Large deviations for a binary collision model: energy evaporation

1.   Introduction

Large deviations associated to Boltzmann-type equations have been the object of recent investigations. The most challenging case of Newtonian dynamics of hard spheres in the Boltzmann-Grad limit has been heuristically discussed in ^[5] and rigorously, by means of cluster expansion, in ^[4]. The case of microscopic stochastic dynamics has been originally analyzed in ^[8], where a large deviation upper bound is derived. In ^[15] a large deviation principle is obtained for a space inhomogeneous model with a finite set of velocities. More recently, in ^[2] it is considered a homogeneous model which conserves momentum but not energy. The large deviation upper bound is achieved, while the lower bound is obtained for a restricted class of paths. In ^[6] the analogous results are provided for the Kac walk, which conserves also the energy.

We emphasize that, except in the case of bounded velocities, a proof of a large deviation principle with matching upper and lower bound is still missing, even in the homogeneous case. A key issue is the possible occurrence of macroscopic paths with finite rate function that violate the conservation of the energy. A class of examples has been constructed in ^[6] by exploiting the solutions of homogeneous Boltzmann equations provided by Lu and Wennberg in ^[10], for which the energy is increasing.

Here we consider a Kac-like microscopic dynamics with discrete energy, that is inspired by the so-called KMP model ^[3,7], and described as follows: $N$ particles, with equally spaced energy levels, evolve via random binary collisions such that in each collision the total energy is preserved. At the kinetic level the one-particle energy distribution evolves according to a discrete homogeneous Boltzmann equation, that is an infinite system of coupled ordinary differential equations. We focus on the large deviation properties of the pair empirical measure and flux and propose a candidate rate function $I$ when the initial distribution of the energies satisfies the micro-canonical constraint, i.e., the total energy is fixed. By the arguments in ^[1,2,6], it can be shown that the large deviation upper bound holds with rate function $I$, and a matching lower bound can be proven for a restricted class of paths which conserve the energy. Our main novel point is the construction of a path $(\bar f, \bar Q)$ which looses the energy and whose probability is exponentially small with rate $I(\bar f, \bar Q)$. This result quantifies the probability to violate the conservation of the energy, which is exponentially small in the number of particles $N$. It can be compared with the result in ^[14] where, in the contest of the derivation of incompressible Navier-Stokes equations from stochastic lattice gas, it is shown that the probability of violating the incompressibility condition is of the order $\mathrm{e}^{-N^2}$.

Referring to ^[1] for a general discussion, we emphasize that, due to the micro-canonical constraint, at the kinetic level the energy cannot increase and the candidate rate function $I$ is different from the one in ^[6,8].

1.1.   The model

Given $N\ge 2$, a configuration is defined by $N$ energies in ${\mathbb N}$. The configuration space is therefore given by $\Sigma_N = {\mathbb N}^{N}$. Elements of $\Sigma_N$ are denoted by ${\boldsymbol{\varepsilon }}: = (\varepsilon_1, .., \varepsilon_N)$ and we denote by $\Sigma_{N, E}$ the configuration space with total energy $E\in {\mathbb N}$, i.e.,

The microscopic dynamics is defining by choosing at random a pair $\{i, j\}$ and redistributing uniformly the corresponding energies. Therefore we consider the Markov processes on $\Sigma_{N}$ whose generator acts on bounded functions $f\colon \Sigma_N\to {\mathbb R}$ as

where the sum is carried over the unordered pairs $\{i, j\}\subset \{1, .., N\}$, $i\neq j$, and

in which

Observe that, for each $E > 0$, the dynamics preserves $\Sigma_{N, E}$. Moreover, it is ergodic when restricted to $\Sigma_{N, E}$ and reversible with respect to the uniform measure on $\Sigma_{N, E}$.

