# Large Deviations of Kac’s Conservative Particle System and Energy Non-
Conserving Solutions to the Boltzmann Equation: A Counterexample to the
Predicted Rate Function

Daniel<EMAIL_ADDRESS>[ Centre for Mathematical Sciences,
Wilberforce Road, Cambridge, CB30WA,

###### Abstract

We consider the dynamic large deviation behaviour of Kac’s collisional process
for a range of initial conditions including equilibrium. We prove an upper
bound with a rate function of the type which has previously been found for
kinetic large deviation problems, and a matching lower bound restricted to a
class of sufficiently good paths. However, we are able to show by an explicit
counterexample that the predicted rate function does not extend to a global
lower bound: even though the particle system almost surely conserves energy,
large deviation behaviour includes solutions to the Boltzmann equation which
do not conserve energy, as found by Lu and Wennberg, and these occur strictly
more rarely than predicted by the proposed rate function. At the level of the
particle system, this occurs because a macroscopic proportion of energy can
concentrate in $\mathfrak{o}(N)$ particles with probability
$e^{-\mathcal{O}(N)}$.

35Q20, 60F10,

00X00,

82C22, 82C26,

Boltzmann Equation,

Large Deviations,

Kac Process,

###### keywords:

[class=MSC2020]

###### keywords:

T1This work was supported by the UK Engineering and Physical Sciences Research
Council (EPSRC) grant EP/L016516/1 for the University of Cambridge Centre for
Doctoral Training, the Cambridge Centre for Analysis

###### Contents

  1. 1 Introduction & Results
    1. 1.1 A Proposed Rate Function
    2. 1.2 Main Results
    3. 1.3 Literature Review & Discussion
  2. 2 Exponential Tightness & Upper Bound
    1. 2.1 Exponential Tightness
    2. 2.2 Variational Formulation of the Rate Function
    3. 2.3 Upper Bound
  3. 3 Properties of the Kac Process & Boltzmann Equation
  4. 4 Restricted Lower Bound
    1. 4.1 Proof of Approximation Lemma
    2. 4.2 Proof of Lemmas
  5. 5 Proof of Theorem 1.3
    1. 5.1 Construction of a change of measure $\mathbb{Q}^{N}$.
    2. 5.2 Law of Large Numbers
    3. 5.3 Proof of Theorem
    4. 5.4 Maxwell Molecules Case
  6. 6 Proof of Corollaries
  7. A Some Properties of Skorohod Paths
  8. B A Singular Girsanov Theorem for Jump Processes

##  1 Introduction & Results

We consider a family of models, of a kind introduced by Kac [28], modelling a
homogenous gas with elastic collisions associated to the homogeneous Boltzmann
equation. We consider an $N$-particle system, where particles of mass $N^{-1}$
have velocities $V^{1}_{t},...V^{N}_{t}\in\mathbb{R}^{d}$; we write
$\mu^{N}_{t}$ for the normalised empirical measure
$\mu^{N}_{t}=N^{-1}\sum_{i}\delta_{V^{i}_{t}}$. The velocities evolve over
time by random, binary collisions, governed by a collision kernel
$B:\mathbb{R}^{d}\times\mathbb{S}^{d-1}\to[0,\infty)$. For every ordered pair
of particles with velocities $v,v_{\star}\in\text{supp}(\mu^{N}_{t})$, the
velocities change to $v^{\prime}=v^{\prime}(v,v_{\star},\sigma)$ and
$v^{\prime}_{\star}=v^{\prime}_{\star}(v,v_{\star},\sigma)$, given by

$v^{\prime}(v,v_{\star},\sigma)=v-((v-v_{\star})\cdot\sigma)\sigma;\hskip
14.22636ptv_{\star}^{\prime}(v,v_{\star},\sigma)=v_{\star}+((v-v_{\star})\cdot\sigma)\sigma$
(1)

at rate $2B(v-v_{\star},\sigma)/N$. At the level of the empirical measure, we
see the change

$\mu\mapsto\mu^{N,v,v_{\star},\Sigma}=\mu+\frac{1}{N}(\delta_{v^{\prime}}+\delta_{v^{\prime}_{\star}}-\delta_{v}-\delta_{v_{\star}})=\mu+\frac{1}{N}\Delta(v,v_{\star},\sigma)$
(2)

We will write $\Delta f(t,v,v_{\star},\sigma)$ for the resulting change in a
function $\Delta
f(t,v,v_{\star},\sigma)=f(t,v^{\prime})+f(t,v_{\star}^{\prime})-f(t,v)-f(t,v_{\star})$
whenever $f:[0,T]\times\mathbb{R}^{d}\to\mathbb{R}$ or
$f:\mathbb{R}^{d}\to\mathbb{R}$. It is straightforward to see that $\Delta
f=0$ for $f=1,v_{i},|v|^{2}$, which means that the quantities $\langle
1,\mu^{N}_{t}\rangle,\langle v,\mu^{N}_{t}\rangle$ and
$\langle|v|^{2},\mu^{N}_{t}\rangle$ are conserved as time runs. We will work
throughout with one of the two kernels

$B(v,\sigma)=\begin{cases}1+|v|&\text{(Regularised Hard Spheres)}\\\
1&\text{(Maxwell Molecules)}.\end{cases}$ (3)

The first case is a ‘regularised’ version of the (true) hard spheres case
$B=|v|$, perturbed away from $0$ so that $v,\sigma\mapsto\log B(v,\sigma)$ is
globally Lipschitz continuous. In both cases, since $B$ is independent of
$\sigma$, we will use the abuse of notation $B(v)$, and the dynamics described
informally above lead to a well-defined Markov process. We will suppose,
throughout, that the initial data $V^{1}_{0},\dots,V^{N}_{0}$ are given by
independent samples from a reference measure $\mu_{0}^{\star}$, which we
normalise so that $\langle
v,\mu^{\star}_{0}\rangle=0,\langle|v|^{2},\mu^{\star}_{0}\rangle=1$. We will
write $\mathcal{E}_{z}(\mu^{\star}_{0})$ for the Gaussian moments

$\mathcal{E}_{z}(\mu^{\star}_{0}):=\int_{\mathbb{R}^{d}}e^{z|v|^{2}}\mu_{0}^{\star}(dv)\in[1,\infty].$

Kac’s process is closely connected to the spatially homogeneous Boltzmann
equation. Indeed, Kac introduced the collisional process as a toy model for
which Boltzmann’s celebrated _Stoßzahlansatz_ could be more easily derived: if
the velocities are initially approximately uncorrelated, then this property is
propagated in time. At the level of empirical measures this is equivalent to a
hydrodynamic limit or a law of large numbers for the empirical measures
$\mu^{N}_{t}$ towards a solution to the Boltzmann equation: for all $t\geq 0$
and $f$ Lipschitz,

$\langle f,\mu_{t}\rangle=\langle f,\mu_{0}\rangle+\int_{0}^{t}\langle
f,Q(\mu_{s})\rangle ds$ (BE)

where $Q(\mu)$ is the Boltzmann collision operator, given for measures with
finite first moment by

$\displaystyle
Q(\mu)=\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{S}^{d-1}}\Delta(v,v_{\star},\sigma)B(v-v_{\star})\mu(dv)\mu(dv_{\star})d\sigma.$
(4)

Pathwise, one can understand the Kac process as a stochastic pertubation of
the Boltzmann equation (BE): for any bounded $f$,

$\langle f,\mu^{N}_{t}\rangle=\langle f,\mu^{N}_{0}\rangle+\int_{0}^{t}\langle
f,Q(\mu^{N}_{s})\rangle ds+M^{N,f}_{t}$

where $M^{N,f}_{t}$ is a martingale of quadratic variation
$[M^{N,f}]_{t}=\mathcal{O}(N^{-1})$; see, for example, Norris [42]. In this
framework, the law of large numbers, for the kernels we consider, is well-
understood; qualitative results go back to Sznitman [sznitman1984equations],
and quantitative results have been obtained by Mischler and Mouhot [37] and
Norris [42]. Following the ideas of Freidlin and Wentzel [24], we consider the
dynamic large deviation behaviour of the Kac process on a fixed time interval
$[0,T]$; throughout, we will use the subscript ∙ to denote processes
indexed111This notation is chosen to minimise ambiguity between functionals
which may either depend on the whole process $\mu_{\bullet}$, or only on a
single measure $\mu$. by this time interval $t\in[0,T]$, so that
$\mu^{N}_{\bullet}=(\mu^{N}_{t})_{0\leq t\leq T}$.

It will be useful to consider the Kac process together with an auxiliary
_empirical flux_ , which records the collision history of the process [44,
45]. We write $E=[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}$ for the
parameter space of collisions and form measures $w^{N}_{t}$ on $E$ by setting
$w^{N}_{0}=0$ and changing, at collisions,

$w^{N}_{t}=w^{N}_{t-}+\frac{1}{N}\delta_{(t,v,v_{\star},\sigma)}$ (5)

at times $t$ where there is a collision, choosing one possible assignment
$(v,v_{\star},\sigma)$ uniformly at random between the four possibilities. In
this way, the pair $(\mu^{N}_{t},w^{N}_{t})$ together form a Markov process,
which are linked by a consistency relation (CE) below, and we write
$w^{N}:=w^{N}_{T}$ for the final measure, containing the entire collisional
history of the process. With this notation, we investigate estimates
informally given by

$\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\approx(\mu_{\bullet},w)\right)\asymp
e^{-N\mathcal{I}(\mu_{\bullet},w)}.$ (6)

Formally, we consider the space $\mathcal{P}_{2}$ of probability measures on
$\mathbb{R}^{d}$ with finite second moment, equipped with the Monge-
Kantorovich-Wasserstein distance

$W(\mu,\nu)=\sup\left\\{\langle f,\mu-\nu\rangle:f\in\mathcal{F}\right\\};$
(7) $\mathcal{F}=\left\\{f:\mathbb{R}^{d}\rightarrow\mathbb{R},\hskip
2.84544pt\|f\|_{\infty}\leq 1,\hskip 2.84544pt\sup_{v\neq
w}\frac{|f(v)-f(w)|}{|v-w|}\leq 1\right\\}$ (8)

and write $\mathcal{P}_{2}^{N}$ for the subspace consisting of empirical
measures on $N$ points, and $\mathcal{D}$ for the Skorokhod space

$\mathcal{D}:=\left\\{\mu_{\bullet}\in
D([0,T],(\mathcal{P}_{2},W)):\sup_{t\leq
T}\langle|v|^{2},\mu_{t}\rangle<\infty\right\\}$ (9)

which we equip with a metric inducing the Skorokhod $J_{1}$-topology. For the
empirical fluxes, we write $\mathcal{M}(E)$ for the space of finite Borel
measures on $E$, which we equip with the Wasserstein1 metric $d$ given
analogously by

$d(w,w^{\prime})=\sup\left\\{\langle f,w-w^{\prime}\rangle:\sup_{E}|g|\leq
1,\sup_{p,q\in E,p\neq q}\frac{|g(p)-g(q)|}{|p-q|}\leq 1\right\\}$ (10)

where $|\cdot|$ is the Euclidean norm on $E\subset\mathbb{R}^{3d+1}$. With
this notation, the pair $(\mu^{N}_{t},w^{N}_{t})$ is a Markov process in
$\mathcal{P}_{2}^{N}\times\mathcal{M}(E)$ with generator given on bounded
functions by

$\begin{split}\mathcal{G}^{N}F(\mu^{N},w^{N})&=N\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{S}^{d-1}}(F(\mu^{N,v,v_{\star},\sigma},w^{N,t,v,v_{\star},\sigma})-F(\mu^{N},w^{N}))\\\
&\hskip 28.45274pt\hskip 28.45274pt\hskip 28.45274pt\hskip 28.45274pt\hskip
28.45274pt\dots
B(v-v_{\star})\mu^{N}(dv)\mu^{N}(dv_{\star})d\sigma\end{split}$ (11)

with

$\mu^{N,v,v_{\star},\sigma}:=\mu^{N}+\frac{1}{N}\Delta(v,v_{\star},\sigma);\qquad
w^{N,v,v_{\star},\sigma}:=w^{N}+\frac{1}{N}\delta_{(t,v,v_{\star},\sigma)}.$
(12)

Formally, (6) means that, for Kac processes $\mu^{N}_{\bullet}$ and associated
empirical fluxes $w^{N}$,

$\limsup_{N}\frac{1}{N}\log\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\in\mathcal{A}\right)\leq-\inf\left\\{\mathcal{I}(\mu_{\bullet},w):(\mu_{\bullet},w)\in\mathcal{A}\right\\}$
(13)

for any $\mathcal{A}\subset\mathcal{D}\times\mathcal{M}(E)$ closed, and

$\liminf_{N}\frac{1}{N}\log\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\in\mathcal{U}\right)\geq-\inf\left\\{\mathcal{I}(\mu_{\bullet},w):(\mu_{\bullet},w)\in\mathcal{U}\right\\}$
(14)

$\mathcal{U}\subset\mathcal{D}\times\mathcal{M}(E)$ open. It is well-known
that such bounds give a precise mathematical meaning to Boltzmann’s notion of
entropy in terms of _“volume of accessible microstates”_ ; see the discussion
in [52]. We will make the following further hypotheses on the reference
measure $\mu_{0}^{\star}$:

######  Hypothesis 1.

  1. i).

_Gaussian upper bound:_ there exists $z_{1}>0$ such that
$\mathcal{E}_{z_{1}}(\mu^{\star}_{0})<\infty$.

  2. ii).

Gaussian Lower Bound: there exists $z_{2}<\infty$ such that
$\mathcal{E}_{z}(\mu_{0}^{\star})<\infty$ for $z<z_{2}$, and
$\mathcal{E}_{z}(\mu_{0}^{\star})\to\infty$ as $z\uparrow z_{2}$.

  3. iii).

_Continuous Density:_ $\mu_{0}^{\star}$ has a continuous density
$f_{0}^{\star}$ with respect to the Lebesgue measure, and for some
$z_{3}\in(0,\infty)$ and $c>0$,

$f_{0}^{\star}\geq ce^{-z_{3}|v|^{2}}.$ (15)

Under the initial conditions described above, Sanov’s Theorem [54] applies to
show that the initial data satisfy a large deviation function in
$(\mathcal{P}_{2},W)$ with rate function

$H(\mu_{0}|\mu_{0}^{\star}):=\begin{cases}\int_{\mathbb{R}^{d}}\frac{d\mu_{0}}{d\mu_{0}^{\star}}\log\left(\frac{d\mu_{0}}{d\mu_{0}^{\star}}\right)\hskip
2.84544pt\mu_{0}^{\star}(dv)&\text{if }\mu_{0}\ll\mu_{0}^{\star};\\\
\infty&\text{else.}\end{cases}$ (16)

###  1.1 A Proposed Rate Function

Let us review a possible rate function identified by Léonard [31] for exactly
this problem. For $\mu_{\bullet}\in\mathcal{D}$ we define
$\overline{m}_{\mu}\in\mathcal{M}(E)$ by

$\overline{m}_{\mu}(dt,dv,dv_{\star},d\sigma)=B(v-v_{\star})dt\mu_{t}(dv)\mu_{t}(dv_{\star})d\sigma.$
(17)

We say that $(\mu_{\bullet},w)\in\mathcal{D}\times\mathcal{M}(E)$ is a
_measure-flux pair_ if $w\ll\overline{m}_{\mu}$ and if they solve the
_continuity equation_ : for all $0\leq t\leq T$,

$\mu_{t}=\mu_{0}+\int_{E}\Delta(v,v_{\star},\sigma)1_{s\leq t}\hskip
2.84544ptw(ds,dv,dv_{\star},d\sigma).$ (CE)

We will use, throughout, the notation $K$ for the density
$\frac{dw}{d\overline{m}_{\mu}}$, which we call a tilting function. With this
notation, if $(\mu_{\bullet},w)$ is a measure-flux pair, then $\mu_{\bullet}$
solves a modified Boltzmann equation

$\begin{split}\mu_{t}&=\mu_{0}+\int_{E}\Delta(v,v_{\star},\sigma)K(s,v,v_{\star},\sigma)B(v-v_{\star})ds\mu_{s}(dv)\mu_{s}(dv_{\star})d\sigma.\end{split}$
(BEK)

Equivalently, given $\mu_{\bullet}$ solving (BEK) for some $K\in
L^{1}(\overline{m}_{\mu})$, one can define $w=K\overline{m}_{\mu}$ and
$(\mu_{\bullet},w)$ is a measure-flux pair. We define the dynamic cost of a
trajectory $(\mu_{\bullet},w)\in\mathcal{D}\times\mathcal{M}(E)$ to be

$\mathcal{J}(\mu_{\bullet},w):=\begin{cases}\int_{E}\tau\left(\frac{dw}{d\overline{m}_{\mu}}\right)\overline{m}_{\mu}(ds,dv,dv_{\star},d\sigma)&\text{
if $(\mu_{\bullet},w)$ is a measure-flux pair;}\\\
\infty&\text{else}\end{cases}$ (18)

where $\tau:[0,\infty]\to[0,\infty]$ is the function $\tau(k)=k\log k-k+1$,
and define the full rate function to be

$\mathcal{I}(\mu_{\bullet},w):=H(\mu_{0}|\mu_{0}^{\star})+\mathcal{J}(\mu_{\bullet},w).$
(19)

An analagous upper bound, which can be obtained from this rate function using
the contraction principle on $(\mu_{\bullet},w)\to\mu_{\bullet}$ is obtained
by Léonard [31] in a different topology, and the same rate function has been
found in other contexts for kinetic large deviations; see the comments in the
literature review.

###  1.2 Main Results

Our first result collects some useful facts on the proposed rate function
$\mathcal{I}$ and on the exponential tightness.

######  Proposition 1.1 (Exponential Tightness and Semicontinuity).

Fix a probability space $(\Omega,\mathfrak{F},\mathbb{P})$. For $N\geq 2$, let
$\mu^{N}_{\bullet}$ be either regularised hard sphere or Maxwell Molecule Kac
processes with initial velocities drawn independently from a measure
$\mu_{0}^{\star}$ satisfying Hypothesis 1i). Then the following hold.

  1. i).

The random variables $(\mu^{N}_{\bullet},w)\in\mathcal{D}\times\mathcal{M}(E)$
are exponentially tight: for any $M>0$, there exists a compact set
$\mathcal{K}\subset\mathcal{D}\times\mathcal{M}(E)$ such that

$\limsup_{N}\frac{1}{N}\log\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\not\in\mathcal{K}\right)\leq-M.$
(20)

  2. ii).

The function $\mathcal{I}$ is lower semicontinuous on
$\mathcal{D}\times\mathcal{M}(E)$: the lower sub-level sets

$\\{(\mu_{\bullet},w)\in\mathcal{D}\times\mathcal{M}(E):\mathcal{I}(\mu_{\bullet},w)\leq
a\\}\subset\mathcal{D}\times\mathcal{M}(E)$ (21)

are closed when $\mathcal{D}$ has the Skorokhod topology and $\mathcal{M}(E)$
the weak topology, metrised by $d$.

We emphasise that we do not claim that $\mathcal{I}$ is ‘good’, in that the
sub-level sets are compact; indeed, Theorem 1.3 suggests that this is false.

The positive result we prove on the large deviations is as follows. We
rederive, in our context, the upper bound with our rate function, which
reproduces the result of Léonard [31], and prove a lower bound with the same
rate function on a restricted set. In this way, the proposed rate function
captures at least some of the correct large deviation behaviour of the Kac
process.

######  Theorem 1.2.

Let $B$ be either the regularised hard spheres or Maxwell molecules kernel,
and for $N\geq 2$ let $(\mu^{N}_{\bullet},w^{N})$ be a Kac process and its
flux, with particles drawn initially from $\mu_{0}^{\star}$ satisfying
Hypothesis 1, and let $\mathcal{I}$ be the rate function given above. Then

  1. i).

For all $\mathcal{A}\subset\mathcal{D}\times\mathcal{M}(E)$ closed, we have

$\limsup_{N}\hskip
2.84544pt\frac{1}{N}\log\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\in\mathcal{A}\right)\leq-\inf\left\\{\mathcal{I}(\mu_{\bullet},w):(\mu_{\bullet},w)\in\mathcal{A}\right\\}.$
(22)

  2. ii).

