}-x_{3})\left|v^{\epsilon,z}(t,x_{1},x_{2})\right|\mathbf{p}_{t}(x_{1},x_{2},x_{3})dtdxdydz$
$\displaystyle\leq
C\iiint_{\mathbb{R}^{3d}}\int_{0}^{T}\theta^{\epsilon}(x_{1}-x_{3})\left|v^{\epsilon,z}(t,x_{1},x_{2})\right|e^{-\sqrt{\sum_{\ell=1}^{3}|x_{\ell}|^{2}}/C}dtdx_{1}dx_{2}dx_{3}$

Integrating in $x_{3}$ first and using
$\int\theta^{\epsilon}(x_{1}-x_{3})dx_{3}=1$, we bound the above integral
above by

$\displaystyle
C\iint_{\mathbb{R}^{2d}}\int_{0}^{T}\left|v^{\epsilon,z}(t,x_{1},x_{2})\right|e^{-\sqrt{|x_{1}|^{2}+|x_{2}|^{2}}/C}dtdx_{1}dx_{2}$

This is the same integral apprearing in the proof of Lemma 12, which has been
shown to tend to zero as $\epsilon\to 0$ followed by $|z|\to 0$. Since the
cardinality in the triple sum of the thesis is (at most) $N^{3}$, this
completes the proof.

###### Remark 16

By Lemma 15, the term $H_{6}$ (30) is negligible in the sense of (41) (by
considering it term by term), after bounding the test functions by constants.

###### Lemma 17

For any finite $T,d\geq 1$, we have that

$\displaystyle\limsup_{N\to\infty}\mathbb{E}\int_{0}^{T}dt\frac{1}{N^{3}}\sum_{i\neq
j\in\mathcal{N}(t)}\theta^{\epsilon}(x_{i}^{N}(t)-x_{j}^{N}(t))\left|v^{\epsilon,z}(t,x_{i}^{N}(t),x_{j}^{N}(t))\right|=0.$

Proof. First switch to the free system. Consider any pair of particles $(i,j)$
with $1\leq i\neq j\leq N$. By (44), we have that

$\displaystyle\mathbb{E}\frac{1}{N}\int_{0}^{T}\theta^{\epsilon}(x_{i}^{f}(t)-x_{j}^{f}(t))\left|v^{\epsilon,z}(t,x_{i}^{f}(t),x_{j}^{f}(t))\right|dt\leq\kappa(\epsilon)\;\mathbb{E}\int_{0}^{T}\theta^{\epsilon}(x_{i}^{f}(t)-x_{j}^{f}(t))dt,$

where

$\displaystyle\kappa(\epsilon):=C\begin{cases}\epsilon^{2-d}N^{-1},\quad d\geq
3\\\ |\log\epsilon|N^{-1},\quad d=2\\\ N^{-1},\quad d=1,\end{cases}$ (56)

which vanishes as $N\to\infty$ by (3). By Proposition 11 with $\ell=2$, we can
write

$\displaystyle\mathbb{E}\int_{0}^{T}\theta^{\epsilon}(x_{i}^{f}(t)-x_{j}^{f}(t))dt$
$\displaystyle=\int_{0}^{T}\iint_{\mathbb{R}^{2d}}\theta^{\epsilon}(x-y)\mathbf{p}_{t}(x,y)dxdydt$
$\displaystyle\leq
CT\iint_{\mathbb{R}^{2d}}\theta^{\epsilon}(x-y)e^{-\sqrt{|x|^{2}+|y|^{2}}/C}dxdy$
$\displaystyle\leq C\int_{\mathbb{R}^{d}}e^{-|x|/C}dx\leq C^{\prime}$

using $\int\theta^{\epsilon}(x-y)dy=1$. Since the double sum has cardinality
at most $N^{2}$, the claim is proved.

###### Remark 18

By Lemma 17, the term $H_{7}$ (31) is negligible in the sense of (41), after
bounding the test functions by constants.

Now we turn to the terms $B_{1},B_{2},B_{3}$ related to the martingales.

###### Lemma 19

For any finite $T,d\geq 1$, we have that

$\displaystyle\lim_{|z|\to
0}\limsup_{N\to\infty}\mathbb{E}\int_{0}^{T}dt\frac{1}{N^{4}}\left(\sum_{i\neq
j\in\mathcal{N}(t)}\left|v^{\epsilon,z}(t,x_{i}^{N}(t),x_{j}^{N}(t))\right|\right)^{2}=0.$

Proof. First switch to the free system. Expanding the square, we have

$\displaystyle\frac{1}{N^{4}}\left(\sum_{i\neq
j=1}^{N}\left|v^{\epsilon,z}(t,x_{i}^{f}(t),x_{j}^{f}(t))\right|\right)^{2}$
$\displaystyle=\frac{1}{N^{4}}\sum_{i\neq j\neq
k\neq\ell=1}^{N}\left|v^{\epsilon,z}(t,x_{i}^{f}(t),x_{j}^{f}(t))v^{\epsilon,z}(t,x_{k}^{f}(t),x_{\ell}^{f}(t))\right|$
$\displaystyle+\frac{1}{N^{4}}\sum_{i\neq
j\neq\ell=1}^{N}\left|v^{\epsilon,z}(t,x_{i}^{f}(t),x_{j}^{f}(t))v^{\epsilon,z}(t,x_{i}^{f}(t),x_{\ell}^{f}(t))\right|$
$\displaystyle+\frac{1}{N^{4}}\sum_{i\neq j\neq
k=1}^{N}\left|v^{\epsilon,z}(t,x_{i}^{f}(t),x_{j}^{f}(t))v^{\epsilon,z}(t,x_{k}^{f}(t),x_{i}^{f}(t))\right|$
$\displaystyle+\frac{1}{N^{4}}\sum_{j\neq
i\neq\ell=1}^{N}\left|v^{\epsilon,z}(t,x_{i}^{f}(t),x_{j}^{f}(t))v^{\epsilon,z}(t,x_{j}^{f}(t),x_{\ell}^{f}(t))\right|$
$\displaystyle+\frac{1}{N^{4}}\sum_{j\neq i\neq
k=1}^{N}\left|v^{\epsilon,z}(t,x_{i}^{f}(t),x_{j}^{f}(t))v^{\epsilon,z}(t,x_{k}^{f}(t),x_{j}^{f}(t))\right|$
$\displaystyle+\frac{1}{N^{4}}\sum_{i\neq
j=1}^{N}\left|v^{\epsilon,z}(t,x_{i}^{f}(t),x_{j}^{f}(t))\right|^{2}$
$\displaystyle+\frac{1}{N^{4}}\sum_{i\neq
j=1}^{N}\left|v^{\epsilon,z}(t,x_{i}^{f}(t),x_{j}^{f}(t))v^{\epsilon,z}(t,x_{j}^{f}(t),x_{i}^{f}(t))\right|$