We denote by $({\boldsymbol{\varepsilon }}(t))_{t\ge 0}$ the continuous time Markov chain generated by ${\mathscr L_N}$. In particular, the path ${\boldsymbol{\varepsilon }}(\cdot)$ is piecewise constant, and the transition probability of its jumps is given in (1.1). Fix hereafter $T > 0$. Given a probability $\nu$ on $\Sigma_{N, E}$ we denote by $\mathbb{P}_{\nu }^{N}$ the law of this chain on the time interval $[0, T]$, when the initial datum is sampled according to $\nu$. Observe that $\mathbb{P}_{\nu }^{N}$ is a probability on the Skorokhod space $D([0, T];\Sigma_{N, E})$. As usual if $\nu = \delta_{{\boldsymbol{\varepsilon }}}$ for some ${\boldsymbol{\varepsilon }} \in \Sigma_{N, E}$, the corresponding law is simply denoted by ${\mathbb P}_{{\boldsymbol{\varepsilon }}}^N$. We refer to ^[9] for a gentle introduction to continuous time Markov chains.

1.2.   Empirical observables

Given $e \in (0, +\infty)$, we denote by ${\mathscr P_e}({\mathbb N})$ the set of probability measures $\pi$ on ${\mathbb N}$ with mean bounded by $e$, i.e., such that $\sum_{ \varepsilon} \varepsilon \pi(\varepsilon)\leq e$. We consider ${\mathscr P_e}({\mathbb N})$ as a closed subset of the space of probability measure on ${\mathbb N}$ equipped with the weak topology. Then ${\mathscr P_e}({\mathbb N})$ endowed with the relative topology is a compact Polish space. Indeed, ${\mathscr P_e}({\mathbb N})$ is the weak closure of the set of probabilities on ${\mathbb N}$ with mean $e$. The empirical measure records the energy of the particles forgetting their labels, for $E = \lfloor N\, e\rfloor$ it is defined as the map $\pi^N\colon \Sigma_{N, E} \to {\mathscr P_e}({\mathbb N})$ given by

Let $D\big([0, T]; {\mathscr P_e}({\mathbb N})\big)$ the set of ${\mathscr P_e}({\mathbb N})$-valued c{á}dl{á}g paths endowed with the Skorokhod topology and the corresponding Borel $\sigma$-algebra. With a slight abuse of notation we denote also by $\pi^N$ the map from $D\big([0, T]; \Sigma_{N, E} \big)$ to $D\big([0, T]; {\mathscr P_e}({\mathbb N})\big)$ defined by $\pi^N_t({\boldsymbol{\varepsilon }}): = \pi^N({\boldsymbol{\varepsilon }}(t))$, $t\in [0, T]$.

We also introduce the empirical flow that records the collisions of the particles forgetting their labels. To this end, we denote by ${\mathscr M}$ the subset of the finite measures $Q$ on $[0, T]\times {\mathbb N}^2\times {\mathbb N}^2$ that satisfy $Q(\mathop{}\!\mathrm{d} t; \varepsilon, \varepsilon_*, \varepsilon', \varepsilon_*') = Q(\mathop{}\!\mathrm{d} t; \varepsilon_*, \varepsilon, \varepsilon', \varepsilon_*') = Q(\mathop{}\!\mathrm{d} t; \varepsilon, \varepsilon_*, \varepsilon'_*, \varepsilon')$. We endow ${\mathscr M}$ with the weak* topology and the associated Borel $\sigma$-algebra. The empirical flow is the map $Q^N\colon D\big([0, T]; \Sigma_{N, E} \big) \to {\mathscr M}$ defined by

where $F\colon [0, T]\times {\mathbb N}^2\times {\mathbb N}^2\to {\mathbb R}$ is continuous, bounded, and $F(t; \varepsilon, \varepsilon_*, \varepsilon', \varepsilon_*') = F(t; \varepsilon_*, \varepsilon, \varepsilon', \varepsilon_*') = F(t; \varepsilon, \varepsilon_*, \varepsilon'_*, \varepsilon')$, while $(\tau^{i, j}_k)_{k\ge 1}$ are the jump times of the pair $(\varepsilon_i, \varepsilon_j)$. Here, $\varepsilon_i(t-) = \lim_{s\uparrow t} \varepsilon_i(s)$. In view of the conservation of the energy, the measure $Q^N(\mathop{}\!\mathrm{d} t; \cdot)$ is supported on ${\mathscr E} : = \{ \varepsilon+ \varepsilon_* = \varepsilon'+ \varepsilon_*'\}\subset {\mathbb N}^2\times {\mathbb N}^2$.