For all $\mathcal{U}\subset\mathcal{D}\times\mathcal{M}(E)$ open, we have

$\liminf_{N}\hskip
2.84544pt\frac{1}{N}\log\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\in\mathcal{U}\right)\geq-\inf\left\\{\mathcal{I}(\mu_{\bullet},w):(\mu_{\bullet},w)\in\mathcal{U}\cap\mathcal{R}\right\\}$
(23)

where
$\mathcal{R}=\\{(\mu_{\bullet},w)\in\mathcal{D}\times\mathcal{M}(E):\langle
1+|v|^{2}+|v_{\star}|^{2},w\rangle<\infty\\}.$

As discussed in the literature review, the proposed rate function is a very
natural candidate for describing the large deviations behaviour, and one might
expect to be able to find a ‘true’ lower bound (14). In this context, the
restricted lower bound presented here is somewhat dissatisfying, as it leaves
open the question of which open sets $\mathcal{U}$ are such that
$\inf_{\mathcal{U}}\mathcal{I}=\inf_{\mathcal{U}\cap\mathcal{R}}\mathcal{I}$,
or the possibility that a better upper bound may be possible. The restriction
to a set $\mathcal{R}$ of ‘good’ paths, as here, is necessary for the paths in
question to be approximated by paths which can be recovered by a Girsanov
transform; see Lemma 4.1. Key to this argument is that these paths should be
uniquely specified by the initial data and tilting $K$, so that the path is
the unique possible hydrodynamic limit of ‘tilted’ dynamics along any
subsequence. However, at the level of the Boltzmann equation (BE), this
uniqueness is known _not_ to hold: solutions with increasing energy have been
constructed by Lu and Wennberg [33]. Since the energy
$\langle|v|^{2},\mu^{N}_{t}\rangle$ is almost surely conserved by the paths of
the stochastic Kac process, one might hope that such solutions are spurious
and can be excluded from the large deviation analysis, so that uniqueness does
hold. However, we prove the following theorem, which shows that such solutions
can be reached with finite exponential cost, and so cannot be excluded from
the large deviation analysis, but the occurrence of such paths is not
correctly predicted by the proposed rate function. Equivalently, this example
can be understood as producing explicitable open sets $\mathcal{U}$ such that
the infima of the rate function over $\mathcal{U}$ and
$\mathcal{U}\cap\mathcal{R}$ do not coincide.

######  Theorem 1.3.

Assume the notation of Proposition 1.1.

  1. i).

Suppose $B$ is the regularised hard spheres kernel, and the reference measure
$\mu_{0}^{\star}$ satisfies Hypothesis 1i-ii). Let $\Theta:[0,T]\to(0,\infty)$
be nondecreasing, nonconstant and left-continuous, with $\Theta(0)=1$ and such
that, for some closed set $E\subset[0,T],0\in E,T\not\in E$ with null
interior, $\Theta$ is locally constant on $[0,T]\setminus E$. For some
constant $\alpha=\alpha(\Theta(T))$, define

$A(t):=\alpha\left(\inf_{s\in E:s<t}(t-s)\right)^{-2}\in(0,\infty]$ (24)

and consider the set $\mathcal{A}_{\Theta}$ given by

$\begin{split}\mathcal{A}_{\Theta}:=&\bigg{\\{}(\mu_{\bullet},w)\in\mathcal{D}\times\mathcal{M}(E):\mu_{\bullet}\text{
solves (\ref{eq: BE}), }w=\overline{m}_{\mu},\hskip
2.84544pt\mu_{0}=\mu^{\star}_{0},\\\ &\hskip 85.35826pt\text{and for all
}t\geq 0,\hskip 2.84544pt\langle|v|^{2},\mu_{t}\rangle=\Theta(t)\text{ and
}\langle|v|^{4},\mu_{t}\rangle\leq A(t)\bigg{\\}}.\end{split}$ (25)

Then $\mathcal{A}_{\Theta}$ is compact, nonempty, and
$\mathcal{I}(\mu_{\bullet},w)=0$ on $\mathcal{A}_{\Theta}$. We have

$\inf_{\mathcal{U}\supset\mathcal{A}_{\Theta}}\liminf_{N}\frac{1}{N}\log\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\in\mathcal{U}\right)\geq-\Theta(T)z_{2}$
(26)

where the infimum runs over all open sets
$\mathcal{U}\subset\mathcal{D}\times\mathcal{M}(E)$ containing
$\mathcal{A}_{\Theta}$, and there exists an open set
$\mathcal{V}\supset\mathcal{A}_{\Theta}$ such that

$\liminf_{N}\frac{1}{N}\log\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\in\mathcal{V}\right)<0.$
(27)

  2. ii).

Suppose instead that $B$ is the cutoff Maxwell Molecules kernel. For all
$\delta>0$ and $\Theta$ as above, and with $A$ as above with $\alpha$
depending on $\delta$ as well as $\Theta(T)$, define the set

$\begin{split}\mathcal{A}_{\Theta,\delta}:=&\bigg{\\{}(\mu_{\bullet},w)\in\mathcal{D}\times\mathcal{M}(E):(\mu_{\bullet},w)\text{
is a measure-flux pair with }K=1+\delta|v-v_{\star}|,\\\ &\hskip
56.9055pt\mu_{0}=\mu^{\star}_{0},\text{ and for all }t>0,\hskip
2.84544pt\langle|v|^{2},\mu_{t}\rangle=\Theta(t)\text{ and
}\langle|v|^{4},\mu_{t}\rangle\leq A(t)\bigg{\\}}.\end{split}$ (28)

The sets $\mathcal{A}_{\Theta,\delta}$ are compact and nonempty, and
$\mathcal{I}(\mu_{\bullet},w)\leq 4\delta^{2}\Theta(T)T$ for all
$(\mu_{\bullet},w)\in\mathcal{A}_{\Theta,\delta}$. We have

$\inf_{\mathcal{U}\supset\mathcal{A}_{\Theta,\delta}}\liminf_{N}\frac{1}{N}\log\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\in\mathcal{U}\right)\geq-\Theta(T)(z_{2}+C\delta)$
(29)

where, as above, the infimum runs over all open sets $\mathcal{U}$ containing
$\mathcal{A}_{\Theta,\delta}$. However, there exist open sets
$\mathcal{V}_{\delta}\supset\mathcal{A}_{\Theta,\delta}$ such that, for any
$\Theta$,

$\limsup_{\delta\downarrow 0}\hskip
2.84544pt\liminf_{N}\frac{1}{N}\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\in\mathcal{V}_{\delta}\right)<0.$
(30)

In both cases, the first point shows that such behaviour cannot be excluded by
superexponential estimates, and so is a form of behaviour which must be taken
into account in the large deviation theory; the second point shows that the
rate function on such paths is not that predicted above. The argument we will
present is a stochastic, large deviation analogue of the construction of Lu
and Wennberg [33], keeping track of the exponential change of measure
necessary. In the first case, we will construct changes of measure
$\mathbb{Q}^{N}\ll\mathbb{P}$, with an exponential cost associated to changing
the initial data and sub-exponential cost associated to modifying the
dynamics, so that $\mathfrak{o}(N)$ particles containing $\mathcal{O}(1)$
energy are temporarily ‘frozen’ and, under these new measures, the Kac
processes concentrate on the set $\mathcal{A}_{\Theta}$ given. The argument
for the Maxwell molecule case is similar, with an additional exponential cost
$\mathcal{O}(e^{N\delta})$ necessary to modify the dynamics. In these cases,
the behaviour of $\mathfrak{o}(N)$ particles has a macroscopic effect on the
evolution of the whole process, meaning that the large deviation behaviour is
not purely captured by the empirical measure and control $K$. Possible
generalisations of this phenomenon will be discussed in Section 1.3.2 below

Let us now examine some consequences of Theorems 1.2, 1.3. One might hope that
it is possible to prove a true large deviation principle under well-chosen
initial conditions where one puts in ‘by hand’ that there is no such
concentration initially. The following easy corollary suggests that, even
under such well-chosen conditions, the same concentration of energy can occur
as a result of the binary collisions.

######  Corollary 1.4.

Let us take $\mu_{0}^{\star}=\gamma$, and fix a decreasing, right-continuous
function $\Theta$, $\Theta(T)=1$, which is locally constant aside from at a
closed set $E\subset[0,T]$ with empty interior, $T\in E,0\not\in E$. For
either Maxwell molecules or hard spheres, there exists an explicitable
function $A$ such

$\begin{split}\mathcal{B}&=\bigg{\\{}(\mu_{\bullet},w)\in\mathcal{D}\times\mathcal{M}(E):\langle|v|^{2},\mu_{t}\rangle=\Theta(t)\text{
for all }t\in[0,T]\text{ and }\langle|v|^{4},\mu_{t}\rangle\leq
A(t)\bigg{\\}}\end{split}$ (31)

satisfies

$\inf_{\mathcal{U}\supset\mathcal{B}}\hskip
2.84544pt\liminf_{N}\frac{1}{N}\log\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\in\mathcal{U}\right)>-\infty.$
(32)

where, as above, the outer infimum runs over open
$\mathcal{U}\subset\mathcal{D}$ containing $\mathcal{B}$.

As a result, it is not possible to find a superexponential estimate to prevent
the accumulation of energy in $\mathfrak{o}(N)$ particles at future times.

Since the stochastic processes $\mu^{N}_{\bullet}$ are exponentially tight in
$\mathcal{D}$, it follows that one can extract subsequences satisfying a
_true_ large deviation principle. As a consequence of Theorem 1.3, no such
subsequence can avoid the bad paths we have constructed.

######  Corollary 1.5.

Let $\mu^{\star}_{0}$ be a reference measure satisfying Hypothesis 1i-ii), and
let $\mu^{N}_{\bullet},w^{N}$ be $N$-particle Kac processes, either for the
regularised hard spheres or Maxwell molecules case. Suppose that
$S\subset\mathbb{N}$ is an infinite subsequence such that
$(\mu^{N}_{\bullet},w^{N})_{N\in S}$ satisfy a large deviation principle in
$\mathcal{D}\times\mathcal{M}(E)$ with some rate function
$\widetilde{\mathcal{I}}$. Then there exists $(\mu_{\bullet},w)$ in
$\mathcal{D}\times\mathcal{M}(E)$ such that
$\widetilde{\mathcal{I}}(\mu_{\bullet},w)<\infty$ but such that
$t\mapsto\langle|v|^{2},\mu_{t}\rangle$ is not constant.

Our final corollary is a positive result, following from Theorem 1.2, which
shows how the entropy plays the role of a _quasipotential_ for the Kac
dynamics. Let us refer the reader to [8, Section 3.3] for a general discussion
of such results.

######  Corollary 1.6 (Entropy as a Quasipotential).

Let $B$ be either the regularised hard spheres or Maxwell molecules kernel,
and fix $\mu\in\mathcal{P}_{2}$. Then

$H(\mu|\gamma)\geq\inf\left\\{H(\nu_{0}|\gamma)+\int_{E}\tau(K)d\overline{m}_{\nu}:\nu\in\mathcal{D},\nu\text{
solves (\ref{eq: BEK})},\nu_{T}=\mu\right\\}$ (33)

and

$\begin{split}H(\mu|\gamma)&\leq\inf\bigg{\\{}H(\nu_{0}|\gamma)+\int_{E}\tau(K)d\overline{m}_{\nu}:\nu\in\mathcal{D},\nu\text{
solves (\ref{eq: BEK})},\nu_{T}=\mu,\\\ &\hskip 199.16928pt\text{ and
}\int_{E}(|v|^{2}+|v_{\star}|^{2})Kd\overline{m}_{\nu}<\infty\bigg{\\}}.\end{split}$
(34)

In this sense, we view $\tau(K)$ as the entropic cost of moving to a higher-
entropy state by tilted Boltzmann dynamics (BEK). The second item generalises
Boltzmann’s famous $H$-Theorem; however, in light of the nonemptiness of the
sets $\mathcal{A}_{\Theta}$ in Theorem 1.3, the upper bound would be false
without the second moment condition.

The paper is structured as follows. In the remainder of this section, we will
review some recent works on large deviations and related problems, and make
some remarks on the hypothesis and functional framework. In Section 2, we
derive an upper bound Theorem 1.2i); in doing so, we will prove Proposition
1.1 via a variational formulation of the rate function $\mathcal{I}$ which
appears in the upper bound. Section 3 gathers some properties of the Kac
process and changes of measure from the literature for later convenience.
Section 4 proves the restricted lower bound Theorem 1.2ii), based on an
approximation argument for paths belonging to $\mathcal{R}$ and a standard
‘tilting’ argument. The proof of Theorem 1.3 is given in Section 5, based on
the properties of the Kac process in Section 3 and a careful analysis of
Cramér bounds, and we deduce the corollaries in Section 6. Finally, Appendix A
is a self-contained appendix on the Skorokhod topology and Appendix B contains
a justification of the change-of-measure formula.

###  1.3 Literature Review & Discussion

#### Large Deviations for Jump Particle Systems

The theory of large deviations for Markov processes in the small-noise limit
goes back to Freidlin and Wentzell [24]. The seminal work of Feng & Kurz [23]
developed tools based on a comparison principle for Hamilton-Jacobi equation
in infinite dimensions, which are general but hard to verify. The analysis is
somewhat different in the case where the dynamics are driven by diffusive
rather than jump noise, see the discussion in Léonard [31]. In this context
let us mention the recent works [3, 10, 11, 40].

Within collisional kinetic theory, previous works have reported upper bounds
of a similar form. The work of Léonard [31] already cited considers the same
case of the energy-preserving Kac model, and produces a rate function exactly
given by $\mathcal{I}(\mu_{\bullet})=\inf_{w}\mathcal{I}(\mu_{\bullet},w)$,
albeit for a different topology. Rezakhanlou [46] considers a collisional
model for a spatially inhomogeneous gas, where the positions take values in
the unit circle $\mathbb{R}/\mathbb{Z}$ and the velocities take values only in
a finite set, and finds an upper bound and a restricted lower bound, where the
infimum runs only over a subset $\mathcal{R}\cap\mathcal{U}$ as in Theorem 1.2
rather than over the full open set $\mathcal{U}$ as in (14), with a rate
function analagous to the variational form (cf Lemma 2.4 or [31, Theorem
7.1]). Bodineau et al. [6] consider the full spatially inhomogeneous
Boltzmann–Grad limit with random initial data and deterministic dynamics for
local interactions; the rate function is again given in a variational form,
and the lower bound is again restricted to sufficiently good paths. Most
recently, a very similar large deviation result for a range of ‘Kac–like’
processes has been found by Basile et al. [4]. The lower bound is again of the
restricted form as in Theorem 1.2, and is proven only in cases where the
collisions preserve momentum but not energy.

Outside of kinetic theory, analagous rate functions have been found for large
deviations of jump processes, for instance [17]. A number of works [44, 43,
45] have considered the case of ‘reaction networks’, which formally includes
the Kac/Boltzmann dynamics considered here by viewing $\mathbb{R}^{d}$ as a
continuum of particle species; these works are the origin of considering the
pair $(\mu^{N}_{\bullet},w^{N})$ which significantly eases the analysis. Other
works [20, 30] in the context of reaction networks or mean-field dynamics
exploit a control representation of the dynamics, leading to an equation
similar to (BEK) with random controls $K$ and the same cost function $\tau$,
and using weak convergence method due to Dupuis [19]. In this weak converence
method, it is essential that the control uniquely determines possible limiting
paths (see a similar argument in the proof of Theorem 1.3ii) in Section 4),
whereas this type of uniqueness is known not to be hold in the Boltzmann case,
even in the most advantageous possible case of Maxwell molecules. In the work
[30], the key to removing the restriction on regular paths is an approximation
argument so that paths are perturbed to lie in the _interior_ of the space of
probability measures on the space of the finite space of species $S$, which is
clearly impossible in the infinite-dimensional setting here.

Let us mention that this is a very natural form for the rate function jump
processes. One recognises $\tau$ as the dynamic cost of controlling a Poisson
random measure: for a Poisson process of unit intensity, a straightforward
argument using Stirling’s formula shows that $\mathbb{P}(N^{-1}Z_{N}\approx
z)\asymp\exp(-N\tau(z))$ in the same sense as (13, 14). Similarly, if one
fixes a finite space $S$ and a probability measure $\mu$ and forms $X_{N}$ as
a Poisson random measure of intensity $N\mu$, then one has the equivalent

$\mathbb{P}\left(N^{-1}X_{N}\approx\nu\right)\asymp\exp\left(-N\sum_{S}\tau\left(\frac{\nu(x)}{\mu(x)}\right)\mu(x)\right)=\exp\left(-N\int_{S}\tau\left(\frac{d\nu}{d\mu}\right)\mu(dx)\right).$
(35)

The proposed rate function above would correspond to the intuition that, given
$\mu^{N}_{t}\approx\mu_{t}$, the instantaneous distribution of jumps is
approximately Poisson, with intensity $\approx
B(v-v_{\star})dt\mu_{t}(dv)\mu_{t}(dv_{\star})d\sigma$.

As remarked above, several other works [17, 46, 6, 4] have encountered the
same problem that the lower bound can only be proven over a class of good
paths. Both the works [17, 46] conjecture that a ‘true’ lower bound should
hold in the respective frameworks. In the works cited above, such a conjecture
has only been proven in the cases of reaction networks with a finite set of
species [30, 44, 43, 45] or mean-field dynamics with finite state space [20],
which are very far from the Kac/Bolzmann dynamics we consider. To the best of
our knowledge, the current work represents the first time that such a
hypothesis has been falsified.

#### Remarks on the Hypotheses & Functional Framework

We make the following remarks on the Hypothesis 1 and on the functional
framework. Firstly, the hypotheses allow the very natural choice of taking
$\mu^{\star}_{0}$ to be the equilibrium distribution

$\mu_{0}^{\star}(dv)=\gamma(dv)=\frac{1}{(2\pi d)^{d/2}}e^{-|v|^{2}/2d}dv$
(36)

but Hypothesis 1ii). disallows measures of the form
$\mu_{0}^{\star}(dv)\propto(1+|v|^{2})^{-m}\gamma(dv),m>\frac{d}{2}$. In
general, the condition that $\mu^{N}_{0}$ be given by drawing particles
independently from a reference measure $\mu_{0}^{\star}$ will not propagate in
time. However, this is natural in order to ensure that $\mu^{N}_{0}$ satisfies
a large deviation principle; elementary counterexamples can be found to show
that the more usual conditions, that the initial data be chaotic or
entropically chaotic [25], do not imply a large deviation principle for
$\mu^{N}_{0}$. Moreover, in the most important case $\mu_{0}^{\star}=\gamma$,
the independence _is_ propagated, as $\gamma^{\otimes N}$ is an equilibrium
distribution for the $N$-particle dynamics.

Regarding the functional framework, while $(\mathcal{P}_{2},W)$ is not
complete, the choice of metric $W$ and Skorokhod space $\mathcal{D}$ are
natural to guarantee that $(\mu^{N}_{\bullet},w^{N})$ are exponentially tight.
One could alternatively equip $\mathcal{P}_{2}$ with the Wasserstein2 metric
$W_{2}$, which makes the map $\mu\mapsto\langle|v|^{2},\mu\rangle$ continuous,
and one can take a limit of the pathwise energy conservation
$\langle|v|^{2},\mu^{N}_{t}\rangle=\langle|v|^{2},\mu^{N}_{0}\rangle$ to
conclude that all possible large deviation paths still conserve energy.
However, carefully following the arguments leading to our counterexamples in
Section 5 proves that the initial measures $\mu^{N}_{0}$ then fail to be
exponentially tight, as does the whole process $(\mu^{N}_{\bullet},w^{N})$.
Since large deviations techniques rely heavily on such tightness to prove the
existence of subsequential limits under the change of measures, we have been
unable to determine $\mathcal{I}$ correctly determines the large deviations in
this framework. In light of this, we interpret Theorem 1.3 as showing the
existence of a different kind of large deviations behaviour, where macroscopic
energy concentrates in $\mathfrak{o}(N)$ particles, which is not captured by
convergence in $(\mathcal{P}_{2},W_{2})$.

In future works, it may be interesting to consider the large deviations in the
functional framework similar to that of Léonard [31]. Let us write
$C_{\mathrm{qu}}(\mathbb{R}^{d})$ for the continuous functions of quadratic
growth, we write $\mathfrak{P}_{2}$ for the space of linear maps
$\mathfrak{m}:C_{\mathrm{qu}}(\mathbb{R}^{d})\to\mathbb{R}$ satisfying
$\mathfrak{m}(1)=1$, $\mathfrak{m}(\varphi)\geq 0$ whenever $\varphi\geq 0$,
and such that there exists $\mu=j[\mathfrak{m}]\in\mathcal{P}_{2}$ with
$\langle\varphi,\mu\rangle=\mathfrak{m}(\varphi)$ for all bounded $\varphi\in
C_{\mathrm{qu}}(\mathbb{R}^{d})$. We then equip $\mathfrak{P}_{2}$ with the
product topology from the inclusion
$\mathfrak{P}_{2}\subset\mathbb{R}^{C_{\mathrm{q}}(\mathbb{R}^{d})}$, and we
can view $\mathcal{P}_{2}\subset\mathfrak{P}_{2}$ via the identification
$\iota:\mathcal{P}_{2}\to\mathfrak{P}_{2}$,
$\iota(\mu)(\varphi):=\langle\varphi,\mu\rangle$, so that the Kac process can
be understood as taking values in $\mathfrak{P}_{2}$. Moreover, thanks to the
classical theorems of Tychonoff or Banach-Alaoglu, the sets

$\mathfrak{K}_{a}=\left\\{\mathfrak{m}\in\mathfrak{P}_{2}:\text{ for all
}\varphi\in C_{\mathrm{qu}}(\mathbb{R}^{d}),|\mathfrak{m}(\varphi)|\leq
a\sup_{v}\frac{|\varphi(v)|}{1+|v|^{2}}\right\\}$ (37)

are compact for all $a\in[0,\infty)$, and cover $\mathfrak{P}_{2}$. In this
framework, one has both exponential tightness, and continuity of the map
$\mathfrak{m}\mapsto\mathfrak{m}(|v|^{2})$. On the other hand, we warn the
reader that elements of $\mathfrak{P}_{2}$ are typically _not_ measures, since
$j[\mathfrak{m}]=\mu$ does not imply that $\mathfrak{m}=\iota(\mu)$. Indeed,
following the construction of the initial data in Section 5 leading to Theorem
1.3 produces limits with
$\mathfrak{m}(|v|^{2})=\Theta(T)>\langle|v|^{2},j[\mathfrak{m}]\rangle=1$.