The last six terms are negligible, since the cardinality in their sums are at
most $N^{3}$ (due to repeated indices), and by the bound (44), each individual
quadratic term, e.g.

$\displaystyle\frac{1}{N}\left|v^{\epsilon,z}(t,x_{i}^{f}(t),x_{j}^{f}(t))v^{\epsilon,z}(t,x_{i}^{f}(t),x_{\ell}^{f}(t))\right|\leq\kappa(\epsilon)\left|v^{\epsilon,z}(t,x_{i}^{f}(t),x_{j}^{f}(t))\right|$
(57)

where $\kappa(\epsilon)$ defined in (56), and hence the second sum above

$\displaystyle\frac{1}{N^{4}}\sum_{i\neq
j\neq\ell=1}^{N}\left|v^{\epsilon,z}(t,x_{i}^{f}(t),x_{j}^{f}(t))v^{\epsilon,z}(t,x_{i}^{f}(t),x_{\ell}^{f}(t))\right|$
$\displaystyle\leq\frac{\kappa(\epsilon)}{N^{3}}\sum_{i\neq
j\neq\ell=1}^{N}\left|v^{\epsilon,z}(t,x_{i}^{f}(t),x_{j}^{f}(t))\right|\leq\frac{\kappa(\epsilon)}{N^{2}}\sum_{i\neq
j=1}^{N}\left|v^{\epsilon,z}(t,x_{i}^{f}(t),x_{j}^{f}(t))\right|.$

Taking expectation and integrating in time, it is negligible by the statement
of Lemma 12. The other sums can be handled similarly.

Now we deal with the first (principle) term, where the cardinality of the sum
is $O(N^{4})$ and the indices are all distinct, thus it suffices to consider
any fixed quadruple of particles $(i,j,k,\ell)$.

$\displaystyle\mathbb{E}\int_{0}^{T}\left|v^{\epsilon,z}(t,x_{i}^{f}(t),x_{j}^{f}(t))v^{\epsilon,z}(t,x_{k}^{f}(t),x_{\ell}^{f}(t))\right|dt.$

By Proposition 11, $(x_{i}^{f}(t),x_{j}^{f}(t),x_{k}^{f}(t),x_{\ell}^{f}(t))$
has a joint density $\mathbf{p}_{t}(x_{1},x_{2},x_{3},x_{4})$ in
$\mathbb{R}^{4d}$ satisfying the bound (50) with $\ell=4$. Thus, we can write

$\displaystyle\mathbb{E}\int_{0}^{T}\left|v^{\epsilon,z}(t,x_{i}^{f}(t),x_{j}^{f}(t))v^{\epsilon,z}(t,x_{k}^{f}(t),x_{\ell}^{f}(t))\right|dt$
$\displaystyle=\int_{\mathbb{R}^{4d}}\int_{0}^{T}\left|v^{\epsilon,z}(t,x_{1},x_{2})v^{\epsilon,z}(t,x_{3},x_{4})\right|\mathbf{p}_{t}(x_{1},x_{2},x_{3},x_{4})dtdx_{1}dx_{2}dx_{3}dx_{4}$
$\displaystyle\leq
C\int_{\mathbb{R}^{4d}}\int_{0}^{T}\left|v^{\epsilon,z}(t,x_{1},x_{2})v^{\epsilon,z}(t,x_{3},x_{4})\right|e^{-\sqrt{\sum_{i=1}^{4}|x_{i}|^{2}}/C}dtdx_{1}dx_{2}dx_{3}dx_{4}$
$\displaystyle\leq
C\int_{0}^{T}\left(\iint_{\mathbb{R}^{2d}}\left|v^{\epsilon,z}(t,x_{1},x_{2})\right|e^{-\frac{1}{\sqrt{2}C}\sqrt{|x_{1}|^{2}+|x_{2}|^{2}}}dx_{1}dx_{2}\right)^{2}dt.$

As shown in the proof of Lemma 12,

$\displaystyle\lim_{|z|\to 0}\limsup_{\epsilon\to
0}\iint_{\mathbb{R}^{2d}}\sup_{t\in[0,T]}\left|v^{\epsilon,z}(t,x_{1},x_{2})\right|e^{-\frac{1}{\sqrt{2}C}\sqrt{|x_{1}|^{2}+|x_{2}|^{2}}}dx_{1}dx_{2}=0,$

and hence, by the preceding inequality, we also have that

$\displaystyle\lim_{|z|\to 0}\limsup_{\epsilon\to
0}\mathbb{E}\int_{0}^{T}\left|v^{\epsilon,z}(t,x_{i}^{f}(t),x_{j}^{f}(t))v^{\epsilon,z}(t,x_{k}^{f}(t),x_{\ell}^{f}(t))\right|dt=0.$

Since the cardinality in the quadruple sum is (at most) $N^{4}$, this
completes the proof.