For each ${\boldsymbol{\varepsilon }} \in \Sigma_{N, E}$, with ${\mathbb P}^N_{{\boldsymbol{\varepsilon }}}$ probability one, the pair $(\pi^N, Q^N)$ satisfies the following balance equation that express the conservation of probability. For each $\phi\colon [0, T]\times {\mathbb N}\to {\mathbb R}$ bounded and continuously differentiable with respect to time

1.3.   Law of large numbers

Fix $m\in {\mathscr P}({\mathbb N})$ and assume one of the following condition: $m$ is a point mass or the support of $m$ does not generate a proper sub-lattice of ${\mathbb Z}$. Note that in the second case $m$ satisfies the condition for the local central limit theorem for i.i.d. lattice random variables, see [13,§ VII.1]. For $\gamma\in {\mathbb R}$ we set

We assume that there exists $\gamma^*\in (0, +\infty]$ such that $Z_\gamma < +\infty$ for $\gamma\in (0, \gamma^*)$ and $Z_\gamma \uparrow + \infty$ for $\gamma \uparrow \gamma^*$. For $e\in (0, +\infty)$, we then define the probability $\mu_{N, e}$ on $\Sigma_{N, \lfloor N e \rfloor}$ by considering i.i.d. $m$-distributed energies and conditioning to the total energy, i.e.,

that will be chosen as the initial distribution of the microscopic dynamics. In the case of point mass, we require that $\{e\}$ is exactly the support of $m$. Observe that, by the equivalence of the ensembles, as $N\to +\infty$ the one-marginal of $\mu_{N, e}$ converges to the probability $m_e$ given by

Denoting by $B$ the collision kernel in (1.1), i.e.,

the law of the large numbers for the empirical measure is described by the following discrete homogeneous Boltzmann equation

More precisely, in probability with respect to ${\mathbb P}^N_{\mu_{N, e}}$, the empirical path $\big(\pi^N_t\big)_{t\in [0, T]}$ converges to $\big (f_t\big)_{t\in [0, T]}$ where $f_t$ is the unique solution of the Cauchy problem associated to (1.9) with initial datum $f_0 = m_e$. As the proof of this statement can be achieved by adapting the chaos propagation arguments in ^[16], we omit the details. The law of the large numbers of the empirical flow $Q^N$ reads

where the convergence is in probability with respect to ${\mathbb P}^N_{\mu_{N, e}}$. We refer to Lemma 3.3 below for the proof.

In the general contest of homogeneous Boltzmann equations, uniqueness of the Cauchy problem associated to (1.9) holds for paths $f_t$ that conserves the energy, see e.g., ^[12]. However in the present case, since the $\sup_{ \varepsilon, \varepsilon_*}\sum_{ \varepsilon', \varepsilon'_*} B(\varepsilon, \varepsilon_*, \varepsilon', \varepsilon'_*)$ is bounded, Gronwall's inequality implies the uniqueness without assuming the energy conservation, see e.g., Lemma 4.1 in ^[2]. In particular, by uniqueness, for this model Lu and Wennberg like solutions do not exist. We finally observe that (1.9) admits a one-parameter family of stationary solutions given by $f_{ \; {\rm{stat}}\; }(\varepsilon) = p(1-p)^{ \varepsilon}$, $p\in (0, 1]$.

1.4.   The candidate rate function

For $e\in (0, +\infty)$, let ${\mathscr S_e}$ be the (closed) subset of $D\big([0, T]; {\mathscr P_e}({\mathbb N})\big)\times {\mathscr M}$ given by elements $(\pi, Q)$ that satisfies the balance equation

for each $\phi:[0, T]\times {\mathbb N}\to {\mathbb R}$ bounded and continuously differentiable in $t$. We consider ${\mathscr S_e}$ endowed with the relative topology and the corresponding Borel $\sigma$-algebra.