#### Other Models & Generality of the Phenomenon

Let us first remark that, although the current work only focusses on the two
kernels identified in (3), identical arguments would apply with
$B(v)=1+|v|^{\gamma}$ in place of the regularised hard spheres kernel, at the
cost of further complications. The modification by adding $1$ is also not
necessary for Theorem 1.3, but eases some technical difficulties in Theorem
1.2ii). Léonard [31] also obtains an upper bound in the case of non-cutoff
Maxwell molecules, where $B(v,\sigma)$ has a non-integrable angular
singularity as $\sigma\to\pm v/|v|$, leading to an abundance of grazing
collisions.

It may be interesting in future to consider the large deviations of other
collisional processes associated to the Boltzmann Equation. In the Nanbu
process [39], the rate of each collision is doubled, but only one particle at
a time is updated. In this case, one can construct a measure dominating
$\mathbb{P}$ under which the jumps are independent, while this is impossible
for the Kac process (see the comments [31, Remark 1.9i]). One could also
consider the large deviations behaviour of the Kac model coupled to a heat
bath [7, 49] which may introduce enough additional randomness to avoid the
kind of behaviour described here. Heuristically, one could already view the
counterexamples in Theorem 1.3 as constructing a heat bath from
$\mathfrak{o}(N)$ particles, which are allowed to drive energy into the
system; it would also be interesting to formalise this connection.

It also seems that the key ingredients of the counterexample Theorem 1.3 may
generalise to other large deviation systems. Although we will not explore the
general case in more detail, the key points we require generalise to an
interacting particle system on a locally compact222for such a space and
$f:S\to\mathbb{R}$, we say that $f\to\infty$ if all preimages $\\{x:f(x)\leq
M\\}\subset S$ are compact. state space $S$ as follows:

  1. 1.

Conserved Quantity:  There exists a continuous $\varphi:S\to[0,\infty)$,
$\varphi\to\infty$ such that, almost surely,
$\langle\varphi,\mu^{N}_{t}\rangle$ is constant along sample paths;

  2. 2.

Criticality:  The initial distributions are such that
$F(z):=\lim_{N}N^{-1}\log\mathbb{E}[e^{Nz\langle\varphi,\mu^{N}_{0}\rangle}]$
exists in $[0,\infty]$ for all $z$. Moreover, the function $F(z)$ is finite on
a neighbourhood $I$ of the origin, but diverges to infinity as $z\uparrow\sup
I<\infty$.

  3. 3.

Delocalisation Mechanism: For some continuous $\psi$ with
$\psi\geq\varphi,\psi/\varphi\to\infty$, one either has

    1. (a)

Uniformly in $N$, for all $t>0$ and all starting points $\mu^{N}_{0}$,
$\mathbb{E}\langle\psi,\mu^{N}_{t}\rangle<\infty$ can be controlled only in
terms of $t$ and $\langle\varphi,\mu^{N}_{0}\rangle$, uniformly in compact
subsets of $t\in(0,\infty)$; or

    2. (b)

For some $\mu^{N}_{0}$ and for all $\delta>0$, one can find changes of measure
$\mathbb{Q}^{N}\ll\mathbb{P}$ by modifying only the dynamics, such that
$\sup_{N}\mathbb{E}_{\mathbb{Q}^{N}}\langle\psi,\mu^{N}_{t}\rangle$ can be
controlled in terms of $t,\langle\varphi,\mu^{N}_{0}\rangle,\delta$, uniformly
in $N$, uniformly in compact subsets of $t\in(0,\infty)$, and the perturbation
is small in the sense that
$\mathbb{Q}^{N}(\frac{d\mathbb{Q}^{N}}{d\mathbb{P}}>e^{N\delta
a})<\frac{1}{2}$ for $N$ large enough, for some $a$ depending only on
$\langle\varphi,\mu^{N}_{0}\rangle$.

In our case, the conserved quantity $\varphi$ is the energy
$\varphi(v)=|v|^{2}$, and this would apply to any system with stochastic,
energy-preserving dynamics. The second point is natural for initial data
sampled from Gibbs distributions in statistical mechanics, where the density
with respect to some Lebesgue measure is given by $\propto
e^{-H}=e^{-N\langle\varphi,\mu^{N}_{0}\rangle}$, and corresponds to Hypothesis
1i-ii). The third point says that, potentially under a small perturbation of
the dynamics, the system rapidly distributes $\varphi$ among all particles. By
item 2, no exponential moments for $\langle\psi,\mu^{N}_{0}\rangle$ can be
hoped for, so that bounds on
$\mathbb{E}_{\mathbb{Q}^{N}}\langle\psi,\mu^{N}_{t}\rangle$ will not hold
under typical changes of measure $\mathbb{Q}^{N}\ll\mathbb{P}$: we only ask
that one such change of measure can be found. In our case, this rôle will be
played by the moment creation property and Povzner estimates with
$\psi=|v|^{p},p>2$, see Proposition 3.1; case a) corresponds to regularised
hard spheres, and case b) to Maxwell molecules.

In either case, since both $\varphi$ is necessarily unbounded, the dynamics
cannot only be captured by a weakly continuous function of the empirical
measure. Item 2 allows cases where a macroscopic pertubation of
$\langle\varphi,\mu^{N}_{0}\rangle$ is achieved with only a small pertubation
of $\mu^{N}_{0}$ in the weak topology, and playing the pathwise conservation
(item 1) against the delocalisation mechanism (item 3) instantaneously spreads
this pertubation to the whole empirical measure. This leads to a law of large
numbers for paths $\mu_{\bullet}$ along which $\langle\varphi,\mu_{t}\rangle$
is not conserved, and is a given, nonconstant function $\Theta(t)$ which is
constant aside from a jump discontinuity at $0$; more general $\Theta$ could
be found with further assumptions on the dynamics. The dynamic cost required
for such paths is either $0$, or $e^{\mathcal{O}(N\delta)}$, by following
exactly the arguments in Section 5; the conclusion that the large deviation
occurrence of such limit paths is not correctly predicted by the naïve rate
function then follows by exploiting the conflict between the non-conservative
limit paths and conservative finite-$N$ paths (item 1).

#### Relationship to Other Problems

We mention some other aspects of the Boltzmann/Kac dynamics which are related
to the current work.

As already mentioned above, large deviations give a probabilistic meaning to
Boltzmann’s Entropy functional $H(\cdot|\gamma)$; the $H$-Theorem, which
guarantees that this decreases along solutions to (BE) or its spatially
inhomogeneous version, goes back to the foundations of kinetic theory.
Quantitative versions of this increase, and hence qualifying the convergence
to equilibrium, have been a major topic in the analysis of the Boltzmann
Equation (among many others, [12, 48, 5, 50, 51, 16]). Let us also mention the
work of Mischler and Mouhot [37], which gives a probabilistic proof of the
$H$-theorem via entropic chaos of the Kac process; however, as remarked above,
entropic chaos does not lead to the large deviations considered here.

Following the seminal work of Jordan, Kinderlehrer and Otto [27], it has been
shown that many equations of mean-field type can be understood as the gradient
flow of the entropy for a metric adapted to the particular problem, so that
the dynamics not only increase entropy, but do so in the most efficient way
possible. Further, it is known that such gradient flow properties can be
derived from large deviation principles [1, 18, 2, 35, 21]. Since such a
gradient descent formulation of the Boltzmann equation is already known [22,
8, 4], we will not explore this here.

Finally, let us refer to the recent work of Bouchet [8] which discusses the
classical paradox of reversibility based on large deviations, with a rate
function analagous to (19) above and the rôle of entropy as a quasipotential
driving the Boltzmann dynamics.

##  2 Exponential Tightness & Upper Bound

In this section, we will prove Propositon 1.1 and the upper bound Theorem
1.2i). We first verify exponential tightness in Subsection 2.1. In Subsection
2.2 we introduce a variational form for the rate function, similar to that of
Léonard [31, Theorem 3.1, Theorem 7.1] and prove equivalence of the two
formulations; this leads to a simple proof of lower semicontinuity in
Proposition 1.1ii), as well as some functional analytical facts which will be
useful later. Finally, we use the variational formulation to prove the upper
bound in Section 2.1, based on standard martingale techniques and a covering
argument.

###  2.1 Exponential Tightness

We first prove that $(\mu^{N}_{\bullet},w^{N})$ are exponentially tight in
$\mathcal{D}\times\mathcal{M}(E)$, which proves the first assertion
Proposition 1.1i.

######  Lemma 2.1 (Verification of Conditions for Exponential Tightness).

For any $M>0$, the following hold.

  1. a).

For $\lambda>0$, set
$\mathcal{P}_{2,\lambda}=\\{\mu\in\mathcal{P}_{2}:\langle|v|^{2},\mu\rangle\leq\lambda\\}$
and
$\mathcal{D}_{\lambda}:=\\{\mu_{\bullet}\in\mathcal{D}:\mu_{t}\in\mathcal{P}_{2,\lambda}\text{
for all }t\\}$. There exists $\lambda\in(0,\infty)$ such that

$\limsup_{N}\frac{1}{N}\log\mathbb{P}\left(\mu^{N}_{\bullet}\not\in\mathcal{D}_{\lambda}\right)\leq-M.$
(38)

  2. b).

For all $\delta>0$, define
$q^{N}(\delta)=\sup(W(\mu^{N}_{s},\mu^{N}_{t}):|s-t|<\delta)$. For all
$\epsilon>0$, there exists $\delta>0$ such that

$\limsup_{N}\frac{1}{N}\log\mathbb{P}\left(q^{N}(\delta)>\epsilon\right)\leq-M.$
(39)

  3. c).

There exists $C>0$ such that

$\limsup_{N}\frac{1}{N}\log\mathbb{P}\left(w^{N}(E)>C\right)\leq-M.$ (40)

Let us remark that the first item proves that, for each fixed $t\in[0,T]$,
$\mu^{N}_{t}$ are exponentially tight because $\mathcal{P}_{2,\lambda}$ are
compact for the metric $W$. Together, the first two conditions verify the
well-known criteria for exponential tightness in the Skorokhod space
$\mathcal{D}$ due to Feng and Kurz [23, Theorem 4.1]. In the third item, the
sets $\\{w\in\mathcal{M}(E):w(E)\leq C\\}$ are compact for the metric $d$,
which induces the weak topology, and hence the third item shows that $w^{N}$
are exponentially tight in $\mathcal{M}(E)$. Together, these prove that the
pair $(\mu^{N}_{\bullet},w^{N})$ together are exponentially tight in
$\mathcal{D}\times\mathcal{M}(E)$.

###### Proof of Lemma 2.1.

Fix $M$ throughout. We start with the first point, and begin by noting that,
thanks to Hypothesis 1i) and a Chebychev bound, for $z_{1}>0$ sufficiently
small and all $\lambda>0$,

$\mathbb{P}\left(\langle|v|^{2},\mu^{N}_{0}\rangle>\lambda\right)\leq
e^{-N\lambda
z_{1}}\mathbb{E}\left[e^{Nz_{1}\langle|v|^{2},\mu^{N}_{0}\rangle}\right]=\exp\left(-N(\lambda
z_{1}-\log\mathcal{E}_{z_{1}}(\mu^{\star}_{0}))\right)$

where, in the right-hand side, we recall that $\mu^{N}_{0}$ is given by
sampling particles independently from $\mu^{\star}_{0}$. Taking
$\lambda_{M}=z_{1}^{-1}(M+\log\mathcal{E}_{z_{1}}(\mu^{\star}_{0}))$, we
conclude that

$\mathbb{P}\left(\mu^{N}_{0}\not\in\mathcal{P}_{2,\lambda_{M}}\right)=\mathbb{P}\left(\langle|v|^{2},\mu^{N}_{0}\rangle>\lambda_{M}\right)\leq
e^{-MN}.$

To extend this to the whole process we note that the kinetic energy
$\langle|v|^{2},\mu^{N}_{t}\rangle$ is constant in time, so that
$\mu^{N}_{t}\in\mathcal{D}_{\lambda_{M}}$ if, and only if,
$\mu^{N}_{0}\in\mathcal{P}_{2,\lambda_{M}}$. Therefore, for any $t\in[0,T]$,

$\mathbb{P}\left(\mu^{N}_{\bullet}\not\in\mathcal{D}_{\lambda_{M}}\right)=\mathbb{P}\left(\mu^{N}_{0}\not\in\mathcal{P}_{2,\lambda_{M}}\right)\leq
e^{-MN}$

and the first item follows. For the second item, we observe that the
instantaneous rate of the Kac process, in either the hard spheres or Maxwell
molecules case, is bounded by

$\begin{split}N\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{S}^{d-1}}B(v-v_{\star})\mu^{N}_{t}(dv)\mu^{N}_{t}(dv_{\star})d\sigma&\leq
N\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(3+|v|^{2}+|v_{\star}|^{2})\mu^{N}_{t}(dv)\mu^{N}_{t}(dv_{\star})\\\\[4.30554pt]
&\leq 3N(1+\langle|v|^{2},\mu^{N}_{0}\rangle)\end{split}$ (41)

where we note that, for either kernel, $B(|v-v_{\star}|)\leq
1+|v|+|v_{\star}|\leq 3+|v|^{2}+|v_{\star}|^{2}$, and in the final inequality,
we recall again that the second moment $\langle|v|^{2},\mu^{N}_{t}\rangle$ is
independent of time. It therefore follows that we can construct a time-
homogenous Poisson process $\widetilde{w}^{N}_{t}$, of constant, random rate
$3N(1+\langle|v|^{2},\mu^{N}_{0}\rangle)$, such that $\widetilde{w}^{N}_{t}$
has jumps on a superset of the times when $\mu^{N}_{t}$ jumps. This leads to
the bound, for any $s\leq t$,

$(w^{N}_{t}-w^{N}_{s})([0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{S}^{d-1})\leq\frac{1}{N}(\widetilde{w}^{N}_{t}-\widetilde{w}^{N}_{s}).$

For each $\delta$, we now pick a partition $0=t_{0}<t_{1}...<t_{m}$ of size
$\lceil T/\delta\rceil$ by taking constant steps of size $\delta$. We now
observe that $W(\mu^{N}_{t},\mu^{N}_{t-})\leq 4/N$ at times when $\mu^{N}_{t}$
jumps, and for any $|s-t|\leq\delta$, the interval $[s,t]$ is contained in at
most two adjacent intervals $[t_{i-1},t_{i+1}]$. Together, we conclude that

$q^{N}(\delta)\leq
8\max_{i}\frac{1}{N}(\widetilde{w}^{N}_{t_{i}}-\widetilde{w}^{N}_{t_{i-1}}).$
(42)

For $\lambda_{M}$ as above and for any $z>0$, we bound

$\begin{split}\mathbb{P}\left(\frac{1}{N}(\widetilde{w}^{N}_{t_{i}}-\widetilde{w}^{N}_{t_{i-1}})>\frac{\epsilon}{8}\hskip
2.84544pt\bigg{|}\hskip
2.84544pt\mu^{N}_{0}\in\mathcal{P}_{2,\lambda_{M}}\right)&\leq
e^{-zN\epsilon/8}\mathbb{E}\left[e^{z(\widetilde{w}^{N}_{t_{i}}-\widetilde{w}^{N}_{t_{i-1}})}\hskip
2.84544pt\bigg{|}\hskip
2.84544pt\mu^{N}_{0}\in\mathcal{P}_{2,\lambda_{M}}\right]\\\
&\leq\exp\left(-N\left(\frac{z\epsilon}{8}-3(1+\lambda_{M})\delta(e^{z}-1)\right)\right)\end{split}$
(43)

where, in the second line, we use the bound that the rate of
$\widetilde{w}^{N}$ is at most $3N(1+\lambda_{M})$ if
$\mu^{N}_{0}\in\mathcal{P}_{2,\lambda_{M}}$. We now choose
$z=8(M+1)/\epsilon$, and $\delta>0$ small enough, depending on
$z,\lambda_{M}$, so that $3(1+\lambda_{M})\delta(e^{z}-1)<1$. For this choice
of $\delta$, the final expression in (43) is $e^{-NM}$, for each interval.
Finally, we take a union bound:

$\\{q^{N}(\delta)>\epsilon\\}\subset\\{\mu^{N}_{0}\not\in\mathcal{P}_{2,\lambda_{M}}\\}\cup\bigcup_{i\leq\lceil
T/\delta\rceil}\left\\{\widetilde{w}^{N}_{t_{i}}-\widetilde{w}^{N}_{t_{i-1}}>\frac{N\epsilon}{8},\hskip
2.84544pt\mu^{N}_{0}\in\mathcal{P}_{2,\lambda_{M}}\right\\}.$

By the choices of $\lambda_{M}$ and $\delta$,

$\mathbb{P}\left(q^{N}(\delta)>\epsilon\right)\leq(1+\lceil
T/\delta\rceil)e^{-NM}$

and the second item now follows. The final item follows in exactly the same
way: following (43), for all $C>0$ we bound

$\begin{split}\mathbb{P}\left(w^{N}(E)>C\big{|}\mu^{N}_{0}\in\mathcal{P}_{2,\lambda_{M}}\right)&\leq\mathbb{P}\left(\widetilde{w}^{N}_{T}>CN\big{|}\mu^{N}_{0}\in\mathcal{P}_{2,\lambda_{M}}\right)\\\\[4.30554pt]
&\leq
e^{-CN}\mathbb{E}\left[e^{\widetilde{w}^{N}_{T}}\big{|}\mu^{N}_{0}\in\mathcal{P}_{2,\lambda_{M}}\right]\\\\[4.30554pt]
&\leq\exp\bigg{(}-N\big{(}C-3(1+\lambda_{M})T(e-1)\big{)}\bigg{)}\end{split}$
(44)

and choosing $C=M+3(1+\lambda_{M})T(e-1)$ makes the final probability at most
$e^{-MN}$. Using a union bound,

$\mathbb{P}\left(w^{N}(E)>C)\right)\leq\mathbb{P}\left(w^{N}(E)>C\big{|}\mu^{N}_{0}\in\mathcal{P}_{2,\lambda_{M}}\right)+\mathbb{P}\left(\mu^{N}_{0}\not\in\mathcal{P}_{2,\lambda_{M}}\right)\leq
2e^{-MN}$ (45)

from which (40) follows. ∎

Let us also record, for later use, the following corollary.

######  Corollary 2.2.

Let $\mathbb{Q}^{N}\ll\mathbb{P}$ be changes of measure such that

$\lim_{a\to\infty}\liminf_{N}\hskip
2.84544pt\mathbb{Q}^{N}\left(\frac{d\mathbb{Q}^{N}}{d\mathbb{P}}\leq
e^{Na}\right)=1.$ (46)

Then the laws of $(\mu^{N}_{\bullet},w^{N})$ under $\mathbb{Q}^{N}$ are tight:
for all $\epsilon>0$ there exists a compact set
$\mathcal{K}\subset\mathcal{D}\times\mathcal{M}(E)$ such that

$\sup_{N}\hskip
2.84544pt\mathbb{Q}^{N}\left((\mu^{N}_{\bullet},w^{N})\not\in\mathcal{K}\right)<\epsilon.$
(47)

###### Proof.

This follows from Proposition 1.1i) and the hypothesis (46) by purely general
considerations. Let us fix $\epsilon>0$; thanks to (46) we can choose $a$ such
that, for all but finitely many $N$,

$\mathbb{Q}^{N}\left(\frac{d\mathbb{Q}^{N}}{d\mathbb{P}}>e^{Na}\right)<\frac{\epsilon}{2}$
(48)

and, changing $a$ if necessary, we can arrange that (48) holds for all $N$. We
now choose $M=a-\log(\epsilon/2)$; by Proposition 1.1i), there exists a
compact set $\mathcal{K}$ such that

$\limsup_{N}N^{-1}\log\mathbb{P}((\mu^{N}_{\bullet},w^{N})\not\in\mathcal{K})<-M$
(49)

which implies that, for all $N$ sufficiently large,

$\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\not\in\mathcal{K}\right)\leq
e^{-MN}.$ (50)

Since the space $\mathcal{D}\times\mathcal{M}(E)$ is a separable metric space,
each $(\mu^{N}_{\bullet},w^{N})$ is tight so $\mathcal{K}$ can be replaced
with a larger compact set such that (50) again holds for all $N$. Together,
(48, 50) imply that

$\begin{split}\mathbb{Q}^{N}\left((\mu^{N}_{\bullet},w^{N})\not\in\mathcal{K}\right)&\leq\mathbb{Q}^{N}\left(\frac{d\mathbb{Q}^{N}}{d\mathbb{P}}\leq
e^{Na},(\mu^{N}_{\bullet},w^{N})\not\in\mathcal{K}\right)+\mathbb{Q}^{N}\left(\frac{d\mathbb{Q}^{N}}{d\mathbb{P}}>e^{Na}\right)\\\\[4.30554pt]
&<e^{Na}\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\not\in\mathcal{K}\right)+\frac{\epsilon}{2}\leq
e^{Na}e^{-N(a+\log(\epsilon/2))}+\frac{\epsilon}{2}\leq\epsilon\end{split}$
(51)

and we are done. ∎

###  2.2 Variational Formulation of the Rate Function

In preparation for the upper bound, we will now present a variational
formulation of the rate function. This will also allow us to prove the lower
semicontinuity in Proposition 1.1. We are aided in this equivalence by the
inclusion of the flux in the large deviation principle: the choice of $K$, if
it exists, is unique, which allows us to significantly simplify the proof of
Léonard [31].

