# Scaling limit for a second-order particle system
with local annihilation111Keywords and phrases: second-order system; scaling
limit; annihilating particles; local interaction; hypoellipticity. 222AMS
subject classification (2020): 60K35, 82C21, 82C40, 35J70.

Ruojun Huang333 Fachbereich Mathematik und Informatik, Universität Münster.
Einsteinstr. 62, 48149 Münster, Germany. E-mail<EMAIL_ADDRESS>
###### Abstract

For a second-order particle system in $\mathbb{R}^{d}$ subject to locally-in-
space pairwise annihilation, we prove a scaling limit for its empirical
measure on position and velocity towards a degenerate elliptic partial
differential equation. Crucial ingredients are Green’s function estimates for
the associated hypoelliptic operator and an Itô-Tanaka trick.

##  1 Introduction

Second-order particle systems comprising of dynamics of positions and
velocities, in the spirit of Newtonian mechanics, give a more complete
description of interacting particles. When the behavior of a first-order
system of positions only is understood, or is insufficient for one’s purpose,
it is natural to look at a second-order system. The technical difficulties
associated with second-order systems are in general greater, some coming from
the degeneracy of the associated operators. Here, we explore a simplest model
of interacting particles with local interactions in $\mathbb{R}^{d}$ written
in second-order, where the interaction occurs not in the equation of the
motion but in pairwise annihilation. Besides being well-related to a large
body of works on annihilating (or proliferating, coagulating, colliding etc)
particle systems and their scaling limits (e.g. [21, 32, 25, 26, 22, 28, 16,
14, 12]) and generally relevant for reaction-diffusion phenomena, one
motivation is also from the technical side, namely, we aim to derive a scaling
limit when the interaction range between particles is truly local (see
Condition 2).

Given a probability space $(\Omega,\mathcal{F},\mathbb{P})$, for any positive
integer $N\in\mathbb{N}$ and $d\geq 1$, consider a system of particles, whose
cardinality at time $t=0$ is $N(0)=N$, and with positions and velocities at
any time $t\geq 0$ denoted by $x_{i}^{N}(t)\in\mathbb{R}^{d}$,
$v_{i}^{N}(t)\in\mathbb{R}^{d}$, $i=1,...,N(t)$, and the interaction comes
from the possibility of pairwise annhilation when spatial positions are close.
More specifically, the position and velocity of each particle, while active,
obey the second-order dynamics

$\displaystyle\begin{cases}x_{i}^{N}(t)=x_{i}(0)+\int_{0}^{t}v_{i}^{N}(s)ds\\\
v_{i}^{N}(t)=v_{i}(0)+B_{i}(t)\end{cases},\quad t\geq 0,\,i\in\mathcal{N}(t),$
(1)

where $\mathcal{N}(t)$ denotes the set of indices of active particles at time
$t$ whose cardinality $|\mathcal{N}(t)|=N(t)\leq N$,
$\\{B_{i}(t)\\}_{i=1}^{\infty}$ is a collection of independent standard
Brownian motions in $\mathbb{R}^{d}$, and the initial conditions
$(x_{i}(0),v_{i}(0))_{i=1}^{\infty}$ are random, chosen i.i.d. with some given
probability density $f_{0}(x,v):\mathbb{R}^{2d}\to\mathbb{R}_{+}$ satisfying
the condition below, independent of the Brownian motions.

###### Condition 1

For every $d\geq 1$, we assume that there exist some finite constants $\Gamma$
and $R$ such that $\|f_{0}(x,v)\|_{L^{\infty}(\mathbb{R}^{2d})}\leq\Gamma$ and
supp$(f_{0})\subset\mathbb{B}_{E}((0,0),R)$, the closed Euclidean ball of
radius $R$ centered at the origin in $\mathbb{R}^{2d}$.

The probability space is endowed with the canonical filtration

$\displaystyle\mathcal{F}_{t}:=\sigma\big{\\{}\\{B_{i}(s):0\leq s\leq
t\\}_{i=1}^{\infty},(x_{i}(0),v_{i}(0))_{i=1}^{\infty}\big{\\}},\quad t\geq
0.$

Interaction between different particles comes by means of pairwise
annhilation, informally described as follows. Let
$\theta(x):\mathbb{R}^{d}\to\mathbb{R}_{+}$ be given, nonnegative, of class
$C^{\infty}$, with $\int\theta dx=1$, $\theta(0)=0$, $\theta(x)=\theta(-x)$
and compact support in $\mathbb{B}_{E}(0,1)$. Then, for any
$\epsilon\in(0,1]$, denote by

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

(For the necessity of the condition $\theta(0)=0$, see Remark 6.) A
configuration of the particle system is given by

$\displaystyle\zeta=(x_{1},v_{1},x_{2},v_{2},...,x_{N},v_{N})\in(\mathbb{R}^{d}\cup\emptyset)^{2N}$

where we adopt the convention that if a particle $i_{0}$ has been annihilated,
we set $x_{i_{0}}=v_{i_{0}}=\emptyset$. It will no more participate in the
dynamics of the system. Now suppose the current configuration is $\zeta$, then
for any pair of particles $i,j$ with $i\neq j$ (each ranging over the index
set $\mathcal{N}(\zeta)$ of active particles), with an infinitesimal rate

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

we remove both particles $i,j$ from the system (i.e. they annihilate with each
other) and obtain a new configuration, denoted $\zeta^{-ij}$ which is equal to
$\zeta$ in all other slots except for $x_{i},v_{i},x_{j},v_{j}$ which have
been set to $\emptyset$. Since the support of the function $\theta^{\epsilon}$
has radius $\epsilon$, interaction happens for the pair $i,j$ only when their
spatial distance is within the critical distance $\epsilon$. In fact, our
interest is in the case where $\epsilon=\epsilon(N)$ depends on $N$ and
converges to zero as $N\to\infty$. An important case to cover is the regime of
local interaction, corresponding to $\epsilon=N^{-1/d}$ in which case an
individual particle typically interacts with a bounded number of others at any
given time. Contrary to lattice models [6, 19] where nearest-neighbor or
bounded-range interactions are standard, in our continuum case such local
interaction has to be enforced by a potential (i.e. $\theta^{\epsilon}$),
which is a source of technical difficulty. One of the key techniques available
to date to handle such local interaction, or sometimes even more stringently
the mean-free path interaction, is due to Hammond and Rezakhanlou [16], see
also [21, 32, 34, 33] for other techniques. On the other hand, mean-field type
models correspond to choosing $\epsilon$ independent of $N$, and in that
setting techniques for proving scaling limits of (both first and second-order)
particle systems have become very advanced, cf. the survey [18], [4], even
when interactions are very singular and influence directly the stochastic
differential equations. Clearly, there is a big technical gap between mean-
field and more local cases, which serves as another motivation for this
research.

###### Condition 2

For any $d\geq 1$, we assume that $\epsilon=\epsilon(N)\to 0$ as $N\to\infty$
and

$\displaystyle\limsup_{N\to\infty}\frac{\epsilon^{-d}}{N}\leq 1.$

When the equality is achieved, we call the interaction local.

We now give formally the infinitesimal generator of the jointly Markov process
$(x_{i}^{N}(t),v_{i}^{N}(t))_{i=1}^{\mathcal{N}{(t)}}$:

$\displaystyle\mathcal{L}_{N}=\mathcal{L}_{N}^{D}+\mathcal{L}_{N}^{J}$ (2)

where the diffusion and respectively, jump (i.e. annhilation) part of the
generator are (acting on test functions $F$)

$\displaystyle\mathcal{L}_{N}^{D}F(\zeta)$
$\displaystyle:=\sum_{i\in\mathcal{N}(\zeta)}\big{[}v_{i}\cdot\nabla_{x_{i}}F(\zeta)+\frac{1}{2}\Delta_{v_{i}}F(\zeta)\big{]}$
$\displaystyle\mathcal{L}_{N}^{J}F(\zeta)$
$\displaystyle:=\sum_{i\in\mathcal{N}(\zeta)}\sum_{j\in\mathcal{N}(\zeta):j\neq
i}\frac{1}{N}\theta^{\epsilon}(x_{i}-x_{j})\big{[}F(\zeta^{-ij})-F(\zeta)\big{]}.$

For every $N$ and $T$, denote the process of joint empirical measure of
positions and velocities of active particles

$\displaystyle\mu_{t}^{N}(dx,dv):=\frac{1}{N}\sum_{i\in\mathcal{N}(t)}\delta_{\left(x_{i}^{N}(t),\,v_{i}^{N}(t)\right)}(dx,dv),\quad
t\in[0,T].$ (3)

For each $t$, $\mu_{t}^{N}(dx,dv)$ is a subprobability measure on
$\mathbb{R}^{2d}$. Denote by $\mathcal{M}_{+,1}(\mathbb{R}^{2d})$ the space of
subprobability measures on $\mathbb{R}^{2d}$ endowed with the weak topology,
and by

$\displaystyle\mathcal{D}_{T}(\mathcal{M}_{+,1}):=\mathcal{D}\big{(}[0,T];\mathcal{M}_{+,1}(\mathbb{R}^{2d})\big{)}$
(4)

the space of càdlàg functions on $[0,T]$ taking values in
$\mathcal{M}_{+,1}(\mathbb{R}^{2d})$ endowed with the Skorohod topology [8,
Chapter 3]. Our goal is to prove that the $N\to\infty$ limit of (3) is given
by the unique weak solution (see Definition 23) of the following degenerate
elliptic partial differential equation (pde)

$\displaystyle\begin{cases}\partial_{t}f(t,x,v)&=-v\cdot\nabla_{x}f+\frac{1}{2}\Delta_{v}f-2\int_{\mathbb{R}^{d}}f(t,x,v)f(t,x,v^{\prime})dv^{\prime}\\\
f|_{t=0}&=f_{0}(x,v)\end{cases},\quad(t,x,v)\in[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}.$
(5)

###### Theorem 3

Under Conditions 1 and 2, for any finite $T$ and $d\geq 1$, the sequence of
empirical measures $\\{\mu^{N}_{t}:t\in[0,T]\\}_{N\in\mathbb{N}}$ converges in
probability in the space $\mathcal{D}_{T}(\mathcal{M}_{+,1})$ towards a limit
$\\{\overline{\mu}_{t}:t\in[0,T]\\}$. For every $t\in[0,T]$, the measure
$\overline{\mu}_{t}(dx,dv)$ has a density $f(t,x,v)$ with respect to Lebesgue
measure and $\\{f(t,x,v):t\in[0,T],x,v\in\mathbb{R}^{d}\\}$ is the unique weak
solution to the pde (5) in the sense of Definition 23.

###### Remark 4

Our setting (1) of a second-order system is almost the simplest possible. The
motivation and challenge is to handle the local interaction. One may notice
that, if substituting the velocity into the first equation (1), then we get a
closed expression for the dynamics of the position

$\displaystyle x_{i}^{N}(t)=x_{i}(0)+v_{i}(0)t+\int_{0}^{t}B_{i}(s)ds.$ (6)

One may ask why not study directly a first-order system of $N$ locally
interacting particles with dynamics given by (6), especially if one is mainly
interested in the limiting density of positions alone

$\displaystyle\rho(t,x):=\int_{\mathbb{R}^{d}}f(t,x,v)dv.$

However, it is not hard to check that $t\mapsto\int_{0}^{t}B(s)ds$ is not a
Markov process, hence working directly with the first-order dynamics is not
necessarily easier. Consistent with that, if we integrate the pde (5) formally
in variable $v$, we do not obtain a closed equation for $\rho(t,x)$. It
remains unclear what (if any) equation $\rho$ should satisfy.

The rest of the article is organized as follows. In Section 2, we derive an
identity satisfied by the empirical measure, and identity the nonlinear term
whose convergence is the main issue. For this purpose, in Section 3 we set up
the Itô-Tanaka trick which involves the solution of an auxiliary pde with a
degenerate Kolmogorov (or Fokker-Planck) operator, see that section for more
on the meaning of this manouvre. It turns out that key to our proof are
pointwise estimates of the Green’s function associated with this operator,
which are recalled and developed in Section 4. In Section 5, we apply these
estimates to show that various terms coming from the Itô-Tanaka trick are
negligible. In Section 6, we check the tightness of the laws of our particle
system, which together with previous sections completes the proof of
subsequential convergence. In the last Section 7, we prove the uniqueness of
the limit pde. The techniques used here, which are very influenced by the
paper of Hammond-Rezakhanlou [16] on particle approximation of the
Smoluchowski coagulation equation, share elements with the author’s previous
works [12, 13], hence certain arguments are made more succinct.

##  2 Identity from the empirical measure

Fixing $T<\infty$ and any $\phi_{t}(x,v)=\phi(t,x,v)\in
C_{c}^{\infty}([0,T]\times\mathbb{R}^{2d})$, we consider the process

$\displaystyle
F_{1}(t,\zeta(t)):=\langle\phi_{t},\mu_{t}^{N}\rangle=\frac{1}{N}\sum_{i\in\mathcal{N}(t)}\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)},\quad
t\geq 0,$ (7)

where $\langle f,\mu\rangle:=\int_{\mathbb{R}^{2d}}f(x,v)\mu(dx,dv)$. Applying
Itô’s formula to (7), we obtain

$\displaystyle\langle\phi_{T},\mu_{T}^{N}\rangle$
$\displaystyle=F_{1}(T,\zeta(T)=F_{1}(0,\zeta(0)+\int_{0}^{T}\left(\partial_{t}+\mathcal{L}_{N}^{D}+\mathcal{L}_{N}^{J}\right)F_{1}(t,\zeta(t))dt+M_{T}^{N,\phi,D}+M_{T}^{N,\phi,J}$
(8)
$\displaystyle=\langle\phi_{0},\mu_{0}^{N}\rangle+\int_{0}^{T}\left\langle\partial_{t}\phi+v\cdot\nabla_{x}\phi+\frac{1}{2}\Delta_{v}\phi,\,\mu_{t}^{N}\right\rangle
dt$
$\displaystyle-\int_{0}^{T}\frac{1}{N^{2}}\sum_{i\in\mathcal{N}(t)}\sum_{j\in\mathcal{N}(t):j\neq
i}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t)\big{)}\left[\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}+\phi\big{(}t,x_{j}^{N}(t),v_{j}^{N}(t)\big{)}\right]\,dt$
$\displaystyle+M_{T}^{N,\phi,D}+M_{T}^{N,\phi,J}.$

More precisely, (7) is a time-dependent functional of the particle
configuration $\zeta(t)$, hence we may act the Markov generator (2) on (7).
The diffusion part of the generator acts individually on each
$\phi\left(t,x_{i}^{N}(t),v_{i}^{N}(t)\right)$ as

$\displaystyle\mathcal{L}_{N}^{D}\phi\left(t,x_{i}^{N}(t),v_{i}^{N}(t)\right)=\left(v_{i}^{N}(t)\cdot\nabla_{x}+\frac{1}{2}\Delta_{v}\right)\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)},$

hence

$\displaystyle\left(\partial_{t}+\mathcal{L}_{N}^{D}\right)F_{1}(t,\zeta(t))$
$\displaystyle=\frac{1}{N}\sum_{i\in\mathcal{N}(t)}\left(\partial_{t}+v_{i}^{N}(t)\cdot\nabla_{x}+\frac{1}{2}\Delta_{v}\right)\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}$
$\displaystyle=\left\langle\partial_{t}\phi+v\cdot\nabla_{x}\phi+v\cdot\nabla_{x}\phi+\frac{1}{2}\Delta_{v}\phi,\,\mu_{t}^{N}\right\rangle.$

Turning to the jump part of the generator, for each distinct pair $i\neq
j\in\mathcal{N}(t)$, we know that at an exponential rate

$\displaystyle
q_{ij}^{N}(t):=\frac{1}{N}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t)\big{)},$
(9)

they annihilate and hence the sum (7) decreases by the amount

$F_{1}(t,\zeta^{-ij}(t))-F_{1}(t,\zeta(t))=-N^{-1}\big{[}\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}+\phi\big{(}t,x_{j}^{N}(t),v_{j}^{N}(t)\big{)}\big{]}.$

Therefore,

$\mathcal{L}_{N}^{J}F_{1}(t,\zeta(t))=-\sum_{i\in\mathcal{N}(t)}\sum_{j:\,j\neq
i}\frac{1}{N}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t)\big{)}\frac{1}{N}\big{[}\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}+\phi\big{(}t,x_{j}^{N}(t),v_{j}^{N}(t)\big{)}\big{]}.$

(From now on, for the ease of notation, all sums are only over active
particles hence we sometimes drop $\mathcal{N}(t)$.) Any inactive particle is
assigned a cemetery state $\emptyset$; it stays put and does not contribute to
the dynamics. We note in passing that even though $v\in\mathbb{R}^{d}$ is an
unbounded variable, $v\cdot\nabla_{x}\phi$ is bounded continuous since here
$v$ is in the compact support of $\phi$, thus in the first line of (8), we
have the form of $\mu_{t}^{N}$ integrated against a bounded continuous test
function.