For $\pi\in D\big([0, T]; {\mathscr P_e}({\mathbb N})\big)$ let $Q^\pi$ be the measure defined by

where $B$ is the collision kernel in (1.8). Observe that $Q^\pi(\mathop{}\!\mathrm{d} t, \cdot)$ is supported on ${\mathscr E}$. Let ${\mathscr S}_{e}^{\mathrm{ac}}$ be the subset of ${\mathscr S_e}$ given by the elements $(\pi, Q)$ such that $\pi\in C\big([0, T];{\mathscr P_e}({\mathbb N})\big)$ and $Q\ll Q^\pi$. The dynamical rate function $J\colon {\mathscr S_e} \to [0, +\infty]$ is defined by

Given two probabilities $\mu_1, \mu_2$, the relative entropy $\mathop{{\; {\rm{Ent}}\; }}\nolimits(\mu_2\vert\mu_1)$ is defined as $\mathop{{\; {\rm{Ent}}\; }}\nolimits(\mu_2\vert\mu_1) = \int d\mu_1 \rho\log\rho$, where $d\mu_2 = \rho\, d\mu_1$, understanding that $\mathop{{\; {\rm{Ent}}\; }}\nolimits(\mu_2|\mu_1) = +\infty$ if $\mu_2$ is not absolutely continuous with respect to $\mu_1$. Let $H_e\colon {\mathscr P_e} ({\mathbb N}) \to [0, +\infty]$ be defined by

where $m_e$ and $\gamma_e$ are as in (1.7). When $m$ is the point mass on $e$, $H_e(\pi)$ is zero when $\pi = \delta_e$ and $+\infty$ otherwise. The candidate large deviation rate function is given by

As discussed in ^[1], the (static) large deviations of the empirical measure with respect to the probability $\mu_{N, e}$ are described by the rate function $H_e$, where the second term on the r.h.s. of (1.14) is the cost of having energy less than $e$. Note that if $\gamma^* = +\infty$, then $H_e(\pi)$ is finite only if the energy of $\pi$ is $e$. A key ingredient in the proof is the local central limit theorem for the sum of independent $m_e$ distributed random variables.

Denote by $\hat {{\mathscr S}}_e$ the subset of ${\mathscr S}$ given by the pair $(\pi, Q)$ such that

If $(\pi, Q)\in \hat {{\mathscr S}}_e$, the balance equation (1.11) implies that the path $\pi_t$ conserves the energy.

As already mentioned, a proof of a large deviations principle for the pair $(\pi^N, Q^N\!)$ with matching upper and lower has not been yet achieved. The analysis in ^[2,6] implies however the large deviation upper bound with rate $I$ with a matching lower bound on the set $\hat {{\mathscr S}}_e$. The precise statement is the following.

Theorem 1.1. Fix $e \in (0, +\infty)$ and let $\mu_{N, e}$ be the family of probabilities on $\Sigma_{N, \lfloor Ne \rfloor}$ defined in (1.6). The family $\{{\mathbb P}^N_{\mu_{N, e}} \circ (\pi^N, Q^N)^{-1}\}$ satisfies a large deviations upper bound with good rate function $I\colon {\mathscr S_e} \to [0, +\infty]$, namely $I$ has compact level sets and for each closed ${\mathcal {C}} \subset {\mathscr S}$

Moreover, for each open ${\mathcal {O}} \subset {\mathscr S_e}$

Referring to ^[2,6] for comments on the technicalities involved in the lower bound, we now turn to the novel point of the present analysis, that is the construction of paths $(\pi, Q)$ – with $\pi$ not energy conserving – whose probability is precisely of the order $\exp\{-N I(\pi, Q)\}$. Since these paths do not belong to $\hat{{\mathscr S}}_e$, this result provides insights on the large deviations properties of Kac's walk not covered by (1.17). As the large deviations upper bound is already covered by (1.16), we focus on the matching lower bound.