We begin with the following construction. Let us write
$C^{1,1}_{0,b}([0,T]\times\mathbb{R}^{d})$ for those functions
$f:[0,T]\times\mathbb{R}^{d}\to\mathbb{R}$ which are bounded and Lipschitz in
the $v$-variable, with a bounded and $v$-Lipschitz time derivative
$\partial_{t}f$, and such that $f_{0}=0$. For $\varphi\in
C_{b}(\mathbb{R}^{d}),f\in C^{1,1}_{0,b}([0,T]\times\mathbb{R}^{d})$ and $g\in
C_{c}(E)$ and $t\in[0,T]$, we define

$\Xi(\mu_{\bullet},w,\varphi,f,g)_{t}=\Xi_{0}(\mu_{\bullet},\varphi)+\Xi_{1,t}(\mu_{\bullet},w,f)+\Xi_{2,t}(\mu_{\bullet},w,g)$
(52)

where

$\Xi_{0}(\mu_{\bullet},\varphi)=\langle\varphi,\mu_{0}\rangle-\log\langle
e^{\varphi},\mu^{\star}_{0}\rangle;$ (53)
$\begin{split}\Xi_{1,t}(\mu_{\bullet},w,f):&=\langle
f_{t},\mu_{t}\rangle-\int_{0}^{t}\langle\partial_{s}f_{s},\mu_{s}\rangle ds\\\
&\hskip
28.45274pt-\int_{(0,t]\times\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{S}^{d-1}}\Delta
f(s,v,v_{\star},\sigma)w(ds,dv,dv_{\star},d\sigma)\end{split}$ (54)

and

$\begin{split}\Xi_{2,t}(\mu_{\bullet},w,g)&:=\int_{E}1(s\leq
t)(g(s,v,v_{\star},\sigma)w(ds,dv,dv_{\star},d\sigma)\\\ &\hskip
28.45274pt-\int_{E}(e^{g}-1)(s,v,v_{\star},\sigma)\overline{m}_{\mu}(ds,dv,dv_{\star},d\sigma)).\end{split}$
(55)

We write $\Xi(\mu_{\bullet},w,\varphi,f,g)$ for the terminal value
$\Xi(\mu_{\bullet},w,\varphi,f,g)=\Xi(\mu_{\bullet},w,\varphi,f,g)_{T}$. Let
us note that these processes make sense at the level of the particle system
$\mu^{N}_{\bullet},w^{N}$. The function $f$ here entering into
$\Xi_{1,t}(\mu_{\bullet},w,f)$ will play the rôle of a Lagrange multiplier to
enforce the constraint of the continuity equation. This is made precise by the
following result.

######  Lemma 2.3.

Fix $(\mu_{\bullet},w)\in\mathcal{D}\times\mathcal{M}(E)$. Then

$\sup\left\\{\Xi_{1,T}(\mu_{\bullet},w,f):f\in
C^{1,1}_{0,b}([0,T]\times\mathbb{R}^{d})\right\\}=\begin{cases}0&\text{if
}(\mu_{\bullet},w)\text{ solves (\ref{eq: CE})};\\\
\infty&\text{else.}\end{cases}$ (56)

###### Proof.

For the case where $(\mu_{\bullet},w)$ solves (CE), we will show that, for all
$f\in C^{1,1}_{0,b}([0,T]\times\mathbb{R}^{d})$ and all $t\in[0,T]$, we have
the time-dependent equivalent of (CE):

$\langle f,\mu_{t}\rangle=\int_{0}^{t}\langle\partial_{s}f_{s},\mu_{s}\rangle
ds+\int_{E}1(s\leq t)\Delta
f(s,v,v_{\star},\sigma)w(ds,dv,dv_{\star},d\sigma).$ (57)

This will immediately imply that $\Xi_{1,t}(\mu_{\bullet},w,f)=0$ for all $f$
and all $t$, which implies the claim. The proof of this formulation is
slightly complicated by the lack of regularity, since we only assume _a
priori_ that $\mu_{\bullet}$ is càdlàg rather than continuous; we will instead
use the facts about cádlág paths from Proposition A.1. Since (57) is linear in
$f$, we can assume that $f_{t},\partial_{t}f_{t}$ belong to the class
$\mathcal{F}$ which are $1$-bounded and $1$-Lipschitz. Fix $t\in[0,T]$,
$\epsilon>0$, $\delta>0$.

Let us write $E\subset[0,T]$ for those $s\in[0,T]$ with
$W(\mu_{s-},\mu_{s})\geq\epsilon$; thanks to Proposition A.1a), $E$ is finite,
and we write $m=|E|<\infty$ for its cardinality. Possibly making $\delta>0$
smaller, Proposition A.1b) guarantees that $\delta$ can be chosen so that any
interval $[u,v)$ of length $\delta$ either contains a point of $E$, or for all
$s\in[u,v)$, $W(\mu_{s},\mu_{u})<\epsilon$. Now, for such $\delta$, we
decompose $(0,t]$ into intervals $(t_{i},t_{i+1}]$ of length at most $\delta$,
and add

$\langle
f_{t_{i+1}},\mu_{t_{i+1}}-\mu_{t_{i}}\rangle=\int_{E}1(s\in(t_{i},t_{i+1}])\Delta
f(t_{i+1},v,v_{\star},\sigma)w(ds,dv,dv_{\star},d\sigma);$ (58) $\langle
f_{t_{i+1}}-f_{t_{i}},\mu_{t_{i}}\rangle=\int_{t_{i}}^{t_{i+1}}\langle\partial_{s}f_{s},\mu_{t_{i}}\rangle
ds$ (59)

to obtain

$\begin{split}\langle f_{t_{i+1}},\mu_{t_{i+1}}\rangle-\langle
f_{t_{i}},\mu_{t_{i}}\rangle&=\int_{t_{i}}^{t_{i+1}}\langle\partial_{s}f_{s},\mu_{t_{i}}\rangle
ds\\\ &+\int_{E}1(s\in(t_{i},t_{i+1}])\Delta
f(t_{i+1},v,v_{\star},\sigma)w(ds,dv,dv_{\star},d\sigma).\end{split}$ (60)

We approximate the two terms by

$\begin{split}\left|\int_{t_{i}}^{t_{i+1}}\langle\partial_{s}f_{s},\mu_{t_{i}}\rangle
ds-\int_{t_{i}}^{t_{i+1}}\langle\partial_{s}f_{s},\mu_{s}\rangle
ds\right|&\leq\int_{t_{i}}^{t_{i+1}}W(\mu_{s},\mu_{t_{i}})ds\\\\[4.30554pt]
&\leq\epsilon(t_{i+1}-t_{i})+2\delta\cdot
1(E\cap[t_{i},t_{i+1})\neq\emptyset)\end{split}$ (61)

since, by the choice of $\delta$, wither $W(\mu_{s},\mu_{t_{i}})\leq\epsilon$
for all $s\in[t_{i},t_{i+1}]$, or there is a point of $E$ in
$[t_{i},t_{i+1})$, in which case we use the trivial bound
$W(\mu_{s},\mu_{t_{i}})\leq 2$ and recall that the interval is of length at
most $\delta$. For the second term

$\begin{split}&\left|\int_{E}1(s\in(t_{i},t_{i+1}])\left(\Delta
f(t_{i+1},v,v_{\star},\sigma)-\Delta
f(s,v,v_{\star},\sigma)\right)w(ds,dv,dv_{\star},d\sigma)\right|\\\ &\hskip
28.45274pt\hskip 28.45274pt\hskip 28.45274pt\hskip
28.45274pt\leq\left|\int_{E}1(s\in(t_{i},t_{i+1}])4\|f_{t_{i+1}}-f_{s}\|_{\infty}w(ds,dv,dv_{\star},d\sigma)\right|\\\\[4.30554pt]
&\hskip 113.81102pt\leq 4\delta
w\left((t_{i},t_{i+1}]\times\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{S}^{d-1}\right)\end{split}$
(62)

where in the final line we recall that we have scaled so that
$\|\partial_{t}f_{t}\|_{\infty}\leq 1$. Adding, we conclude that

$\begin{split}&\bigg{|}\langle f_{t_{i+1}},\mu_{t_{i+1}}\rangle-\langle
f_{t_{i}},\mu_{t_{i}}\rangle-\int_{t_{i}}^{t_{i+1}}\langle\partial_{s}f_{s},\mu_{s}\rangle\\\
&\hskip 113.81102pt-\int_{E}1(s\in(t_{i},t_{i+1}])\Delta
f(s,v,v_{\star},\sigma)w(ds,dv,dv_{\star},d\sigma)\bigg{|}\\\
&\leq\epsilon(t_{i+1}-t_{i})+2\delta\cdot
1\left(E\cap[t_{i},t_{i+1})\neq\emptyset\right)+4\delta
w\left((t_{i},t_{i+1}]\times\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{S}^{d-1}\right).\end{split}$
(63)

Summing over all such intervals $(t_{i},t_{i+1}]$ covering $(0,t]$, and
recalling that $f_{0}\equiv 0$, we obtain

$\begin{split}&\left|\langle
f_{t},\mu_{t}\rangle-\int_{0}^{t}\langle\partial_{s}f_{s},\mu_{s}\rangle
ds-\int_{E}1(s\leq t)\Delta
f(s,v,v_{\star},\sigma)w(ds,dv,dv_{\star},d\sigma)\right|\\\ &\hskip
85.35826pt\leq\epsilon t+2m\delta+4\delta w(E)\end{split}$ (64)

and the right-hand side can be made arbitrarily small by taking
$\epsilon,\delta\to 0$, recalling that $w$ is a finite measure by hypothesis,
so the claim (57) is proven, and we conclude that
$\sup_{f}\Xi_{1,T}(\mu_{\bullet},w,f)=0$ as claimed. Otherwise, if (CE) fails,
there exists some $t_{0}\in(0,T]$ and some $g\in
C^{\infty}_{c}(\mathbb{R}^{d})$ such that

$\langle g,\mu_{t_{0}}\rangle-\langle g,\mu_{0}\rangle-\int_{E}1(s\leq
t_{0})\Delta g(v,v_{\star},\sigma)w(ds,dv,dv_{\star},d\sigma)>1.$ (65)

Let us assume that $t_{0}\in(0,T)$; the proof $t_{0}=T$ is similar and
strictly easier. We now fix a smooth, increasing function
$\chi:\mathbb{R}\to\mathbb{R}$ such that $\chi=0$ on $(-\infty,0]$ and
$\chi=1$ on $[1,\infty)$, and for $0<\delta<\min(t_{0},T-t_{0})$, we construct
$f^{\delta}\in C^{1,1}_{0,b}([0,T]\times\mathbb{R}^{d})$ by defining

$f^{\delta}_{t}(v):=\begin{cases}\chi(t/\delta)g(v)&\text{if
}t\in[0,\delta];\\\ g(v)&\text{if }\delta<t\leq t_{0};\\\
\chi(1-(t-t_{0})/\delta)g(v)&\text{if }t_{0}<t<t_{0}+\delta\\\
0&\text{else}.\end{cases}$ (66)

Thanks to right-continuity of $\mu_{t}$, we observe that
$\int_{0}^{T}\langle\partial_{t}f^{\delta}_{t},\mu_{t}\rangle dt\to\langle
g,\mu_{0}\rangle-\langle g,\mu_{t_{0}}\rangle$, and using dominated
convergence,

$\int_{E}\Delta
f^{\delta}(s,v,v_{\star},\sigma)w(ds,dv,dv_{\star},d\sigma)\to\int_{E}1(s\leq
t_{0})\Delta g(v,v_{\star},\sigma)w(ds,dv,dv_{\star},d\sigma).$ (67)

Therefore, $\Xi_{1,T}(\mu_{\bullet},w,f^{\delta})$ converges to

$\Xi_{1,T}(\mu_{\bullet},w,f^{\delta})\to\langle g,\mu_{t_{0}}\rangle-\langle
g,\mu_{0}\rangle-\int_{E}1(s\leq t_{0})\Delta
g(v,v_{\star},\sigma)w(ds,dv,dv_{\star},d\sigma)$ (68)

and in particular, we can choose $\delta>0$ small enough that
$\Xi_{1,T}(\mu_{\bullet},w,f^{\delta})>1$. By linearity, for all $\lambda>0$,
$\Xi_{1,T}(\mu_{\bullet},w,\lambda f^{\delta})>\lambda$, and so the supremum
is infinite, as claimed. ∎

We now use this equality to show that the functions $\Xi$ above give a
variational formulation of the rate function $\mathcal{I}$ given in the
introduction.

######  Lemma 2.4.

For $(\mu_{\bullet},w)\in\mathcal{D}\times\mathcal{M}(E)$, we have

$\mathcal{I}(\mu_{\bullet},w)=\sup\left\\{\Xi(\mu_{\bullet},w,\varphi,f,g):\varphi\in
C_{b}(\mathbb{R}^{d}),f\in C^{1,1}_{0,b}([0,T]\times\mathbb{R}^{d}),g\in
C_{c}(E)\right\\}.$ (69)

###### Proof.

Let us write $\widetilde{\mathcal{I}}$ for the right-hand side. Since
$\Xi_{0}$ depends only on $\varphi$, $\Xi_{1}$ only on $f$ and $\Xi_{2}$ only
on $g$, the supremum decomposes as

$\widetilde{\mathcal{I}}(\mu_{\bullet},w)=\sup_{\varphi}\Xi_{0}(\mu_{\bullet},\varphi)+\sup_{f}\Xi_{1,T}(\mu_{\bullet},w,f)+\sup_{g}\Xi_{2,T}(\mu_{\bullet},w,g)$
(70)

where the suprema run over the same sets as above. Optimising over $\varphi$
produces the well-known variational formulation
$\sup_{\varphi}\Xi_{0}(\mu_{0},\varphi)=H(\mu_{0}|\mu_{0}^{\star})$ of the
relative entropy. This identity can be found in [29, Appendix 1], or derived
using essentially the same argument as for $\Xi_{2,T}$ below. Thanks to Lemma
2.3, the supremum over $f$ is infinite unless the continuity equation (CE)
holds, in which case this term vanishes.

We now deal with the third term. If $w\not\ll\overline{m}_{\mu}$, there is a
compact set $E^{\prime}\subset E$ with $w(E^{\prime})>0$ but
$\overline{m}_{\mu}(E^{\prime})=0$, and by since $E$ is a metric space and $w$
is a Borel measure, we can find open $U_{n}\downarrow E^{\prime}$ and closed
$U_{n}\supset A_{n}\supset E^{\prime}$ with
$\overline{m}_{\mu}(U_{n})\downarrow 0$. We now choose $g_{n}\in C_{c}(E)$ so
that $0\leq g_{n}\leq 1$, $g_{n}=1$ on $A_{n}$, and $=0$ except on $U_{n}$,
and bound for $\lambda>0$,

$\int_{E}\lambda g_{n}w(ds,dv,dv_{\star},d\sigma)\geq\lambda w(E^{\prime});$
(71) $\int_{E}(e^{\lambda
g_{n}}-1)\overline{m}_{\mu}(ds,dv,dv_{\star},d\sigma)\leq(e^{\lambda}-1)\overline{m}_{\mu}(U_{n})$
(72)

so that

$\Xi_{2,T}(\mu_{\bullet},w,\lambda g_{n})\geq\lambda
w(E^{\prime})-(e^{\lambda}-1)\overline{m}_{\mu}(U_{n}).$ (73)

By taking $\lambda=\lambda_{n}\to\infty$ slowly enough, the right-hand side
can be made arbitrarily large as $n\to\infty$, so in this case
$\sup_{g}\Xi_{2,T}(\mu_{\bullet},w,g)=\infty$. On the other hand, if
$w\ll\overline{m}_{\mu}$, let us write $K$ for the tilting function
$\frac{dw}{d\overline{m}_{\mu}}$, so that

$\Xi_{2,T}(\mu_{\bullet},w,g)=\int_{E}(Kg-e^{g}+1)(s,v,v_{\star},\sigma)\overline{m}_{\mu}(ds,dv,dv_{\star},d\sigma).$
(74)

Observing that, for all $x\in\mathbb{R},y\geq 0$, it holds that
$xy\leq(e^{x}-1)+(y\ln y-y+1)=(e^{x}-1)+\tau(y)$, the first term $\int Kg$ can
be bounded by

$\begin{split}\int_{E}Kg(s,v,v_{\star},\sigma)\overline{m}_{\mu}(ds,dv,dv_{\star},d\sigma)&\leq\int_{E}\tau(K)\overline{m}_{\mu}(ds,dv,dv_{\star},d\sigma)\\\
&+\int_{E}(e^{g}-1)\overline{m}_{\mu}(ds,dv,dv_{\star},d\sigma)\end{split}$
(75)

which leads to the bound, uniformly in $g\in C_{c}(E)$,

$\Xi_{2,T}(\mu_{\bullet},w,g)\leq\int_{E}\tau(K(s,v,v_{\star},\sigma))\overline{m}_{\mu}(ds,dv,dv_{\star},d\sigma)$
(76)

whether or not the right-hand side is finite. On the other hand, let us fix
$M$. By Lusin’s theorem, we can construct continuous, bounded $g_{n}\in
C_{c}(E)$ with $g_{n}\to\ln K\land M$ for $\overline{m}_{\mu}$-almost all
$(t,v,v_{\star},\sigma)$, so that

$Kg_{n}-e^{g_{n}}+1\leq MK+1$ (77)

and

$Kg_{n}-e^{g_{n}}+1\to K(\ln K\land M)-(K\land e^{M})+1$ (78)

$\overline{m}_{\mu}(ds,dv,dv_{\star},d\sigma)$-almost everywhere. Since $K\in
L^{1}(\overline{m}_{\mu})$, we can use dominated convergence to obtain

$\Xi_{2,T}(\mu_{\bullet},w,g_{n})\to\int_{E}(K(\ln K\land M)-(K\land
e^{M})+1)\overline{m}_{\mu}(ds,dv,dv_{\star},d\sigma)$ (79)

and the supremum is at least the right-hand side. The integrand is increasing
in $M$, and converges to $\tau(K)$ pointwise, so the whole integral converges
to $\int_{E}\tau(K)d\overline{m}_{\mu}$. We conclude that

$\sup\left\\{\Xi_{2,T}(\mu_{\bullet},w,g):g\in
C_{c}(E)\right\\}\geq\int_{E}\tau(K(s,v,v_{\star},\sigma))\overline{m}_{\mu}(ds,dv,dv_{\star},d\sigma)$
(80)

and (76) shows that this is an equality. Putting everything together, we have
shown that

$\begin{split}&\sup_{f,g}\left\\{\Xi_{1,T}(\mu_{\bullet},w,f)+\Xi_{2,T}(\mu_{\bullet},w,g)\right\\}\\\
&\hskip
56.9055pt=\begin{cases}\int_{E}\tau(K)\overline{m}_{\mu}(ds,dv,dv_{\star},d\sigma)&\text{if
}(\mu_{\bullet},w)\text{is a measure flux pair;}\\\
\infty&\text{else}\end{cases}\end{split}$ (81)

and the right-hand side is exactly the definition of
$\mathcal{J}(\mu_{\bullet},w)$; returning to (70), we have proven that

$\widetilde{\mathcal{I}}(\mu_{\bullet},w)=H(\mu_{0}|\mu^{\star}_{0})+\mathcal{J}(\mu_{\bullet},w)=\mathcal{I}(\mu_{\bullet},w)$
(82)

as desired.∎

Thanks to this variational form, we readily obtain the lower semi-continuity
claimed in Proposition 1.1. We first record, as a separate lemma, a result
which will be helpful later.

######  Lemma 2.5.

For fixed $f\in L^{\infty}([0,T],C_{b}(\mathbb{R}^{d}))$ and $g\in C_{c}(E)$,
the maps

$\mu_{\bullet}\mapsto\int_{0}^{T}\langle f_{t},\mu_{t}\rangle
dt;\qquad\mu_{\bullet}\mapsto\int_{E}g\overline{m}_{\mu}(ds,dv,dv_{\star},d\sigma)$
(83)

are continuous in the topology of $\mathcal{D}$.