To compensate for the action of the generator,
$M_{t}^{N,\phi,D},M_{t}^{N,\phi,J}$ are two martingales due to diffusion and
jump (i.e. annhilation), respectively. We can bound the martingales via
quadratic variation: indeed,

$\displaystyle\mathbb{E}\big{[}|M_{T}^{N,\phi,D}|^{2}\big{]}$
$\displaystyle=\mathbb{E}\Big{[}\big{|}\frac{1}{N}\int_{0}^{T}\sum_{i\in\mathcal{N}(t)}\nabla_{v}\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\cdot
dB_{i}(t)\big{|}^{2}\Big{]}$
$\displaystyle=\frac{1}{N^{2}}\mathbb{E}\int_{0}^{T}\sum_{i\in\mathcal{N}(t)}\big{|}\nabla_{v}\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\big{|}^{2}dt$
$\displaystyle\leq\frac{\|\phi\|_{C^{1}}^{2}}{N^{2}}\mathbb{E}\int_{0}^{T}\mathcal{N}(t)dt\leq\frac{\|\phi\|_{C^{1}}^{2}T}{N},$
(10)

and by [5, Proposition 8.7] (the quantity $\alpha$ therein, so-called carré-
du-champ, is the sum over all possible jumps of the particle configuration
$\zeta(t)$, of the squared change of the value of the functional
$F_{1}(t,\zeta(t))$, weighted by the rate $q_{ij}^{N}(t)$ of such a change),
see also [19, Appendix 1, Lemma 5.1],

$\displaystyle\mathbb{E}\big{[}|M_{T}^{N,\phi,J}|^{2}\big{]}$
$\displaystyle\leq\mathbb{E}\int_{0}^{T}\sum_{i\in\mathcal{N}(t)}\sum_{j:j\neq
i}q_{ij}^{N}(t)\left[F_{1}(t,\zeta^{-ij}(t))-F_{1}(t,\zeta(t))\right]^{2}dt$
$\displaystyle\leq\mathbb{E}\int_{0}^{T}\sum_{i\in\mathcal{N}(t)}\sum_{j:j\neq
i}\frac{1}{N}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t)\big{)}\left[\frac{1}{N}\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}+\frac{1}{N}\phi\big{(}t,x_{j}^{N}(t),v_{j}^{N}(t)\big{)}\right]^{2}dt$
$\displaystyle=\frac{\|\phi\|_{L^{\infty}}^{2}}{N^{3}}\mathbb{E}\int_{0}^{T}\sum_{i\in\mathcal{N}(t)}\sum_{j:j\neq
i}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t)\big{)}dt.$

To bound the remaining quantity, applying Itô’s formula to $\mathcal{N}(t)$
and then taking expectation, we obtain (cf. [16, Proof of Lemma 3.1] for a
similar context)

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

hence we see that

$\displaystyle\mathbb{E}\int_{0}^{T}\sum_{i\in\mathcal{N}(t)}\sum_{j:j\neq
i}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t)\big{)}dt\leq\frac{N^{2}}{2},$
(11)

whereby

$\displaystyle\mathbb{E}\big{[}|M_{T}^{\phi,J}|^{2}\big{]}\leq\frac{\|\phi\|_{L^{\infty}}^{2}}{2N}.$
(12)

To handle the nonlinearity, we first fix another test function $\psi(x,v)\in
C_{c}^{\infty}(\mathbb{R}^{2d})$ and consider

$\displaystyle\int_{0}^{T}\frac{1}{N^{2}}\sum_{i\in\mathcal{N}(t)}\sum_{j:j\neq
i}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t)\big{)}\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\psi\big{(}x_{j}^{N}(t),v_{j}^{N}(t)\big{)}\,dt.$
(13)

Let us denote by $K=K^{\phi,\psi}$ the maximal radius of the compact supports
of $\phi$ and $\psi$, and by

$\displaystyle\chi^{K}(\xi):\mathbb{R}^{d}\to[0,1]$ (14)

a compactly-supported function of class $C^{\infty}$ that is identically $1$
for $|\xi|\leq 2K$ and $0$ for $|\xi|\geq 4K$. Then, (13) is equal to

$\displaystyle\int_{0}^{T}\frac{1}{N^{2}}\sum_{i\in\mathcal{N}(t)}\sum_{j:j\neq
i}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t)\big{)}\chi^{K}\big{(}v_{i}^{N}(t)-v_{j}^{N}(t)\big{)}\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\psi\big{(}x_{j}^{N}(t),v_{j}^{N}(t)\big{)}\,dt$
(15)

Indeed, since $v_{i}^{N}(t),v_{j}^{N}(t)$ are in the arguments of $\phi,\psi$,
their moduli are less than $K$, whereby $|v_{i}^{N}(t)-v_{j}^{N}(t)|\leq 2K$
otherwise one of $\phi,\psi$ is zero. Hence, adding the function $\chi^{K}$ to
(13) does not change its value.

In the next two sections we concentrate on proving

###### Proposition 5

For any finite $T$ and $d\geq 1$,

$\displaystyle\lim_{|z|\to
0}\limsup_{N\to\infty}\mathbb{E}\Big{|}\int_{0}^{T}\frac{1}{N^{2}}\sum_{i\in\mathcal{N}(t)}\sum_{j:j\neq
i}$
$\displaystyle\big{[}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t)+z\big{)}-\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t)\big{)}\big{]}$
$\displaystyle\cdot\chi^{K}\big{(}v_{i}^{N}(t)-v_{j}^{N}(t)\big{)}\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\psi\big{(}x_{j}^{N}(t),v_{j}^{N}(t)\big{)}\,dt\Big{|}=0.$

This continuity claim should be seen as a form of local equilibrium (cf. [19])
for our continuum particle system at two macroscopically separated sites (here
$z$ is a macroscopic variable whereas $\epsilon$ is microscopic or
mesoscopic). Since for $\epsilon$ small, the function $\theta^{\epsilon}$ is
very close to a Dirac delta function (hence singular), this continuity claim
is difficult to prove directly. The main bulk of subsequent work is to apply
the so-called Itô-Tanaka trick to (15), so that instead of working with the
function $\theta^{\epsilon}$, we work equivalently with a more regular
function $U^{\epsilon}$ (more regular in the sense not for fixed $\epsilon$
but for $\epsilon$ tending to zero). Roughly speaking, it is given by a
convolution of $\theta^{\epsilon}$ with the Green’s function of a hypoelliptic
operator, see (38). Since the Green’s function is not integrable at infinity
and $\theta^{\epsilon}$ is a function of $x$ only, we seek additional
localization in the variable $v$. This is the reason we introduce the function
$\chi^{K}$, as it will be part of an auxiliary pde (33) we set up for
$U^{\epsilon}$.

###### Remark 6

By the assumption $\theta(0)=0$, we can have a version of Proposition 5 where
the double sum is over all $i,j$ (including repeated indices $i=j$). Indeed,
there are at most $N$ diagonal terms, and after bounding the test functions by
constants, we would like their contributions, which is at most

$\lim_{|z|\to
0}\limsup_{N\to\infty}\frac{C}{N^{2}}N\epsilon^{-d}|\theta(\epsilon^{-1}z)-\theta(0)|,$

to be zero. Recall that at the local scaling, $\epsilon^{-d}=N$ and if
$\theta(0)>0$ the above quantity is not negligible. If $\theta(0)=0$, then
since $\epsilon=\epsilon(N)\to 0$ first and $\theta$ has compact support, we
have $\theta(\epsilon^{-1}z)\to 0$ as $\epsilon\to 0$ for any fixed $z$. The
above quantity is thus $0$.

Given Proposition 5, one can argue as in [16, pp. 42-43] or [12, pp. 631-633]
to derive the following Proposition 7, that is key to showing the convergence
of the nonlinearity. Let $\eta(x):\mathbb{R}^{d}\to\mathbb{R}_{+}$ be given,
nonnegative, of class $C^{\infty}$ with $\int\eta dx=1$, $\eta(x)=\eta(-x)$
and compactly supported in $\mathbb{B}_{E}(0,1)$. Then, for any
$\delta\in(0,1]$, denote by

$\displaystyle\eta^{\delta}(x):=\delta^{-d}\eta(\delta^{-1}x).$

###### Proposition 7

For any finite $T$ and $d\geq 1$,

$\displaystyle\lim_{\delta\to 0}\limsup_{N\to\infty}$
$\displaystyle\,\mathbb{E}\Big{|}\int_{0}^{T}\frac{1}{N^{2}}\sum_{i\in\mathcal{N}(t)}\sum_{j\in\mathcal{N}(t):j\neq
i}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t)\big{)}\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\psi\big{(}x_{j}^{N}(t),v_{j}^{N}(t)\big{)}\,dt$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{R}^{d}}dw\left\langle\eta^{\delta}(w-x_{1})\eta^{\delta}(w-x_{2})\phi(t,w,v_{1})\psi(w,v_{2}),\;\mu_{t}^{N}(dx_{1},dv_{1})\mu_{t}^{N}(dx_{2},dv_{2})\right\rangle
dt\Big{|}=0.$

Proof. We shorten some parts of the proof as it is similar to arguments
detailed in [16, 12, 13]. In the sequel, we use Err$(N,\delta)$ to denote a
stochastic error that can change from line to line and which vanishes in the
following limit

$\displaystyle\lim_{\delta\to
0}\limsup_{N\to\infty}\mathbb{E}|\text{Err}(N,\delta)|=0.$

By the assumption $\theta(0)=0$, Proposition 5 (we can remove the function
$\chi^{K}$ without changing the value, as explained near (15)) and Remark 6,
we firstly have that

$\displaystyle\int_{0}^{T}\frac{1}{N^{2}}\sum_{i\in\mathcal{N}(t)}\sum_{j:j\neq
i}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t)\big{)}\phi\big{(}x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\psi\big{(}x_{j}^{N}(t),v_{j}^{N}(t)\big{)}\,dt$
$\displaystyle=\int_{0}^{T}\frac{1}{N^{2}}\sum_{i,j\in\mathcal{N}(t)}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t)\big{)}\phi\big{(}x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\psi\big{(}x_{j}^{N}(t),v_{j}^{N}(t)\big{)}\,dt$
$\displaystyle=\int_{0}^{T}\iint_{\mathbb{R}^{2d}}\eta^{\delta}(z_{1})\eta^{\delta}(z_{2})dz_{1}dz_{2}\frac{1}{N^{2}}\sum_{i,j\in\mathcal{N}(t)}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t)-z_{1}+z_{2}\big{)}$
$\displaystyle\quad\quad\quad\cdot\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\psi\big{(}x_{j}^{N}(t),v_{j}^{N}(t)\big{)}\,dt+\text{Err}(N,\delta).$

Indeed, any $z_{1},z_{2}$ in the support of $\eta^{\delta}$ in the last line
have modulus less than $\delta$, hence the stochastic error depends only on
$N$ and $\delta$. Next, since $\phi,\psi$ are test functions, shifting their
$x$-slot by $z_{1},z_{2}$ respectively causes a stochastic error on the order
of $\delta$ (here recall (11)), hence the preceding display is equal to

$\displaystyle\int_{0}^{T}\iint_{\mathbb{R}^{2d}}\eta^{\delta}(z_{1})\eta^{\delta}(z_{2})dz_{1}dz_{2}\frac{1}{N^{2}}\sum_{i,j\in\mathcal{N}(t)}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t)-z_{1}+z_{2}\big{)}$
$\displaystyle\quad\quad\quad\cdot\phi\big{(}t,x_{i}^{N}(t)-z_{1},v_{i}^{N}(t)\big{)}\psi\big{(}x_{j}^{N}(t)-z_{2},v_{j}^{N}(t)\big{)}\,dt+\text{Err}(N,\delta).$

By a change of variables $w_{1}=x_{i}^{N}(t)-z_{1}$,
$w_{2}=x_{j}^{N}(t)-z_{2}$, we can rewrite the above as

$\displaystyle\int_{0}^{T}\iint_{\mathbb{R}^{2d}}\frac{1}{N^{2}}\sum_{i,j\in\mathcal{N}(t)}\eta^{\delta}\left(x_{i}^{N}(t)-w_{1}\right)\eta^{\delta}\big{(}x_{j}^{N}(t)-w_{2}\big{)}\theta^{\epsilon}\big{(}w_{1}-w_{2}\big{)}$
$\displaystyle\quad\quad\quad\cdot\phi\big{(}t,w_{1},v_{i}^{N}(t)\big{)}\psi\big{(}w_{2},v_{j}^{N}(t)\big{)}\,dw_{1}dw_{2}dt+\text{Err}(N,\delta).$

Since $|w_{1}-w_{2}|<\epsilon$ in the support of $\theta^{\epsilon}$, we shift
the $w_{2}$ in $\psi\big{(}w_{2},v_{j}^{N}(t)\big{)}$ to $w_{1}$ causing an
error on the order of $\epsilon\delta^{-2d}$. Since in our claim $\epsilon\to
0$ before $\delta\to 0$, this error is negligible and can be combined into
Err$(N,\delta)$. Similar, shifting the $w_{2}$ in the argument of
$\eta^{\delta}\big{(}x_{j}^{N}(t)-w_{2}\big{)}$ to $w_{1}$ causes an error of
order $\epsilon\delta^{-1-2d}$, also negligible. Thus the preceding display is
equal to

$\displaystyle\int_{0}^{T}\iint_{\mathbb{R}^{2d}}\frac{1}{N^{2}}\sum_{i,j\in\mathcal{N}(t)}\eta^{\delta}\left(x_{i}^{N}(t)-w_{1}\right)\eta^{\delta}\big{(}x_{j}^{N}(t)-w_{1}\big{)}\theta^{\epsilon}\big{(}w_{1}-w_{2}\big{)}$
$\displaystyle\quad\quad\quad\cdot\phi\big{(}t,w_{1},v_{i}^{N}(t)\big{)}\psi\big{(}w_{1},v_{j}^{N}(t)\big{)}\,)dw_{1}dw_{2}dt+\text{Err}(N,\delta)$
$\displaystyle=\int_{0}^{T}\int_{\mathbb{R}^{d}}\frac{1}{N^{2}}\sum_{i,j\in\mathcal{N}(t)}\eta^{\delta}\left(x_{i}^{N}(t)-w_{1}\right)\eta^{\delta}\big{(}x_{j}^{N}(t)-w_{1}\big{)}\phi\big{(}t,w_{1},v_{i}^{N}(t)\big{)}\psi\big{(}w_{1},v_{j}^{N}(t)\big{)}\,dw_{1}dt+\text{Err}(N,\delta)$
$\displaystyle=\int_{0}^{T}\int_{\mathbb{R}^{d}}dw_{1}\left\langle\eta^{\delta}(x_{1}-w_{1})\eta^{\delta}(x_{2}-w_{1})\phi(t,w_{1},v_{1})\psi(w_{1},v_{2}),\;\mu_{t}^{N}(dx_{1},dv_{1})\mu_{t}^{N}(dx_{2},dv_{2})\right\rangle
dt+\text{Err}(N,\delta),$

where one of the integral in $dw_{2}$ is removed using the property
$\int\theta^{\epsilon}(w_{1}-w_{2})dw_{2}=1$ for any $w_{1}$. The last
equality is a rewritting in terms of the empirical measure $\mu^{N}_{t}$.

##  3 The Itô-Tanaka trick

In this section, we apply the so-called Itô-Tanaka trick to (15). The trick
itself is a classical one in stochastic analysis, related to Tanaka’s formula
in local time, see e.g. [11, pp. 4] for an account, nevertheless its use in a
particle system context may be quite recent, which we learned from Hammond and
Rezakhanlou [16]. Roughly speaking, its aim is to replace a term of low
regularity equivalently by a series of other terms involving functions of
better regularity, by applying inversely the Itô’s formula.