Theorem 1.2. Fix $e \in (0, +\infty)$ and let $\mu_{N, e}$ be the family of probabilities on $\Sigma_{N, \lfloor Ne \rfloor}$ defined in (1.6). For each $t^*\in (0, T)$ there exists a path $(\bar f, \bar Q)$, satisfying $\sum_{ \varepsilon} \bar f_t(\varepsilon) \varepsilon = e$ for $t\in [0, t^*)$ and $\sum_{ \varepsilon} \bar f_t(\varepsilon) \varepsilon < e$ for $t\in [t^*, T]$, such that $I(\bar f, \bar Q) < +\infty$ and

for any open neighborhood $\mathcal O\ni (\bar f, \bar Q)$.

We will provide a self-contained proof of this statement that do not rely on Theorem 1.1. In the argument we take advantage of the fact that the energies are in ${\mathbb N}$. However we expect that the strategy can be extended also to the continuous case. In Section 2 we construct a path $(\bar f, \bar Q)$ satisfying the above requirements, i.e., with evaporating energy for $t > t^*$ and such that $I(\bar f, \bar Q) < +\infty$. The lower bound (1.18) is then proven in Section 3. For the sake of concreteness, the proposed path $(\bar f, \bar Q)$ has zero energy for $t\in(t_*, T]$.

2.   Perturbed Boltzmann equation

Fix $t^*\in (0, T)$. In order to construct a path $(\bar f, \bar Q)$, satisfying $\sum_{ \varepsilon} \bar f_t(\varepsilon) \varepsilon = e$ for $t\in [0, t^*)$ and $\sum_{ \varepsilon} \bar f_t(\varepsilon) \varepsilon < e$ for $t\in (t^*, T]$, we start by considering a solution to a perturbed Boltzmann equation, namely a Boltzmann equation with a suitable modified collision kernel.

Consider the collision kernel $\tilde B$ given by

that describes a scenario in which only particles with the same energy collide, and in each collision the whole energy is transferred to a single particle. The Cauchy problem for the corresponding modified homogeneous Boltzmann equation reads

Proposition 2.1. Assume that $m$ has energy $e$, and let $f$ be the unique solution to (2.2). Then its energy is conserved, i.e., for any $t\in [0, +\infty)$ $\sum_{ \varepsilon \ge 0} f_t(\varepsilon) \varepsilon = e$, while $f_t$ weakly converges to $\delta_0$, as $t\to +\infty$. Moreover, for every $t\geq 0$,

where $c = c(e)$ does not depend on $t$ and the initial datum $m$.

Proof. We prove Eq (2.3), from which the convergence of $f_t$ follows. The modified Boltzmann equation reads as

Note that the equation for $f_t(\varepsilon)$ involves only $f_t(\varepsilon')$ with $\varepsilon'\le \varepsilon$, then the system has global and unique solution. If $\varepsilon$ is odd,

If $\varepsilon$ is even, set $\xi_t(\varepsilon) = (1+t) f_t(\varepsilon)$. Let $T_0$ be the first time $t$ such that $\xi_t(\varepsilon') = 2$ for some $\varepsilon'\le \varepsilon$. The time $T_0$ is strictly positive, since $\xi_0(\varepsilon') \le 1$ for any $\varepsilon'$. For $t < T_0$ it holds

then $T_0 = +\infty$, and this concludes the proof of $i)$ in (2.3).

In order to prove $ii)$ and $iii)$ we first note that

and then $\sum_0^{+\infty} \varepsilon f_t(\varepsilon) \le e$. Inequality $ii)$ and $iii)$ follows by using this fact and the Chebyshev's inequality. We conclude the proof by noticing that the energy is in fact conserved, since $f_t(\varepsilon) \le e/ \varepsilon$ and then for any $h\ge 1$,

which assures that the right-hand-side of Eq (2.5), is vanishing as $n\to +\infty$.