###### Proof.

Noting that the topology of $\mathcal{D}$ is induced by a metric, it is
sufficient to prove sequential continuity: let us fix
$\mu^{(n)}_{\bullet}\to\mu_{\bullet}$. By Proposition A.2, it follows that
$W(\mu^{(n)}_{t},\mu_{t})\to 0$ for $dt$-almost all $t$, and for all such $t$,
we also have the weak convergence
$\mu^{(n)}_{t}\otimes\mu^{(n)}_{t}\to\mu_{t}\otimes\mu_{t}$. Since $g$ has
compact support in $E$, for any fixed $\sigma$ and for such $t$, the map
$(v,v_{\star})\mapsto g(t,v,v_{\star},\sigma)B(v-v_{\star})$ is bounded and
continuous, and so we have the convergences

$\langle f_{t},\mu^{(n)}_{t}\rangle\to\langle f_{t},\mu_{t}\rangle;$ (84)
$\begin{split}&\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}g(t,v,v_{\star},\sigma)B(v-v_{\star})\mu^{(n)}_{t}(dv)\mu^{(n)}_{t}(dv_{\star})\\\
&\hskip
142.26378pt\to\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}g(t,v,v_{\star},\sigma)B(v-v_{\star})\mu_{t}(dv)\mu_{t}(dv_{\star}).\end{split}$
(85)

Since these hold for all $\sigma$ and $dt$-almost all $t$, we can integrate
and use bounded convergence to find that

$\int_{0}^{T}\langle f_{t},\mu^{(n)}_{t}\rangle dt\to\int_{0}^{T}\langle
f_{t},\mu_{t}\rangle dt;$ (86)

and

$\int_{E}g\overline{m}_{\mu^{(n)}}(ds,dv,dv_{\star},d\sigma)\to\int_{E}g\overline{m}_{\mu}(ds,dv,dv_{\star},d\sigma)$
(87)

and we are done. ∎

######  Lemma 2.6.

For fixed $\varphi\in C_{b}(\mathbb{R}^{d}),f\in
C^{1,1}_{0,b}([0,T]\times\mathbb{R}^{d})$ and $g\in C_{c}(\mathbb{R}^{d})$,
the maps

$(\mu_{\bullet},w)\to\Xi_{0}(\mu_{\bullet},\varphi);\qquad(\mu_{\bullet},w)\to\Xi_{1,T}(\mu_{\bullet},\varphi);\qquad(\mu_{\bullet},w)\to\Xi_{2,T}(\mu_{\bullet},\varphi);$
(88) $(\mu_{\bullet},w)\to\Xi(\mu_{\bullet},w,\varphi,f,g)$ (89)

are continuous for the topology of $\mathcal{D}\times\mathcal{M}(E)$. In
particular, the sub-level sets $\\{\mathcal{J}\leq a\\},\\{\mathcal{I}\leq
a\\}$ are closed in $\mathcal{D}\times\mathcal{M}(E)$ for all
$a\in[0,\infty)$, as is the set of pairs $(\mu_{\bullet},w)$ for which (CE)
holds, and $\\{\mu\in\mathcal{P}_{2}:H(\mu|\mu_{0}^{\star})\leq a\\}$ in the
topology of $(\mathcal{P}_{2},W)$.

###### Proof.

With the choices of topologies on $\mathcal{D},\mathcal{M}(E)$, the maps

$w\mapsto\int_{E}gw(ds,dv,dv_{\star},d\sigma);\qquad w\mapsto\int_{E}\Delta
f(s,v,v_{\star},\sigma)w(ds,dv,dv_{\star},d\sigma)$

are immediately continuous, and thanks to Proposition A.2a), so are
$\mu_{\bullet}\mapsto\langle\varphi,\mu_{0}\rangle;\mu_{\bullet}\mapsto\langle
f_{T},\mu_{T}\rangle$. Combining with Lemma 2.5, with $f_{t}$ replaced by
$\partial_{t}f_{t}$, each expression appearing in the definitions of
$\Xi_{i,T}$ is continuous, and we conclude the claimed continuity of the
stated maps. For the second point, we use Lemma 2.4 to write the sublevel
sets, for any $a\in[0,\infty]$, as

$\\{\mathcal{I}\leq a\\}=\bigcap_{\varphi\in C_{b}(\mathbb{R}^{d}),f\in
C^{1,1}_{0,b}([0,T]\times\mathbb{R}^{d}),g\in
C_{c}(E)}\left\\{(\mu_{\bullet},w):\Xi(\mu_{\bullet},w,\varphi,f,g)_{T}\leq
a\right\\}.$ (90)

Each set in the intersection is closed, and hence so is the left-hand side,
which proves lower semi-continuity; the assertions for $\mathcal{J}$ and
$H(\cdot|\mu^{\star}_{0})$ are identical, recalling (81) and that
$H(\mu|\mu^{\star}_{0})=\sup_{\varphi}\Xi_{0}(\varphi,\mu_{0})$. The remaining
assertion is similar: using Lemma 2.3,

$\left\\{(\mu_{\bullet},w):\text{(\ref{eq: CE}) holds}\right\\}=\bigcap_{f\in
C^{1,1}_{0,b}([0,T]\times\mathbb{R}^{d})}\left\\{(\mu_{\bullet},w):\Xi_{1,T}(\mu_{\bullet},w,f)=0\right\\}$
(91)

which is an intersection of closed sets, and hence closed. ∎

###  2.3 Upper Bound

Using the variational formulation above, we now prove the upper bound in
Theorem 1.2. We begin with a local version of the result.

######  Lemma 2.7.

Fix $(\mu_{\bullet},w)\in\mathcal{D}\times\mathcal{M}(E)$ with finite rate
$\mathcal{I}(\mu_{\bullet},w)<\infty$, and fix $\epsilon>0$. Then there exists
an open set $\mathcal{U}\ni(\mu_{\bullet},w)$ such that

$\limsup_{N}\frac{1}{N}\log\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\in\mathcal{U}\right)\leq-\mathcal{I}(\mu_{\bullet},w)+\epsilon.$
(92)

If instead $\mathcal{I}(\mu_{\bullet},w)=\infty$ and $M<\infty$ then there
exists an open set $\mathcal{U}\ni(\mu_{\bullet},w)$ such that

$\limsup_{N}\frac{1}{N}\log\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\in\mathcal{U}\right)\leq-M.$
(93)

###### Proof.

Let us consider the first case; the second is essentially identical. Thanks to
Lemma 2.4, we can choose $\varphi\in C_{b}(\mathbb{R}^{d}),f\in
C^{1,1}_{0,b}([0,T]\times\mathbb{R}^{d})$ and $g\in C_{c}(E)$ such that

$\mathcal{I}(\mu_{\bullet},w)<\Xi(\mu_{\bullet},w,\varphi,f,g)+\frac{\epsilon}{2}$
(94)

and, thanks to Lemma 2.6, we can find open $\mathcal{U}\ni(\mu_{\bullet},w)$
such that, for all $(\mu^{\prime}_{\bullet},w^{\prime})\in\mathcal{U}$, we
have

$\Xi(\mu^{\prime}_{\bullet},w^{\prime}\varphi,f,g)>\Xi(\mu_{\bullet},w,\varphi,f,g)-\frac{\epsilon}{2}>\mathcal{I}(\mu_{\bullet},w)-\epsilon.$
(95)

We consider the processes

$Z^{N}_{t}:=\exp\left(N\Xi(\mu^{N}_{\bullet},w,\varphi,f,g)_{t}\right).$ (96)

We first observe that, since $\mu^{N}_{\bullet},w^{N}$ satisfy the continuity
equation (CE), Lemma 2.3 shows that, for all $t\geq 0$,

$\Xi_{1,t}(\mu^{N}_{\bullet},w^{N},\varphi,f,g)=0.$ (97)

Next, we show that $Z^{N}$ is a martingale, following arguments of [13]. We
observe that at points $(t,v,v_{\star},\sigma)$ of $w^{N}$, $Z^{N}_{t}$ jumps
by

$Z^{N}_{t}-Z^{N}_{t-}=Z^{N}_{t-}\left(e^{g(t,v,v_{\star},\sigma)}-1\right)$
(98)

while between jumps, $Z^{N}_{t}$ is differentiable and

$\partial_{t}Z^{N}_{t}=-N\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{S}^{d-1}}Z^{N}_{t}\left(e^{g(t,v,v_{\star},\sigma)}-1\right)B(v-v_{\star})\mu^{N}_{t}(dv)\mu^{N}_{t}(dv_{\star})d\sigma.$
(99)

Together, $Z^{N}_{t}$ admits the representation

$Z^{N}_{t}=Z^{N}_{0}+\int_{(0,t]\times\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{S}^{d-1}}Z^{N}_{s-}\left(e^{g(s,v,v_{\star},\sigma)}-1\right)(w^{N}-\overline{m}_{\mu^{N}})(ds,dv,dv_{\star},d\sigma).$
(100)

Recalling the generator (11), $Z^{N}_{t}$ is a local martingale, and since it
is clearly positive, a supermartingale, and at time $0$,

$\mathbb{E}\left[e^{N\Xi_{0}(\mu_{\bullet},\varphi)}\right]=\frac{\mathbb{E}\left[e^{N\langle\varphi,\mu^{N}_{0}\rangle}\right]}{\langle
e^{\varphi},\mu_{0}^{\star}\rangle^{N}}=1$ (101)

where we recall that $\mu^{N}_{0}$ is formed by independent samples from
$\mu_{0}^{\star}$. We now take the expectation of

$\begin{split}1\left((\mu^{N}_{\bullet},w^{N})\in\mathcal{U}\right)&\leq
Z^{N}_{T}\exp\left(-N\inf\left\\{\Xi(\mu_{\bullet},w,\varphi,f,g):(\mu_{\bullet},w)\in\mathcal{U}\right\\}\right)\\\
&\leq
Z^{N}_{T}\exp\left(-N(\mathcal{I}(\mu_{\bullet},w)-\epsilon)\right)\end{split}$
(102)

to obtain

$\begin{split}\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\in\mathcal{U}\right)&\leq\mathbb{E}[Z^{N}_{T}]\exp\left(-N(\mathcal{I}(\mu_{\bullet},w)-\epsilon)\right)\\\
&\leq\exp\left(-N(\mathcal{I}(\mu_{\bullet},w)-\epsilon)\right)\end{split}$
(103)

to produce the desired result. The case where
$\mathcal{I}(\mu_{\bullet},w)=\infty$ is essentially identical. ∎

We now give the proof of the global upper bound.

###### Proof of Theorem 1.3i).

Let $\mathcal{A}$ be any closed subset of $\mathcal{D}\times\mathcal{M}(E)$
and fix $\epsilon\in(0,1]$. Let us assume that $\mathcal{A}$ is nonempty, and
that $\inf_{\mathcal{A}}\mathcal{I}<\infty$. Choosing
$M=\inf_{\mathcal{A}}\mathcal{I}+1$, by Proposition 1.1i) there exists a
compact set $\mathcal{K}\subset\mathcal{D}\times\mathcal{M}(E)$ such that

$\limsup_{N}\frac{1}{N}\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\not\in\mathcal{K}\right)\leq-M.$
(104)

Now, $\mathcal{A}\cap\mathcal{K}$ is compact, since $\mathcal{A}$ was assumed
to be closed. For all $(\mu_{\bullet},w)\in\mathcal{A}$, we now use Lemma 2.7
to construct $\mathcal{U}(\mu_{\bullet},w)\ni(\mu_{\bullet},w)$: if
$\mathcal{I}(\mu_{\bullet},w)<\infty$ then choose
$\mathcal{U}(\mu_{\bullet},w)$ such that

$\limsup_{N}\frac{1}{N}\log\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\in\mathcal{U}(\mu_{\bullet},w)\right)\leq-\mathcal{I}(\mu_{\bullet},w)+\epsilon$
(105)

or if $\mathcal{I}(\mu_{\bullet},w)=\infty$, then choose
$\mathcal{U}(\mu_{\bullet},w)$ such that

$\limsup_{N}\frac{1}{N}\log\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\in\mathcal{U}(\mu_{\bullet},w)\right)\leq-M.$
(106)

The sets
$\\{\mathcal{U}(\mu_{\bullet},w):(\mu_{\bullet},w)\in\mathcal{A}\cap\mathcal{K}\\}$
are an open cover of $\mathcal{A}\cap\mathcal{K}$, so by compactness we can
find $n<\infty$ and
$(\mu^{(i)}_{\bullet},w^{(i)})\in\mathcal{A}\cap\mathcal{K}$ such that
$\mathcal{A}\cap\mathcal{K}$ is covered by
$\mathcal{U}_{i}=\mathcal{U}(\mu^{(i)}_{\bullet},w^{(i)}),i\leq n$ and
conclude that for each $N$,

$\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\in\mathcal{A}\right)\leq\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\not\in\mathcal{K}\right)+\sum_{i=1}^{n}\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\in\mathcal{U}_{i}\right).$
(107)

It follows that

$\begin{split}&\limsup_{N}\frac{1}{N}\log\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\in\mathcal{A}\right)\\\
&\hskip
56.9055pt\leq\max\left(\limsup\frac{1}{N}\log\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\in\mathcal{V}\right):\mathcal{V}=\mathcal{K}^{\mathrm{c}},\mathcal{U}_{1},...,\mathcal{U}_{n}\right).\end{split}$
(108)

The terms appearing in the right hand side are all _either_ bounded by
$-M\leq-\inf_{\mathcal{A}}\mathcal{I}$, for the cases where
$\mathcal{V}=\mathcal{K}^{\mathrm{c}}$ or $\mathcal{V}=\mathcal{U}_{i}$, for a
path $(\mu^{(i)}_{\bullet},w^{(i)})$ with
$\mathcal{I}(\mu^{(i)}_{\bullet},w^{(i)})=\infty$, _or_ at most
$-\mathcal{I}(\mu^{(i)}_{\bullet},w^{(i)})+\epsilon\leq-\inf_{\mathcal{A}}\mathcal{I}+\epsilon$.
All together, we conclude that

$\limsup_{N}\frac{1}{N}\log\mathbb{P}\left(\mu^{N}_{\bullet},w^{N})\not\in\mathcal{A}\right)\leq-\inf\\{\mathcal{I}(\mu_{\bullet},w):(\mu_{\bullet},w)\in\mathcal{A}\\}+\epsilon$
(109)

and taking $\epsilon\to 0$ concludes the proof in the case where the infimum
is finite. The case where the infimum is infinite is essentially identical: we
now keep $M$ as a free parameter, choose a compact set $\mathcal{K}$ such that
$\limsup_{N}N^{-1}\log\mathbb{P}((\mu^{N}_{\bullet},w^{N})\not\in\mathcal{K})\leq-M$,
and cover $\mathcal{A}\cap\mathcal{K}$ with open sets
$\mathcal{U}(\mu_{\bullet},w)\ni(\mu_{\bullet},w)$ satisfying (106). The same
covering argument then gives

$\limsup_{N}\frac{1}{N}\log\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\not\in\mathcal{A}\right)\leq-M$
(110)

and the conclusion follows by taking $M\to\infty$. ∎

##  3 Properties of the Kac Process & Boltzmann Equation

In order to prove Theorem 1.3 and its consequences, we will use some facts
about the Kac process, including behaviour under changes of measure. We recall
first the following moment creation property, both for the _sample paths_ of
the Kac process and for the Boltmzann equation, and a result on the energy of
the Boltzmann equation. [42].

######  Proposition 3.1 (Moments of the Kac process and Boltzmann Equation).

Let $B$ be a kernel of the form $B(v)=1+\delta|v|$ for some $\delta>0$.

  1. i).

Fix $p\geq 2$, and let $\mu^{N}_{t}$ be a Kac process with collision kernel
$B$. Then there exists a constant $C=C(p,\delta)<\infty$ such that, for all
$0<s<t$,

$\mathbb{E}\left[\sup_{u\in[s,t]}\langle|v|^{p},\mu^{N}_{t}\rangle\right]\leq
C(1+t)s^{2-p}\mathbb{E}\left[\langle|v|^{2},\mu^{N}_{0}\rangle\right].$ (111)

  2. ii).

For the same $B,p$ as in item i), if $(\mu_{t})_{t\geq 0}$ is a solution to
the Boltzmann equation (BE) and the energy $\langle|v|^{2},\mu_{t}\rangle$ is
constant, then for all $s>0$,

$\langle|v|^{p},\mu_{t}\rangle\leq Cs^{2-p}\langle|v|^{2},\mu_{0}\rangle.$
(112)

  3. iii).

For any solution $(\mu_{t})_{t\geq 0}$ to the Boltzmann equation (BE), the
energy $\langle|v|^{2},\mu_{t}\rangle$ is nondecreasing in time.

Moment estimates similar to the first two items go back to Desvillettes [15]
and Wennberg [36, 53], based on Povzner esimtates; the same methods were
applied to the Kac process for single fixed times by Mischler and Mouhot [37],
and the pathwise estimate on compact time intervals of the kind given here was
proven by Norris [42]. The monotonicity of the energy can be found in works by
Mischler and Wennberg [36] and Lu [32].

The other property we will need are the changes of probability measure
necessary to perturb the initial data and dynamics. The changes of measure we
will use are as follows.

######  Proposition 3.2 (Kac process under change of measure).

Let $\mu^{N}_{t}$ be a Kac process with collision kernel $B$, and velocities
initially sampled independently from $\mu_{0}^{\star}$, which is a Markov
process on a filtered probability space
$(\Omega,\mathfrak{F},(\mathfrak{F}_{t})_{t\geq 0},\mathbb{P})$, and let
$w^{N}_{t}$ be the associated empirical flux. Let
$\varphi:\mathbb{R}^{d}\to\mathbb{R}$ be such that $\int
e^{\varphi(v)}\mu_{0}^{\star}(dv)=1$, and let $K:\mathcal{P}_{2}^{N}\times
E\to[0,\infty)$ be measurable and such that $K/(1+|v|+|v_{\star}|)$ is
uniformly bounded. Define a new measure $\mathbb{Q}$ by

$\begin{split}\frac{d\mathbb{Q}}{d\mathbb{P}}=&\exp\bigg{(}N\langle
f,\mu^{N}_{0}\rangle+N\langle\log K(\mu^{N}_{0},\cdot),w^{N}\rangle\\\ &\hskip
56.9055pt-N\int_{0}^{T}\int_{E}(K-1)(\mu^{N}_{0},t,v,v_{\star},\sigma)\overline{m}_{\mu^{N}}(dt,dv,dv_{\star},d\sigma)\bigg{)}\end{split}$
(113)

where we understand the right-hand side to be $0$ if
$\text{supp}(w^{N})\cap\\{K=0\\}\neq\emptyset$. Then $\mathbb{Q}$ is a
probability measure, under which $\mu^{N}_{0}$ is given as the empirical
measure of $N$ independent draws from $e^{\varphi(v)}\mu_{0}^{\star}(dv)$, and
under which $(\mu^{N}_{0},\mu^{N}_{t},w^{N}_{t})$ is a time-inhomogeneous
Markov process, with time-dependent generator, for bounded
$F:\mathcal{P}_{2}^{N}\times\mathcal{P}_{2}^{N}\times\mathcal{M}(E)\to\mathbb{R}$,

$\begin{split}\mathcal{G}_{t}F(\nu,\mu^{N},w^{N})=&N\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{S}^{d-1}}(F(\nu,\mu^{N,v,v_{\star},\sigma},w^{N,t,v,v_{\star},\sigma})-F(\nu,\mu^{N},w^{N}))\\\
&\hskip 28.45274pt\hskip 28.45274pt\hskip 28.45274pt\dots\times
K(\nu,t,v,v_{\star},\sigma)B(v-v_{\star})\mu^{N}(dv)\mu^{N}(dv_{\star})d\sigma.\end{split}$
(114)

This is a version of the standard Girsanov theorem for jump processes; see,
for example, [29, Appendix 1, Theorem 7.3], which is tailor-made for our
purposes. The hypotheses on the growth of $K$ are probably not the most
general possible, but are sufficient for the applications in this paper in
Sections 4, 5. Since this particular form does not appear to be standard, a
proof is given in Appendix B.

##  4 Restricted Lower Bound

We now give a proof of the lower bound with the additional integrability
hypothesis. The restricted lower bound is based on the following approximation
lemma.

######  Lemma 4.1 (Approximation by Regular Paths).

Let $(\mu_{\bullet},w)$ be a measure-flux pair such that

$\mathcal{I}(\mu_{\bullet},w)<\infty;\qquad\langle
1+|v|^{2}+|v_{\star}|^{2},w\rangle<\infty.$

Then there exists a sequence $(\mu^{(n)}_{\bullet},w^{(n)})$ of measure-flux
pairs whose tilting functions $K^{(n)}$, are continuous, such that and
$K^{(n)}(t,v,v_{\star},\sigma)B(v-v_{\star},\sigma)$ is bounded and bounded
away from $0$, such that $\mu^{(n)}_{0}$ admits a bounded density with respect
to $\mu^{\star}_{0}$, and such that

$\sup_{t\geq 0}W(\mu^{(n)}_{t},\mu_{t})+d(w^{(n)},w)\to
0;\qquad\limsup_{n}\mathcal{I}(\mu^{(n)}_{\bullet},w)\leq\mathcal{I}(\mu_{\bullet},w).$
(115)

Moreover, each $(\mu^{(n)}_{\bullet},w^{(n)})$ is the unique measure-flux pair
starting from $\mu^{(n)}_{0}$ and with tilting function $K^{(n)}$.