For the above-fixed test functions $\phi,\psi$, and for any fixed
$z\in\mathbb{R}^{d}$, consider the process

$\displaystyle\mathcal{Y}_{t}:=\frac{1}{N^{2}}\sum_{i\in\mathcal{N}(t)}\sum_{j\in\mathcal{N}(t):\,j\neq
i}H^{\epsilon,z}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t),v_{i}^{N}(t)-v_{j}^{N}(t)\big{)}\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\psi\big{(}x_{j}^{N}(t),v_{j}^{N}(t)\big{)},\quad
t\geq 0$ (16)

which is also a functional of the configuration $\zeta(t)$, where

$\displaystyle H^{\epsilon,z}(x,v):=U^{\epsilon}(x+z,v)-U^{\epsilon}(x,v)$
(17)

and $U^{\epsilon}(x,v):\mathbb{R}^{2d}\to\mathbb{R}$ is a function of class
$C^{\infty}$ to be defined in (38). Applying Itô’s formula to
$\mathcal{Y}_{t}$, we obtain that

$\displaystyle\mathcal{Y}_{T}-\mathcal{Y}_{0}$ (18)
$\displaystyle=\int_{0}^{T}\frac{1}{N^{2}}\sum_{i\in\mathcal{N}(t)}\sum_{j:j\neq
i}\big{(}v_{i}^{N}(t)-v_{j}^{N}(t)\big{)}\cdot\nabla_{x}H^{\epsilon,z}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t),v_{i}^{N}(t)-v_{j}^{N}(t)\big{)}\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\psi\big{(}x_{j}^{N}(t),v_{j}^{N}(t)\big{)}\,dt$
(19)
$\displaystyle+\int_{0}^{T}\frac{1}{N^{2}}\sum_{i\in\mathcal{N}(t)}\sum_{j:j\neq
i}\Delta_{v}H^{\epsilon,z}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t),v_{i}^{N}(t)-v_{j}^{N}(t)\big{)}\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\psi\big{(}x_{j}^{N}(t),v_{j}^{N}(t)\big{)}\,dt$
(20)
$\displaystyle+\int_{0}^{t}\frac{1}{N^{2}}\sum_{i\in\mathcal{N}(s)}\sum_{j:j\neq
i}\partial_{t}\phi\big{(}t,x_{i}^{N}(s),v_{i}^{N}(s)\big{)}\psi\big{(}x_{j}^{N}(s),v_{j}^{N}(s)\big{)}H^{\epsilon,z}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t),v_{i}^{N}(t)-v_{j}^{N}(t)\big{)}\,ds$
(21)
$\displaystyle+\int_{0}^{T}\frac{1}{N^{2}}\sum_{i\in\mathcal{N}(t)}\sum_{j:j\neq
i}\Big{[}v_{i}^{N}(t)\cdot\nabla_{x}\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\psi+v_{j}^{N}(t)\cdot\nabla_{x}\psi\big{(}x_{j}^{N}(t),v_{j}^{N}(t)\big{)}\phi\Big{]}H^{\epsilon,z}\,dt$
(22)
$\displaystyle+\int_{0}^{T}\frac{1}{2N^{2}}\sum_{i\in\mathcal{N}(t)}\sum_{j:j\neq
i}\Big{[}\Delta_{v}\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\psi\big{(}x_{j}^{N}(t),v_{j}^{N}(t)\big{)}+\Delta_{v}\psi\big{(}x_{j}^{N}(t),v_{j}^{N}(t)\big{)}\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\Big{]}H^{\epsilon,z}\,dt$
(23)
$\displaystyle-\int_{0}^{T}\frac{2}{N^{3}}\sum_{i\in\mathcal{N}(t)}\sum_{k:k\neq
i}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{k}^{N}(t)\big{)}\sum_{j:j\neq
i,k}H^{\epsilon,z}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t),v_{i}^{N}(t)-v_{j}^{N}(t)\big{)}\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\psi\big{(}x_{j}^{N}(t),v_{j}^{N}(t)\big{)}\,dt$
(24)
$\displaystyle-\int_{0}^{T}\frac{2}{N^{3}}\sum_{i\in\mathcal{N}(t)}\sum_{k:k\neq
i}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{k}^{N}(t)\big{)}\sum_{j:j\neq
i,k}H^{\epsilon,z}\big{(}x_{j}^{N}(t)-x_{i}^{N}(t),v_{j}^{N}(t)-v_{i}^{N}(t)\big{)}\phi\big{(}t,x_{j}^{N}(t),v_{j}^{N}(t)\big{)}\psi\big{(}x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\,dt$
(25)
$\displaystyle-\int_{0}^{T}\frac{2}{N^{3}}\sum_{i\in\mathcal{N}(t)}\sum_{k:k\neq
i}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{k}^{N}(t)\big{)}H^{\epsilon,z}\big{(}x_{i}^{N}(t)-x_{k}^{N}(t),v_{i}^{N}(t)-v_{k}^{N}(t)\big{)}\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\psi\big{(}x_{k}^{N}(t),v_{k}^{N}(t)\big{)}\,dt$
(26) $\displaystyle+\widetilde{M}_{T}^{D}+\widetilde{M}_{T}^{J},$ (27)

where we omitted some arguments for brevity as they are self-evident.
Specifically, in (24)-(26) we used the fact that if a pair $(i,k)$ of distinct
particles coagulate at time $t$ (which occurs at rate $q_{ik}^{N}(t)$ (9)),
the double sum (16) decreases by an amount of

$\displaystyle\frac{1}{N^{2}}\sum_{j\neq
i,k}H^{\epsilon,z}\left(x_{i}^{N}(t)-x_{j}^{N}(t),v_{i}^{N}(t)-v_{j}^{N}(t)\right)\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\psi\big{(}x_{j}^{N}(t),v_{j}^{N}(t)\big{)}$
$\displaystyle+\frac{1}{N^{2}}\sum_{j\neq
i,k}H^{\epsilon,z}\left(x_{k}^{N}(t)-x_{j}^{N}(t),v_{k}^{N}(t)-v_{j}^{N}(t)\right)\phi\big{(}t,x_{k}^{N}(t),v_{k}^{N}(t)\big{)}\psi\big{(}x_{j}^{N}(t),v_{j}^{N}(t)\big{)}$
$\displaystyle+\frac{1}{N^{2}}\sum_{j\neq
i,k}H^{\epsilon,z}\left(x_{j}^{N}(t)-x_{i}^{N}(t),v_{j}^{N}(t)-v_{i}^{N}(t)\right)\phi\big{(}t,x_{j}^{N}(t),v_{j}^{N}(t)\big{)}\psi\big{(}x_{i}^{N}(t),v_{i}^{N}(t)\big{)}$
$\displaystyle+\frac{1}{N^{2}}\sum_{j\neq
i,k}H^{\epsilon,z}\left(x_{j}^{N}(t)-x_{k}^{N}(t),v_{j}^{N}(t)-v_{k}^{N}(t)\right)\phi\big{(}t,x_{j}^{N}(t),v_{j}^{N}(t)\big{)}\psi\big{(}x_{k}^{N}(t),v_{k}^{N}(t)\big{)}$
$\displaystyle+\frac{1}{N^{2}}H^{\epsilon,z}\left(x_{i}^{N}(t)-x_{k}^{N}(t),v_{i}^{N}(t)-v_{k}^{N}(t)\right)\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\psi\big{(}x_{k}^{N}(t),v_{k}^{N}(t)\big{)}$
$\displaystyle+\frac{1}{N^{2}}H^{\epsilon,z}\left(x_{k}^{N}(t)-x_{i}^{N}(t),v_{k}^{N}(t)-v_{i}^{N}(t)\right)\phi\big{(}t,x_{k}^{N}(t),v_{k}^{N}(t)\big{)}\psi\big{(}x_{i}^{N}(t),v_{i}^{N}(t)\big{)}.$

Since $\theta$ is symmetric, hence
$\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{k}^{N}(t)\big{)}=\theta^{\epsilon}\big{(}x_{k}^{N}(t)-x_{i}^{N}(t)\big{)}$,
after summing over all distinct pairs $(i,k)$ of the above six terms, we can
switch the indices of $i$ and $k$ and regroup them into the claimed three
sums.

The martingale associated with diffusion is

$\displaystyle\widetilde{M}_{T}^{D}:=\frac{1}{N^{2}}\int_{0}^{T}\sum_{i\in\mathcal{N}(t)}\sum_{j:j\neq
i}(\nabla_{v}H^{\epsilon,z}\phi\psi+H^{\epsilon,z}\psi\nabla_{v}\phi)\,dB_{i}(t)+(-\nabla_{v}H^{\epsilon,z}\phi\psi+H^{\epsilon,z}\phi\nabla_{v}\psi)\,dB_{j}(t),$

hence

$\displaystyle\frac{1}{2}\mathbb{E}\big{[}|\widetilde{M}_{T}^{D}|^{2}\big{]}\leq$
$\displaystyle\mathbb{E}\frac{1}{N^{4}}\int_{0}^{T}\sum_{i\in\mathcal{N}(t)}\Big{|}\sum_{j:j\neq
i}(\nabla_{v}H^{\epsilon,z}\phi\psi+H^{\epsilon,z}\psi\nabla_{v}\phi)\Big{|}^{2}dt$
(28)
$\displaystyle+\mathbb{E}\frac{1}{N^{4}}\int_{0}^{T}\sum_{j\in\mathcal{N}(t)}\Big{|}\sum_{i:i\neq
j}(-\nabla_{v}H^{\epsilon,z}\phi\psi+H^{\epsilon,z}\phi\nabla_{v}\psi)\Big{|}^{2}dt.$
(29)

The martingale associated with jumps can be bounded via

$\displaystyle\frac{1}{4}\mathbb{E}\big{[}|\widetilde{M}_{T}^{J}|^{2}\big{]}$
$\displaystyle\leq\mathbb{E}\int_{0}^{T}\sum_{i\in\mathcal{N}(t)}\sum_{k:k\neq
i}\frac{1}{N}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{k}^{N}(t)\big{)}\Big{[}\frac{1}{N^{2}}\sum_{j:j\neq
i,k}H^{\epsilon,z}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t),v_{i}^{N}(t)-v_{j}^{N}(t)\big{)}\phi\psi\Big{]}^{2}dt$
(30)
$\displaystyle+\mathbb{E}\int_{0}^{T}\sum_{i\in\mathcal{N}(t)}\sum_{k:k\neq
i}\frac{1}{N}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{k}^{N}(t)\big{)}\Big{[}\frac{1}{N^{2}}\sum_{j:j\neq
i,k}H^{\epsilon,z}\big{(}x_{j}^{N}(t)-x_{i}^{N}(t),v_{j}^{N}(t)-v_{i}^{N}(t)\big{)}\phi\psi\Big{]}^{2}dt$
(31)
$\displaystyle+\mathbb{E}\int_{0}^{T}\sum_{i\in\mathcal{N}(t)}\sum_{k:k\neq
i}\frac{1}{N}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{k}^{N}(t)\big{)}\Big{[}\frac{1}{N^{2}}H^{\epsilon,z}\big{(}x_{i}^{N}(t)-x_{k}^{N}(t),v_{i}^{N}(t)-v_{k}^{N}(t)\big{)}\phi\psi\Big{]}^{2}dt,$
(32)

cf. [5, Proposition 8.7].

The only term with second derivatives of $H^{\epsilon,z}$ from the above
development is (20). In order to substitute the highest derivative (also
together with (19)), we set up the following subelliptic equation for
$U^{\epsilon}:\mathbb{R}^{2d}\to\mathbb{R}$:

$\displaystyle
v\cdot\nabla_{x}U^{\epsilon}(x,v)+\Delta_{v}U^{\epsilon}(x,v)=-\theta^{\epsilon}(x)\chi^{K}(v).$
(33)

We will choose in (38) a particular solution of (33), which is the convolution
of the rhs with the subelliptic Green’s function. Recall in (17) we have
defined

$H^{\epsilon,z}(x,v):=U^{\epsilon}(x+z,v)-U^{\epsilon}(x,v).$

(33) yields, by linearity, the equation satisfied by
$H^{\epsilon,z}:\mathbb{R}^{2d}\to\mathbb{R}$, for any fixed $z$:

$\displaystyle
v\cdot\nabla_{x}H^{\epsilon,z}(x,v)+\Delta_{v}H^{\epsilon,z}(x,v)=\big{[}\theta^{\epsilon}(x)-\theta^{\epsilon}(x+z)\big{]}\chi^{K}(v).$
(34)

Consequently, we can substitute (19)-(20) as follows

$\displaystyle\eqref{2nd-order-1}+\eqref{2nd-order-2}=$
$\displaystyle\int_{0}^{T}\frac{1}{N^{2}}\sum_{i\in\mathcal{N}(t)}\sum_{j:j\neq
i}$
$\displaystyle\cdot\big{[}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t)\big{)}-\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t)+z\big{)}\big{]}\chi^{K}\big{(}v_{i}^{N}(t)-v_{j}^{N}(t)\big{)}\phi\big{(}x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\psi\big{(}x_{j}^{N}(t),v_{j}^{N}(t)\big{)}.$

The rhs is precisely the expression appearing in Proposition 5.

We remark at this stage that $\nabla_{x}H^{\epsilon,z}$ does not appear any
more in the rest of terms coming from the Itô’s formula applied to
$\mathcal{Y}_{t}$, after it is absorbed into the operator on the lhs of (34).
Its estimate is expected to be bad due to the degeneracy of the subellitic
operator in the direction of $x$. What do appear in the rest of the terms are
$\nabla_{v}H^{\epsilon,z}$ and $H^{\epsilon,z}$ itself, whose estimates are
nice enough. We will show in the next sections that all the other terms (18),
(21)-(27) (i.e. apart from (19)-(20)) are negligible in $L^{1}(\Omega)$ in the
order of limits $N\to\infty$ followed by $|z|\to 0$, as required for
Proposition 5.

We can thus summarise the use of Itô-Tanaka trick (33) in our context as
follows. It allows to transfer proving a continuity claim Proposition 5
involving a nearly singular function $\theta^{\epsilon}$ (for $\epsilon$
small), into proving similar claims involving more regular functions
$\nabla_{v}H^{\epsilon,z}$ and $H^{\epsilon,z}$. The estimates for the latter
can be equivalently obtained from estimates for $U^{\epsilon}$, which is a
convolution of $\theta^{\epsilon}(x)\chi^{K}(v)$ with a subelliptic Green’s
function. This convolution is a sort of regularization. Indeed, in the limit
$\epsilon\to 0$, $U^{\epsilon}$ does not tend to a singular object, but rather
to a pointwise evaluation (in $x$) of the Green’s function (suitably
integrated in $v$). This is the reason why pointwise estimates and gradient
estimates (in $v$) of the associated Green’s function are key to our analysis.

##  4 The hypoelliptic Green’s function

Denote the operator

$\displaystyle L:=v\cdot\nabla_{x}+\Delta_{v}$ (35)

acting on appropriate functions on $\mathbb{R}^{2d}$, and we rewrite the pde
(33) more compactly as

$\displaystyle LU^{\epsilon}=-\theta^{\epsilon}(x)\chi^{K}(v).$ (36)

It is well-known that $L$ (35) is an hypoelliptic operator in the sense of
Hörmander [17], and its parabolic version (i.e. $\partial_{t}-L$) was earlier
studied by Kolmogorov [20] who wrote down its kernel explicitly. cf. [1] for a
recent survey. Crucial for our purpose are pointwise estimates for the Green’s
function of $-L$.

To briefly recall that $L$ is hypoelliptic, let us write

$\displaystyle L=A_{0}+\sum_{i=1}^{d}A_{i}^{2}$ (37)

where

$\displaystyle A_{i}=\partial_{v_{i}},\quad i=1,...,d;\quad
A_{0}=\sum_{j=1}^{d}v_{j}\partial_{x_{j}}.$

Since the commutators

$\displaystyle[A_{0},A_{i}]=\sum_{j=1}^{d}v_{j}\partial_{x_{j}}(\partial_{v_{i}})-\partial_{v_{i}}(\sum_{j=1}^{d}v_{j}\partial_{x_{j}})=-\partial_{x_{i}},\quad
i=1,...,d$

we see that at every point in $\mathbb{R}^{2d}$,

$\displaystyle\text{span}\big{\\{}A_{i},[A_{0},A_{i}],\;i=1,...,d\big{\\}}=\mathbb{R}^{2d}.$

Thus $L$ satisfies Hörmander’s condition [17, Theorem 1.1], [24, pp. 128],
implying that it is an hypoelliptic operator; that is, if $u$ is a
distribution such that $Lu$ is $C^{\infty}$, then $u$ must be $C^{\infty}$ as
well, cf. [17, pp. 147], [24, pp. 129]. Since the rhs of (36) is of class
$C^{\infty}$ with compact support (for every fixed $\epsilon>0$), we can find
a solution of the form

$\displaystyle
U^{\epsilon}(x,v)=\iint_{\mathbb{R}^{2d}}G\big{(}(x,v),(y,w)\big{)}\theta^{\epsilon}(y)\chi^{K}(w)dydw,$
(38)

where $G=G_{L}$ is a distribution of class $C^{\infty}$ away from the
diagonal, called the Green’s function of $-L$, see e.g. [30, pp. 143], [10,
pp. 247]. The definition of hypoellipticity implies that $U^{\epsilon}$ is of
class $C^{\infty}$ for every fixed $\epsilon$; in particular, this is the
function that is used in Section 3 to define $\mathcal{Y}_{t}$.

To state the classical results of [23, 30, 10] on the estimates for the
Green’s function of general hypoelliptic operators, we recall that they are
expressed in terms of a control metric (or Carnot-Carathéodory metric)
$\rho_{L}(X,Y)$ canonically attached to $L$ (to avoid confusion, we use
capital letters $X=(x,v),Y=(y,w),...$ to denote variables in
$\mathbb{R}^{2d}$). We also set here

$\displaystyle D:=2d.$

Let (in our case) $q=1+2d$ and $\\{V_{j},j=1,...,q\\}$ are an enumeration of

$\displaystyle V_{1}:=A_{0},\;V_{i+1}:=A_{i},\;V_{i+d+1}:=[A_{0},A_{i}],\quad
i=1,...,d.$

As explained in [23, pp.107], [29, Section 18], for operators of the form (37)
which are not just a sum-of-squares but with $A_{0}$, the degree of $A_{0}$
counts $2$ while the degree of each $A_{i}$ counts $1$, and likewise for their
commutators. Hence, in our case we have that

$\displaystyle d_{1}=2,\;d_{i+1}=1,\;d_{i+d+1}=3,\quad i=1,...,d,$

where $d_{j}$ is the degree of $V_{j}$.

######  Definition 8 ([23, Definition 1.1])

Let $C(\delta)$ denote the class of absolutely continuous mappings
$\phi:[0,1]\to\mathbb{R}^{D}$ which almost everywhere satisfy the differential
equation

$\displaystyle\phi^{\prime}(t)=\sum_{j=1}^{q}a_{j}(t)V_{j}(\phi(t))$ (39)

with $|a_{j}(t)|\leq\delta^{d_{j}}$. Then define

$\displaystyle\rho_{L}(X,Y):=\inf\big{\\{}\delta>0:\;\exists\phi\in
C(\delta),\phi(0)=X,\phi(1)=Y\big{\\}}.$

###### Remark 9

The geometric meaning of this definition is as follows. The ball of radius
$\delta$ around the point $X$ in the control metric $\rho_{L}$ consists of
those points $Y$ that can be reached at time $t=1$ by a “particle” (whose
trajectory is denoted $\phi(t)$, $t\in[0,1]$), that starts at $t=0$ from $X$
and travels with velocity the rhs of (39). The velocity at time $t$ has
direction controlled by the vector fields $V_{j}$ evaluated at “particle”
position, and modulus controlled together by the strength of the vector fields
and the scalar multipliers $a_{j}(t)$.

