A sufficient condition for the quasipotential to be the rate function of the invariant measure of countable-state mean-field interacting particle systems

Consider the graph $(\mathcal{Z},\mathcal{E}_{Q})$ whose edge set $\mathcal{E}_{Q}$ consists of forward edges $\{(z,z+1),z\in\mathcal{Z}\}$ and backward edges $\{(z,z-1),z\in\mathcal{Z}\setminus\{0\}\}$ (see Figure 1). Let $\lambda_{f}$ and $\lambda_{b}$ be two positive numbers. Consider the generator $L^{Q}$ acting on functions $f$ on $\mathcal{Z}$ by

$\displaystyle L^{Q}f(z)\coloneqq\sum_{z^{\prime}:(z,z^{\prime})\in\mathcal{E}_{Q}}\lambda_{z,z^{\prime}}(f(z^{\prime})-f(z)),\,z\in\mathcal{Z},$

where $\lambda_{z,z+1}=\lambda_{f}$ for each $z\in\mathcal{Z}$ and $\lambda_{z,z-1}=\lambda_{b}$ for each $z\in\mathcal{Z}\setminus\{0\}$. When $\lambda_{f}<\lambda_{b}$, the invariant probability measure associated with this Markov process is

$\displaystyle\xi^{*}_{Q}(z)\coloneqq\left(1-\frac{\lambda_{f}}{\lambda_{b}}\right)\left(\frac{\lambda_{f}}{\lambda_{b}}\right)^{z},\,z\in\mathcal{Z}.$

For each $N\geq 1$, we consider $N$ particles, each of which evolves independently as a Markov process on $\mathcal{Z}$ with the infinitesimal generator $L^{Q}$. That is, the particles are independent M/M/1 queues. It is easy to check that the empirical measure of the system of particles is also a Markov process on the state space $\mathcal{M}_{1}^{N}(\mathcal{Z})$ and it possesses a unique invariant probability measure, which we denote by $\wp^{N}_{Q}$.

On one hand, it is straightforward to see that the family $\{\wp^{N}_{Q},N\geq 1\}$ satisfies the LDP on $\mathcal{M}_{1}(\mathcal{Z})$. Indeed, under stationarity, the state of each particle is distributed as $\xi^{*}_{Q}$. As a consequence, $\wp^{N}_{Q}$ is the law of the random variable $\frac{1}{N}\sum_{n=1}^{N}\delta_{\zeta_{n}}$ on $\mathcal{M}_{1}(\mathcal{Z})$, where $\zeta_{1},\ldots,\zeta_{N}$ are independent and identically distributed (i.i.d.) as $\xi^{*}_{Q}$. Therefore, by Sanov’s theorem [13, Theorem 6.2.10], $\{\wp^{N}_{Q},N\geq 1\}$ satisfies the LDP with the rate function $I(\cdot\|\xi_{Q}^{*})$, where $I:\mathcal{M}_{1}(\mathcal{Z})\times\mathcal{M}_{1}(\mathcal{Z})\to[0,\infty]$ is the relative entropy defined by³³3We use the convention $0\log 0=0$.

$\displaystyle I(\zeta\|\nu)\coloneqq\left\{\begin{aligned} &\sum_{z\in\mathcal{Z}}\zeta(z)\log\left(\frac{\zeta(z)}{\nu(z)}\right),&\text{ if }\zeta\ll\nu,\\ &\infty,&\text{ otherwise.}\end{aligned}\right.$

(1.4)

On the other hand, it is natural to conjecture that the rate function for the family $\{\wp^{N}_{Q},N\geq 1\}$ is given by the quasipotential (1.3) with $\xi^{*}$ replaced by $\xi_{Q}^{*}$. However, as discussed in the next paragraph, the quasipotential is not the same as $I(\cdot\|\xi_{Q}^{*})$. Hence, from the uniqueness of the large deviations rate function [13, Lemma 4.1.4], the quasipotential does not govern the rate function for the family $\{\wp^{N}_{Q},N\geq 1\}$.

We now provide some intuition on why the quasipotential is not the rate function in the example under consideration. For a formal proof, see Section 8. We first introduce some notation. Let $\mathbb{R}^{\infty}$ denote the infinite product of $\mathbb{R}$ equipped with the product topology. We view $\mathcal{M}_{1}(\mathcal{Z})$ as the subset $\{x\in\mathbb{R}^{\infty}:x_{i}\geq 0\,\forall i,\sum_{i\geq 0}x_{i}=1\}$ of $\mathbb{R}^{\infty}$ with the subspace topology (e.g., see [15, Chapter 3, Section 2]). If $\xi,f\in\mathbb{R}^{\infty}$, we define

$\displaystyle\langle\xi,f\rangle\coloneqq\lim_{m\to\infty}\sum_{z=0}^{m}\xi(z)f(z),$

(1.5)

whenever the limit exists. Also, define $\vartheta:\mathcal{Z}\to\mathbb{R}_{+}$ by