Throughout, we write the indexes (n) in the superscripts in brackets, to
distinguish them from similar notation for the Kac process
$\mu^{N}_{\bullet},w^{N}$.The proof of this lemma is rather technical, and so
is deferred until Subsection 4.1. Once this lemma is in hand, the restricted
lower bound Theorem 1.2ii) follows straightforwardly from standard ‘tilting’
arguments, using the change-of-measure given in Proposition 3.2 via the
following law of large numbers.

######  Lemma 4.2.

Let $(\mu_{\bullet},w)$ be a measure-flux pair whose tilting function $K$ is
continuous and such that $KB(v-v_{\star})$ is bounded and bounded away from
$0$, and which is the unique measure-flux pair with this tilting function and
this value of $\mu_{0}$. Let $\mathbb{Q}^{N}$ be the measures given by
Proposition 3.2 with $\varphi=\log\frac{d\mu_{0}}{d\mu^{\star}_{0}}$ and
$K:E\to(0,\infty)$ the tilting function for $(\mu_{\bullet},w)$. Then for all
open sets $\mathcal{U}\ni(\mu_{\bullet},w)$ and $\epsilon>0$, we have

$\mathbb{Q}^{N}\left((\mu^{N}_{\bullet},w^{N})\in\mathcal{U},\left|\frac{1}{N}\log\frac{d\mathbb{Q}^{N}}{d\mathbb{P}}-\mathcal{I}(\mu_{\bullet},w)\right|<\epsilon\right)\to
1.$ (116)

###### Proof.

We start by applying Proposition 3.2. Since $K$ is a function only
$E\to\mathbb{R}$, $(\mu^{N}_{t},w^{N}_{t})$ is a Markov process with generator
given by (114) applied to functions
$F:\mathcal{P}_{2}^{N}\times\mathcal{M}(E)\to\mathbb{R}$. For the initial
data, $\mu^{N}_{0}$ is given, under $\mathbb{Q}^{N}$, by sampling $N$
particles independently with common law $e^{\log
d\mu_{0}/d\mu_{0}^{\star}}\mu_{0}^{\star}=\mu_{0}$.

#### Step 1: Functional Law of Large Numbers

We begin by show that, under $\mathbb{Q}^{N}$, the pairs
$(\mu^{N}_{\bullet},w^{N})$ converge in probability to $(\mu_{\bullet},w)$.
Since $K,\varphi$ are bounded, Corollary 2.2 applies and the laws
$\mathbb{Q}^{N}\circ(\mu^{N}_{\bullet},w^{N})^{-1}$ are tight on
$\mathcal{D}\times\mathcal{M}(E)$, so every subsequence has a further
subsequence converging weakly on $\mathcal{D}\times\mathcal{M}(E)$. We will
now prove that the only possible subsequential limit is
$\delta_{(\mu_{\bullet},w)}$, which implies that the whole sequence
$\mathbb{Q}^{N}\circ(\mu^{N}_{\bullet},w^{N})^{-1}$ converges weakly to this
limit, and hence

$\mathbb{Q}^{N}\left((\mu^{N}_{\bullet},w^{N})\in\mathcal{U}\right)\to 1.$
(117)

Let $S\subset\mathbb{N}$ be any subsequence along which
$\mathbb{Q}^{N}\circ(\mu^{N}_{\bullet},w^{N})^{-1}$ converges weakly. thanks
to Skorokhod’s representation theorem, we can realise all
$(\mu^{N}_{\bullet},w^{N}),N\in S$ with these laws on a common probability
space, with probability measure $\mathbb{Q}$, converging $\mathbb{Q}$-almost
surely to a limit $(\widetilde{\mu}_{\bullet},\widetilde{w})$. For each $N$,
$(\mu^{N}_{\bullet},w^{N})$ almost surely lies in the set of pairs satisfies
the continuity equation, which is closed by Lemma 2.6, and hence
$(\widetilde{\mu}_{\bullet},\widetilde{w})$ almost surely satisfies (CE). We
now show that the limit is almost surely a measure-flux pair with tilting $K$:
for all $g\in C_{c}(E)$, the processes

$\begin{split}M^{N,g}_{t}&=\langle
g,w^{N}_{t}\rangle-\int_{(0,t]\times\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{S}^{d-1}}g(s,v,v_{\star},\sigma)K(s,v,v_{\star},\sigma)\\\
&\hskip 170.71652pt\dots
B(v-v_{\star})ds\mu^{N}_{s}(dv)\mu^{N}_{s}(dv_{\star})d\sigma\end{split}$
(118)

is a càdlàg martingale, with previsible, increasing quadratic variation

$\begin{split}[M^{N,g}]_{t}&=\frac{1}{N}\int_{(0,t]\times\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{S}^{d-1}}g^{2}K(s,v,v_{\star},\sigma)B(v-v_{\star})ds\mu^{N}_{s}(dv)\mu^{N}_{s}(dv_{\star})d\sigma\\\\[8.61108pt]
&\hskip
28.45274pt\leq\|g\|_{\infty}^{2}\sup_{E}(B(v-v_{\star})K)T/N,\end{split}$
(119)

see, for instance, [13, 42]. In particular, since $B(v-v_{\star})K$ is bounded
by construction, the constant in the final expression is finite. Therefore,
for all such $g$,

$\mathbb{E}_{\mathbb{Q}}\left|\langle
g,w^{N}\rangle-\int_{E}gK(s,v,v_{\star},\sigma)B(v-v_{\star})ds\mu^{N}_{s}(dv)\mu^{N}_{s}(dv_{\star})d\sigma\right|\leq\frac{C_{g}}{\sqrt{N}}$
(120)

for some constant $C_{g}$. Taking $N\to\infty$ through $S$, the first term in
the expectation converges almost surely to $\langle g,\widetilde{w}\rangle$,
and the second term converges to $\int gKd\overline{m}_{\widetilde{\mu}}$ by
Lemma 2.5 applied to $gK$. We now take $N\to\infty$ through $S$ to obtain

$\mathbb{E}_{\mathbb{Q}}\left|\langle
g,\widetilde{w}\rangle-\int_{E}g(s,v,v_{\star},\sigma)K(s,v,v_{\star},\sigma)\overline{m}_{\widetilde{\mu}}(ds,dv,dv_{\star},d\sigma)\right|=0$
(121)

and so the integrand is $0$, $\mathbb{Q}$-almost surely. Taking a union bound
over a countable dense set in $C_{c}(E)$, we conclude that
$\widetilde{w}=K\overline{m}_{\widetilde{\mu}}$ almost surely, and the limit
is a measure-flux pair with the prescribed rate function $K$, and the
convergence $\mu^{N}_{0}\to\widetilde{\mu}_{0}$ implies that
$\mu_{0}=\widetilde{\mu}_{0}$. By hypothesis, these properties uniquely
characterise the desired limit $(\mu_{\bullet},w)$, so
$\mathbb{Q}((\widetilde{\mu}_{\bullet},\widetilde{w})=(\mu_{\bullet},w))=1$
and the step is complete.

#### Step 2: Law of Large Numbers for the Dynamic Cost

We will now show that the (random) exponential cost induced by the change of
measure (113) converges under $\mathbb{Q}^{N}$: for all $\epsilon>0$,

$\mathbb{Q}^{N}\left(\left|\frac{1}{N}\log\frac{d\mathbb{Q}^{N}}{d\mathbb{P}}-\mathcal{I}(\mu_{\bullet},w)\right|>\epsilon\right)\to
0.$ (122)

We begin by using the definitions of $\mathcal{I}$ and (113) to rewrite the
difference as

$\begin{split}\frac{1}{N}\log\frac{d\mathbb{Q}^{N}}{d\mathbb{P}}-\mathcal{I}(\mu_{\bullet},w)&=\langle\varphi,\mu^{N}_{0}\rangle-H(\mu_{0}|\mu^{\star}_{0})+\langle\log
K,w^{N}-w\rangle\\\
&-\int_{E}(K-1)B(v-v_{\star})(\mu^{N}_{t}(dv)\mu^{N}_{t}(dv_{\star})-\mu_{t}(dv)\mu_{t}(dv_{\star})d\sigma\end{split}$
(123)

and examine the terms one by one. Fix, throughout,
$\epsilon,\epsilon^{\prime}>0$.

#### Step 2a: Cost of the Initial Data

For the cost of the initial data, $\langle\varphi,\mu^{N}_{0}\rangle$ is the
empirical mean of $\log\frac{d\mu^{\prime}_{0}}{d\mu^{\star}_{0}}$, sampled at
$N$ independent draws from $\mu_{0}$. The mean of each draw is exactly
$\int_{\mathbb{R}^{d}}\log\frac{d\mu_{0}}{d\mu^{\star}_{0}}(v)\mu_{0}(dv)=:H(\mu_{0}|\mu_{0}^{\star})$,
so by the weak law of large numbers, for all $N$ large enough

$\mathbb{Q}^{N}\left(|\langle\varphi,\mu^{N}_{0}\rangle-H(\mu_{0}|\mu^{\star}_{0})|>\epsilon/4\right)<\epsilon^{\prime}/3.$
(124)

#### Step 2b: Integral against Empirical Flux

Let us now examine the second term. By the choice of $K$, $\log K$ is
continuous, bounded above, and bounded below by $\log(c/B(v-v_{\star}))$ for
some constant $c>0$. We can further bound this below by

$\log\frac{1}{B(v-v_{\star})}\geq\log\frac{1}{(1+|v|)(1+|v_{\star})}\geq-c(|v|+|v_{\star}|)$
(125)

for a new constant $c$: in particular, $|\log K|\leq C(1+|v|+|v_{\star}|)$ is
continuous, and of at most linear growth. Recalling that $B(v-v_{\star})K$ is
bounded, we also estimate, uniformly in $N$,

$\begin{split}\mathbb{E}_{\mathbb{Q}^{N}}\left[\langle
1+|v|^{2}+|v_{\star}|^{2},w^{N}\rangle\right]&=\mathbb{E}_{\mathbb{Q}^{N}}\left[\int_{E}(1+|v|^{2}+|v_{\star}|^{2})B(v-v_{\star})K\mu^{N}_{t}(dv)\mu^{N}_{t}(dv_{\star})d\sigma\right]\\\\[4.30554pt]
&\leq C\mathbb{E}_{\mathbb{Q}^{N}}\langle
1+|v|^{2},\mu^{N}_{0}\rangle=C\langle 1+|v|^{2},\mu_{0}\rangle.\end{split}$
(126)

Elementary Chebychev estimates produce $R<\infty$ such that, uniformly in $N$,

$\mathbb{Q}^{N}\left(\langle(1+|v|+|v_{\star}|)1(|v|>R\text{ or
}|v_{\star}|>R),w^{N}\rangle>\epsilon/12C\right)<\epsilon^{\prime}/6;$ (127)

and similarly, using the boundedness of $B(v-v_{\star})K$ and finiteness of
the second moments, the second moment $\langle
1+|v|^{2}+|v_{\star}|^{2},w\rangle<\infty$ is also finite, and so we can
additionally choose $R$ so that

$\langle(1+|v|+|v_{\star}|)1(|v|>R\text{ or
}|v_{\star}|>R),w\rangle<\frac{\epsilon}{12C}$ (128)

and construct a continuous, compactly supported function $g:E\to\mathbb{R}$
such that $|g-\log K|\leq C(1+|v|+|v_{\star}|)$ and which agrees with $\log K$
when both $|v|,|v_{\star}|\leq R$. We therefore find from (127) that

$\mathbb{Q}^{N}\left(\langle|g-\log
K|,w^{N}\rangle>\epsilon/12\right)<\epsilon^{\prime}/6;\qquad\langle|g-\log
K|,w\rangle<\frac{\epsilon}{12}.$ (129)

Thanks to the convergence in distribution, for $N$ large enough,

$\mathbb{Q}^{N}\left(|\langle
g,w^{N}-w\rangle|>\epsilon/12\right)<\epsilon^{\prime}/6$ (130)

and we find from (129,130) that

$\mathbb{Q}^{N}\left(\left|\langle\log
K,w^{N}-w\rangle\right|>\epsilon/4\right)<\epsilon^{\prime}/3.$ (131)

#### Step 2c: Integral against Compensator

We finally deal with the third term in (123). Since $B(v-v_{\star})K$ is
bounded and $K$ is continuous, it follows that $B(v-v_{\star})(K-1)$ is of at
most linear growth, so there exists $C$ such that $|B(v-v_{\star})(K-1)|\leq
C(1+|v|+|v_{\star}|)$, and as in the previous step, we can choose $R$ such
that, uniformly in $N$,

$\mathbb{Q}^{N}\left(\sup_{t\leq
T}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(1+|v|+|v_{\star}|)1(|v|>R\text{ or
}|v_{\star}|>R)\mu^{N}_{t}(dv)\mu^{N}_{t}(dv_{\star})>\epsilon/12CT\right)<\epsilon^{\prime}/6$
(132)

and similarly such that

$\sup_{t\leq
T}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(1+|v|+|v_{\star}|)1(|v|>R\text{ or
}|v_{\star}|>R)\mu^{\prime}_{t}(dv)\mu^{\prime}_{t}(dv_{\star})\leq\epsilon/12CT.$
(133)

We again truncate, with a proxy $h:E\to\mathbb{E}$ which is continuous,
compactly supported, agrees with $(K-1)B(v-v_{\star})$ if both
$|v|,|v_{\star}|\leq R$, and such that $|(K-1)B(v-v_{\star})-h|\leq
C(1+|v|+|v_{\star}|)$ for the same constant $C$. Using Lemma 2.5 again,

$\mathbb{Q}^{N}\left(\left|\int_{E}h(t,v,v_{\star},\sigma)dt(\mu^{N}_{t}(dv)\mu^{N}_{t}(dv_{\star})-\mu_{t}(dv)\mu_{t}(dv_{\star}))d\sigma\right|>\epsilon/12\right)<\epsilon^{\prime}/6$
(134)

for $N$ large enough, while (132) implies that

$\mathbb{Q}^{N}\left(\left|\int_{E}(h-B(v-v_{\star})(K-1))dt\mu^{N}_{t}(dv)\mu^{N}_{t}(dv_{\star})d\sigma\right|>\frac{\epsilon}{12}\right)<\epsilon^{\prime}/6$
(135)

and (133) implies that

$\left|\int_{E}(h-B(v-v_{\star})(K-1))dt\mu_{t}(dv)\mu_{t}(dv_{\star})\right|<\frac{\epsilon}{12}.$
(136)

Gathering (134, 135, 136), we conclude that, for $N$ large enough,

$\mathbb{Q}^{N}\left(\left|\int_{E}(K-1)B(v-v_{\star})dt(\mu^{N}_{t}(dv)\mu^{N}_{t}(dv_{\star})-\mu_{t}(dv)\mu_{t}(dv_{\star}))d\sigma\right|>\epsilon/4\right)<\epsilon^{\prime}/3.$
(137)

Returning to (123), we combine (124, 131, 137) to obtain, for all
$\epsilon,\epsilon^{\prime}>0$, and all $N$ large enough, depending on
$\epsilon,\epsilon^{\prime}$,

$\mathbb{Q}^{N}\left(\left|\frac{1}{N}\log\frac{d\mathbb{Q}^{N}}{d\mathbb{P}}-\mathcal{I}(\mu_{\bullet},w)\right|>\epsilon\right)<\epsilon^{\prime}$
(138)

and we have proven the desired convergence (122). Together with the previous
step, the proof is complete. ∎

We can now prove the restricted lower bound.

###### Proof of Theorem 1.2ii).

Let us fix a Skorokhod-open set $\mathcal{U}$, a path
$(\mu_{\bullet},w)\in\mathcal{U}\cap\mathcal{R}$, and $\epsilon>0$. Let us
assume that $\mathcal{I}(\mu_{\bullet},w)<\infty$. Thanks to Lemma 4.1, there
exists a measure-flux pair $(\mu^{\prime}_{\bullet},w^{\prime})\in\mathcal{U}$
with overall cost
$\mathcal{I}(\mu^{\prime}_{\bullet},w^{\prime})<\mathcal{I}(\mu_{\bullet},w)+\epsilon$,
satisfying the conclusions of Lemma 4.1. For the changes of measure
$\mathbb{Q}^{N}\ll\mathbb{P}$ as in Lemma 4.2, we then have, for all $N$ large
enough,

$\mathbb{Q}^{N}\left((\mu^{N}_{\bullet},w^{N})\in\mathcal{U},\frac{d\mathbb{Q}^{N}}{d\mathbb{P}}\leq\exp\left(N(\mathcal{I}(\mu^{\prime}_{\bullet},w^{\prime})+\epsilon)\right)\right)\geq\frac{1}{2}$
(139)

which implies that

$\begin{split}\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\in\mathcal{U}\right)&\geq\mathbb{E}_{\mathbb{Q}^{N}}\left[\left(\frac{d\mathbb{Q}^{N}}{d\mathbb{P}}\right)^{-1}1\left(\frac{d\mathbb{Q}^{N}}{d\mathbb{P}}\leq\exp\left(N(\mathcal{I}(\mu^{\prime}_{\bullet},w^{\prime})+\epsilon)\right),\hskip
2.84544pt(\mu^{N}_{\bullet},w^{N})\in\mathcal{U}\right)\right]\\\\[8.61108pt]
&\geq\frac{1}{2}\exp\left(-N(\mathcal{I}(\mu^{\prime}_{\bullet},w^{\prime})+\epsilon)\right)\geq\frac{1}{2}\exp\left(-N(\mathcal{I}(\mu_{\bullet},w)+2\epsilon)\right).\end{split}$
(140)

Taking the logarithm and the limit $N\to\infty$, and then $\epsilon\to 0$,

$\liminf_{N}\frac{1}{N}\log\mathbb{P}\left((\mu^{N}_{\bullet},w^{N})\in\mathcal{U}\right)\geq-\mathcal{I}(\mu_{\bullet},w).$
(141)

Of course, (141) trivially holds if $\mathcal{I}(\mu_{\bullet},w)=\infty$, and
so applies to any $(\mu_{\bullet},w)\in\mathcal{U}\cap\mathcal{R}$, and the
result is proven.∎

###  4.1 Proof of Approximation Lemma

We will now present the proof of the approximation lemma as a number of
intermediate steps. We will present the statements here, to give an overview
of the proof of the overall approximation lemma, and the proofs in Subsection
4.2. We begin with the following definition.

######  Definition 4.3.

Let $(\mu_{\bullet},w)$ be a measure-flux pair, and $\lambda>0$. Let
$g_{\lambda}$ be the Gaussian in $\mathbb{R}^{d}$

$g_{\lambda}(x):=\frac{1}{(2\pi\lambda)^{d/2}}\exp\left(-|x|^{2}/2\lambda\right).$
(142)

We define the convolutions $(g_{\lambda}\star\mu_{\bullet}),g_{\lambda}\star
w$ by

$(g_{\lambda}\star\mu_{\bullet})_{t}(dv)=(g_{\lambda}\star\mu_{t})(dv)=\int_{\mathbb{R}^{d}}g_{\lambda}(v-u)\mu_{t}(du)dv;$
(143) $(g_{\lambda}\star
w)(dt,dv,dv_{\star},d\sigma)=\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}g_{\lambda}(v-u)g_{\lambda}(v_{\star}-u_{\star})w(dt,du,du_{\star},d\sigma).$
(144)

The measures $g_{\lambda}\star\mu_{t}$ are absolutely continuous with respect
to the Lebesgue measure; we will alternatively use the notation
$g_{\lambda}\star\mu_{t}$ for their density on $\mathbb{R}^{d}$.

######  Remark 4.4.

Let us note that the choice of Gaussian mollification is essential here, as it
is the unique mollifier which is invariant under changing between the pre- and
post- collisional velocities.

######  Lemma 4.5 (Approximation by Convolution).

Suppose $(\mu_{\bullet},w)$ is a measure-flux pair with a bounded tilting
function $K$, such that

$\langle|v|^{2}+|v_{\star}|^{2},w\rangle<\infty;\qquad\mathcal{J}(\mu_{\bullet},w)<\infty;\qquad\sup_{t\leq
T}\langle|v|^{2},\mu_{t}\rangle=\langle|v|^{2},\mu_{0}\rangle<\infty.$ (145)

Then, for all $\lambda>0$, $(\mu_{\bullet}\star g_{\lambda},w\star
g_{\lambda})$ is a measure-flux pair. Furthermore, there exists a continuous
function $\vartheta:[0,1]\to[0,\infty)$, which is continuous at $0$ and
$\vartheta(0)=0$, and a constant $C$, _which only depends on upper bounds for
the quantities in ( 145)_ and not the boundedness of $K$, such that for all
$\lambda\in(0,1]$,

$\mathcal{J}(g_{\lambda}\star\mu_{\bullet},g_{\lambda}\star
w)\leq\mathcal{J}(\mu_{\bullet},w)+C\vartheta(\lambda).$ (146)

Finally, the tilting function $K^{\lambda}$ satisfies

$\sup_{t,v,v_{\star},\sigma}K^{\lambda}(t,v,v_{\star},\sigma)B(v-v_{\star})\leq\sup_{t,v,v_{\star},\sigma}K(t,v,v_{\star},\sigma)B(v-v_{\star})$
(147)

We now apply this to produce some approximation results.