It is known that $\rho_{L}$ is a metric and in our case, it compares with the
Euclidean distance $|\cdot|$ in the following way [23, Proposition 1.1]: for
any compact set $\mathbb{K}\subset\mathbb{R}^{D}$, there are constants
$C_{i}=C_{i}(\mathbb{K},d)$, $i=1,2$, such that for any $X,Y\in\mathbb{K}$ we
have that

$\displaystyle C_{1}|X-Y|\leq\rho_{L}(X,Y)\leq C_{2}|X-Y|^{1/3}.$ (40)

Given the control metric $\rho_{L}$ we define balls in this metric of radius
$\delta>0$ by

$\displaystyle\mathbb{B}_{L}(X,\delta):=\big{\\{}Y\in\mathbb{R}^{D}:\rho_{L}(X,Y)\leq\delta\big{\\}}.$

These balls are locally volume doubling as proved in [23].

######  Theorem 10 ([23, Theorem 1, pp. 110])

For every compact set $\mathbb{K}\subset\mathbb{R}^{D}$, there exist constants
$C_{1},C_{2}$ so that for all $X\in\mathbb{K}$,

$\displaystyle
0<C_{1}\leq\frac{|\mathbb{B}_{L}(X,\delta)|}{\Lambda(X,\delta)}\leq
C_{2}<\infty,$

where

$\displaystyle\Lambda(X,\delta):=\sum_{1\leq i_{1},...,i_{D}\leq
q}\big{|}\text{det}(V_{i_{1}},...,V_{i_{D}})(X)\big{|}\delta^{d_{i_{1}}+...+d_{i_{D}}}.$

Here, the differential operators $V_{i}$, $i=1,...,q$, are also viewed as
vector fields, that is,

$V_{1}(X)=(v_{1},...,v_{d},0,...0),\quad V_{i+1}(X)=e_{i+d},\quad
V_{i+d+1}(X)=-e_{i},\quad i=1,...,d,\,X:=(x_{1},...,x_{d},v_{1},...,v_{d}).$

where $\\{e_{i},i=1,...,D\\}$ is the canonical basis of $\mathbb{R}^{D}$. Thus
the above determinant for a $D\times D$ matrix. In our case, it is
straightforward to check that we have

$\displaystyle\Lambda(X,\delta)=\sum_{i=1}^{d}|v_{i}|\delta^{4d-1}+\delta^{4d},\quad
X:=(x_{1},...,x_{d},v_{1},...,v_{d}).$ (41)

It is then clear that

$\displaystyle\frac{|\mathbb{B}_{L}(X,2\delta)|}{|\mathbb{B}_{L}(X,\delta)|}\leq\frac{C_{2}\Lambda(X,2\delta)}{C_{1}\Lambda(X,\delta)}\leq\frac{C_{2}}{C_{1}}2^{4d},$
(42)

where the rhs is uniform for all $X\in\mathbb{K}$.

We can now state the pointwise estimates on Green’s function and its partial
derivatives in $v$, in the case of our operator $L$. As already remarked
before, we do not need estimates for partial derivatives in $x$, which are
worse.

######  Theorem 11 ([23, Corollary, pp. 114], see also [10, Theorem 1])

Suppose that $D=2d\geq 2$. For any compact set
$\mathbb{K}\subset\mathbb{R}^{D}$, there exist some finite constants
$C_{i}=C_{i}(\mathbb{K},d)$, $i=1,2$, such that for any
$X=(x,v),Y=(y,w)\in\mathbb{K}$, we have that

$\displaystyle|G(X,Y)|$ $\displaystyle\leq
C_{1}\frac{\rho_{L}(X,Y)^{2}}{|\mathbb{B}_{L}\big{(}X,\rho_{L}(X,Y)\big{)}|},$
(43) $\displaystyle|\nabla_{v}G(X,Y)|$ $\displaystyle\leq
C_{2}\frac{\rho_{L}(X,Y)}{|\mathbb{B}_{L}\big{(}X,\rho_{L}(X,Y)\big{)}|}.$
(44)

We check that (43) and (44) are sufficiently integrable near its singularity.

###### Proposition 12

Suppose that $D=2d\geq 2$. There exists some $\eta=\eta(d)>0$ such that for
any compact set $\mathbb{K}\subset\mathbb{R}^{D}$, and any
$X=(x,v)\in\mathbb{K}$, $\delta\in(0,\infty)$, there are finite constants
$C_{i}(\delta,\mathbb{K},d)$, $i=1,2$ such that

$\displaystyle\int_{\mathbb{B}_{L}(X,\delta)}|G(X,Y)|^{1+\eta}dY$
$\displaystyle\leq C_{1}$ (45)
$\displaystyle\int_{\mathbb{B}_{L}(X,\delta)}|\nabla_{v}G(X,Y)|^{1+\eta}dY$
$\displaystyle\leq C_{2}.$ (46)

Proof. We prove (46) while the proof of (45) is similar. For
$X\in\mathbb{K}\subset\mathbb{R}^{D}$ compact and
$Y\in\mathbb{B}_{L}(X,\delta)$ with $\delta\in(0,\infty)$, they both belong to
some other compact set $\mathbb{K}^{\prime}$ that depends only on
$\mathbb{K},\delta$. This fact can be inferred from the meaning of the control
metric Definition 8 in the case of our operator $L$. By (44), there exists
some $C=C(\mathbb{K}^{\prime},d)$ such that for any $\eta>0$,

$\displaystyle\int_{\mathbb{B}_{L}(X,\delta)}|\nabla_{v}G(X,Y)|^{1+\eta}dY$
$\displaystyle\leq
C\int_{\mathbb{B}_{L}(X,\delta)}\frac{\rho_{L}(X,Y)^{1+\eta}}{|\mathbb{B}_{L}\big{(}X,\rho_{L}(X,Y)\big{)}|^{1+\eta}}dY$
$\displaystyle=C\sum_{k=0}^{\infty}\int_{\mathbb{B}_{L}(X,\delta
2^{-k})\backslash\mathbb{B}_{L}(X,\delta
2^{-k-1})}\frac{\rho_{L}(X,Y)^{1+\eta}}{|\mathbb{B}_{L}\big{(}X,\rho_{L}(X,Y)\big{)}|^{1+\eta}}dY$
$\displaystyle\leq C\sum_{k=0}^{\infty}\frac{(\delta
2^{-k})^{1+\eta}}{|\mathbb{B}_{L}\big{(}X,\delta
2^{-k-1}\big{)}|^{1+\eta}}|\mathbb{B}_{L}(X,\delta
2^{-k})\backslash\mathbb{B}_{L}(X,\delta 2^{-k-1})|$ $\displaystyle\leq
C\sum_{k=0}^{\infty}\frac{(\delta
2^{-k})^{1+\eta}}{|\mathbb{B}_{L}\big{(}X,\delta
2^{-k-1}\big{)}|^{\eta}}\frac{|\mathbb{B}_{L}(X,\delta
2^{-k}))|}{|\mathbb{B}_{L}\big{(}X,\delta 2^{-k-1}\big{)}|}.$

By (41),

$\displaystyle|\mathbb{B}_{L}\big{(}X,\delta 2^{-k-1}\big{)}|\geq
C_{1}(\mathbb{K})\Lambda(X,\delta 2^{-k-1})\geq C_{1}(\delta
2^{-(k+1)})^{4d}.$

Combined with the local volume doubling property (42) of
$\mathbb{B}_{L}(\cdot)$, we conclude that

$\displaystyle\int_{\mathbb{B}_{L}(X,\delta)}|\nabla_{v}G(X,Y)|^{1+\eta}dY\leq
C^{\prime}\sum_{k=0}^{\infty}2^{-k(1+\eta-4d\eta)},$

for some finite constant $C^{\prime}=C^{\prime}(\mathbb{K},d,\delta)$.
Choosing $\eta>0$ small enough such that $1+\eta-4d\eta>1/2$, the sum on the
rhs is convergent.

Another remark is that in the case of our operator $-L$ (35), its Green’s
function is translation invariant in $x$ (but not in $v$). That is, we claim
that

$\displaystyle G\big{(}(x,v),(y,w)\big{)}=G\big{(}(0,v),(y-x,w)\big{)}.$ (47)

This can be seen starting from the explicit kernel
$\mathsf{P}_{t}\big{(}(x,v),(y,w)\big{)}$ associated with the parabolic
operator $\partial_{t}-L$. Since it is probabilistically linked to Gaussian
processes, it can be explicitly calculated that, cf. [20], [9, Eq.
(3.24)-(3.25)], [15, Eq. (1.4)],

$\displaystyle\mathsf{P}_{t}\big{(}(x,v),(y,w)\big{)}$
$\displaystyle=\left(\frac{\sqrt{3}}{2\pi
t^{2}}\right)^{d}\exp\Big{\\{}-\frac{|v-w|^{2}}{t}-3\frac{(v-w)\cdot(x-y-
tv)}{t^{2}}-3\frac{|x-y-tv|^{2}}{t^{3}}\Big{\\}}$
$\displaystyle=\left(\frac{\sqrt{3}}{2\pi
t^{2}}\right)^{d}\exp\Big{\\{}-3\big{|}\frac{v-w}{2t^{1/2}}+\frac{x-y-
tv}{t^{3/2}}\big{|}^{2}-\frac{|v-w|^{2}}{4t}\Big{\\}}.$ (48)

which depends on $x,y$ through $y-x$ only. Since the Green’s function is the
time-integral of the parabolic kernel (cf. [35, pp. 109], [2, pp. 220], [31,
pp. 276]):

$\displaystyle
G\big{(}(x,v),(y,w)\big{)}=\int_{0}^{\infty}\mathsf{P}_{t}\big{(}(x,v),(y,w)\big{)}\,dt,$

the same translation invariance with respect to $x$ transfers to $G$ and thus
we have (47).

##  5 The control of negligible terms

We proceed to bound all the terms coming out of Itô’s formula applied to
$\mathcal{Y}_{t}$ in Section 3, that are expected to be negligible, thereby
completing the proof of Proposition 5. By a slight abuse of notation, we
define on the same probability space $(\Omega,\mathcal{F},\mathbb{P})$

$\displaystyle
X_{i}(t):=(x_{i}(t),v_{i}(t)):=\left(x_{i}(0)+v_{i}(0)t+\int_{0}^{t}B_{i}(s)ds,\,v_{i}(0)+B_{i}(t)\right),\quad
i\in\mathbb{N},\;t\geq 0,$ (49)

a countable collection of independent particles, with the same Brownian
motions $\\{B_{i}(t)\\}_{i=1}^{\infty}$ as (1) and same initial condition for
$(x_{i}(0),v_{i}(0))_{i=1}^{\infty}$ with density $f_{0}(x,v)$ satisfying
Condition 1. The only difference with (1) is that the particles in (49) are
non-interacting (i.e. they do not annihilate), hence they are independent,
each with infinite lifespan. Further, there is an obvious a.s. domination
between (1) and (49) such that whenever $(x_{i}^{N}(t),v_{i}^{N}(t))$ is
active, its trajectory coincides with that of $(x_{i}(t),v_{i}(t))$.

###### Lemma 13

Let $d\geq 1$. For any $t\geq 0$, the density of $X_{1}(t)-X_{2}(t)$ is
uniformly bounded by $\Gamma$, the uniform bound on $f_{0}$ in Condition 1.
Further, the density $\mathsf{p}(t,x,v)$ of one particle $X_{1}(t)$ satisfies
for every $t\in[0,T]$, $x,v\in\mathbb{R}^{d}$, the bound

$\displaystyle\mathsf{p}(t,x,v)\leq Ce^{-|v|/C}1_{\\{|v|\geq C\\}}+\Gamma
1_{\\{|v|<C\\}},$ (50)

for some constant $C=C(R,\Gamma,T,d)$, where $R$ is the maximal radius of the
compact support of $f_{0}$ in Condition 1.

Proof. The Markov generator of a single-particle process $X_{1}(t)$ is

$L:=v\cdot\nabla_{x}+\frac{1}{2}\Delta_{v}.$

(Here we abuse notation and use the same letter $L$ even though the operator
(35) discussed in Section 4 differs by $1/2$ in front of the Laplacian.) The
hypoellipticity of $L$ implies that $X_{1}(t)$, $t>0$, has a smooth density
$\mathsf{p}(t,x,v)$ with respect to Lebesgue measure of $\mathbb{R}^{2d}$ [24,
Theorems 2.3.2, 2.3.3], and it is the unique solution of the Fokker-Planck
equation

$\displaystyle\partial_{t}\mathsf{p}(t,x,v)$
$\displaystyle=L^{*}\mathsf{p}(t,x,v)$ (51) $\displaystyle\mathsf{p}|_{t=0}$
$\displaystyle=f_{0}(x,v),$

where $L^{*}$ is the adjoint of $L$

$\displaystyle L^{*}=-v\cdot\nabla_{x}+\frac{1}{2}\Delta_{v}.$

One can write $\mathsf{p}(t,x,v)$ explicitly as

$\mathsf{p}(t,x,v)=\iint
f_{0}(x_{0},v_{0})\mathsf{P}^{\star}_{t}\big{(}(x_{0},v_{0}),(x,v)\big{)}dx_{0}dv_{0},$

where the kernel $\mathsf{P}^{\star}_{t}$ is given by a formula similar to (4)
but with certain sign and constants changes:

$\displaystyle\mathsf{P}^{\star}_{t}\big{(}(x_{0},v_{0}),(x,v)\big{)}:=\left(\frac{\sqrt{3}}{\pi
t^{2}}\right)^{d}\exp\Big{\\{}-6\big{|}\frac{v-v_{0}}{2t^{1/2}}+\frac{x-x_{0}-tv_{0}}{t^{3/2}}\big{|}^{2}-\frac{|v-v_{0}|^{2}}{2t}\Big{\\}}.$
(52)

It then follows (e.g. from probabilistic representation of the solution of
(51), cf. [27, pp. 254, Eq. (21)], [15, Lemma 2.4 (ii)]) that

$\displaystyle\left\|\mathsf{p}(t,\cdot)\right\|_{L^{\infty}}\leq\|f_{0}\|_{L^{\infty}}\leq\Gamma.$

Since the density of $X_{1}(t)-X_{2}(t)$ is a convolution between the
respective densities of $X_{1}(t)$ and $-X_{2}(t)$, the two being independent,
we conclude that it is also uniformly bounded by $\Gamma$.

Turning to a finer bound on $\mathsf{p}(t,x,v)$ when $|v|$ is large compared
to $T$ and $R$, we note that by (52),

$\displaystyle\mathsf{p}(t,x,v)$
$\displaystyle=\iint_{\mathbb{R}^{2d}}f_{0}(x_{0},v_{0})\mathsf{P}^{\star}_{t}\big{(}(x_{0},v_{0}),(x,v)\big{)}dx_{0}dv_{0}$
$\displaystyle\leq C(\Gamma)\iint_{|x_{0}|,|v_{0}|\leq
R}\frac{1}{t^{2d}}\exp\left\\{-\frac{|v-v_{0}|^{2}}{4t}\right\\}dx_{0}dv_{0}.$

If $|v_{0}|\leq R$ and

$\displaystyle|v|\geq 2R+T\geq t,$

then $|v-v_{0}|\geq|v|/2$ and $\lambda:=|v|^{2}/t\geq|v|$. Hence, for any such
$v$ we have that

$\displaystyle\mathsf{p}(t,x,v)$ $\displaystyle\leq
C(\Gamma)R^{2d}\frac{|v|^{4d}}{t^{2d}}e^{-\frac{|v|^{2}}{16t}}|v|^{-4d}$
$\displaystyle\leq C(\Gamma,R,T,d)\lambda^{2d}e^{-\lambda/16}.$

Since $\lambda\mapsto\lambda^{2d}e^{-\lambda/16}$ is decreasing for
$\lambda\geq|v|\geq\lambda_{0}$ for some $\lambda_{0}(d)<\infty$, we get the
bound

$\displaystyle\mathsf{p}(t,x,v)\leq C|v|^{2d}e^{-|v|/C},$

for any $|v|\geq C$, $x\in\mathbb{R}^{d}$ and $t\in[0,T]$, where
$C=C(\Gamma,R,T,d)$ is a finite constant. Further adjusting the constant $C$
to absorb the polynomial factor, we get (50).

The following proposition bounds (22). We note at this stage that since the
independent system (49) dominates our true system (1) in terms of lifespan, it
suffices to prove all the following propositions for (49).