For the modified Boltzmann equation with rate in (2.1) the energy vanishes for dispersion to infinity as $t\to +\infty$. We reparametrize the time so that this happens at a finite time. Fixed $t^*\in (0, T)$, let $\alpha \colon [0, t^*)\to [0, +\infty)$ given by $\alpha(t) = \frac t {1-t/t^*}$. Letting $f$ the solution to (2.2), set

which satisfies the homogeneous Boltzmann equation with time dependent collision kernel

We define the corresponding flux $d\bar Q = \mathop{}\!\mathrm{d} t \, \bar q_t$, where

so that the pair $(\bar f, \bar Q)$ satisfies the balance equation (1.11). Observe that, by construction, $\sum_{ \varepsilon} \bar f_t(\varepsilon) \varepsilon = e$ for $t\in [0, t^*)$ and $\sum_{ \varepsilon} \bar f_t(\varepsilon) \varepsilon = 0$ for $t\in [t^*, T]$. We now show that the pair $(\bar f, \bar Q)$ is such that $I(\bar f, \bar Q) < +\infty$. Since $\bar f_0 = m$, it is enough to show $J(\bar f, \bar Q) < +\infty$. This is stated in the next Proposition.

Proposition 2.2. For $(\bar f, \bar Q)$ defined above the dynamical rate function $J(\bar f, \bar Q)$ is finite.

Proof. Since $(\bar f, \bar Q)\in {\mathscr S_e} ^{\mathrm{ac}}$, the dynamical rate function defined in Eq (1.13) is given by

For $t < t^*$ we have that

Since $\log \dot \alpha = 2 \log (1 + \alpha/t^*)$, the first integral is

Using (2.3) we bound this term by

where $c$ depend only on $e$ and $t^*$.

The second integral in Eq (2.9) is

which completes the proof.

3.   Large deviations lower bound

In order to explain the strategy to prove (1.18), we recall some basic facts on the large deviations lower bound. Let $\{P_n\}$ be a sequence of probabilities on a topological space ${\mathscr X}$. Fix $x\in {\mathscr X}$ and a open neighborhood $\mathcal O\ni x$. To obtain a lower bound for $P_n({\mathcal O})$ we modify the probability $P_n$ so that $x$ becomes the typical behavior. If we are able to do so by paying – as measured by the relative entropy with respect to $P_n$ – not too much then we obtain a good lower bound. The precise statement is summarized in the next lemma, see e.g., ^[11] for its proof.

Lemma 3.1. Let $\{P_n\}_{n\in {\mathbb N}}$ be a sequence of probabilities on a completely regular topological space ${\mathscr X}$ and fix $x\in {\mathscr X}$. Assume that there exists a sequence $\{P_n^x\}$ weakly convergent to $\delta_x$ and such that

for some $I\colon {\mathscr X}\to [0, +\infty]$. Then for any open neighborhood $\mathcal O\ni x$

In most of the applications, and indeed also in our case, the strategy suggested by the above lemma is implemented together with a density argument. The family of perturbed probabilities $P_n^x$ is not constructed for the point $x$ itself but rather for an approximation $x_k$; if the function $I$ is continuous along the sequence $x_k$ then this will do as well. We emphasize that in typical infinite dimensional applications – as in the present case – the rate function $I$ is only lower semicontinuous so the sequence $x_k$ has to be properly chosen. We summarize the argument in the next statement which is deduced from Lemma 3.1 by a straightforward diagonal argument.

Lemma 3.2. Let $\{P_n\}_{n\in {\mathbb N}}$ be a sequence of probabilities on a completely regular topological space ${\mathscr X}$, fix $x\in {\mathscr X}$, and a sequence $x_k\to x$. Assume that there exists $I\colon {\mathscr X}\to [0, +\infty]$ meeting the following conditions:

$(i)$ for each $k\in {\mathbb N}$ there exists a family $\{P_n^{x_k}\}_{n\in{\mathbb N}}$ satisfying the conditions in Lemma 3.1;

$(ii)$ $\varlimsup_k\, I(x_k) \le I(x)$.