###### Lemma 20

For any finite $T,d\geq 1$, we have that

$\displaystyle\lim_{|z|\to
0}\limsup_{N\to\infty}\mathbb{E}\int_{0}^{T}dt\frac{1}{N^{4}}\left(\sum_{i\neq
j\in\mathcal{N}(t)}\left|\nabla
v^{\epsilon,z}(t,x_{i}^{N}(t),x_{j}^{N}(t))\right|\right)^{2}=0.$

Proof. First switch to the free system. The proof is analogous to that of
Lemmas 19, just using the estimate (45) instead of (44), and Lemma 13 instead
of Lemma 12. Just note that when performing the analogous step to (57) to
control the negligible terms, by (45)

$\displaystyle\frac{1}{N}\left|\nabla
v^{\epsilon,z}(t,x_{i}^{f}(t),x_{j}^{f}(t))\right|\left|\nabla
v^{\epsilon,z}(t,x_{i}^{f}(t),x_{\ell}^{f}(t))\right|\leq\widetilde{\kappa}(\epsilon)\left|\nabla
v^{\epsilon,z}(t,x_{i}^{f}(t),x_{j}^{f}(t))\right|$

where now

$\displaystyle\widetilde{\kappa}(\epsilon):=C\begin{cases}\epsilon^{1-d}N^{-1},\quad
d\geq 2\\\ |\log\epsilon|N^{-1},\quad d=1\end{cases}$

is uniformly bounded by (3).

###### Remark 21

By Lemmas 19 and 20, the term $B_{2}$ (33) is negligible in the sense of (41),
after bounding the test functions and $\sup_{k\in K}\|\sigma_{k}\|_{\infty}$
by constants. They also show that $B_{1}$ (32) is negligible, since

$\displaystyle\sum_{i\in\mathcal{N}(t)}\left(\sum_{j\in\mathcal{N}(t),j\neq
i}\left|v^{\epsilon,z}(t,x_{i}^{N}(t),x_{j}^{N}(t))\right|\right)^{2}$
$\displaystyle\leq\left(\sum_{i\neq
j\in\mathcal{N}(t)}\left|v^{\epsilon,z}(t,x_{i}^{N}(t),x_{j}^{N}(t))\right|\right)^{2}$
$\displaystyle\sum_{i\in\mathcal{N}(t)}\left(\sum_{j\in\mathcal{N}(t),j\neq
i}\left|\nabla v^{\epsilon,z}(t,x_{i}^{N}(t),x_{j}^{N}(t))\right|\right)^{2}$
$\displaystyle\leq\left(\sum_{i\neq j\in\mathcal{N}(t)}\left|\nabla
v^{\epsilon,z}(t,x_{i}^{N}(t),x_{j}^{N}(t))\right|\right)^{2}$

###### Lemma 22

For any finite $T,d\geq 1$, we have that

$\displaystyle\lim_{|z|\to
0}\limsup_{N\to\infty}\mathbb{E}\int_{0}^{T}dt\frac{1}{N^{5}}\sum_{i\neq
k\in\mathcal{N}(t)}\theta^{\epsilon}(x_{i}^{N}(t)-x_{k}^{N}(t))\left[\sum_{j\in\mathcal{N}(t),j\neq
i,k}v^{\epsilon,z}(t,x_{i}^{N}(t),x_{j}^{N}(t))\right]^{2}=0.$

Proof. By the elementary inequality $(\sum_{i=1}^{L}a_{i})^{2}\leq
L\sum_{i=1}^{L}a_{i}^{2}$ for any $L\in\mathbb{N}$, and (44), we have that

$\displaystyle\frac{1}{N^{5}}\sum_{i\neq
k\in\mathcal{N}(t)}\theta^{\epsilon}(x_{i}^{N}(t)-x_{k}^{N}(t))\left[\sum_{j\in\mathcal{N}(t),j\neq
i,k}v^{\epsilon,z}(t,x_{i}^{N}(t),x_{j}^{N}(t))\right]^{2}$
$\displaystyle\leq\frac{1}{N^{4}}\sum_{i\neq
k\in\mathcal{N}(t)}\theta^{\epsilon}(x_{i}^{N}(t)-x_{k}^{N}(t))\sum_{j\in\mathcal{N}(t),j\neq
i,k}\left|v^{\epsilon,z}(t,x_{i}^{N}(t),x_{j}^{N}(t))\right|^{2}$
$\displaystyle\leq\frac{\kappa(\epsilon)}{N^{3}}\sum_{i\neq
k\in\mathcal{N}(t)}\theta^{\epsilon}(x_{i}^{N}(t)-x_{k}^{N}(t))\sum_{j\in\mathcal{N}(t),j\neq
i,k}\left|v^{\epsilon,z}(t,x_{i}^{N}(t),x_{j}^{N}(t))\right|$

where $\kappa(\epsilon)\to 0$ is as in (56). The conclusion then follows from
the statement of Lemma 15.

###### Remark 23

By Lemma 22, the term $B_{3}$ (34) is negligible in the sense of (41). Indeed,
there are five terms inside the square, which we first use the elementary
inequality $(\sum_{i=1}^{5}a_{i})^{2}\leq 5\sum_{i=1}^{5}a_{i}^{2}$, then we
handle term by term, of which the first four are of the form in the lemma
(after bounding the test functions by constants), and the last one by Lemma 17
together with the fact
$N^{-2}|v^{\epsilon,z}(\cdot)|^{2}\leq\kappa(\epsilon)^{2}$.

To conclude, we have thus far shown that Proposition 5 holds, by the
discussion around (41).