###### Proposition 14

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

$\displaystyle\lim_{|z|\to
0}\limsup_{N\to\infty}\mathbb{E}\int_{0}^{T}dt\frac{1}{N^{2}}\sum_{i\neq
j=1}^{N}\Big{|}v_{i}(t)\cdot\nabla_{x}\phi\big{(}t,x_{i}(t),v_{i}(t)\big{)}\psi\big{(}x_{j}(t),v_{j}(t)\big{)}H^{\epsilon,z}\big{(}x_{i}(t)-x_{j}(t),v_{i}(t)-v_{j}(t)\big{)}\Big{|}=0.$

Proof. Recall that we have denoted by $K=K^{\phi,\psi}$ the maximal radius of
the compact supports of $\phi,\psi$, and defined a compactly-supported
function $\chi^{K}:\mathbb{R}^{d}\to[0,1]$ (14). For the quantity in the
thesis, it suffices to consider

$\displaystyle|x_{i}(t)|,|x_{j}(t)|,|v_{i}(t)|,|v_{j}(t)|\leq K$

(otherwise one of the arguments of $\phi$ and $\psi$ is zero). When this is
the case, we have that

$\displaystyle|x_{i}(t)-x_{j}(t)|,|v_{i}(t)-v_{j}(t)|\leq 2K.$

We control each term of the double sum in the thesis, and by exchangeability
it suffices to focus on the term containing the particles with index $1$ and
$2$:

$\displaystyle\mathbb{E}\int_{0}^{T}dt\Big{|}v_{1}(t)\cdot\nabla_{x}\phi\big{(}t,x_{1}(t),v_{1}(t)\big{)}\psi\big{(}x_{2}(t),v_{2}(t)\big{)}H^{\epsilon,z}\big{(}x_{1}(t)-x_{2}(t),v_{1}(t)-v_{2}(t)\big{)}\Big{|}$
$\displaystyle\leq
K\|\phi\|_{C^{1}}\|\psi\|_{L^{\infty}}\mathbb{E}\int_{0}^{T}dt\big{|}H^{\epsilon,z}\big{(}x_{1}(t)-x_{2}(t),v_{1}(t)-v_{2}(t)\big{)}\big{|}\chi^{K}\big{(}x_{1}(t)-x_{2}(t)\big{)}\chi^{K}\big{(}v_{1}(t)-v_{2}(t)\big{)}.$

Denote by $\mathsf{p}^{12}(t,x,v)$ the density of
$\big{(}x_{1}(t)-x_{2}(t),v_{1}(t)-v_{2}(t)\big{)}$, which by Lemma 13 is
uniformly bounded by $\Gamma$. We can write the expectation above explicitly
as

$\displaystyle\int_{0}^{T}dt\iint_{\mathbb{R}^{2d}}|H^{\epsilon,z}(x,v)|\chi^{K}(x)\chi^{K}(v)\mathsf{p}^{12}(t,x,v)dxdv.$
(53)

By (38), (47), a change of variables $\widetilde{y}=y-x$ and the symmetry of
$\theta^{\epsilon}$, we have that

$\displaystyle H^{\epsilon,z}(x,v)$
$\displaystyle=\iint_{\mathbb{R}^{2d}}G\big{(}(x,v),(y,w)\big{)}\big{[}\theta^{\epsilon}(y)-\theta^{\epsilon}(y+z)\big{]}\chi^{K}(w)dydw$
$\displaystyle=\iint_{\mathbb{R}^{2d}}G\big{(}(0,v),(y-x,w)\big{)}\big{[}\theta^{\epsilon}(y)-\theta^{\epsilon}(y+z)\big{]}\chi^{K}(w)dydw$
$\displaystyle=\iint_{\mathbb{R}^{2d}}G\big{(}(0,v),(\widetilde{y},w)\big{)}\big{[}\theta^{\epsilon}(\widetilde{y}+x)-\theta^{\epsilon}(\widetilde{y}+x+z)\big{]}\chi^{K}(w)d\widetilde{y}dw$
$\displaystyle=\iint_{\mathbb{R}^{2d}}G\big{(}(0,v),(\widetilde{y},w)\big{)}\big{[}\theta^{\epsilon}(-\widetilde{y}-x)-\theta^{\epsilon}(-\widetilde{y}-x-z)\big{]}\chi^{K}(w)d\widetilde{y}dw$
$\displaystyle=\iint_{\mathbb{R}^{2d}}\big{[}G\big{(}(0,v),(\widetilde{y},w)\big{)}-G\big{(}(0,v),(\widetilde{y}-z,w)\big{)}\big{]}\theta^{\epsilon}(-\widetilde{y}-x)\chi^{K}(w)d\widetilde{y}dw.$
(54)

It yields that

$\displaystyle\iint_{\mathbb{R}^{2d}}|H^{\epsilon,z}(x,v)|\chi^{K}(x)\chi^{K}(v)\mathsf{p}^{12}(t,x,v)dxdv$
$\displaystyle\leq\iiint_{\mathbb{R}^{3d}}\big{|}G\big{(}(0,v),(\widetilde{y},w)\big{)}-G\big{(}(0,v),(\widetilde{y}-z,w)\big{)}\big{|}\chi^{K}(w)\Big{(}\int_{\mathbb{R}^{d}}\theta^{\epsilon}(-\widetilde{y}-x)\chi^{K}(x)\chi^{K}(v)\mathsf{p}^{12}(t,x,v)dx\Big{)}d\widetilde{y}dwdv$
$\displaystyle\overset{\epsilon\to
0}{\to}\iiint_{\mathbb{R}^{3d}}\big{|}G\big{(}(0,v),(\widetilde{y},w)\big{)}-G\big{(}(0,v),(\widetilde{y}-z,w)\big{)}\big{|}\chi^{K}(w)\chi^{K}(-\widetilde{y})\chi^{K}(v)\mathsf{p}^{12}(t,-\widetilde{y},v)d\widetilde{y}dwdv.$

Since the rhs no longer depends on $\epsilon$, and also $\mathsf{p}^{12}$ is
bounded uniformly by $\Gamma$, it suffices to prove that the following
converges to $0$ as $|z|\to 0$:

$\displaystyle\iiint_{\mathbb{R}^{3d}}\big{|}G\big{(}(0,v),(\widetilde{y},w)\big{)}-G\big{(}(0,v),(\widetilde{y}-z,w)\big{)}\big{|}\chi^{K}(w)\chi^{K}(-\widetilde{y})\chi^{K}(v)d\widetilde{y}dwdv.$

By the continuity of $G$, the integrand converges to $0$ as $|z|\to 0$
Lebesgue a.e. Further, due to the compact support of $w,\widetilde{y},v$ we
can apply the finite measure case of Vitali’s convergence theorem, if we can
check uniform integrability. By the de la Vallée-Poussin theorem it is enough
to check that there exists some $\eta=\eta(d)>0$ and $C=C(K,d)$ finite such
that

$\displaystyle\sup_{|z|\leq
1}\iiint_{\mathbb{R}^{3d}}\big{|}G\big{(}(0,v),(\widetilde{y}-z,w)\big{)}\big{|}^{1+\eta}\chi^{K}(w)\chi^{K}(-\widetilde{y})\chi^{K}(v)d\widetilde{y}dwdv<C.$
(55)

Note that the support of $(0,v)$ is in a compact set, the Euclidean ball
$\mathbb{B}_{E}((0,0),2K)$ in $\mathbb{R}^{2d}$. By (40) there exists a ball
in the metric $\rho_{L}$ of some radius $K^{\prime}$ (depending only on $K$
and $d$) such that for any $|z|\leq 1$ and $-\widetilde{y},w$ in the compact
support of $\chi^{K}$,

$\displaystyle(\widetilde{y}-z,w)\in\mathbb{B}_{E}\big{(}(0,v),16K\big{)}\cap\mathbb{B}_{E}\big{(}(0,0),32K\big{)}\subset\mathbb{B}_{L}\big{(}(0,v),K^{\prime}\big{)}.$

Hence, we are in the setting of Proposition 12. By (45) there exists some
$\eta=\eta(d)>0$ and constant $C=C(K,K^{\prime},d)$ finite such that for any
fixed $v$ with $|v|\leq 2K$ we have that

$\displaystyle\iint_{\mathbb{B}_{L}\left((0,v),K^{\prime}\right)}\big{|}G\big{(}(0,v),(\widetilde{y}-z,w)\big{)}\big{|}^{1+\eta}d\widetilde{y}dw\leq
C.$

Integrating further in $|v|\leq 2K$ yields a different constant which proves
(55), hence the whole thesis.

###### Remark 15

An analogous proof as in Proposition 14 can treat (18), (21) and (23) and
prove their negligibility. Indeed, they are all of the form of an averaged (by
$N^{-2}$) double sum over distinct pairs of particles $i\neq j$, of an
expression that consists of $H^{\epsilon,z}(x_{i}-x_{j},v_{i}-v_{j})$
multipled by two compact-supported smooth test functions localizing all four
variables $x_{i},x_{j},v_{i},v_{j}$. Since it is an averaged double sum, we
can focus on controlling the term of a particular pair $i=1,j=2$. Despite the
test functions slightly differ, we only use that they are bounded and that
they make the variables compactly supported. Finally, the uniform boundedness
of the particle joint densities $\mathsf{p}^{12}$ suggests that the time-
integral $\int_{0}^{T}$ is not important, and the same proof works for the
terms (18), corresponding to the process $\mathcal{Y}_{t}$ at initial and
terminal times.

To prepare for the next proposition, we require a crude pointwise bound on
$H^{\epsilon,z}$ and its partial derivative in $v$. It suffices to derive such
for the function $U^{\epsilon}$.

###### Lemma 16

Suppose that $D=2d\geq 2$. For any compact set
$\mathbb{K}\subset\mathbb{R}^{D}$, there exists some finite constant
$C=C(\mathbb{K},K,\theta,d)$ such that for any $(x,v)\in\mathbb{K}$ we have
that

$\displaystyle|U^{\epsilon}(x,v)|,|\nabla_{v}U^{\epsilon}(x,v)|\leq
C\epsilon^{-d}.$ (56)

Proof. We prove the thesis for the gradient, the other part being similar.
Recall (38), we have that

$\displaystyle\nabla_{v}U^{\epsilon}(x,v)=\int_{\mathbb{R}^{2d}}\nabla_{v}G\big{(}(x,v),(y,w)\big{)}\theta^{\epsilon}(y)\chi^{K}(w)dydw.$

Note that, with the function $\chi^{1}$ defined in (14), we can write

$\displaystyle\theta^{\epsilon}(y)\leq\epsilon^{-d}\|\theta\|_{L^{\infty}}\chi^{1}(y).$

Hence, by (46) (with $\eta=0$), there exists some finite constant
$C=C(\mathbb{K},K,\theta,d)$ such that for any $(x,v)\in\mathbb{K}$, we have

$\displaystyle|\nabla_{v}U^{\epsilon}(x,v)|\leq\epsilon^{-d}\|\theta\|_{L^{\infty}}\int_{\mathbb{R}^{2d}}|\nabla_{v}G\big{(}(x,v),(y,w)\big{)}|\chi^{1}(y)\chi^{K}(w)dydw\leq\epsilon^{-d}C.$

Indeed, the range of integration of $(y,w)$ is in $\mathbb{B}_{E}((0,0),8K)$,
which is inside $\mathbb{B}_{L}((x,v),K^{\prime})$ for some
$K^{\prime}<\infty$ depending on $\mathbb{K},K$, uniformly for all
$(x,v)\in\mathbb{K}$.

The following proposition bounds (28). An analogous proof can treat (29) and
prove its negligibility. Indeed, note that the only difference between (28)
and (29) are their test functions, i.e. instead of $\psi$ we have
$\nabla_{v}\psi$ or instead of $\nabla_{v}\phi$ we have $\phi$. The proof only
used that they are smooth bounded and they localize the variables
$x_{i},x_{j},v_{i},v_{j}$ in compact sets.

###### Proposition 17

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

$\displaystyle\lim_{|z|\to
0}\limsup_{N\to\infty}\mathbb{E}\int_{0}^{T}dt\frac{1}{N^{4}}\sum_{i=1}^{N}\Big{|}\sum_{j\neq
i}\nabla_{v}H^{\epsilon,z}\big{(}x_{i}(t)-x_{j}(t),v_{i}(t)-v_{j}(t)\big{)}\phi\big{(}t,x_{i}(t),v_{i}(t)\big{)}\psi\big{(}x_{j}(t),v_{j}(t)\big{)}$
$\displaystyle+\sum_{j\neq
i}H^{\epsilon,z}\big{(}x_{i}(t)-x_{j}(t),v_{i}(t)-v_{j}(t)\big{)}\psi\big{(}x_{j}(t),v_{j}(t)\big{)}\nabla_{v}\phi\big{(}x_{i}(t),v_{i}(t)\big{)}\Big{|}^{2}=0.$

Proof. First, using the elementary inequality $|\sum_{k=1}^{M}a_{k}|^{2}\leq
M\sum_{k=1}^{M}|a_{k}|^{2}$, it suffices to bound separately

$\displaystyle\mathbb{E}\int_{0}^{T}dt\frac{1}{N^{3}}\sum_{i,j=1}^{N}\Big{|}\nabla_{v}H^{\epsilon,z}\big{(}x_{i}(t)-x_{j}(t),v_{i}(t)-v_{j}(t)\big{)}\phi\big{(}t,x_{i}(t),v_{i}(t)\big{)}\psi\big{(}x_{j}(t),v_{j}(t)\big{)}\Big{|}^{2}$
(57)

and

$\displaystyle\mathbb{E}\int_{0}^{T}dt\frac{1}{N^{3}}\sum_{i,j=1}^{N}\Big{|}H^{\epsilon,z}\big{(}x_{i}(t)-x_{j}(t),v_{i}(t)-v_{j}(t)\big{)}\psi\big{(}x_{j}(t),v_{j}(t)\big{)}\nabla_{v}\phi\big{(}t,x_{i}(t),v_{i}(t)\big{)}\Big{|}^{2}.$
(58)

With the arguments of $H^{\epsilon,z}$ in a compact set, we can bound one term
of the square in (58) by the crude bound (56) in Lemma 16, then using the
relation $\epsilon^{-d}\leq N$ in Condition 2, we get that

$\displaystyle\eqref{sub-item-2}\leq
C(K,d,\theta)\|\phi\|_{C^{1}}\|\psi\|_{L^{\infty}}\mathbb{E}\int_{0}^{T}dt\frac{1}{N^{2}}\sum_{i,j=1}^{N}\Big{|}H^{\epsilon,z}\big{(}x_{i}(t)-x_{j}(t),v_{i}(t)-v_{j}(t)\big{)}\psi\big{(}x_{j}(t),v_{j}(t)\big{)}\nabla_{v}\phi\big{(}t,x_{i}(t),v_{i}(t)\big{)}\Big{|}.$

This expression is negligible which can be proved with the same argument as
Proposition 14, as it involves the same type of double sum. Only the text
functions look different, but all we use of the test functions are their
boundedness and that they localize the variable in a compact set.

Turning to (57), again by (56) and Condition 2, it is sufficient to bound

$\displaystyle\mathbb{E}\int_{0}^{T}dt\frac{1}{N^{2}}\sum_{i,j=1}^{N}\Big{|}\nabla_{v}H^{\epsilon,z}\big{(}x_{i}(t)-x_{j}(t),v_{i}(t)-v_{j}(t)\big{)}\psi\big{(}x_{j}(t),v_{j}(t)\big{)}\phi\big{(}t,x_{i}(t),v_{i}(t)\big{)}\Big{|}.$

The proof of the negligibility of the above quantity is analogous to the proof
presented in Proposition 14, just using the bound (46) for the gradient
instead of (45), hence we only sketch the idea. Indeed, as in (5), we have
that

$\displaystyle\nabla_{v}H^{\epsilon,z}(x,v)=\iint_{\mathbb{R}^{2d}}\big{[}\nabla_{v}G\big{(}(0,v),(\widetilde{y},w)\big{)}-\nabla_{v}G\big{(}(0,v),(\widetilde{y}-z,w)\big{)}\big{]}\theta^{\epsilon}(-\widetilde{y}-x)\chi^{K}(w)d\widetilde{y}dw.$

One can now take $\epsilon\to 0$ to obtain

$\displaystyle\limsup_{\epsilon\to
0}\iint_{\mathbb{R}^{2d}}|\nabla_{v}H^{\epsilon,z}(x,v)|\chi^{K}(x)\chi^{K}(v)\mathsf{p}^{12}(t,x,v)dxdv$
$\displaystyle\leq\iiint_{\mathbb{R}^{3d}}\big{|}\nabla_{v}G\big{(}(0,v),(\widetilde{y},w)\big{)}-\nabla_{v}G\big{(}(0,v),(\widetilde{y}-z,w)\big{)}\big{|}\chi^{K}(w)\chi^{K}(-\widetilde{y})\chi^{K}(v)\mathsf{p}^{12}(t,-\widetilde{y},v)d\widetilde{y}dwdv.$

In order to use Vitali convergence theorem to take $|z|\to 0$, we need to
check uniform integrability similar to (55). In this case, the estimate (46)
guarantees the existence of some $\eta=\eta(d)>0$ and $C=C(K,d)$ finite such
that

$\displaystyle\sup_{|z|\leq
1}\iiint_{\mathbb{R}^{3d}}\big{|}\nabla_{v}G\big{(}(0,v),(\widetilde{y}-z,w)\big{)}\big{|}^{1+\eta}\chi^{K}(w)\chi^{K}(-\widetilde{y})\chi^{K}(v)d\widetilde{y}dwdv<C.$

The following proposition bounds (24). The same proof can treat (25), indeed
its structure is the same as (24) except the roles of $\phi$ and $\psi$ are
permuted. Since they are both smooth test functions, this fact does not alter
the proof in an essential way. For (26), note that upon bounding
$\|\theta^{\epsilon}\|_{L^{\infty}}\leq\epsilon^{-d}\|\theta\|_{L^{\infty}}$
and using $\epsilon^{-d}\leq N$, it reduces to Proposition 14.