$\displaystyle\vartheta(z)\coloneqq z\log z,\,z\in\mathcal{Z},$

(1.6)

with the convention that $0\log 0=0$, and define $\iota(z)\coloneqq z$, $z\in\mathcal{Z}$. Using the fact that $\xi^{*}_{Q}$ has geometric decay, it can be checked that $I(\xi\|\xi^{*}_{Q})$ is finite if and only if the first moment of $\xi$ (i.e., $\langle\xi,\iota\rangle$) is finite. However it turns out that $V(\xi)$ (i.e., the quantity in (1.3) with $\xi^{*}$ replaced by $\xi^{*}_{Q}$) is finite if and only if the $\vartheta$-moment of $\xi$ (i.e., $\langle\xi,\vartheta\rangle$) is finite. In particular, if we consider a $\xi\in\mathcal{M}_{1}(\mathcal{Z})$ whose first moment is finite but $\vartheta$-moment is infinite then $V(\xi)\neq I(\xi\|\xi^{*}_{Q})$. Let $\varepsilon>0$, $\xi\in\mathcal{M}_{1}(\mathcal{Z})$ be such that $\langle\xi,\iota\rangle<\infty$ but $\langle\xi,\vartheta\rangle=\infty$, and consider the $\varepsilon$-neighbourhood of $\xi$ in $\mathcal{M}_{1}(\mathcal{Z})$. By Sanov’s theorem, the probability of this neighbourhood under $\wp^{N}_{Q}$ is of the form $\exp\{-N(I(\xi\|\xi^{*}_{Q})+o(1))\}$. For a fixed $T>0$, let us now try to estimate the probability of $\mu^{N}_{\nu_{N}}(T)$ being in this neighbourhood when $\nu_{N}$ is in a small neighbourhood of $\xi^{*}_{Q}$. If the process $\mu^{N}$ is initiated at a $\nu_{N}$ near $\xi^{*}_{Q}$, then the probability that the random variable $\mu^{N}_{\nu_{N}}(T)$ is in the $\varepsilon$-neighbourhood of $\xi$ is at most

$\displaystyle\exp\left\{-N\left(\inf_{\{\xi^{\prime}:d(\xi,\xi^{\prime})\leq\varepsilon\}}V(\xi^{\prime})+o(1)\right)\right\}.$

Since $V$ is lower semicontinuous (we prove this in Lemma 5.4), we must have

$\displaystyle\inf_{\{\xi^{\prime}:d(\xi,\xi^{\prime})\leq\varepsilon\}}V(\xi^{\prime})\to\infty\text{ as }\varepsilon\to 0.$

Hence we can choose an $\varepsilon$ small enough so that $\inf_{\{\xi^{\prime}:d(\xi,\xi^{\prime})\leq\varepsilon\}}V(\xi^{\prime})>2I(\xi\|\xi^{*}_{Q})$. For this $\varepsilon$, the probability that $\mu^{N}_{\nu_{N}}(T)$ lies is the $\varepsilon$-neighbourhood of $\xi$ is upper bounded by $\exp\{-N\times(2I(\xi\|\xi^{*}_{Q})+o(1))\}$, which is smaller than $\exp\{-N(I(\xi\|\xi^{*}_{Q})+o(1))\}$, even in the exponential scale, for large enough $N$. That is, for any arbitrary but fixed $T$, we can find a small neighbourhood of $\xi$ such that the probability that $\mu^{N}_{\nu_{N}}(T)$ lies in that neighbourhood is smaller than what we expect to see in the stationary regime. In other words, there are some barriers in $\mathcal{M}_{1}(\mathcal{Z})$ that cannot be surmounted in any finite time, yet these barriers can be crossed in the stationary regime. These barriers indicate that, to obtain the correct stationary regime probability of a small neighbourhood of $\xi$ using the dynamics of $\mu^{N}_{\nu_{N}}$, one should wait longer than any fixed time horizon. That is, one should consider the random variable $\mu^{N}_{\nu_{N}}(T(N))$, where $T(N)$ is a suitable function of $N$, and estimate the probability that $\mu^{N}_{\nu_{N}}(T(N))$ belongs to a small neighbourhood of $\xi$. However it is not straightforward to obtain such estimates from the process-level large deviation estimates of $\mu^{N}_{\nu_{N}}$ since the latter are usually available for a fixed time duration.

There are natural barriers in the context of finite-state mean-field models when the limiting dynamical system has multiple (but finitely many) stable equilibria [36]. In such situations, passages from a neighbourhood of one equilibrium to a neighbourhood of another take place over time durations of the form $\exp\{N\times O(1)\}$ where $N$ is the number of particles⁴⁴4$O(1)$ refers to a bounded sequence, and $\omega(1)$ refers to a sequence that goes to $\infty$.. Interestingly, these barriers can be surmounted using trajectories of finite time durations; i.e., for any fixed $T$, the probability that the empirical measure process reaches a neighbourhood of an equilibrium at time $T$ when it is initiated in a small neighbourhood of another equilibrium is of the form $\exp\{-N\times O(1)\}$. In contrast, in the case of the above counterexample, the barriers cannot be surmounted in finite time durations; for any fixed $T$, the probability that $\mu^{N}(T)$ reaches a small neighbourhood of a point in $\mathcal{M}_{1}(\mathcal{Z})$ with finite first moment but infinite $\vartheta$-moment when it is initiated from a neighbourhood of $\xi^{*}_{Q}$ is of the form $\exp\{-N\times\omega(1)\}$. Hence we anticipate that the barriers that we encounter in the above counterexample are somehow more difficult to surmount than those that arise in the case of finite-state mean-field models with multiple stable equilibria.