######  Lemma 4.6.

Let $\mu_{\bullet},w$ be as in Lemma 4.1. Then there exist measure-flux pairs
$\mu^{(n)}_{\bullet},w^{(n)}$ such that

$\mathcal{J}(\mu^{(n)}_{\bullet},w^{(n)})\to\mathcal{J}(\mu_{\bullet},w);\qquad\sup_{t\leq
T}\|(1+|v|^{2})(\mu^{(n)}_{t}-\mu_{t})\|_{\mathrm{TV}}+\|(1+|v|^{2}+|v_{\star}|^{2})(w^{(n)}-w)\|_{\mathrm{TV}}\to
0$ (148)

and, for each $n$, the tilting function $K^{(n)}$ is such that
$K^{(n)}(t,v,v_{\star},\sigma)B(v-v_{\star})$ is bounded and $K^{(n)}$ is
continuous in $v,v_{\star}$. Furthermore, the starting points $\mu^{(n)}_{0}$
can be taken to be of the form

$\mu^{(n)}_{0}(dv)=c_{n}^{-1}\left(\mu_{0}(dv)+\nu^{(n)}(dv)\right)$ (149)

for a suitable normalising constant $c_{n}\to 1$ and measures $\nu^{(n)}$ with
$\langle 1+|v|^{2},\nu^{(n)}\rangle\to 0$.

######  Lemma 4.7.

Let $\mu_{\bullet},w$ be as in Lemma 4.1. Then there exist measure-flux pairs
$\mu^{(n)}_{\bullet},w^{(n)}$ such that

$\mathcal{I}(\mu^{(n)}_{\bullet},w^{(n)})\to\mathcal{I}(\mu_{\bullet},w);\qquad\sup_{t\leq
T}\hskip 2.84544ptW\left(\mu^{(n)}_{t},\mu_{t}\right)+d(w^{(n)},w)\to 0$ (150)

and, for each $n$, the tilting function $K^{(n)}$ is such that
$K^{(n)}(t,v,v_{\star},\sigma)B(v-v_{\star})$ is bounded and $K^{(n)}$ is
continuous in $v,v_{\star}$.

######  Lemma 4.8.

Let $\mu_{\bullet},w$ be a measure-flux pair with finite rate
$\mathcal{I}(\mu_{\bullet},w)<\infty$, such that
$K(t,v,v_{\star},\sigma)B(v-v_{\star})$ is bounded and $K$ is continuous in
$v,v_{\star}$. Then there exist measure-flux pairs
$\mu^{(n)}_{\bullet},w^{(n)}$ with

$\mathcal{I}(\mu^{(n)}_{\bullet},w)\to\mathcal{I}(\mu_{\bullet},w);\qquad\sup_{t\leq
T}\|\mu^{(n)}_{t}-\mu_{t}\|_{\mathrm{TV}}+\|w^{(n)}-w\|_{\mathrm{TV}}\to 0$
(151)

and additionally, for each $n$,

$\sup_{(t,v,v_{\star},\sigma)}K^{(n)}(t,v,v_{\star},\sigma)B(v-v_{\star})\leq\sup_{(t,v,v_{\star},\sigma)}K(t,v,v_{\star},\sigma)B(v-v_{\star})+1;$
(152)
$\inf_{(t,v,v_{\star},\sigma)}K^{(n)}(t,v,v_{\star},\sigma)B(v-v_{\star})>0;$
(153) $\sup\frac{d\mu^{(n)}_{0}}{d\mu^{\star}_{0}}<\infty$ (154)

and such that $K^{(n)}$ are continuous functions on $E$. Moreover, the
approximations are uniquely characterised among measure-flux pairs by the
initial value $\mu^{(n)}_{0}$ and the tilting function $K^{(n)}$.

Equipped with these lemmas, the stated result Lemma 4.1 follows by a standard
diagonal argument.

###### Proof of Lemma 4.1.

Let us fix $\mu_{\bullet},w$ as given, and construct a sequence of
approximating measure-flux pairs $\mu^{(n)}_{\bullet},w^{(n)}$ as follows. By
Lemma 4.7, there exists a pair $\mu^{(n,1)}_{\bullet},w^{(n,1)}$ whose tilting
function $K^{(n,1)}$ is continuous in $v,v_{\star}$ and
$K^{(n,1)}B(v-v_{\star})$ is bounded, such that $\mu^{(n,1)}_{0}$ admits a
continuous and positive density, and such that

$\sup_{t\leq
T}W\left(\mu^{(n,1)}_{t},\mu_{t}\right)+d(w^{(n,1)},w)+\left|\mathcal{I}(\mu^{(n,1)}_{\bullet},w^{(n,1)})-\mathcal{I}(\mu_{\bullet},w)\right|<\frac{1}{n}.$
(155)

Thanks to Lemma 4.8, we can approximate $\mu^{(n,1)}_{\bullet},w^{(n,1)}$ by a
further pair $\mu^{(n,2)}_{\bullet},w^{(n,2)}$, whose tilting function
$K^{(n,2)}$ is continuous and so that $K^{(n,2)}B(v-v_{\star})$ is still
bounded and bounded away from $0$, and where $\mu^{(n,2)}_{0}$ has a bounded
density with respect to $\mu^{\star}_{0}$, which is uniquely characterised
among measure-flux pairs by the initial data $\mu^{(n,2)}_{0}$ and tilting
function $K^{(n,2)}$, with further error

$\sup_{t\leq
T}\|\mu^{(n,2)}_{t}-\mu^{(n,1)}_{t}\|_{\mathrm{TV}}+\|w^{(n,2)}-w^{(n,1)}\|_{\mathrm{TV}}+\left|\mathcal{I}(\mu^{(n,2)}_{\bullet},w^{(n,2)})-\mathcal{I}(\mu_{\bullet}^{(n,1)},w^{(n,1)})\right|<\frac{1}{n}.$
(156)

Combining (155, 156), we recall that the total variation distance on measures
on $\mathbb{R}^{d}$, respectively $E$, dominates the Wasserstein1 distance
$W$, respectively $d$, so the sequence
$\mu^{(n)}_{\bullet}=\mu^{(n,2)}_{\bullet},w^{(n)}=w^{(n,2)}$ has the desired
properties. ∎

###  4.2 Proof of Lemmas

We start with the convolution lemma, which is the most difficult step.

###### Proof of Lemma 4.5.

We divide the proof into several steps. Throughout, $C$ will denote a
constant, which may vary from line to line, but is allowed to depend _only_ on
the quantities specified in (145).

#### Step 1: $(g_{\lambda}\star\mu_{\bullet},g_{\lambda}\star w)$ solves the
continuity equation

This property is fairly well-known, see Erbar [22] or Basile [4], and we
include a proof for completeness. If we fix $f\in C_{b}(\mathbb{R}^{d})$, let
us denote $g_{\lambda}\star f$ the convolution $(g_{\lambda}\star
f)(v):=\int_{\mathbb{R}^{d}}g_{\lambda}(v-w)f(w)dw$, and observe that $\langle
f,g_{\lambda}\star\mu_{t}\rangle=\langle g_{\lambda}\star f,\mu_{t}\rangle$
for all $t\in[0,T].$ Now, using the continuity equation for
$(\mu_{\bullet},w)$ with the test function $g_{\lambda}\star f$, for all
$t\in[0,T]$,

$\begin{split}\langle f,g_{\lambda}\star\mu_{t}\rangle-\langle
f,g_{\lambda}\star\mu_{0}\rangle&=\langle g_{\lambda}\star
f,\mu_{t}\rangle-\langle g_{\lambda}\star f,\mu_{0}\rangle\\\
&=\int_{E}\Delta(g_{\lambda}\star f)(v,v_{\star},\sigma)1_{s\leq
t}w(ds,dv,dv_{\star},d\sigma).\end{split}$ (157)

Let us now fix $v,v_{\star},\sigma$, and observe that the map
$\mathcal{T}_{\sigma}:(v,v_{\star})\to(v^{\prime},v^{\prime}_{\star})$ is a
linear isometry of Euclidean distance on $(\mathbb{R}^{d})^{2}$; for variables
$(u^{\prime},u_{\star}^{\prime})\in(\mathbb{R}^{d})^{2}$, let us write
$(u,u_{\star})$ for the preimage under $\mathcal{T}_{\sigma}$. We therefore
have

$\begin{split}&(g_{\lambda}\star f)(v^{\prime})+(g_{\lambda}\star
f)(v^{\prime}_{\star})\\\ &\hskip
28.45274pt=\frac{1}{(2\pi\lambda)^{d}}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(f(u^{\prime})+f(u_{\star}^{\prime}))\exp\left(-\frac{|u^{\prime}-v^{\prime}|^{2}+|u_{\star}^{\prime}-v_{\star}^{\prime}|^{2}}{2\lambda}\right)du^{\prime}du_{\star}^{\prime}\\\\[8.61108pt]
&\hskip
28.45274pt=\frac{1}{(2\pi\lambda)^{d}}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(f(u^{\prime})+f(u_{\star}^{\prime}))\exp\left(-\frac{|u-v|^{2}+|u_{\star}-v_{\star}|^{2}}{2\lambda}\right)du^{\prime}du_{\star}^{\prime}\\\\[8.61108pt]
&\hskip
28.45274pt=\frac{1}{(2\pi\lambda)^{d}}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(f(u^{\prime})+f(u_{\star}^{\prime}))\exp\left(-\frac{|u-v|^{2}+|u_{\star}-v_{\star}|^{2}}{2\lambda}\right)dudu_{\star}\\\\[8.61108pt]
&\hskip
28.45274pt=\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(f(u^{\prime})+f(u_{\star}^{\prime}))g_{\lambda}(u-v)g_{\lambda}(u_{\star}-v_{\star})dudu_{\star}\end{split}$
(158)

where the penultimate line makes the change of variables
$(u^{\prime},u_{\star}^{\prime})=\mathcal{T}_{\sigma}(u,u_{\star})$, with unit
determinant. We now substitute the resulting identity

$\begin{split}\Delta(g_{\lambda}\star
f)(v,v_{\star},\sigma)&=\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(f(u^{\prime})+f(u_{\star}^{\prime})-f(u)-f(u_{\star}))g_{\lambda}(u-v)g_{\lambda}(u_{\star}-v_{\star})dudu_{\star}\\\
&=\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\Delta
f)(u,u_{\star},\sigma)g_{\lambda}(u-v)g_{\lambda}(u_{\star}-v_{\star})dudu_{\star}\end{split}$
(159)

into (157) to obtain

$\begin{split}\langle f,g_{\lambda}\star\mu_{t}\rangle-\langle
f,g_{\lambda}\star\mu_{0}\rangle&=\int_{E}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\Delta
f)(u,u_{\star},\sigma)1_{s\leq
t}g_{\lambda}(u-v)g_{\lambda}(u_{\star}-v_{\star})w(ds,dv,dv_{\star},d\sigma)\\\
&=:\int_{E}(\Delta f)(u,u_{\star},\sigma)1_{s\leq t}(g_{\lambda}\star
w)(ds,du,du_{\star},d\sigma)\end{split}$ (160)

and, since $f$ is arbitrary, we conclude that
$(g_{\lambda}\star\mu_{\bullet},g_{\lambda}\star w)$ satisfies the continuity
equation (CE) as desired.

#### Step 2: Identification of the Tilting Function

To show that $(\mu_{\bullet},w)$ is a measure-flux pair, and in preparation
for estimating the dynamic cost, we will now explicitly find a tilting
function. Let us write $K$ for the tilting function for the pair
$(\mu_{\bullet},w)$, so that $w=K\overline{m}_{\mu}$. For any Borel subset
$A\subset E$, we observe that

$\begin{split}(g_{\lambda}\star
w)(A)=&\int_{E}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}1_{A}(t,u,u_{\star},\sigma)g_{\lambda}(u-v)g_{\lambda}(u_{\star}-v_{\star})K(t,v,v_{\star},\sigma)\\\
&\hskip
170.71652pt\dots\times\overline{m}_{\mu}(dt,dv,dv_{\star},d\sigma)dudu_{\star}\\\
&=\int_{E}1_{A}(t,u,u_{\star},\sigma)\left(\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}g_{\lambda}(u-v)g_{\lambda}(u_{\star}-v_{\star})K(t,v,v_{\star},\sigma)B(v-v_{\star})\mu_{t}(dv)\mu_{t}(dv_{\star})\right)\\\
&\hskip 170.71652pt\dots\times dudu_{\star}dtd\sigma.\end{split}$ (161)

Now, let us define $K^{\lambda}$ by

$K^{\lambda}(t,u,u_{\star},\sigma):=\frac{\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}g_{\lambda}(u-v)g_{\lambda}(u_{\star}-v_{\star})K(t,v,v_{\star},\sigma)B(v-v_{\star})\mu_{t}(dv)\mu_{t}(dv_{\star})}{B(u-u_{\star})(g_{\lambda}\star\mu_{t})(u)(g_{\lambda}\star\mu_{t})(u_{\star})}$
(162)

and observe that this is a well-defined function, since $K$ was assumed to be
bounded and $B$ is bounded away from $0$, and where in the denominator
$(g_{\lambda}\star\mu_{t})(u)>0$ denotes the density of the measure
$g_{\lambda}\star\mu_{t}$ with respect to the Lebesgue measure. Returning to
(161), this definition yields

$\begin{split}(g_{\lambda}\star
w)(A)&=\int_{E}1_{A}(t,u,u_{\star},\sigma)K^{\lambda}(t,u,u_{\star},\sigma)B(u-u_{\star})(g_{\lambda}\star\mu_{t})(u)du(g_{\lambda}\star\mu_{t})(u_{\star})du_{\star}dtd\sigma\\\
&=\int_{E}1_{A}(t,u,u_{\star},\sigma)K^{\lambda}(t,u,u_{\star},\sigma)\overline{m}_{g_{\lambda}\star\mu}(dt,du,du_{\star},d\sigma).\end{split}$
(163)

We conclude that $K^{\lambda}$ is a tilting function for
$(g_{\lambda}\star\mu_{\bullet},g_{\lambda}\star w)$, and so this is a
measure-flux pair as claimed, and the bound (147) is immediate. For future
convenience, we will now define

$\overline{r}^{\lambda}_{t}(u,u_{\star}):=\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}g_{\lambda}(u-v)g_{\lambda}(u_{\star}-v_{\star})B(v-v_{\star})\mu_{t}(dv)\mu_{t}(dv_{\star})$
(164)

and introduce the proxy to $K^{\lambda}$ given by

$\begin{split}\overline{K}^{\lambda}(t,u,u_{\star},\sigma):&=\frac{\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}g_{\lambda}(u-v)g_{\lambda}(u_{\star}-v_{\star})K(t,v,v_{\star},\sigma)B(v-v_{\star})\mu_{t}(dv)\mu_{t}(dv_{\star})}{\overline{r}^{\lambda}_{t}(u,u_{\star})}\\\\[8.61108pt]
&=\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}K(t,v,v_{\star},\sigma)\nu^{\lambda}_{(t,u,u_{\star})}(dv,dv_{\star})\end{split}$
(165)

which we have written in terms of the integral against the probability measure

$\nu^{\lambda}_{(t,u,u_{\star})}(dv,dv_{\star})=\frac{B(v-v_{\star})g_{\lambda}(v-u)\mu_{t}(dv)g_{\lambda}(v_{\star}-u_{\star})\mu_{t}(dv_{\star})}{\overline{r}^{\lambda}_{t}(u,u_{\star})}.$
(166)

For any $u,u_{\star},\sigma$, the quotient is given by the function

$\begin{split}\frac{K^{\lambda}}{\overline{K}^{\lambda}}(t,u,u_{\star},\sigma)&=\psi_{\lambda}(t,u,u_{\star}):=\frac{\overline{r}^{\lambda}_{t}(u,u_{\star})}{B(u-u_{\star})(g_{\lambda}\star\mu_{t})(u)(g_{\lambda}\star\mu_{t})(u_{\star})}\\\\[12.91663pt]
&=\frac{1}{B(u-u_{\star})}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}B(v-v_{\star})\frac{g_{\lambda}(v-u)\mu_{t}(dv)}{(g_{\lambda}\star\mu_{t})(u)}\frac{g_{\lambda}(v_{\star}-u_{\star})\mu_{t}(dv_{\star})}{(g_{\lambda}\star\mu_{t})(u_{\star})}.\end{split}$
(167)

which depends only on $\mu_{t}$, and not on $w$.

#### Step 3: Decomposition of the Rate Function

We now break the rate function up into several parts which can be more easily
manipulated; for the rest of the proof, let us consider only $0<\lambda\leq
1$. We start from

$\mathcal{J}(g_{\lambda}\star\mu_{\bullet},g_{\lambda}\star
w)=\int_{E}(\tau-1)(K^{\lambda}(t,v,v_{\star},\sigma))\overline{m}_{g_{\lambda}\star\mu}(dt,dv,dv_{\star},d\sigma)+\overline{m}_{g_{\lambda}\star\mu}(E)$
(168)

where we recall that $(\tau-1)(x)=x\log x-x$ is a convex function on
$[0,\infty)$. Next, we observe that

$(\tau-1)(K^{\lambda})=K^{\lambda}\log\psi_{\lambda}+\psi_{\lambda}(\tau-1)(\overline{K}^{\lambda}).$
(169)

For the first term, we start with the observation that

$\begin{split}&K^{\lambda}(t,u,u_{\star},\sigma)\overline{m}_{g_{\lambda}\star\mu}(dt,du,du_{\star},d\sigma)\\\
&\hskip
28.45274pt=\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}K(t,v,v_{\star},\sigma)B(v-v_{\star})\mu_{t}(dv)\mu_{t}(dv_{\star})g_{\lambda}(u-v)g_{\lambda}(u_{\star}-v_{\star})dtd\sigma
dudu_{\star}\end{split}$ (170)

from which it follows that we can rewrite the integral of the first term as an
error term

$\begin{split}\mathcal{T}_{\lambda}(\mu_{\bullet},w)&:=\int_{E}K^{\lambda}(t,u,u_{\star},\sigma)\log\psi_{\lambda}(t,u,u_{\star})\overline{m}_{g_{\lambda}\star\mu}(dt,du,du_{\star},d\sigma)\\\
&=\int_{E\times\mathbb{R}^{d}\times\mathbb{R}^{d}}K(t,v,v_{\star},\sigma)B(v-v_{\star})g_{\lambda}(u-v)g_{\lambda}(u_{\star}-v_{\star})\log\psi_{\lambda}(t,u,u_{\star})\\\
&\hskip 142.26378pt\dots\times dt\mu_{t}(dv)\mu_{t}(dv_{\star})d\sigma
dudu_{\star}.\end{split}$ (171)

To integrate the second term, we note that

$\begin{split}\psi_{\lambda}(t,u,u_{\star})\overline{m}_{g_{\lambda}\star\mu}(dt,du,du_{\star},d\sigma)&=\overline{r}_{t}(u,u_{\star})dtdudu_{\star}d\sigma.\end{split}$
(172)

Since $\nu^{\lambda}_{(t,u,u_{\star})}$ are probability measures, we can apply
Jensen to the convex function $\tau-1$ to find

$\begin{split}(\tau-1)(\overline{K}^{\lambda}(t,u,u_{\star},\sigma))&\leq\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\tau-1)(K(t,v,v_{\star},\sigma))\nu_{(t,u,u_{\star})}(dv,dv_{\star})\\\
&=\frac{1}{\overline{r}^{\lambda}_{t}(u,u_{\star})}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\tau-1)(K(t,v,v_{\star},\sigma))B(v-v_{\star})g_{\lambda}(u-v)g_{\lambda}(u_{\star}-v_{\star})\\\
&\hskip 170.71652pt\dots\times\mu_{t}(dv)\mu_{t}(dv_{\star}).\end{split}$
(173)

Gathering (172, 173), we obtain

$\begin{split}&\int_{E}\psi_{\lambda}(t,u,u_{\star})(\tau-1)(\overline{K}^{\lambda})(t,u,u_{\star},\sigma)\overline{m}_{g_{\lambda}\star\mu}(dt,du,du_{\star},d\sigma)\\\\[4.30554pt]
&\hskip
28.45274pt\leq\int_{E\times\mathbb{R}^{d}\times\mathbb{R}^{d}}(\tau-1)(K(t,v,v_{\star},\sigma))B(v-v_{\star})g_{\lambda}(u-v)g_{\lambda}(u_{\star}-v_{\star})dt\mu_{t}(dv)\mu_{t}(dv_{\star})d\sigma\\\\[4.30554pt]
&\hskip
28.45274pt=\int_{E}(\tau-1)(K(t,v,v_{\star},\sigma))B(v-v_{\star})dt\mu_{t}(dv)\mu_{t}(dv_{\star})d\sigma\\\\[4.30554pt]
&\hskip
28.45274pt=\int_{E}(\tau-1)(K(t,v,v_{\star},\sigma))\overline{m}_{\mu}(dt,dv,dv_{\star},d\sigma).\end{split}$
(174)

Returning to (168) and using the analagous equation for $\mu_{\bullet}$, we
finally obtain the decomposition

$\mathcal{J}(g_{\lambda}\star\mu_{\bullet},g_{\lambda}\star
w)\leq\mathcal{J}(\mu_{\bullet},w)+(\overline{m}_{g_{\lambda}\star\mu}-\overline{m}_{\mu})(E)+\mathcal{T}_{\lambda}(\mu_{\bullet},w).$
(175)

It is very straightforward to show that the second term converges to $0$ as
$\lambda\to 0$, with a rate depending only on
$\sup_{t}\langle|v|^{2},\mu_{t}\rangle$.