###### Proposition 18

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

$\displaystyle\lim_{|z|\to
0}\limsup_{N\to\infty}\mathbb{E}\int_{0}^{T}dt\frac{1}{N^{3}}\sum_{i\neq
j,i\neq k=1}^{N}$
$\displaystyle\theta^{\epsilon}\big{(}x_{i}(t)-x_{k}(t)\big{)}$
$\displaystyle\cdot\Big{|}H^{\epsilon,z}\big{(}x_{i}(t)-x_{j}(t),v_{i}(t)-v_{j}(t)\big{)}\phi\big{(}t,x_{i}(t),v_{i}(t)\big{)}\psi\big{(}x_{j}(t),v_{j}(t)\big{)}\Big{|}=0.$

Proof. First, we note that for $j=k$, we can bound
$\|\theta^{\epsilon}\|_{L^{\infty}}\leq\epsilon^{-d}\|\theta\|_{L^{\infty}}$
and with the condition $\epsilon^{-d}\leq N$, we have a contribution

$\displaystyle\mathbb{E}\int_{0}^{T}dt\frac{1}{N^{3}}\sum_{i\neq
j=1}^{N}\theta^{\epsilon}\big{(}x_{i}(t)-x_{j}(t)\big{)}\Big{|}H^{\epsilon,z}\big{(}x_{i}(t)-x_{j}(t),v_{i}(t)-v_{j}(t)\big{)}\phi\big{(}t,x_{i}(t),v_{i}(t)\big{)}\psi\big{(}x_{j}(t),v_{j}(t)\big{)}\Big{|}$
$\displaystyle\leq\mathbb{E}\int_{0}^{T}dt\frac{1}{N^{2}}\sum_{i\neq
j=1}^{N}\Big{|}H^{\epsilon,z}\big{(}x_{i}(t)-x_{j}(t),v_{i}(t)-v_{j}(t)\big{)}\phi\big{(}t,x_{i}(t),v_{i}(t)\big{)}\psi\big{(}x_{j}(t),v_{j}(t)\big{)}\Big{|},$

whose negligibility follows from Proposition 14. Hence, it suffices to
consider the contribution of $i\neq j\neq k$ all distinct. By similar
considerations as in the beginning of the proof of Proposition 14, it suffices
to bound each term of the form

$\displaystyle\mathbb{E}\int_{0}^{T}dt\;\theta^{\epsilon}\big{(}x_{1}(t)-x_{3}(t)\big{)}\Big{|}H^{\epsilon,z}\big{(}x_{1}(t)-x_{2}(t),v_{1}(t)-v_{2}(t)\big{)}\Big{|}\chi^{K}(x_{1}(t))\chi^{K}(x_{2}(t))\chi^{K}(v_{1}(t))\chi^{K}(v_{2}(t)).$

By independence, the joint density of $(x_{i}(t),v_{i}(t))_{i=1}^{3}$ is given
by $\prod_{i=1}^{3}\mathsf{p}(t,x_{i},v_{i})$. We write the above expectation
explicitly

$\displaystyle\int_{0}^{T}dt\int_{\mathbb{R}^{6d}}\theta^{\epsilon}(x_{1}-x_{3})\Big{|}H^{\epsilon,z}\big{(}x_{1}-x_{2},v_{1}-v_{2}\big{)}\Big{|}\chi^{K}(x_{1}))\chi^{K}(x_{2})\chi^{K}(v_{1})\chi^{K}(v_{2})\prod_{i=1}^{3}\mathsf{p}(t,x_{i},v_{i})dx_{i}dv_{i}.$

By (50) estimate for the one-particle density, we integrate first in
$x_{3},v_{3}$

$\displaystyle\int_{\mathbb{R}^{2d}}\theta^{\epsilon}(x_{1}-x_{3})\mathsf{p}(t,x_{3},v_{3})dx_{3}dv_{3}$
$\displaystyle\leq\int_{\mathbb{R}^{2d}}\theta^{\epsilon}(x_{1}-x_{3})\big{[}Ce^{-|v|/C}1_{\\{|v|\geq
C\\}}+\Gamma 1_{\\{|v|<C\\}}\big{]}dx_{3}dv_{3}\leq C^{\prime}(\Gamma,R,T,d),$

where we used that $\int\theta^{\epsilon}(x_{1}-x_{3})dx_{3}=1$ for any
$x_{1}$. We are left to bound

$\displaystyle\int_{0}^{T}dt\int_{\mathbb{R}^{4d}}\Big{|}H^{\epsilon,z}\big{(}x_{1}-x_{2},v_{1}-v_{2}\big{)}\Big{|}\chi^{K}(x_{1}))\chi^{K}(x_{2})\chi^{K}(v_{1})\chi^{K}(v_{2})\prod_{i=1}^{2}\mathsf{p}(t,x_{i},v_{i})dx_{i}dv_{i}.$

Recall the definition of $\chi^{K}$ (14) that
$x_{1},x_{2},v_{1},v_{2}\in\text{supp}(\chi^{K})$ implies that
$x_{1}-x_{2},v_{1}-v_{2}\in\text{supp}(\chi^{4K})$, and
$\chi^{4K}(x_{1}-x_{2})=\chi^{4K}(v_{1}-v_{2})=1$. Also
$\mathsf{p}(t,x_{i},v_{i})\leq\Gamma$, $i=1,2$, hence the preceding display
can be bnounded above by

$\displaystyle
T\Gamma^{2}\int_{\mathbb{R}^{4d}}\Big{|}H^{\epsilon,z}\big{(}x_{1}-x_{2},v_{1}-v_{2}\big{)}\Big{|}\chi^{4K}(x_{1}-x_{2})\chi^{4K}(v_{1}-v_{2})\chi^{K}(x_{1}))\chi^{K}(x_{2})\chi^{K}(v_{1})\chi^{K}(v_{2})dx_{1}dv_{1}dx_{2}dv_{2}$
$\displaystyle\stackrel{{\scriptstyle[}}{{\widetilde{}}}v=v_{1}-v_{2}]{\widetilde{x}=x_{1}-x_{2}}{=}T\Gamma^{2}\int_{\mathbb{R}^{4d}}\Big{|}H^{\epsilon,z}\big{(}\widetilde{x},\widetilde{v}\big{)}\Big{|}\chi^{4K}(\widetilde{x})\chi^{4K}(\widetilde{v})\chi^{K}(\widetilde{x}+x_{2}))\chi^{K}(x_{2})\chi^{K}(\widetilde{v}+v_{2})\chi^{K}(v_{2})d\widetilde{x}d\widetilde{v}dx_{2}dv_{2}$
$\displaystyle\leq
T\Gamma^{2}\int_{\mathbb{R}^{4d}}\Big{|}H^{\epsilon,z}\big{(}\widetilde{x},\widetilde{v}\big{)}\Big{|}\chi^{4K}(\widetilde{x})\chi^{4K}(\widetilde{v})\chi^{K}(x_{2})\chi^{K}(v_{2})d\widetilde{x}d\widetilde{v}dx_{2}dv_{2}$
$\displaystyle\leq
C(K)T\Gamma^{2}\iint_{\mathbb{R}^{2d}}\Big{|}H^{\epsilon,z}\big{(}\widetilde{x},\widetilde{v}\big{)}\Big{|}\chi^{4K}(\widetilde{x})\chi^{4K}(\widetilde{v})d\widetilde{x}d\widetilde{v},$

where the second line is due to a change of variable, the third line due to
$\chi^{K}\in[0,1]$ and the last line is because the integration in variables
$x_{2},v_{2}$ is over the compact support of $\chi^{K}$.

The resulting expression can be treated in the same way as (53) in the proof
of Proposition 14, and its negligibility in the limit $N\to\infty$ followed by
$|z|\to 0$ follows therefrom.

Finally, we remark that the control of (30), (31), (32) reduces to
Propositions 18 and 14. Indeed, take (30) for instance, using the elementary
inequality $|\sum_{k=1}^{M}a_{k}|^{2}\leq M\sum_{k=1}^{M}|a_{k}|^{2}$ and the
bound (56), we have that

$\displaystyle\eqref{mg-triple-1}$
$\displaystyle=\mathbb{E}\int_{0}^{T}dt\sum_{i\neq
k=1}^{N}\frac{1}{N^{5}}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{k}^{N}(t)\big{)}\Big{|}\sum_{j:j\neq
i,k}H^{\epsilon,z}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t),v_{i}^{N}(t)-v_{j}^{N}(t)\big{)}\phi\psi\Big{|}^{2}$
$\displaystyle\leq\mathbb{E}\int_{0}^{T}dt\sum_{i\neq j\neq
k=1}^{N}\frac{1}{N^{4}}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{k}^{N}(t)\big{)}\Big{|}H^{\epsilon,z}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t),v_{i}^{N}(t)-v_{j}^{N}(t)\big{)}\phi\psi\Big{|}^{2}$
$\displaystyle\leq
C\|\phi\|_{L^{\infty}}\|\psi\|_{L^{\infty}}\mathbb{E}\int_{0}^{T}dt\sum_{i\neq
j\neq
k=1}^{N}\frac{1}{N^{3}}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{k}^{N}(t)\big{)}\Big{|}H^{\epsilon,z}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t),v_{i}^{N}(t)-v_{j}^{N}(t)\big{)}\phi\psi\Big{|}.$

It then reduces to Propositions 18.

We have controlled all the negligible terms (18), (21)-(27), and by the
discussions in Section 3, we have established Proposition 5.

##  6 Tightness and subsequential convergence

We show that for any finite $T$, the laws of the sequence
$\\{\mu_{t}^{N}(dx,dv):t\in[0,T]\\}_{N\in\mathbb{N}}$ (3) of
$\mathcal{D}_{T}(\mathcal{M}_{+,1})$-valued random variables is tight, hence
weakly relatively compact by Prokhorov’s theorem, where
$\mathcal{D}_{T}(\mathcal{M}_{+,1})=\mathcal{D}([0,T];\mathcal{M}_{+,1}(\mathbb{R}^{2d}))$
is the Skorohod space (4). The following is a sufficient condition.

###### Proposition 19

If (a). there exists constant finite $C_{1}$ such that for any
$N\in\mathbb{N}$ and $t\in[0,T]$ we have that

$\displaystyle\mathbb{E}\int_{\mathbb{R}^{2d}}\big{(}|x|+|v|\big{)}\mu_{t}^{N}(dx,dv)\leq
C_{1};$ (59)

and (b). there exists some $\beta>0$ such that for every $\phi\in
C_{c}^{\infty}(\mathbb{R}^{2d})$ there exists $C(\phi)<\infty$ such that for
all $N$ and all stopping times $\tau\leq T$, $\gamma>0$ small enough and
$0<\sigma<\gamma$,

$\displaystyle\mathbb{E}\big{|}\langle\mu_{\tau+\sigma}^{N},\,\phi\rangle-\langle\mu_{\tau}^{N},\,\phi\rangle\big{|}\leq
C(\phi)\,\gamma^{\beta};$ (60)

then the laws $\\{\mathsf{P}_{N}\\}_{N}$ of the sequence
$\\{\mu_{t}^{N}:t\in[0,T]\\}_{N\in\mathbb{N}}$ is tight.

Proof. Tightness of a collection of laws on a Skorohod space
$\mathcal{D}([0,T];E)$, where $E$ is a complete separable metric space, is
well-studied in the literature; it builds on the characterization of compact
sets in these spaces, e.g. [3, Theorem 3.12.3], [8, Theorem 3.6.3], [19,
Proposition 4.1.2].

Specializing to our case, $E=\mathcal{M}_{+,1}(\mathbb{R}^{2d})$, endowed with
the following metric that metrizes weak convergence, cf. [19, pp. 49, Eq.
(1.1)], [13, Section 5.1]

$\displaystyle\delta(\mu,\nu):=\sum_{k=1}^{\infty}\frac{1}{2^{k}}\left[\left|\langle\mu,\phi_{k}\rangle-\langle\nu,\phi_{k}\rangle\right|\wedge
1\right],\quad\mu,\nu\in\mathcal{M}_{+.1}(\mathbb{R}^{2d}),$ (61)

where $\\{\phi_{k}\\}_{k=1}^{\infty}$ is a fixed dense sequence in
$C_{c}^{\infty}(\mathbb{R}^{2d})$. Recall [19, Theorem 4.1.3, Proposition
4.1.6] that a sequence of probability law $\\{\mathsf{P}_{N}\\}_{N}$ on
$\mathcal{D}_{T}(\mathcal{M}_{+,1})$ is relatively compact if the following
two conditions hold: (a’). For every $t>0$ and $\iota>0$, there exists a
compact set $K(t,\iota)\subset\mathcal{M}_{+,1}$ such that

$\displaystyle\sup_{N}\mathsf{P}_{N}\left(\mu_{t}\not\in
K(t,\iota)\right)<\iota.$ (62)

(b’). For every $\iota>0$,

$\displaystyle\lim_{\gamma\to
0}\limsup_{N\to\infty}\sup_{\tau\in\mathcal{T}_{T},\,\sigma\leq\gamma}\mathsf{P}_{N}\left(\delta(\mu_{\tau},\mu_{\tau+\sigma})>\iota\right)=0,$
(63)

where $\mathcal{T}_{T}$ is the set of stopping times $\leq T$.

Now recall that given $R<\infty$, the set

$\mathcal{K}^{R}:=\left\\{\mu\in\mathcal{M}_{+,1}(\mathbb{R}^{2d}):\,\int_{\mathbb{R}^{2d}}\left(|x|+|v|\right)\mu(dx,dv)\leq
R\right\\}$

is relatively compact in $\mathcal{M}_{+,1}(\mathbb{R}^{2d})$, since it
consists of a tight collection of subprobability measures on
$\mathbb{R}^{2d}$. Indeed, for any $A>0$ we have that

$\sup_{\mu\in\mathcal{K}^{R}}\mu\left((x,v)\in\mathbb{R}^{2d}:|x|+|v|>A\right)\leq\sup_{\mu\in\mathcal{K}^{R}}\frac{1}{A}\int_{\mathbb{R}^{2d}}\left(|x|+|v|\right)\mu(dx,dv)\leq\frac{R}{A}.$

Therefore, condition (59) implies that

$\mathsf{P}_{N}\left(\mu_{t}\not\in\overline{\mathcal{K}^{R}}\right)\leq\mathbb{P}\left(\int_{\mathbb{R}^{2d}}\left(|x|+|v|\right)\mu(dx,dv)>R\right)\leq\frac{1}{R}\mathbb{E}\int_{\mathbb{R}^{2d}}\left(|x|+|v|\right)\mu(dx,dv)\leq\frac{C_{1}}{R},$

by Markov’s inequality, and the rhs can be made as small as one wants,
uniformly in $N$ (and $t$) upon choosing $R$ large enough. This yields
condition (62).

Turning to consider (60), let us denote $c_{k}=C(\phi_{k})$ the constant on
the rhs of (60). Given any $\iota>0$, let $k_{0}=k_{0}(\iota)$ be the largest
integer $k^{\prime}$ such that
$\sum_{k=k^{\prime}}^{\infty}2^{-k}\geq\iota/2$. By the definition of (61), we
have that

$\displaystyle\delta\left(\mu_{\tau},\mu_{\tau+\sigma}\right)\leq\sum_{k=1}^{k_{0}}\frac{1}{2^{k}}\left|\langle\mu_{\tau},\phi_{k}\rangle-\langle\mu_{\tau+\sigma},\phi_{k}\rangle\right|+\frac{\iota}{2}$

Hence,

$\displaystyle\mathsf{P}_{N}\left(\delta\left(\mu_{\tau},\mu_{\tau+\sigma}\right)>\iota\right)$
$\displaystyle\leq\mathsf{P}_{N}\left(\sum_{k=1}^{k_{0}}\frac{1}{2^{k}}\left|\langle\mu_{\tau},\phi_{k}\rangle-\langle\mu_{\tau+\sigma},\phi_{k}\rangle\right|>\frac{\iota}{2}\right)$
$\displaystyle\leq\sum_{k=1}^{k_{0}}\mathsf{P}_{N}\left(\left|\langle\mu_{\tau},\phi_{k}\rangle-\langle\mu_{\tau+\sigma},\phi_{k}\rangle\right|>\frac{\iota}{2}\right).$

By (60) and Markov’s inequality, for every $k\leq k_{0}$ and
$\tau\in\mathcal{T}_{t}$, $\sigma\leq\gamma$,

$\mathsf{P}_{N}\left(\left|\langle\mu_{\tau}-\phi_{k}\rangle-\langle\mu_{\tau+\sigma},\phi_{k}\rangle\right|>\frac{\iota}{2}\right)\leq\frac{2}{\iota}\mathbb{E}\left|\langle\mu^{N}_{\tau},\phi_{k}\rangle-\langle\mu^{N}_{\tau+\sigma},\phi_{k}\rangle\right|\leq\frac{2c_{k}\gamma^{\beta}}{\iota}.$

Given any $\lambda>0$, there exists
$\gamma_{0}=\gamma_{0}(\lambda,\iota,c_{1},...,c_{k_{0}})>0$ such that
$2c_{k}\gamma^{\beta}/\iota<\lambda/2^{k}$ for all $k\leq k_{0}$ and
$\gamma\in(0,\gamma_{0})$. Thus, for all $N$ and $\tau\in\mathcal{T}_{T}$,
$\sigma\leq\gamma$, $\gamma<\gamma_{0}$, we have that

$\mathsf{P}_{N}\left(\delta\left(\mu_{\tau},\mu_{\tau+\sigma}\right)>\iota\right)\leq\sum_{k=1}^{k_{0}}\frac{\lambda}{2^{k}}<\lambda.$

Since $\lambda>0$ is arbitrary, we get (63). This completes the proof that
(59)-(60) is a sufficient condition for the sequence of laws $\mathsf{P}_{N}$
to be tight.