##  5 Pathwise uniqueness of the SPDE (8) and regularity of its solutions

Consider the system

$\displaystyle du_{m}\left(t,x\right)$
$\displaystyle=\left(\mathcal{L}u_{m}\left(t,x\right)+F_{m}\left(u\left(t,x\right)\right)\right)dt+\sum_{k\in
K}\sigma_{k}\cdot\nabla u_{m}\left(t,x\right)dW_{t}^{k}$ $\displaystyle m$
$\displaystyle=1,...,M,\qquad u=\left(u_{1},...,u_{M}\right)$

where

$\displaystyle F_{m}\left(u\left(t,x\right)\right)$
$\displaystyle=\sum_{n=1}^{m-1}u_{n}\left(t,x\right)u_{m-n}\left(t,x\right)-2u_{m}\left(t,x\right)\sum_{n=1}^{M}u_{n}\left(t,x\right)$
$\displaystyle\mathcal{L}u_{m}$ $\displaystyle=\frac{\lambda^{2}}{2}\Delta
u_{m}+\frac{1}{2}\operatorname{div}\left(Q\left(x,x\right)\nabla u_{m}\right)$
(58) $\displaystyle Q\left(x,y\right)$ $\displaystyle=\sum_{k\in
K}\sigma_{k}\left(x\right)\otimes\sigma_{k}\left(y\right)$

with initial condition $\left(r_{1}p_{1},...,r_{M}p_{M}\right)$, where
$\sum_{m=1}^{M}r_{m}=1$, satisfying

$0\leq p_{m}\leq C,\qquad\int p_{m}\left(x\right)dx\leq 1$

for every $m=1,...,M$. Notice that, also

$\int p_{m}^{2}\left(x\right)dx\leq C\int p_{m}\left(x\right)dx\leq C,$

property often used below also for $u_{m}\left(t,x\right)$.

In the equations above (58), $\mathcal{L}$ is the resulting elliptic operator
after the reformulation of Stratonovich in Itô form. Assume $\sigma_{k}\in
C_{b}^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d})$,
$\operatorname{div}\sigma_{k}=0$.

###### Definition 24

Given a filtered probability space
$(\Omega,\mathcal{F},\\{\mathcal{G}_{t}\\}_{t\geq 0},\mathbb{P})$ and Brownian
motions $\\{W_{t}^{k}\\}_{k\in K}$, by very weak solution we mean a
progressively measurable process $u\left(t,x\right)$ such that, for some
constant $U>0$,

$\mathbb{P}\left(0\leq u_{m}\left(t,x\right)\leq U\text{ for all
}\left(t,x\right)\right)=1\text{ for all }m=1,...,M$ $\mathbb{P}\left(\int
u_{m}\left(t,x\right)dx\leq 1\text{ for all }t\right)=1\text{ for all
}m=1,...,M$ $\displaystyle\left\langle u_{m}\left(t\right),\phi\right\rangle$
$\displaystyle=\left\langle
r_{m}p_{m},\phi\right\rangle+\int_{0}^{t}\left\langle
u_{m}\left(s\right),\mathcal{L}^{\ast}\phi\right\rangle
ds+\int_{0}^{t}\left\langle
F_{m}\left(u\left(s\right)\right),\phi\right\rangle ds$
$\displaystyle+\sum_{k\in K}\int_{0}^{t}\left\langle
u_{m}\left(s\right),\sigma_{k}\cdot\nabla\phi\right\rangle dW_{s}^{k}$

for all $\phi\in C_{c}^{\infty}$, $\mathbb{P}$-a.s. If in addition they
satisfy

$\max_{m=1,...,M}\mathbb{E}\int_{0}^{T}\int\left|\nabla
u_{m}\left(t,x\right)\right|^{2}dxdt<\infty$

then they are called weak solutions.

As already remarked for $p_{m}$, from the assumptions it follows that $\int
u_{m}^{2}\left(t,x\right)dx\leq U$ a.s., for every $m=1,...,M$.

###### Lemma 25

Very weak solutions are also weak solutions.

###### Corollary 26

Weak solutions are pathwise unique.

###### Proposition 27

Let $n>d/4$ be an integer and let $\left(u_{m}\right)_{m=1,...,M}$ be the
unique solution given by Corollary 26. If the initial conditions
$u_{m}\left(0\right)$ belong to $W^{2n,2}\left(\mathbb{R}^{d}\right)$,
$m=1,...,M$, then $\left(u_{m}\right)_{m=1,...,M}$ has the following
regularity:

$\mathbb{E}\left[\sup_{t\in\left[0,T\right]}\left\|u_{m}\left(t,\cdot\right)\right\|_{W^{2n,2}}^{2}\right]+\mathbb{E}\int_{0}^{T}\left\|u_{m}\left(t,\cdot\right)\right\|_{W^{2n+1,2}}^{2}dt<\infty$

for every $m=1,...,M$. In particular, if $u_{m}\left(0\right)\in
C^{\infty}\left(\mathbb{R}^{d}\right)$, $m=1,...,M$, with square integrable
derivatives of all orders, then $\mathbb{P}$-a.s. one has
$u_{m}\left(t\right)\in C^{\infty}\left(\mathbb{R}^{d}\right)$ for all
$t\in\left[0,T\right]$ and $m=1,...,M$.

The proof of Proposition 27 is postponed to Section 8, since it shares some
technical ingredients with the proofs presented in that section.

###  5.1 Proof of Lemma 25

####  5.1.1 Preparation

One can prove (cf. [10], [12]) that there exists $\eta<1$ such that

$\sum_{k\in K}\left\|\sigma_{k}\cdot\nabla
f\right\|_{L^{2}}^{2}\leq-2\eta\left\langle\mathcal{L}f,f\right\rangle$

for all $f\in C_{c}^{\infty}$. We want to exploit this property by means of an
energy inequality. For this purpose we need to apply Itô formula but $u_{m}$
does not have the necessary regularity, in particular the term
$\left\langle\mathcal{L}u_{m}\left(t\right),u_{m}\left(t\right)\right\rangle$
is not well defined (in a sense its good definition is our thesis). Thus we
take a smooth symmetric density $\rho$, with compact support in the unitary
ball $B\left(0,1\right)$, we define
$\rho_{\epsilon}\left(x\right)=\epsilon^{-d}\rho\left(\epsilon^{-1}x\right)$
and set

$u_{m}^{\epsilon}\left(t,x\right)=\left(\rho_{\epsilon}\ast
u_{m}\left(t\right)\right)\left(x\right)=\int\rho_{\epsilon}\left(x-y\right)u_{m}\left(t,y\right)dy.$

The process $u_{m}^{\epsilon}\left(t,x\right)$ is smooth in $x$; since $\int
u_{m}^{2}\left(t,x\right)dx\leq U$ a.s., from the smoothness and compact
support of $\rho$ plus Young’s inequality for convolutions, it follows in
particular $u_{m}^{\epsilon}\left(t\right)\in W^{2,2}$ a.s. (but the
$W^{2,2}$-norm depends on $\epsilon$), hence the term
$\left\langle\mathcal{L}u_{m}^{\epsilon}\left(t\right),u_{m}^{\epsilon}\left(t\right)\right\rangle$
is well defined.