Then, for any open neighborhoods $\mathcal O\ni x$

To implement condition (i) and (ii) in the previous lemma, for $0 < \delta < t_*$, define the pair $(\bar f^\delta, \bar Q^\delta)$ by

and $d\bar Q^\delta = \mathop{}\!\mathrm{d} t \, \bar q^\delta_t$ with $\bar q^\delta_t = \bar q_t {1 \mskip -5mu {{\; {\rm{I}}\; }}}_{[0, t_*-\delta)}(t)$. Let $\mu_{N, e}$ as in the statement of Theorem 1.2, and denote by $\bar{{\mathbb P}}^{N, \delta}_{\mu_{N, e}}$ the law of the microscopic dynamics with the perturbed collision kernel $\bar B^\delta = \bar B {1 \mskip -5mu {{\; {\rm{I}}\; }}}_{[0, t_*-\delta)}(t)$, $\bar B$ in (2.7). Then the following two Lemmata imply the large deviation lower bound (1.18).

Lemma 3.3. For each $\delta\in (0, t_*)$ as $N\to +\infty$ the pair $(\pi^N, Q^N)$ converges in $\bar{{\mathbb P}}^{N, \delta}_{\mu_{N, e}}$ probability to $(\bar f^\delta, \bar Q^\delta)$. Furthermore,

Lemma 3.4. As $\delta\downarrow 0$ we have $(\bar f^\delta, \bar Q^\delta)\to (\bar f, \bar Q)$ and $I(\bar f^\delta, \bar Q^\delta)\to I(\bar f, \bar Q)$.

Proof of Lemma 3.3. By definition of $\bar B^\delta$, $\sup_{t}\sup_{ \varepsilon, \varepsilon'}\sum_{ \varepsilon', \varepsilon'_*}\bar B_t^\delta(\varepsilon, \varepsilon', \varepsilon'_*)\leq c_\delta$. Therefore, by classical chaos propagation argument, $\pi^N$ converges in $\bar{{\mathbb P}}^{N, \delta}_{\mu_{N, e}}$ probability to $\bar f^\delta$. To deduce the convergence of the empirical flow, it is enough to observe that for each bounded $F_t(\varepsilon, \varepsilon_*, \varepsilon', \varepsilon'_*)$

is a $\bar{{\mathbb P}}^{N, \delta}_{\mu_{N, e}}$ martingale with predictable quadratic variation

Set $F^\delta_t(\varepsilon, \varepsilon_*, \varepsilon', \varepsilon'_*) = \log(\bar B^\delta_t/B)$. By standard Markov chain computation, the relative entropy of $\bar{{\mathbb P}} ^{N, \delta}_{\mu_{N, e}}$ with respect to ${\mathbb P} ^N_{\mu_{N, e}}$ is given by

Since $\sup_{t}\sup_{ \varepsilon, \varepsilon'}\sum_{ \varepsilon', \varepsilon'_*}\bar B_t^\delta(\varepsilon, \varepsilon', \varepsilon'_*)\leq c_\delta$, by the law of large numbers

while the last term on the right hand side of (3.5) vanishes as $N$ diverges. Again, by the law of large numbers, in order to prove $\bar{{\mathbb E}}^{N, \delta}_{\mu_{N, e}}\Big (Q^N (F^\delta)\Big) \to \bar Q ^\delta(F^\delta)$ it is enough to show the uniform integrability of $Q^N(F^\delta)$ with respect to $\bar{{\mathbb P}}^{N, \delta}_{\mu_{N, e}}$. By exploiting the martingale decomposition (3.4), since $|F^\delta|\leq c_\delta(1+\log(1+ \varepsilon+ \varepsilon_*))$, a direct computation yields $\bar{{\mathbb E}}^{N, \delta}_{\mu_{N, e}}\Big(Q^N(F^\delta)^2\Big) \leq c_\delta$, which implies the requested uniform integrability. Observing that $H_e(\bar f^\delta_0) = 0$, and recalling (1.13) and (1.15), the proof is concluded.

Proof of Lemma 3.4. The convergence of $(\bar f^\delta, \bar Q^\delta)$ to $(\bar f, \bar Q)$ follows from (3.2), the continuity of $t\mapsto \bar f_t$, and the integrability of $\bar q_t$. The convergence of the rate function is achieved by the arguments in the proof of Proposition 2.2 and dominated convergence.

Conflict of interest

The authors declare no conflict of interest.