Now we check the two conditions in our case. Despite that (60) involves
stopping times, with proper conditioning on the stopped filtration at $\tau$
and using the strong Markov property, it is enough to write the proof for
deterministic times $s,t$.

For (a), by the exchangeability of independent particles
$\\{x_{i}(t),v_{i}(t)\\}_{i=1}^{N}$ (49) whose trajectories coincide with the
active particles in the true system $\\{x_{i}^{N}(t)\\}_{i=1}^{N(t)}$ (1), we
have that

$\displaystyle\mathbb{E}\int_{\mathbb{R}^{2d}}\big{(}|x|+|v|\big{)}\mu_{t}^{N}(dx,dv)$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N(t)}\mathbb{E}\big{(}|x^{N}_{i}(t)|+|v_{i}^{N}(t)|\big{)}$
$\displaystyle\leq\frac{1}{N}\sum_{i=1}^{N}\mathbb{E}\big{(}|x_{i}(t)|+|v_{i}(t)|\big{)}=\mathbb{E}\big{(}|x_{1}(t)|+|v_{1}(t)|\big{)}.$

Since

$\displaystyle(x_{1}(t),v_{1}(t))=\big{(}x_{1}(0)+v_{1}(0)t+\int_{0}^{t}B_{1}(s)ds,\,v_{1}(0)+B_{1}(t)\big{)}$

for any $t\in[0,T]$ and $(x_{1}(0),v_{1}(0))$ is compactly supported in
$\mathbb{B}_{E}(0,R)$ by Condition 1, we have that

$\displaystyle\mathbb{E}|v_{1}(t)|$ $\displaystyle\leq
R+\mathbb{E}|B_{1}(t)|\leq R+c_{d}\sqrt{T}$
$\displaystyle\mathbb{E}|x_{1}(t)|$ $\displaystyle\leq
R+RT+\mathbb{E}\int_{0}^{t}|B_{1}(s)|ds\leq R(1+T)+c_{d}T^{3/2}.$

For (b), by (8) for any $0\leq s<t\leq T$ we have that

$\displaystyle\mathbb{E}\big{|}\langle\mu_{t}^{N},\,\phi\rangle-\langle\mu_{s}^{N},\,\phi\rangle\big{|}$
$\displaystyle\leq\int_{s}^{t}\big{|}v\cdot\nabla_{x}\phi+\frac{1}{2}\Delta_{v}\phi,\mu_{u}^{N}\rangle\big{|}du$
$\displaystyle+\mathbb{E}\Big{|}\int_{s}^{t}\frac{1}{N^{2}}\sum_{i\in\mathcal{N}(u)}\sum_{j:j\neq
i}\theta^{\epsilon}\big{(}x_{i}^{N}(u)-x_{j}^{N}(u)\big{)}\big{[}\phi\big{(}x_{i}^{N}(u),v_{i}^{N}(u)\big{)}+\phi\big{(}x_{j}^{N}(u),v_{j}^{N}(u)\big{)}\big{]}\,du\Big{|}$
$\displaystyle+\big{(}\mathbb{E}|M_{t}^{\phi,D}-M_{s}^{\phi,D}|^{2}\big{)}^{1/2}+\big{(}\mathbb{E}|M_{t}^{\phi,J}-M_{s}^{\phi,J}|^{2}\big{)}^{1/2},$

where, as in the analysis of Section 2 we can bound

$\displaystyle\int_{s}^{t}\big{|}v\cdot\nabla_{x}\phi+\frac{1}{2}\Delta_{v}\phi,\mu_{u}^{N}\rangle\big{|}du$
$\displaystyle\leq(K^{\phi}\|\phi\|_{C^{1}}+\|\phi\|_{C^{2}})|t-s|,$
$\displaystyle\mathbb{E}|M_{t}^{\phi,D}-M_{s}^{\phi,D}|^{2}$
$\displaystyle\leq\frac{1}{N^{2}}\mathbb{E}\int_{s}^{t}\sum_{i\in\mathcal{N}(u)}|\nabla_{v_{i}}\phi(x_{i}^{N}(u),v_{i}^{N}(u))|^{2}du\leq\frac{\|\phi\|_{C^{1}}^{2}}{N}|t-s|,$
$\displaystyle\frac{1}{4}\mathbb{E}|M_{t}^{\phi,J}-M_{s}^{\phi,J}|^{2}$
$\displaystyle\leq\frac{\|\phi\|_{L^{\infty}}^{2}}{N^{3}}\mathbb{E}\int_{s}^{t}\sum_{i\in\mathcal{N}(u)}\sum_{j:j\neq
i}\theta^{\epsilon}(x_{i}^{N}(u)-x_{j}^{N}(u))du$
$\displaystyle\leq\frac{\epsilon^{-d}\|\theta\|_{L^{\infty}}\|\phi\|_{L^{\infty}}^{2}}{N}|t-s|\leq\|\theta\|_{L^{\infty}}\|\phi\|_{L^{\infty}}^{2}|t-s|,$

where $K^{\phi}$ denotes the maximal radius of the compact support of $\phi$.
Further, upon dominating by the independent system, by (50) we have that

$\displaystyle\mathbb{E}\Big{|}\int_{s}^{t}\frac{1}{N^{2}}\sum_{i\in\mathcal{N}(u)}\sum_{j:j\neq
i}\theta^{\epsilon}\big{(}x_{i}^{N}(u)-x_{j}^{N}(u)\big{)}\phi\big{(}x_{i}^{N}(u),v_{i}^{N}(u)\big{)}\,du\Big{|}$
$\displaystyle\leq\mathbb{E}\int_{s}^{t}\theta^{\epsilon}(x_{1}(u)-x_{2}(u))|\phi(x_{1}(u),v_{1}(u))|du$
$\displaystyle=\Gamma|t-s|\int_{\mathbb{R}^{4d}}\theta^{\epsilon}(x_{1}-x_{2})|\phi(x_{1},v_{1})\big{[}Ce^{-|v_{2}|/C}1_{\\{|v_{2}|\geq
C\\}}+\Gamma 1_{\\{|v_{2}|<C\\}}\big{]}dx_{1}dx_{2}dv_{1}dv_{2}$
$\displaystyle\leq C(\Gamma,R,\phi)|t-s|,$

due to $\int\theta^{\epsilon}(x_{1}-x_{2})dx_{2}=1$ for any $x_{1}$ and then
$\phi(x_{1},v_{1})$ has compact support. Thus, we verified (b) with
$\beta=1/2$.

We now show that any weak subsequential limit of $\\{\mu_{t}^{N}:t\in[0,T]\\}$
is supported on the subset of $\mathcal{D}_{T}(\mathcal{M}_{+,1})$ that are
absolutely continuous with respect to Lebesgue measure of $\mathbb{R}^{2d}$,
for every $t\in[0,T]$, and its density is dominated from above by
$\mathsf{p}(t,x,v)$ of (50). To this end, we consider also the empirical
measure of the independent system (49):

$\displaystyle\mu_{t}^{N,f}(dx,dv):=\frac{1}{N}\sum_{i=1}^{N}\delta_{\left(x_{i}(t),v_{i}(t)\right)}(dx,dv),\quad
t\in[0,T].$

Since the continuous paths $\left(x_{i}(t),v_{i}(t),t\geq
0\right)_{i=1}^{\infty}$ are independent and identically distributed, by
Varadarajan’s theorem ([7, Theorem 11.4.1]), a.s. for every $t\in[0,T]$

$\displaystyle\mu_{t}^{N,f}(dx,dv)\overset{N\to\infty}{\to}\overline{\mu}_{t}^{f}(dx,dv):=\mathsf{p}(t,x,v)dxdv,$

where $\mathsf{p}(t,x,v)$ is the one-particle density (51), and
$\overline{\mu}_{t}^{f}$ is deterministic. Consider now the joint process of
empirical measures

$\displaystyle\left\\{\mu_{t}^{N},\mu_{t}^{N,f}:t\in[0,T]\right\\}_{N\in\mathbb{N}}.$

Fix any weak subseqential limit $\overline{\mu}_{t}$ of
$\\{\mu_{t}^{N}\\}_{N}$ along some subsequences $\\{N_{k}\\}_{k}$. Since
$\\{\mu_{t}^{N,f}\\}$ converges a.s. to $\overline{\mu}_{t}^{f}$ along the
full sequence, we have that the pair converges weakly along $\\{N_{k}\\}$:

$\displaystyle\big{(}\mu_{t}^{N_{k}},\mu_{t}^{N_{k},f}:t\in[0,T]\big{)}\overset{k\to\infty}{\to}\big{(}\overline{\mu}_{t},\overline{\mu}_{t}^{f}:t\in[0,T]\big{)}.$
(64)

By Skorohod’s representation theorem, there exists an auxiliary probability
space $(\widetilde{\Omega},\widetilde{\mathcal{F}},\widetilde{\mathbb{P}})$
and a sequence of random variables defined thereon with the same distributions
as the ones in (64) (by an abuse of notation we denote hereafter by the same
letters) such that the convergence in (64) is a.s. By the domination of paths
between true and independent systems, a.s. for any open set
$A\subset\mathbb{R}^{2d}$, $t\geq 0$ and $k\in\mathbb{N}$,

$\displaystyle 0\leq\mu_{t}^{N_{k}}(A)\leq\mu_{t}^{N_{k},f}(A).$

As $k\to\infty$, convergence in Skorohod topology implies that a.s. for any
open set $A$, $t\in[0,T]$,

$\displaystyle 0\leq\overline{\mu}_{t}(A)\leq\overline{\mu}_{t}^{f}(A).$

Approximating a Borel set $B\subset\mathbb{R}^{2d}$ by a sequence of open sets
$A_{\ell}\downarrow B$, we have that a.s. for any Borel sets $B$ and
$t\in[0,T]$,

$\displaystyle 0\leq\overline{\mu}_{t}(B)\leq\overline{\mu}_{t}^{f}(B).$

Since for every $t$, $\overline{\mu}_{t}^{f}$ is absolutely continuous with
respect to Lebesgue measure, the same holds for $\overline{\mu}_{t}$, a.s.
Further, the density of $\overline{\mu}_{t}^{f}$ is a.s. dominated by
$\mathsf{p}(t,x,v)$, the density of $\overline{\mu}_{t}^{f}$. Hence, we have
proven the following proposition.

###### Proposition 20

For any finite $T$ and $d\geq 1$, the sequence
$\\{\mu_{t}^{N}(dx,dv):t\in[0,T]\\}_{N\in\mathbb{N}}$ (3) of
$\mathcal{D}_{T}(\mathcal{M}_{+,1})$-valued random variables is weakly
relatively compact, and any subsequential limit law is concentrated on
absolutely continuous paths, with nonnegative densities bounded above by
$\mathsf{p}(t,x,v)$ of (50).

Combining (8), (2), (12), Proposition 7 and Proposition 20, standard arguments
can yield that any weak subsequential limit of
$\\{\mu_{t}^{N}(dx,dv):t\in[0,T]\\}_{N\in\mathbb{N}}$ has a density
$\overline{f}(t,x,v)$ which is a (possibly random) weak solution of the pde
(5), in the sense of Definition 23, provided we can remove the extra test
function $\psi$ from the statement of Proposition 7, which is not present in
the equation (8) satisfied by the empirical measure. We will also show in
Section 7 that weak solutions to (5) are unique hence all subsequential limits
must coincide and be nonrandom.

To this end, in place of $\psi(x,v)$ in (13), consider a sequence
$\\{\psi^{\Lambda}(x,v)\\}_{\Lambda\in\mathbb{N}}$ of the form
$\psi^{\Lambda}(x,v)=\psi_{1}^{\Lambda}(x)\psi_{2}^{\Lambda}(v)$, where for
$i=1,2$,

$\displaystyle\psi_{i}^{\Lambda}:\mathbb{R}^{d}\to[0,1]$

is of class $C_{c}^{\infty}$ that is identically $1$ in
$\mathbb{B}_{E}(0,\Lambda)$ and identically $0$ in
$\mathbb{B}_{E}(0,2\Lambda)^{c}$. For any $i\neq j$, we have that

$\displaystyle\int_{0}^{T}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t)\big{)}\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\,dt-\int_{0}^{T}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t)\big{)}\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\psi^{\Lambda}\big{(}x_{j}^{N}(t),v_{j}^{N}(t)\big{)}\,dt$
$\displaystyle=\int_{0}^{T}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t)\big{)}\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\big{[}1-\psi_{1}^{\Lambda}(x_{j}^{N}(t))\psi_{2}^{\Lambda}(v_{j}^{N}(t))\big{]}\,dt.$
(65)

Since $|x_{i}^{N}(t)-x_{j}^{N}(t)|\leq\epsilon$ in the support of
$\theta^{\epsilon}$ and $x_{i}^{N}(t)$ is in the compact support of $\phi$,
for $\Lambda$ large enough we have $\psi_{1}^{\Lambda}(x_{j}^{N}(t))=1$ and
therefore

$\displaystyle\eqref{cut-
remove}=\int_{0}^{T}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t)\big{)}\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\big{[}1-\psi_{2}^{\Lambda}(v_{j}^{N}(t))\big{]}\,dt.$

Furthermore, we have that

###### Lemma 21

For any finite $T$ and $d\geq 1$,

$\displaystyle\lim_{\Lambda\to\infty}\sup_{N\in\mathbb{N}}\mathbb{E}\int_{0}^{T}\frac{1}{N^{2}}\sum_{i\in\mathcal{N}(t)}\sum_{j:j\neq
i}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t)\big{)}\big{|}\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\big{[}1-\psi_{2}^{\Lambda}(v_{j}^{N}(t))\big{]}\big{|}dt=0.$

Proof. We can upper bound by the independent system (49) and consider each
term of the form

$\displaystyle\mathbb{E}\int_{0}^{T}\theta^{\epsilon}(x_{1}(t)-x_{2}(t))\big{|}\phi\big{(}t,x_{1}(t),v_{1}(t)\big{)}\big{[}1-\psi_{2}^{\Lambda}(v_{2}(t))\big{]}\big{|}dt.$

Since $1-\psi_{2}^{\Lambda}(v_{2}(t))$ is identically $0$ in
$\mathbb{B}_{E}(0,\Lambda)$ and bounded above by $1$ elsewhere, utilizing (50)
we have that

$\displaystyle\mathbb{E}\int_{0}^{T}\theta^{\epsilon}(x_{1}(t)-x_{2}(t))\big{|}\phi\big{(}t,x_{1}(t),v_{1}(t)\big{)}\big{[}1-\psi_{2}^{\Lambda}(v_{2}(t))\big{]}\big{|}dt$
$\displaystyle\leq
C(\Gamma,R,T)\int_{\mathbb{R}^{4d}}\theta^{\epsilon}(x_{1}-x_{2})1_{\\{|x_{1}|,|v_{1}|\leq
K^{\phi}\\}}e^{-|v_{2}|/C}1_{\\{|v_{2}|\geq\Lambda\\}}\,dx_{1}dx_{2}dv_{1}dv_{2}$
$\displaystyle\leq
C(\Gamma,R,T,\phi)\int_{\mathbb{R}^{d}}e^{-|v_{2}|/C}1_{\\{|v_{2}|\geq\Lambda\\}}\,dv_{2}$
$\displaystyle\leq Ce^{-\Lambda/C},$

where $K^{\phi}$ denotes the maximal radius of the compact support of $\phi$,
and $\int\theta^{\epsilon}(x_{1}-x_{2})dx_{2}=1$ for every $x_{1}$. Note also
$1/N^{2}$ cancels with the cardinality of the double sum, and the limit as
$\Lambda\to\infty$ of the above expression is $0$, uniformly in $N$.

Since Proposition 7 applies for $\psi=\psi^{\Lambda}$ for each $\Lambda>0$,
(6) and Lemma 21 yield a version of Proposition 7 without the extra test
function.