From the weak formulation, using a test function of the form
$\rho_{\epsilon}\ast\phi$ and the arbitrariety of $\phi$, we easily get

$\displaystyle u_{m}^{\epsilon}\left(t\right)$
$\displaystyle=u_{m}^{\epsilon}\left(0\right)+\int_{0}^{t}\left(\mathcal{L}u_{m}^{\epsilon}\left(s\right)+\rho_{\epsilon}\ast
F_{m}\left(u\left(s\right)\right)\right)ds$ $\displaystyle+\sum_{k\in
K}\int_{0}^{t}\rho_{\epsilon}\ast\left(\sigma_{k}\cdot\nabla
u_{m}\left(s\right)\right)dW_{s}^{k}$

where all processes can be interpreted, for instance, as $L^{2}$-valued
processes and where the term $\rho_{\epsilon}\ast\left(\sigma_{k}\cdot\nabla
u_{m}\left(t\right)\right)$ is a short notation for

$\left[\rho_{\epsilon}\ast\left(\sigma_{k}\cdot\nabla
u_{m}\left(t\right)\right)\right]\left(x\right):=\int\nabla_{x}\rho_{\epsilon}\left(x-y\right)\cdot\sigma_{k}\left(y\right)u_{m}\left(t,y\right)dy.$

Now $u_{m}^{\epsilon}\left(t\right)$ is regular enough to apply Itô formula
(cf. [31], [39], [40]):

$\displaystyle d\left\|u_{m}^{\epsilon}\left(t\right)\right\|_{L^{2}}^{2}$
$\displaystyle=2\left\langle\mathcal{L}u_{m}^{\epsilon}\left(t\right),u_{m}^{\epsilon}\left(t\right)\right\rangle
dt+2\left\langle\rho_{\epsilon}\ast
F_{m}\left(u\left(t\right)\right),u_{m}^{\epsilon}\left(t\right)\right\rangle
dt$ $\displaystyle+2\sum_{k\in
K}\left\langle\rho_{\epsilon}\ast\left(\sigma_{k}\cdot\nabla
u_{m}\left(t\right)\right),u_{m}^{\epsilon}\left(t\right)\right\rangle
dW_{t}^{k}+\sum_{k\in K}\left\|\rho_{\epsilon}\ast\left(\sigma_{k}\cdot\nabla
u_{m}\left(t\right)\right)\right\|_{L^{2}}^{2}dt.$

Then we introduce the commutators. Let us write

$\rho_{\epsilon}\ast\left(\sigma_{k}\cdot\nabla
u_{m}\left(t\right)\right)=\sigma_{k}\cdot\nabla
u_{m}^{\epsilon}\left(t\right)+R_{m,k}^{\epsilon}\left(t\right)$

where $R_{m,k}^{\epsilon}\left(t\right)$ is defined by the identity. We have

$\displaystyle d\left\|u_{m}^{\epsilon}\left(t\right)\right\|_{L^{2}}^{2}$
$\displaystyle=2\left\langle\mathcal{L}u_{m}^{\epsilon}\left(t\right),u_{m}^{\epsilon}\left(t\right)\right\rangle
dt+2\left\langle\rho_{\epsilon}\ast
F_{m}\left(u\left(t\right)\right),u_{m}^{\epsilon}\left(t\right)\right\rangle
dt$ $\displaystyle+2\sum_{k\in K}\left\langle
R_{m,k}^{\epsilon}\left(t\right),u_{m}^{\epsilon}\left(t\right)\right\rangle
dW_{t}^{k}+\sum_{k\in K}\left\|\sigma_{k}\cdot\nabla
u_{m}^{\epsilon}\left(t\right)+R_{m,k}^{\epsilon}\left(t\right)\right\|_{L^{2}}^{2}dt$

where we have used the fact that, being $\operatorname{div}\sigma_{k}=0$,

$\left\langle\sigma_{k}\cdot\nabla
u_{m}^{\epsilon}\left(t\right),u_{m}^{\epsilon}\left(t\right)\right\rangle=0.$

For every $\delta>0$ we have $2ab\leq\delta a^{2}+\delta^{-1}b^{2}$, hence

$\displaystyle\sum_{k\in K}\left\|\sigma_{k}\cdot\nabla
u_{m}^{\epsilon}\left(t\right)+R_{m,k}^{\epsilon}\left(t\right)\right\|_{L^{2}}^{2}$
$\displaystyle\leq\left(1+\delta\right)\sum_{k\in
K}\left\|\sigma_{k}\cdot\nabla
u_{m}^{\epsilon}\left(t\right)\right\|_{L^{2}}^{2}$
$\displaystyle+\left(1+\delta^{-1}\right)\sum_{k\in
K}\left\|R_{m,k}^{\epsilon}\left(t\right)\right\|_{L^{2}}^{2}$
$\leq-\left(1+\delta\right)2\eta\left\langle\mathcal{L}u_{m}^{\epsilon}\left(t\right),u_{m}^{\epsilon}\left(t\right)\right\rangle+\left(1+\delta^{-1}\right)\sum_{k\in
K}\left\|R_{m,k}^{\epsilon}\left(t\right)\right\|_{L^{2}}^{2}.$

Choose $\delta>0$ such that $\left(1+\delta\right)2\eta=2-\zeta$ for some
$\zeta>0$. We get