#### Step 4: Analysis of $\psi_{\lambda}$

We now turn to the error term $\mathcal{T}_{\lambda}$ identified in (171),
which depends on the continuity of $\log B$. We remark first that this term
cannot be avoided purely on general considerations; consider, for example, the
kernel $B(v)=1_{|v|\geq 1}$ in which case
$\mathcal{J}(g_{\lambda}\star\mu_{\bullet},g_{\lambda}\star w)$ can become
infinite due to contributions in the region $\\{|v-v_{\star}|<1\\}$.

We start with an upper bound for $\psi_{\lambda}$. We define first the
measures

$\xi^{\lambda}_{(t,u)}(dv):=\frac{g_{\lambda}(v-u)\mu_{t}(dv)}{(g_{\lambda}\star\mu_{t})(u)}.$
(176)

Setting $R:=\sqrt{2\sup_{s}\langle|v|^{2},\mu_{s}\rangle}$, a simple Chebychev
inequality shows that $\mu_{t}(|v|\leq R)\geq\frac{1}{2}$ for all $t$, which
leads to the lower bound

$(g_{\lambda}\star\mu_{t})(u)\geq\frac{1}{2}\exp\left(-(|u|+R)^{2})/2\lambda\right)/(2\pi\lambda)^{d/2}.$
(177)

We now estimate

$\begin{split}\left\langle|v|^{2}1[|v-u|>R+|u|],\xi^{\lambda}_{(t,u)}\right\rangle&=\frac{\int_{|v-u|>R+|u|}e^{-|v-u|^{2}/2\lambda}|v|^{2}\mu_{t}(dv)/(2\pi\lambda)^{d/2}}{(g_{\lambda}\star\mu_{t})(u)}\\\\[8.61108pt]
&\leq\frac{e^{-(|u|+R)^{2}/2\lambda}/(2\pi\lambda)^{d/2}}{(g_{\lambda}\star\mu_{t})(u)}\int_{|v-u|>R+|u|}|v|^{2}\mu_{t}(dv)\\\\[12.91663pt]
&\leq 2\sup_{s}\langle|v|^{2},\mu_{s}\rangle=R^{2}.\end{split}$ (178)

Together with a trivial bound for the remaining region, we conclude that

$\begin{split}\left\langle|v|^{2},\xi^{\lambda}_{(t,u)}\right\rangle\leq
3R^{2}+4|u|^{2}\leq C(1+|u|^{2})\end{split}$ (179)

since $R\geq 1$ depends only on the quantities in (145). Using the lower bound
$B(u-u_{\star})\geq 1$ and the upper bound $B(v-v_{\star})\leq
1+|v|+|v_{\star}|\leq C(1+|v|^{2}+|v_{\star}|^{2})$, we now return to (167) to
obtain

$\psi_{\lambda}(t,u,u_{\star})\leq C(1+|u|^{2}+|u_{\star}|^{2}).$ (180)

This bound will be useful in general, for $(u,u_{\star})$ where
$\psi_{\lambda}$ cannot be shown to be close to $1$. We complement this with a
bound which will show that, for most points $(u,u_{\star})$ in the support of
$(g_{\lambda}\star\mu_{t})^{\otimes 2}$, $\psi_{\lambda}$ is not too much
bigger than 1. We will exploit, repeatedly, the observation that, for the
kernels given in (3),

$B(v-v_{\star})\leq(1+|v-u|+|v_{\star}-u_{\star}|)B(u-u_{\star}).$ (181)

Fix $t\geq 0$, and suppose that $v^{0},v^{0}_{\star}$ are such that there
exists a sets $U\ni v^{0},U_{\star}\ni v^{0}_{\star}$ of diameter
$c\sqrt{\lambda}$ and
$\mu_{t}(U),\mu_{t}(U_{\star})\geq\epsilon\lambda^{d/2}$, and $(u,u_{\star})$
is such that $|(u,u_{\star})-(v^{0},v^{0}_{\star})|<x\sqrt{\lambda}$, for some
constant $c$ and parameters $\epsilon>0,x<\infty$ to be chosen later. In this
case, we bound the denominator below by observing that

$(g_{\lambda}\star\mu_{t})(u)\geq\frac{1}{(2\pi\lambda)^{d/2}}e^{-(x\sqrt{\lambda}+c\sqrt{\lambda})^{2}/2\lambda}\mu(U)\geq\frac{\epsilon}{(2\pi)^{d/2}}\exp\left(-x^{2}-c^{2}\right)$
(182)

and similarly for $u_{\star}$. We now return to (167) and split the integral
defining $\psi_{\lambda}$ into the regions
$E_{1}=\\{(v,v_{\star}):|(v,v_{\star})-(u,u_{\star})|<\lambda^{1/3}+3x\sqrt{\lambda}\\}$
and its complement $E_{2}$. On $E_{1}$,

$B(v-v_{\star})\leq(1+2\lambda^{1/3}+6x\sqrt{\lambda})B(u-u_{\star})$ (183)

and so

$\begin{split}&\int_{E_{1}}\hskip
2.84544pt\frac{B(v-v_{\star})}{B(u-u_{\star})}\hskip
2.84544pt\frac{g_{\lambda}(v-u)\mu_{t}(dv)}{(g_{\lambda}\star\mu_{t})(u)}\hskip
2.84544pt\frac{g_{\lambda}(v_{\star}-u_{\star})\mu_{t}(dv_{\star})}{(g_{\lambda}\star\mu_{t})(u_{\star})}\leq
1+2\lambda^{1/3}+6x\sqrt{\lambda}.\end{split}$ (184)

On the other hand, using (183) and the trivial bound
$B(v-v_{\star})/B(u-u_{\star})\leq C(1+|v|+|v_{\star}|)$, we bound the term
from the second region by

$\begin{split}&\int_{E_{2}}\hskip
2.84544pt\frac{B(v-v_{\star})}{B(u-u_{\star})}\hskip
2.84544pt\frac{g_{\lambda}(v-u)\mu_{t}(dv)}{(g_{\lambda}\star\mu_{t})(u)}\hskip
2.84544pt\frac{g_{\lambda}(v_{\star}-u_{\star})\mu_{t}(dv_{\star})}{(g_{\lambda}\star\mu_{t})(u_{\star})}\\\\[4.30554pt]
&\hskip 85.35826pt\leq
C\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(1+|v|+|v_{\star}|)\lambda^{-d}\exp\left(-\frac{1}{2\lambda}(9x^{2}\lambda+\lambda^{2/3})\right)\epsilon^{-2}\\\
&\hskip
142.26378pt\dots\times\exp(2x^{2}+2c^{2})\mu_{t}(dv)\mu_{t}(dv_{\star})\\\\[8.61108pt]
&\hskip 28.45274pt\hskip 28.45274pt\hskip 28.45274pt\leq
C\exp\left(2c^{2}-x^{2}\right)\epsilon^{-2}.\end{split}$ (185)

where, in the final line, we recall that
$\lambda^{-d}\exp(-\lambda^{-1/6}/{2})$ is uniformly bounded on $(0,\infty)$,
and that the remaining integral is controlled in terms of the second moments
of $\mu_{t}$, so can be absorbed into $C$. Gathering
(LABEL:eq:_analysis_of_fd_7, 185), we conclude that for
$(u,u_{\star}),x,\epsilon$ as above,

$\begin{split}\psi_{\lambda}(t,u,u_{\star})&\leq
1+2\lambda^{1/3}+6x\sqrt{\lambda}+C\exp(2c^{2}-x^{2})\epsilon^{-2}\\\
&\leq(1+2\lambda^{1/3}+6x\sqrt{\lambda})\left(1+C\exp(2c^{2}-x^{2})\epsilon^{-2}\right).\end{split}$
(186)

#### Step 5: Analysis of $\mathcal{T}_{\lambda}$

Equipped with this preliminary analysis of $\psi_{\lambda}$ in the previous
step, we bound the final term $\mathcal{T}_{\lambda}$ appearing in (175).
Together with the observation under (175), this suffices to prove (146) and
finish the proof of the lemma.

We break up the integration space $E\times\mathbb{R}^{d}\times\mathbb{R}^{d}$
in the definition of $\mathcal{T}_{\lambda}$ as follows. For $M\geq
2,R\in[3,\infty)\cap\sqrt{\lambda}\mathbb{N}$,
$x\in(0,\infty),\epsilon\in(0,\infty)$ to be chosen later, we form a partition
$\mathfrak{P}$ of $(-R,R]^{d}$ into $(2R/\sqrt{\lambda})^{d}$ translates of
$(0,\sqrt{\lambda}]^{d}$, and for $v\in(-R,R]^{d}$, write $\mathcal{B}(v)$ for
the unique $\mathcal{B}\in\mathfrak{P}$ containing $v$. We now consider the
partition of $E\times\mathbb{R}^{d}\times\mathbb{R}^{d}$ given by

$A_{1}:=\left\\{v,v_{\star}\in(-R,R]^{d},K\leq
M,\mu_{t}(\mathcal{B}(w))\geq\epsilon\lambda^{d/2},\mu_{t}(\mathcal{B}(v_{\star}))\geq\epsilon\lambda^{d/2},|(u,u_{\star})-(v,v_{\star})|<x\sqrt{\lambda}\right\\};$
$A_{2}:=\left\\{v,v_{\star}\in(-R,R]^{d},K\leq
M,\mu_{t}(\mathcal{B}(w))\geq\epsilon\lambda^{d/2},\mu_{t}(\mathcal{B}(v_{\star}))\geq\epsilon\lambda^{d/2},|(u,u_{\star})-(v,v_{\star})|\geq
x\sqrt{\lambda}\right\\};$ $A_{3}:=\left\\{v,v_{\star}\in(-R,R]^{d},K\leq
M,\mu_{t}(\mathcal{B}(v))<\epsilon\lambda^{d/2}\text{ or
}\mu_{t}(\mathcal{B}(v_{\star}))<\epsilon\lambda^{d/2}\right\\};$
$A_{4}:=\left\\{v,v_{\star}\in(-R,R]^{d},K(t,v,v_{\star},\sigma)>M\right\\};$
$A_{5}:=\left\\{(v,v_{\star})\not\in(-R,R]^{2d}\right\\}.$ (187)

We analyse the contributions from these regions one-by-one. Roughly, $A_{1}$
is the ‘good’ region, containing most of the contributions from the
integrating measure, where $\log\psi_{\lambda}$ is small by (186), and the
remaining terms are small, depending on the parameters $M,R,x,\epsilon$; at
the end, we will optimise, so that $M,R,x\to\infty$ and $\epsilon\to 0$ as
functions of $\lambda\to 0$.

#### Step 5a: Contribution from $A_{1}$

For the region $A_{1}$, we observe that the hypotheses leading to (186) hold,
with $U=\mathcal{B}(v),U_{\star}=\mathcal{B}(v_{\star})$ and $c=\sqrt{d}$ is
an absolute constant. Further, $K<M$ and $B(v-v_{\star})\leq 1+2R\leq CR$, so
we integrate (186) to find

$\begin{split}&\int_{A_{1}}K(t,v,v_{\star},\sigma)B(v-v_{\star})g_{\lambda}(u-v)g_{\lambda}(u_{\star}-v_{\star})\log\psi_{\lambda}(t,u,u_{\star})dt\mu_{t}(dv)\mu_{t}(dv_{\star})d\sigma
dudu_{\star}\\\ &\hskip 28.45274pt\hskip 28.45274pt\hskip 28.45274pt\hskip
28.45274pt\hskip 28.45274pt\leq
CMR\left(\log(1+2\lambda^{1/3}+6x\sqrt{\lambda})+\log(1+\epsilon^{-2}e^{-x^{2}})\right).\end{split}$
(188)

#### Step 5b: Contribution from $A_{2}$

For $A_{2}$, we use the general upper bound (180), which is valid without
restriction on $u,u_{\star}$. For fixed $v,v_{\star}$ we use Hölder’s
inequality to see that

$\begin{split}&\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\log\psi_{\lambda}(t,u,u_{\star})1[|(u,u_{\star})-(v,v_{\star})|\geq
x\sqrt{\lambda}]g_{\lambda}(u-v)g_{\lambda}(u_{\star}-v_{\star})dudu_{\star}\\\
&\hskip 28.45274pt\hskip 28.45274pt\hskip 28.45274pt\leq
C\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\log(1+|u|^{2}+|u_{\star}|^{2})1[|(u,u_{\star})-(v,v_{\star})|\geq
x\sqrt{\lambda}]\\\ &\hskip 142.26378pt\dots\times
g_{\lambda}(u-v)g_{\lambda}(u_{\star}-v_{\star})dudu_{\star}\\\ &\hskip
28.45274pt\hskip 28.45274pt\leq
C\left(\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(1+|u|^{2}+|u_{\star}|^{2})g_{\lambda}(u-v)g_{\lambda}(u_{\star}-v_{\star})dudu_{\star}\right)^{1/2}\\\
&\hskip 28.45274pt\hskip 28.45274pt\hskip 28.45274pt\hskip 28.45274pt\hskip
28.45274pt\dots\times\left(\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}1[|(u^{\prime},u^{\prime}_{\star})|>x\sqrt{\lambda}]g_{\lambda}(u^{\prime})g_{\lambda}(u^{\prime}_{\star})du^{\prime}du^{\prime}_{\star}\right)^{1/2}\\\\[8.61108pt]
&\hskip 28.45274pt\hskip 28.45274pt\leq
C(1+|v|+|v_{\star}|)\exp\left(-x^{2}/16d\right)\end{split}$ (189)

where for the first factor in the final line, we integrated
$\int|u|^{2}g_{\lambda}(u-v)du=|u|^{2}+d\lambda^{2}\leq|v|^{2}+d$, and for the
second factor we used standard tail estimates for the normal distribution,
absorbing constants into the prefactor $C$. Bounding $B(v-v_{\star}),K$ as
above, and integrating over $t,v,v_{\star},\sigma$, we find

$\begin{split}&\int_{A_{2}}K(t,v,v_{\star},\sigma)B(v-v_{\star})g_{\lambda}(u-v)g_{\lambda}(u_{\star}-v_{\star})\log\psi_{\lambda}(t,u,u_{\star})dt\mu_{t}(dv)\mu_{t}(dv_{\star})d\sigma
dudu_{\star}\\\ &\hskip 142.26378pt\leq
CMR\exp\left(-x^{2}/16d\right).\end{split}$ (190)

#### Step 5c: Contribution from $A_{3}$

Similarly to the previous step, we start from a bound on the integrals over
$u,u_{\star}$, with $t$ and $v,v_{\star}$ fixed. We split the integral over
$u,u_{\star}$ into
$\\{|(u,u_{\star})-(v,v_{\star})|\leq(1+|v|+|v_{\star}|)\\}$ and
$\\{|(u,u_{\star})-(v,v_{\star})|>(1+|v|+|v_{\star}|)\\}$. In the first
region, thanks to (181),

$\log\psi_{\lambda}(t,u,u_{\star})\leq C\log(1+|u|^{2}+|u_{\star}|^{2})\leq
C\log(1+|v|^{2}+|v_{\star}|^{2})$ (191)

while the contribution from the second region is controlled by using Hölder’s
inequality in the same way as (189) to obtain

$\begin{split}&\int_{|(u,u_{\star})-(v,v_{\star})|>(1+|v|+|v_{\star}|)}\log\psi_{\lambda}(t,u,u_{\star})g_{\lambda}(u-v)g_{\lambda}(u_{\star}-v_{\star})dudu_{\star}\\\
&\hskip 142.26378pt\leq
C(1+|v|+|v_{\star}|)\exp(-(1+|v|+|v_{\star}|)^{2}/16d).\end{split}$ (192)

This term can be absorbed into the contribution from (191), and we conclude
that, for all $t$ and $v,v_{\star}\in\mathbb{R}^{d}$,

$\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\log\psi_{\lambda}(t,u,u_{\star})g_{\lambda}(u-v)g_{\lambda}(u_{\star}-v_{\star})dudu_{\star}\leq
C\log(1+|v|^{2}+|v_{\star}|^{2}).$ (193)

In particular, when $v,v_{\star}\in(-R,R]^{d}$, the right-hand side can be
replaced by $C\log R$, and $B(v-v_{\star})\leq CR$. We now integrate over
$A_{3}$ to find

$\begin{split}&\int_{A_{3}}K(t,v,v_{\star},\sigma)B(v-v_{\star})g_{\lambda}(u-v)g_{\lambda}(u_{\star}-v_{\star})\log\psi_{\lambda}(t,u,u_{\star})dt\mu_{t}(dv)\mu_{t}(dv_{\star})d\sigma
dudu_{\star}\\\ &\leq CM(R\log R)\int_{0}^{T}\mu_{t}^{\otimes
2}\left((v,v_{\star})\in(-R,R]^{2d}:\mu_{t}(\mathcal{B}(v))<\epsilon\lambda^{d/2}\text{
or }\mu_{t}(\mathcal{B}(v_{\star}))<\epsilon\lambda^{d/2}\right)dt\\\ &\hskip
28.45274pt\hskip 28.45274pt\leq CM(R\log
R)\int_{0}^{T}\mu_{t}\left(v\in(-R,R]^{d}:\mu_{t}(\mathcal{B}(v))<\epsilon\lambda^{d/2}\right)dt\end{split}$
(194)

where the last line follows using a union bound, absorbing the factor of $2$
into $C$. The integrand is now

$\begin{split}\mu_{t}\left(v\in(-R,R]^{d}:\mu_{t}(\mathcal{B}(v))<\epsilon\lambda^{d/2}\right)&=\sum_{\mathcal{B}\in\mathfrak{P}}\mu_{t}(\mathcal{B})1(\mu_{t}(\mathcal{B})<\epsilon\lambda^{d/2})\\\\[8.61108pt]
&\leq\epsilon\lambda^{d/2}(\\#\mathfrak{P})=\epsilon(2R)^{d}\end{split}$ (195)

recalling that $\\#\mathfrak{P}=(2R/\sqrt{\lambda})^{d}$. Substituting this
bound back into (194) we conclude that

$\begin{split}&\int_{A_{3}}K(t,v,v_{\star},\sigma)B(v-v_{\star})g_{\lambda}(u-v)g_{\lambda}(u_{\star}-v_{\star})\log\psi_{\lambda}(t,u,u_{\star})dt\mu_{t}(dv)\mu_{t}(dv_{\star})d\sigma
dudu_{\star}\\\ &\hskip 28.45274pt\hskip 28.45274pt\leq CM(R^{d+1}\log
R)\epsilon.\end{split}$ (196)

#### Step 5d: Contribution from $A_{4}$

In $A_{4}$, we use the same bound (193) on
$\int\log\psi_{\lambda}g_{\lambda}(u-v)g_{\lambda}(u_{\star}-v_{\star})dudu_{\star}$,
and observe that, on $A_{4}$,
$K(t,v,v_{\star},\sigma)\leq\frac{M}{\tau(M)}\tau(K)$. Integrating, it follows
that

$\begin{split}&\int_{A_{4}}K(t,v,v_{\star},\sigma)B(v-v_{\star})g_{\lambda}(u-v)g_{\lambda}(u_{\star}-v_{\star})\log\psi_{\lambda}(t,u,u_{\star})dt\mu_{t}(du)\mu_{t}(dv_{\star})d\sigma
dudu_{\star}\\\ &\hskip 28.45274pt\leq C(\log
R)\left(\frac{M}{\tau(M)}\right)\int_{E}\tau(K(t,v,v_{\star},\sigma))\overline{m}_{\mu}(dt,dv,dv_{\star},d\sigma)\\\\[8.61108pt]
&\hskip 28.45274pt=C(\log
R)\left(\frac{M}{\tau(M)}\right)\mathcal{J}(\mu_{\bullet},w)=C(\log
R)\left(\frac{M}{\tau(M)}\right)\end{split}$ (197)

since $C$ is allowed to depend on an upper bound for
$\mathcal{J}(\mu_{\bullet},w)$.

#### Step 5e: Contribution from $A_{5}$

We finally turn to the contribution from $A_{5}$. Thanks to (193), for any
$(v,v_{\star})\not\in(-R,R]^{2d}$, we have

$\begin{split}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\log\psi_{\lambda}(t,u,u_{\star})g_{\lambda}(u-v)g_{\lambda}(u_{\star}-v_{\star})dudu_{\star}&\leq
C\log(1+|v|^{2}+|v_{\star}|^{2})\\\ &\leq
C\frac{\log(R^{2})}{R^{2}}(1+|v|^{2}+|v_{\star}|^{2})\end{split}$ (198)

as $(\log x)/x$ is decreasing on $[R,\infty)\subset[e,\infty)$, and
$1+|v|^{2}+|v_{\star}|^{2}\geq R^{2}$. Integrating over
$t,v,v_{\star},\sigma$, we obtain