###### Proposition 22

For any finite $T$ and $d\geq 1$,

$\displaystyle\lim_{\delta\to 0}\limsup_{N\to\infty}$
$\displaystyle\,\mathbb{E}\Big{|}\int_{0}^{T}\frac{1}{N^{2}}\sum_{i\in\mathcal{N}(t)}\sum_{j:j\neq
i}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t)\big{)}\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\,dt$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{R}^{d}}dw\left\langle\eta^{\delta}(w-x_{1})\eta^{\delta}(w-x_{2})\phi(t,w,v_{1}),\;\mu_{t}^{N}(dx_{1},dv_{1})\mu_{t}^{N}(dx_{2},dv_{2})\right\rangle
dt\Big{|}=0.$

Equipped with (8) and Proposition 20, we complete the proof of subsequential
convergence of the empirical measures. Since the sequence of laws
$\\{\mathsf{P}_{N}\\}_{N}$ is tight, we consider any weakly converging
subsequence $\mathsf{P}_{N_{k}}\Rightarrow\mathsf{P}^{*}$. Let us denote by
$\mu^{*}$ a random variable taking values in
$\mathcal{D}_{T}([0,T];\mathcal{M}_{+,1})$, having the law $\mathsf{P}^{*}$.
By Proposition 20, we know that $\mu^{*}_{t}(dx,dv)=f^{*}(t,x,v)dxdv$ for some
density $f^{*}(t,x,v)$ that is dominated from above by the function
$\mathsf{p}(t,x,v)$ (50). Given any $\iota>0$ and for any $\phi\in
C_{c}^{\infty}([0,T]\times\mathbb{R}^{2d})$, we have that

$\displaystyle\mathbb{P}\Big{(}\Big{|}-\langle\phi_{T},\mu_{T}^{*}\rangle+\langle\phi_{0},\mu_{0}^{*}\rangle+\int_{0}^{T}\left\langle\partial_{t}\phi+v\cdot\nabla_{x}\phi+\frac{1}{2}\Delta_{v}\phi,\,\mu_{t}^{*}\right\rangle
dt$
$\displaystyle\quad\quad-\int_{0}^{T}\int_{\mathbb{R}^{d}}dw\left\langle\eta^{\delta}(w-x_{1})\eta^{\delta}(w-x_{2})\phi(t,w,v_{1}),\;\mu_{t}^{*}(dx_{1},dv_{1})\mu_{t}^{*}(dx_{2},dv_{2})\right\rangle
dt\Big{|}>3\iota\Big{)}$
$\displaystyle\leq\liminf_{N\to\infty}\mathbb{P}\Big{(}\Big{|}-\langle\phi_{T},\mu_{T}^{N}\rangle+\langle\phi_{0},\mu_{0}^{N}\rangle+\int_{0}^{T}\left\langle\partial_{t}\phi+v\cdot\nabla_{x}\phi+\frac{1}{2}\Delta_{v}\phi,\,\mu_{t}^{N}\right\rangle
dt$
$\displaystyle\quad\quad-\int_{0}^{T}\int_{\mathbb{R}^{d}}dw\left\langle\eta^{\delta}(w-x_{1})\eta^{\delta}(w-x_{2})\phi(t,w,v_{1}),\;\mu_{t}^{N}(dx_{1},dv_{1})\mu_{t}^{N}(dx_{2},dv_{2})\right\rangle
dt\Big{|}>3\iota\Big{)}$
$\displaystyle\leq\liminf_{N}\mathbb{P}\left(|M_{T}^{N,\phi,D}|>\iota\right)+\liminf_{N}\mathbb{P}\left(|M_{T}^{N,\phi,J}|>\iota\right)$
$\displaystyle\quad\quad+\liminf_{N}\mathbb{P}\Big{(}\Big{|}\int_{0}^{T}\frac{1}{N^{2}}\sum_{i\in\mathcal{N}(t)}\sum_{j:j\neq
i}\theta^{\epsilon}\big{(}x_{i}^{N}(t)-x_{j}^{N}(t)\big{)}\phi\big{(}t,x_{i}^{N}(t),v_{i}^{N}(t)\big{)}\,dt$
$\displaystyle\quad\quad-\int_{0}^{T}\int_{\mathbb{R}^{d}}dw\left\langle\eta^{\delta}(w-x_{1})\eta^{\delta}(w-x_{2})\phi(t,w,v_{1}),\;\mu_{t}^{N}(dx_{1},dv_{1})\mu_{t}^{N}(dx_{2},dv_{2})\right\rangle
dt\Big{|}>\iota\Big{)},$ (66)

where we used the identity (8). The first two limits are zero by (2), (12).
Upon taking another limit $\delta\to 0$ the third term also is zero, by
Proposition 22. It remains to show that as $\delta\to 0$,

$\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{d}}dw\left\langle\eta^{\delta}(w-x_{1})\eta^{\delta}(w-x_{2})\phi(t,w,v_{1}),\;\mu_{t}^{*}(dx_{1},dv_{1})\mu_{t}^{*}(dx_{2},dv_{2})\right\rangle
dt$
$\displaystyle\to\int_{0}^{T}\int_{\mathbb{R}^{3d}}\phi(t,w,v_{1})f^{*}(t,w,v_{1})f^{*}(t,w,v_{2})dwdv_{1}dv_{2}dt,$
(67)

which we will do by dominated convergence. We note that for any $\delta>0$ and
$w\in\mathbb{R}^{d}$,

$\displaystyle\left|\left\langle\eta^{\delta}(w-x_{1})\eta^{\delta}(w-x_{2})\phi(t,w,v_{1}),\;\mu_{t}^{*}(dx_{1},dv_{1})\mu_{t}^{*}(dx_{2},dv_{2})\right\rangle\right|$
$\displaystyle\leq\int_{\mathbb{R}^{4d}}\eta^{\delta}(w-x_{1})\eta^{\delta}(w-x_{2})|\phi(t,w,v_{1})|f^{*}(t,x_{1},v_{1})f^{*}(t,x_{2},v_{2})dx_{1}dv_{1}dx_{2}dv_{2}$
$\displaystyle\leq
C\|\phi\|_{L^{\infty}}\int_{\mathbb{R}^{4d}}\eta^{\delta}(w-x_{1})\eta^{\delta}(w-x_{2})1_{\left\\{w\in\text{supp}(\phi)\right\\}}e^{-(|v_{1}|+|v_{2}|)/C}dx_{1}dv_{1}dx_{2}dv_{2}$
$\displaystyle\leq
C\|\phi\|_{L^{\infty}}1_{\left\\{w\in\text{supp}(\phi)\right\\}}$

where we used that $f^{*}(t,x,v)\leq\mathsf{p}(t,x,v)$ which has exponential
tail in $|v|$, uniformly in $x$. We also used that
$\int_{\mathbb{R}^{d}}\eta^{\delta}(w-x_{i})dx_{i}=1$, $i=1,2$, for any fixed
$w$. The rhs of the preceding display is integrable in $dwdt$,

On the other hand, as $\delta\to 0$ the integrand of the lhs of (6)

$\left\langle\eta^{\delta}(w-x_{1})\eta^{\delta}(w-x_{2})\phi(t,w,v_{1}),\;\mu_{t}^{*}(dx_{1},dv_{1})\mu_{t}^{*}(dx_{2},dv_{2})\right\rangle\to\int\phi(t,w,v_{1})f^{*}(t,w,v_{1})f^{*}(t,w,v_{2})dwdv_{1}dv_{2}$

for every fixed $t,w$, hence by the dominated convergence theorem we arrive at
(6).

Taking the limit as $\delta\to 0$ on both sides of (6), we conclude that
$\mathbb{P}$-a.s., $f^{*}(t,x,v)$ satisfies

$\displaystyle\langle\phi_{T},f^{*}(T,x,v)\rangle=$
$\displaystyle\langle\phi_{0},f^{*}(0,x,v)\rangle+\int_{0}^{T}\left\langle\partial_{t}\phi+v\cdot\nabla_{x}\phi+\frac{1}{2}\Delta_{v}\phi,\,f^{*}(t,x,v)\right\rangle
dt$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{R}^{3d}}\phi(t,w,v_{1})f^{*}(t,w,v_{1})f^{*}(t,w,v_{2})dwdv_{1}dv_{2}dt,$

which is the weak formulation of (5). The $\mathbb{P}$-null set can be taken
to be independent of $\phi$ by applying the above argument only to a countable
dense subset of $C_{c}^{\infty}$.

##  7 Uniqueness of weak solution

###### Definition 23

Assume $f_{0}\in L^{1}_{+}(\mathbb{R}^{2d})$ (the space of nonnegative
integrable functions) satisfies Condition 1, we say that

$\displaystyle f\in L^{\infty}\big{(}[0,T],L^{1}_{+}(\mathbb{R}^{2d})\big{)}$
(68)

is a weak solution to (5)

$\displaystyle\begin{cases}\partial_{t}f(t,x,v)&=-v\cdot\nabla_{x}f+\frac{1}{2}\Delta_{v}f-2\int_{\mathbb{R}^{d}}f(t,x,v)f(t,x,v^{\prime})dv^{\prime}\\\
f|_{t=0}&=f_{0}(x,v)\end{cases},\quad(t,x,v)\in[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}$

if for any $\phi(t,x,v)\in C_{c}^{\infty}([0,T]\times\mathbb{R}^{2d})$ and
$t\in[0,T]$ we have that

$\displaystyle\int_{\mathbb{R}^{2d}}f(t,x,v)\phi(t,x,v)dxdv$
$\displaystyle=\int_{\mathbb{R}^{2d}}f_{0}(x,v)\phi(0,x,v)dxdv$
$\displaystyle+\int_{0}^{t}\int_{\mathbb{R}^{2d}}\big{(}\partial_{s}+v\cdot\nabla_{x}+\frac{1}{2}\Delta_{v}\big{)}\phi(s,x,v)f(s,x,v)dxdvds$
$\displaystyle-2\int_{0}^{t}\int_{\mathbb{R}^{3d}}f(s,x,v)f(s,x,v^{\prime})\phi(s,x,v)dxdvdv^{\prime}dt.$

###### Proposition 24

For every finite $T$ and $d\geq 1$, weak solitions to (5) are unique.

Proof. As in Lemma 13, we denote the operator

$\displaystyle L^{*}=-v\cdot\nabla_{x}+\frac{1}{2}\Delta_{v},$

and by $\\{P(t)\\}_{t>0}$ the Markov semigroup associated with $L^{*}$. It is
a strongly continuous semigroup on $L^{p}(\mathbb{R}^{2d})$, for $1\leq
p<\infty$, see [15, Corollary 2.6], which we also denote by $P(t)=e^{tL^{*}}$.
Let $f_{1},f_{2}$ be two weak solutions of (5) in the sense of Definition 23,
and set

$\overline{f}:=f_{1}-f_{2}.$

Then, $\overline{f}$ satisfies in weak form

$\displaystyle\begin{cases}(\partial_{t}-L^{*})\overline{f}(t,x,v)&=-2\int_{\mathbb{R}^{d}}f_{1}(t,x,v)\overline{f}(t,x,v^{\prime})dv^{\prime}-2\int_{\mathbb{R}^{d}}\overline{f}(t,x,v)f_{2}(t,x,v^{\prime})dv^{\prime}\\\
\overline{f}|_{t=0}&=0.\end{cases}$

That is, for any $\phi(t,x,v)\in C_{c}^{\infty}([0,T]\times\mathbb{R}^{2d})$
and $t\in[0,T]$, we have that

$\displaystyle\int_{\mathbb{R}^{2d}}\overline{f}(t,x,v)\phi(t,x,v)dxdv$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}^{2d}}\big{(}\partial_{s}+L\big{)}\phi(s,x,v)\overline{f}(s,x,v)dxdvds$
$\displaystyle-2\int_{0}^{t}\int_{\mathbb{R}^{3d}}\left[f_{1}(s,x,v)\overline{f}(s,x,v^{\prime})+\overline{f}(s,x,v)f_{2}(s,x,v^{\prime})\right]\phi(s,x,v)\,dv^{\prime}dxdvds.$

A weak solution is also a mild solution. Indeed, for each $t$ taking
$\phi(s,x,v)=\left(e^{(t-s)L}\widetilde{\phi}\right)(x,v)$ for
$\widetilde{\phi}(x,v)\in C_{c}^{\infty}(\mathbb{R}^{2d})$, we obtain that

$\displaystyle\int_{\mathbb{R}^{2d}}\overline{f}(t,x,v)\widetilde{\phi}(x,v)dxdv=-2\int_{0}^{t}\int_{\mathbb{R}^{3d}}\left[f_{1}(s,x,v)\overline{f}(s,x,v^{\prime})+\overline{f}(s,x,v)f_{2}(s,x,v)\right]\left(e^{(t-s)L}\widetilde{\phi}\right)(x,v)dxdvdv^{\prime}ds$
$\displaystyle=-2\int_{0}^{t}\int_{\mathbb{R}^{2d}}e^{(t-s)L^{*}}\left(\int_{\mathbb{R}^{d}}f_{1}(s,\cdot,\cdot)\overline{f}(s,\cdot,v^{\prime})dv^{\prime}+\int_{\mathbb{R}^{d}}\overline{f}(s,\cdot,\cdot)f_{2}(s,\cdot,v^{\prime})dv^{\prime}\right)(x,v)\,\widetilde{\phi}(x,v)\,dxdvds.$

By the arbitrariness of $\widetilde{\phi}$, we see that $\overline{f}$
satisfies

$\displaystyle\overline{f}(t,x,v)=-2\int_{0}^{t}e^{(t-s)L^{*}}\left(\int_{\mathbb{R}^{d}}f_{1}(s,\cdot,\cdot)\overline{f}(s,\cdot,v^{\prime})dv^{\prime}+\int_{\mathbb{R}^{d}}\overline{f}(s,\cdot,\cdot)f_{2}(s,\cdot,v^{\prime})dv^{\prime}\right)(x,v)ds.$
(69)

Since $P(t)=e^{tL^{*}}$ is a contraction in $L^{p}(\mathbb{R}^{2d})$, $1\leq
p\leq\infty$, cf. [27, pp. 255, last line], [15, Lemma 2.4 (iv)], in
particular when $p=1$, we have by taking $L^{1}(\mathbb{R}^{2d})$-norm of (69)
that for any $t\in[0,T]$,

$\displaystyle\int_{\mathbb{R}^{2d}}\left|\overline{f}(t,x,v)\right|dxdv$
$\displaystyle\leq
2\int_{0}^{t}\left\|e^{(t-s)L^{*}}\left(\int_{\mathbb{R}^{d}}f_{1}(s,\cdot,\cdot)\overline{f}(s,\cdot,v^{\prime})dv^{\prime}+\int_{\mathbb{R}^{d}}\overline{f}(s,\cdot,\cdot)f_{2}(s,\cdot,v^{\prime})dv^{\prime}\right)\right\|_{L^{1}(\mathbb{R}^{2d})}ds$
$\displaystyle\leq
2\int_{0}^{t}\left\|\int_{\mathbb{R}^{d}}f_{1}(s,\cdot,\cdot)\overline{f}(s,\cdot,v^{\prime})dv^{\prime}+\int_{\mathbb{R}^{d}}\overline{f}(s,\cdot,\cdot)f_{2}(s,\cdot,v^{\prime})dv^{\prime}\right\|_{L^{1}(\mathbb{R}^{2d})}ds$
$\displaystyle\leq
2\int_{0}^{t}\int_{\mathbb{R}^{3d}}f_{1}(s,x,v)\left|\overline{f}(s,x,v^{\prime})\right|dxdvdv^{\prime}ds+2\int_{0}^{t}\int_{\mathbb{R}^{3d}}\left|\overline{f}(s,x,v)\right|f_{2}(s,x,v^{\prime})dxdvdv^{\prime}ds.$
(70)

By a simple comparison, we have that

$\displaystyle 0\leq f_{i}(t,x,v)\leq g(t,x,v),\quad i=1,2,$

where $g$ is the unique (weak) solution to

$\displaystyle(\partial_{t}-L^{*})g(t,x,v)$ $\displaystyle=0.$ $\displaystyle
g|_{t=0}$ $\displaystyle=f_{0}(x,v).$

Further, $g(t)=P(t)f_{0}$ is precisely the function $\mathsf{p}(t,x,v)$ in
(50) hence, since $f_{0}$ satisfies Condition 1, we have that for some finite
constant $C(T,\Gamma,R,d)$ and uniformly for all $t\in[0,T]$ and
$x\in\mathbb{R}^{d}$

$\displaystyle\int_{\mathbb{R}^{d}}f_{i}(t,x,v)dv\leq\int_{\mathbb{R}^{d}}g(t,x,v)dv\leq
C(T,\Gamma,R,d),\quad i=1,2.$ (71)

Hence, we obtain by (7)-(71) that for any $t\in[0,T]$,

$\displaystyle\left\|\overline{f}(t)\right\|_{L^{1}(\mathbb{R}^{2d})}$
$\displaystyle\leq
C\int_{0}^{t}\int_{\mathbb{R}^{2d}}\left|\overline{f}(s,x,v^{\prime})\right|dxdv^{\prime}ds+C\int_{0}^{t}\int_{\mathbb{R}^{2d}}\left|\overline{f}(s,x,v)\right|dxdvds$
$\displaystyle=C\int_{0}^{t}\left\|\overline{f}(s)\right\|_{L^{1}(\mathbb{R}^{2d})}ds.$

By Gronwall’s lemma, we have that $\overline{f}=f_{1}-f_{2}=0$ in the class
(68).

Since any weak subsequential limit obtained in Proposition 20 is supported in
the class (68), by Proposition 24 we have uniqueness of subsequential limits
hence convergence in law of the full sequence
$\\{\mu^{N}_{t}\\}_{N\in\mathbb{N}}$ to the unique weak solution of the pde,
as stated in Theorem 3. Since the limit is deterministic, we actually have
convergence in probability.

Acknowlegments. The author thanks Prof. F. Flandoli for valuable discussions,
corrections and pointers. The author thanks two anonymous referees whose
constructive feedback significantly improved the presentation of the paper.
This research was supported in part by a CRM-Pisa Junior Visiting Position
2020-22.

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