$\displaystyle d\left\|u_{m}^{\epsilon}\left(t\right)\right\|_{L^{2}}^{2}$
$\displaystyle\leq\zeta\left\langle\mathcal{L}u_{m}^{\epsilon}\left(t\right),u_{m}^{\epsilon}\left(t\right)\right\rangle
dt+2\left\langle\rho_{\epsilon}\ast
F_{m}\left(u\left(t\right)\right),u_{m}^{\epsilon}\left(t\right)\right\rangle
dt$ $\displaystyle+2\sum_{k\in K}\left\langle
R_{m,k}^{\epsilon}\left(t\right),u_{m}^{\epsilon}\left(t\right)\right\rangle
dW_{t}^{k}+\left(1+\delta^{-1}\right)\sum_{k\in
K}\left\|R_{m,k}^{\epsilon}\left(t\right)\right\|_{L^{2}}^{2}dt.$

One has

$2\left\langle\rho_{\epsilon}\ast
F_{m}\left(u\left(t\right)\right),u_{m}^{\epsilon}\left(t\right)\right\rangle\leq\left\|\rho_{\epsilon}\ast
F_{m}\left(u\left(t\right)\right)\right\|_{L^{2}}^{2}+\left\|u_{m}^{\epsilon}\left(t\right)\right\|_{L^{2}}^{2}$

and

$\displaystyle\left\|\rho_{\epsilon}\ast
F_{m}\left(u\left(t\right)\right)\right\|_{L^{2}}^{2}$
$\displaystyle\leq\left\|F_{m}\left(u\left(t\right)\right)\right\|_{L^{2}}^{2}\leq\left\|F_{m}\left(u\left(t\right)\right)\right\|_{\infty}\left\|F_{m}\left(u\left(t\right)\right)\right\|_{L^{1}}$
$\displaystyle\leq C\left\|F_{m}\left(u\left(t\right)\right)\right\|_{L^{1}}$

for some deterministic constant $C$, since $0\leq u_{m}\left(t,x\right)\leq U$
a.s. for every $m$; moreover each term of $F_{m}\left(u\left(t\right)\right)$
has $L^{1}$-norm bounded by $U$: indeed a.s.

$\int u_{k}\left(t,x\right)u_{h}\left(t,x\right)dx\leq U.$

We have found

$\displaystyle
d\left\|u_{m}^{\epsilon}\left(t\right)\right\|_{L^{2}}^{2}+\zeta\int\left|\nabla
u_{m}^{\epsilon}\left(t,x\right)\right|^{2}dxdt$ $\displaystyle\leq
Cdt+\left\|u_{m}^{\epsilon}\left(t\right)\right\|_{L^{2}}^{2}dt+2\sum_{k\in
K}\left\langle
R_{m,k}^{\epsilon}\left(t\right),u_{m}^{\epsilon}\left(t\right)\right\rangle
dW_{t}^{k}+\left(1+\delta^{-1}\right)\sum_{k\in
K}\left\|R_{m,k}^{\epsilon}\left(t\right)\right\|_{L^{2}}^{2}dt.$

for some constant $C>0$.

####  5.1.2 Commutator estimate

Let us prove:

###### Lemma 28

$\sum_{k\in K}\left\|R_{m,k}^{\epsilon}\left(t\right)\right\|_{L^{2}}^{2}\leq
C_{Q}\int\left|u_{m}\left(t,x\right)\right|^{2}dx.$

for a suitable constant $C_{Q}>0$.

Proof. Collecting the definitions, we have

$\displaystyle R_{m,k}^{\epsilon}\left(t\right)$
$\displaystyle=\int\nabla_{x}\rho_{\epsilon}\left(x-y\right)\cdot\sigma_{k}\left(y\right)u_{m}\left(t,y\right)dy-\sigma_{k}\left(x\right)\cdot\nabla\int\rho_{\epsilon}\left(x-y\right)u_{m}\left(t,y\right)dy$
$\displaystyle=\int\nabla_{x}\rho_{\epsilon}\left(x-y\right)\cdot\left(\sigma_{k}\left(y\right)-\sigma_{k}\left(x\right)\right)u_{m}\left(t,y\right)dy.$

One has

$\displaystyle\sum_{k\in
K}\left\|R_{m,k}^{\epsilon}\left(t\right)\right\|_{L^{2}}^{2}$
$\displaystyle=\sum_{k\in
K}\iiint\nabla_{x}\rho_{\epsilon}\left(x-y\right)\cdot\left(\sigma_{k}\left(y\right)-\sigma_{k}\left(x\right)\right)u_{m}\left(t,y\right)$
$\displaystyle\cdot\nabla_{x}\rho_{\epsilon}\left(x-y^{\prime}\right)\cdot\left(\sigma_{k}\left(y^{\prime}\right)-\sigma_{k}\left(x\right)\right)u_{m}\left(t,y^{\prime}\right)dydy^{\prime}dx$
$\displaystyle\leq U^{2}\iiint\left\|\sum_{k\in
K}\left(\sigma_{k}\left(y\right)-\sigma_{k}\left(x\right)\right)\otimes\left(\sigma_{k}\left(y^{\prime}\right)-\sigma_{k}\left(x\right)\right)\right\|$
$\displaystyle\cdot\left|\nabla_{x}\rho_{\epsilon}\left(x-y\right)\right|\left|\nabla_{x}\rho_{\epsilon}\left(x-y^{\prime}\right)\right|\left|u_{m}\left(t,y\right)\right|\left|u_{m}\left(t,y^{\prime}\right)\right|dydy^{\prime}dx$

where $\left\|\cdot\right\|$ here denotes the Euclidean matrix norm. We show
below that

$\left\|\sum_{k\in
K}\left(\sigma_{k}\left(y\right)-\sigma_{k}\left(x\right)\right)\otimes\left(\sigma_{k}\left(y^{\prime}\right)-\sigma_{k}\left(x\right)\right)\right\|\leq
C_{Q}\left|x-y\right|\left|x-y^{\prime}\right|$ (59)

for some constant $C_{Q}$. Recall the support property of $\rho$; it implies
that $\rho_{\epsilon}$ has support in the ball $B\left(0,\epsilon\right)$,
hence the previous expression is bounded by

$\displaystyle\leq C_{Q}\epsilon^{-2}\int
dx\int_{B\left(x,\epsilon\right)}dy\int_{B\left(x,\epsilon\right)}dy^{\prime}\epsilon^{-2d}\left|\left(\nabla\rho\right)\left(\frac{x-y}{\epsilon}\right)\right|\left|\left(\nabla\rho\right)\left(\frac{x-y^{\prime}}{\epsilon}\right)\right|$
$\displaystyle\cdot\left|x-y\right|\left|x-y^{\prime}\right|\left|u_{m}\left(t,y\right)\right|\left|u_{m}\left(t,y^{\prime}\right)\right|$
$\displaystyle\leq C_{Q}\iiint
dxdydy^{\prime}\epsilon^{-2d}\left|\left(\nabla\rho\right)\left(\frac{x-y}{\epsilon}\right)\right|\left|\left(\nabla\rho\right)\left(\frac{x-y^{\prime}}{\epsilon}\right)\right|\left|u_{m}\left(t,y\right)\right|\left|u_{m}\left(t,y^{\prime}\right)\right|$
$\displaystyle=C_{Q}\int\left(\int\epsilon^{-d}\left|\left(\nabla\rho\right)\left(\frac{x-y}{\epsilon}\right)\right|\left|u_{m}\left(t,y\right)\right|dy\right)^{2}dx.$

By Young’s inequality for convolutions, this is bounded by

$\leq C_{Q}\int\left|u_{m}\left(t,x\right)\right|^{2}dx.$

It remains to prove (59). It is equivalent to

$\left\|Q\left(y,y^{\prime}\right)-Q\left(y,x\right)-Q\left(x,y^{\prime}\right)+Q\left(x,x\right)\right\|\leq
C_{Q}\left|x-y\right|\left|x-y^{\prime}\right|.$

It is sufficient to prove a similar estimate componentwise, for the matrix-
value function $Q$. Now

$\displaystyle Q_{ij}\left(y,y^{\prime}\right)-Q_{ij}\left(y,x\right)$
$\displaystyle=\int_{0}^{1}\nabla_{2}Q_{ij}\left(y,\alpha\left(y-y^{\prime}\right)+\left(1-\alpha\right)\left(y-x\right)\right)d\alpha\cdot\left(x-y^{\prime}\right)$
$Q_{ij}\left(x,y^{\prime}\right)-Q_{ij}\left(x,x\right)=\int_{0}^{1}\nabla_{2}Q_{ij}\left(x,\alpha\left(x-y^{\prime}\right)\right)d\alpha\cdot\left(x-y^{\prime}\right)$
$\displaystyle\partial_{h}Q_{ij}\left(y,\alpha\left(y-y^{\prime}\right)+\left(1-\alpha\right)\left(y-x\right)\right)-\partial_{h}Q_{ij}\left(x,\alpha\left(x-y^{\prime}\right)\right)$
$\displaystyle=\int_{0}^{1}\nabla_{2}\partial_{h}Q_{ij}\left(y,\beta\left(\alpha\left(y-y^{\prime}\right)+\left(1-\alpha\right)\left(y-x\right)\right)+\left(1-\beta\right)\left(\alpha\left(x-y^{\prime}\right)\right)\right)d\beta\cdot\left(y-x\right)$
$\displaystyle+\int_{0}^{1}\nabla_{1}\partial_{h}Q_{ij}(\beta
y+(1-\beta)x,\alpha(x-y^{\prime}))d\beta\cdot(y-x).$

Collecting these identities, we get the required bound.

####  5.1.3 Conclusion

From Lemma 28 and the a priori bounds on $u_{m}$ we get

$\sum_{k\in K}\left\|R_{m,k}^{\epsilon}\left(t\right)\right\|_{L^{2}}^{2}\leq
C_{Q}U\int\left|u_{m}\left(t,x\right)\right|dx\leq C_{Q}U.$

Hence

$d\left\|u_{m}^{\epsilon}\left(t\right)\right\|_{L^{2}}^{2}+\zeta\int\left|\nabla
u_{m}^{\epsilon}\left(t,x\right)\right|^{2}dxdt\leq C^{\prime}dt+2\sum_{k\in
K}\left\langle
R_{m,k}^{\epsilon}\left(t\right),u_{m}^{\epsilon}\left(t\right)\right\rangle
dW_{t}^{k}$

for a new constant $C^{\prime}>0$. It follows

$\displaystyle\zeta\mathbb{E}\int_{0}^{T}\int\left|\nabla
u_{m}^{\epsilon}\left(t,x\right)\right|^{2}dxdt$ $\displaystyle\leq
C^{\prime}T+2\left(\sum_{k\in K}\mathbb{E}\int_{0}^{T}\left\langle
R_{m,k}^{\epsilon}\left(t\right),u_{m}^{\epsilon}\left(t\right)\right\rangle^{2}dt\right)^{1/2}$
$\displaystyle\leq C^{\prime}T+2\left(\sum_{k\in
K}\mathbb{E}\int_{0}^{T}\left\|R_{m,k}^{\epsilon}\left(t\right)\right\|_{L^{2}}^{2}\left\|u_{m}^{\epsilon}\left(t\right)\right\|_{L^{2}}^{2}dt\right)^{1/2}.$

As above, $\left\|u_{m}^{\epsilon}\left(t\right)\right\|_{L^{2}}^{2}\leq U$,
and $\sum_{k\in
K}\left\|R_{m,k}^{\epsilon}\left(t\right)\right\|_{L^{2}}^{2}\leq C_{Q}U$,
hence

$\zeta\mathbb{E}\int_{0}^{T}\int\left|\nabla
u_{m}^{\epsilon}\left(t,x\right)\right|^{2}dxdt\leq
C^{\prime}T+2C_{Q}^{1/2}UT^{1/2}.$

The proof of the lemma is complete.

###  5.2 Proof of Corollary 26

Let $u=\left(u_{1},...,u_{M}\right)$,
$u^{\prime}=\left(u_{1}^{\prime},...,u_{M}^{\prime}\right)$ be two weak
solutions. Set $v\left(t\right)=u\left(t\right)-u^{\prime}\left(t\right)$,
$v_{m}\left(t\right)=u_{m}\left(t\right)-u_{m}^{\prime}\left(t\right)$. The
regularity of $v$ is sufficient to apply Itô formula (cf. [31], [39], [40]):

$\displaystyle d\left\|v_{m}\left(t\right)\right\|_{L^{2}}^{2}$
$\displaystyle=2\left\langle\mathcal{L}v_{m}\left(t\right),v_{m}\left(t\right)\right\rangle
dt+2\left\langle
F_{m}\left(u\left(t\right)\right)-F_{m}\left(u^{\prime}\left(t\right)\right),v_{m}\left(t\right)\right\rangle
dt$ $\displaystyle+2\sum_{k\in K}\left\langle\sigma_{k}\cdot\nabla
v_{m}\left(t\right),v_{m}\left(t\right)\right\rangle dW_{t}^{k}+\sum_{k\in
K}\left\|\sigma_{k}\cdot\nabla v_{m}\left(t\right)\right\|_{L^{2}}^{2}dt.$

Since $\left\langle\sigma_{k}\cdot\nabla
v_{m}\left(t\right),v_{m}\left(t\right)\right\rangle=0$ and $\sum_{k\in
K}\left\|\sigma_{k}\cdot\nabla v_{m}\left(t\right)\right\|_{L^{2}}^{2}$ is
bounded above by
$-2\left\langle\mathcal{L}v_{m}\left(t\right),v_{m}\left(t\right)\right\rangle$,
we get

$d\left\|v_{m}\left(t\right)\right\|_{L^{2}}^{2}\leq 2\left\langle
F_{m}\left(u\left(t\right)\right)-F_{m}\left(u^{\prime}\left(t\right)\right),v_{m}\left(t\right)\right\rangle
dt.$

Now

$F_{m}\left(u\right)-F_{m}\left(u^{\prime}\right)=\sum_{n=1}^{m-1}\left(u_{n}u_{m-n}-u_{n}^{\prime}u_{m-n}^{\prime}\right)-2\left(u_{m}\sum_{n=1}^{M}u_{n}-u_{m}^{\prime}\sum_{n=1}^{M}u_{n}^{\prime}\right)$

and each term of the form $u_{h}u_{k}-u_{h}^{\prime}u_{k}^{\prime}$ can be
estimated as

$\displaystyle\left|u_{h}u_{k}-u_{h}^{\prime}u_{k}^{\prime}\right|$
$\displaystyle\leq\left|u_{h}u_{k}-u_{h}u_{k}^{\prime}\right|+\left|u_{h}u_{k}^{\prime}-u_{h}^{\prime}u_{k}^{\prime}\right|$
$\displaystyle\leq U\left|v_{k}\right|+U\left|v_{h}\right|$

hence

$\left|F_{m}\left(u\right)-F_{m}\left(u^{\prime}\right)\right|\leq
C\sum_{n=1}^{M}\left|v_{n}\right|$

where $C$ depends on $U$ and $M$. It follows

$\displaystyle\frac{d}{dt}\left\|v_{m}\left(t\right)\right\|_{L^{2}}^{2}$
$\displaystyle\leq
2C\sum_{n=1}^{M}\int\left|v_{n}\left(t,x\right)\right|\left|v_{m}\left(t,x\right)\right|dx$
$\displaystyle\leq
2C\sum_{n=1}^{M}\left\|v_{n}\left(t\right)\right\|_{L^{2}}^{2}+2CM\left\|v_{m}\left(t\right)\right\|_{L^{2}}^{2}.$

Summing over $m$ and applying Gronwall lemma, we deduce
$\left\|v_{m}\left(t\right)\right\|_{L^{2}}^{2}=0$ for every $m$ and $t$,
which is pathwise uniqueness.

##  6 Relative compactness of the empirical measure

In this section, we show the tightness of the sequence of laws of
$\\{\mu^{N,m}\\}_{m\leq M}$ taking values in
$\mathcal{D}_{T}(\mathcal{M}_{+,1})^{M}$.

###  6.1 A general compactness criterion

Let $\mathcal{M}_{+,1}\left(\mathbb{R}^{d}\right)$ be the set of positive
Borel measures on $\mathbb{R}^{d}$ with mass $\leq 1$. Recall that, given
$R>0$, the set
$\mathcal{K}_{R}\subset\mathcal{M}_{+,1}\left(\mathbb{R}^{d}\right)$ defined
as

$\displaystyle\mathcal{K}_{R}=\Big{\\{}\mu\in\mathcal{M}_{+,1}\left(\mathbb{R}^{d}\right):\int_{\mathbb{R}^{d}}\left|x\right|\mu\left(dx\right)\leq
R\Big{\\}}$

is relatively compact in $\mathcal{M}_{+,1}\left(\mathbb{R}^{d}\right)$
endowed with the topology of weak convergence of measures.

The weak convergence on $\mathcal{M}_{+,1}\left(\mathbb{R}^{d}\right)$ can be
metrized in the following way. For every compact set $K\subset\mathbb{R}^{d}$,
the space $C\left(K\right)$ is separable; let
$\left(f_{n}^{K}\right)_{n\in\mathbb{N}}$ be a dense sequence in
$C\left(K\right)$ and define the function
$\delta_{K}:\mathcal{M}_{+,1}\left(K\right)^{2}\rightarrow[0,\infty)$ as

$\displaystyle\delta_{K}\left(\mu,\nu\right)=\sum_{n=1}^{\infty}2^{-n}\left(\left|\left\langle\mu,f_{n}^{K}\right\rangle-\left\langle\nu,f_{n}^{K}\right\rangle\right|\wedge
1\right).$

This is a metric on $\mathcal{M}_{+,1}\left(K\right)$ and the metric space
$\left(\mathcal{M}_{+,1}\left(K\right),d_{K}\right)$ is complete and