We provide another counterexample where the issue is similar. Consider the graph $(\mathcal{Z},\mathcal{E}_{W})$ whose edge set $\mathcal{E}_{W}$ consists of forward edges $\{(z,z+1),z\in\mathcal{Z}\}$ and backward edges $\{(z,0),z\in\mathcal{Z}\setminus\{0\}\}$ (see Figure 2). Let $\lambda_{f}$ and $\lambda_{b}$ be positive numbers. Consider the generator $L^{W}$ acting on functions $f$ on $\mathcal{Z}$ by

$\displaystyle L^{W}f(z)\coloneqq\sum_{z^{\prime}:(z,z^{\prime})\in\mathcal{E}_{W}}\lambda_{z,z^{\prime}}(f(z^{\prime})-f(z)),z\in\mathcal{Z},$

where $\lambda_{z,z+1}=\lambda_{f}$ for each $z\in\mathcal{Z}$ and $\lambda_{z,0}=\lambda_{b}$ for each $z\in\mathcal{Z}\setminus\{0\}$. The invariant probability measure associated with this Markov process is

$\displaystyle\xi^{*}_{W}(z)\coloneqq\frac{\lambda_{b}}{\lambda_{f}+\lambda_{b}}\left(\frac{\lambda_{f}}{\lambda_{f}+\lambda_{b}}\right)^{z},\,z\in\mathcal{Z}.$

Similar to the previous example, for each $N\geq 1$, we consider $N$ particles, each of which evolves independently as a Markov process on $\mathcal{Z}$ with the infinitesimal generator $L^{W}$. It is easy to check that the empirical measure of the system of particles possesses a unique invariant probability measure, which we denote by $\wp_{W}^{N}$. Under stationarity, the state of each particle is distributed as $\xi^{*}_{W}$. As a consequence, $\wp^{N}_{W}$ is the law of the random variable $\frac{1}{N}\sum_{n=1}^{N}\delta_{\zeta_{n}}$ on $\mathcal{M}_{1}(\mathcal{Z})$, where $\zeta_{1},\ldots,\zeta_{N}$ are i.i.d. $\xi^{*}_{W}$. Hence, by Sanov’s theorem, the family $\{\wp^{N}_{W},N\geq 1\}$ satisfies the LDP with the rate function $I(\cdot\|\xi^{*}_{W})$. As we show in Section 8, in this example too, the quasipotential (1.3) with $\xi^{*}$ replaced by $\xi_{W}^{*}$ is not the same as $I(\cdot\|\xi^{*}_{W})$. As in the previous example, there are points $\xi$ where $V(\xi)=\infty$ but $I(\xi\|\xi_{Q}^{*})<\infty$, points $\xi$ that have a finite first moment but infinite $\vartheta$-moment. Once again, the quasipotential does not govern the rate function for the family $\{\wp^{N}_{W},N\geq 1\}$.

Our first set of assumptions is on the mean-field interacting particle system (i.e., on the generator $\mathscr{L}^{N}$ defined in (1.1)).

1. (A1)

   The edge set is given by $\mathcal{E}=\{(z,z+1),z\in\mathcal{Z}\}\cup\{(z,0),z\in\mathcal{Z}\setminus\{0\}\}.$

2. (A2)

   There exist positive constants $\overline{\lambda}$ and $\underline{\lambda}$ such that

   $\displaystyle\frac{\underline{\lambda}}{z+1}\leq\lambda_{z,z+1}(\xi)\leq\frac{\overline{\lambda}}{z+1},\text{ and }\underline{\lambda}\leq\lambda_{z,0}(\xi)\leq\overline{\lambda},$

   for all $\xi\in\mathcal{M}_{1}(\mathcal{Z})$.

3. (A3)

   The functions $(z+1)\lambda_{z,z+1}(\cdot)$, $z\in\mathcal{Z},$ and $\lambda_{z,0}(\cdot)$, $z\in\mathcal{Z}\setminus\{0\}$, are uniformly Lipschitz continuous on $\mathcal{M}_{1}(\mathcal{Z})$.

Note that assumption (A1) considers a specific transition graph (Figure 2) for each particle. This graph arises in the contexts of random backoff algorithms for medium access in wireless local area networks [20] and decentralised control of loads in a smart grid [26]. Assumption (A2) ensures that the forward transition rates at state $z$ decays as $1/z$. This key assumption cuts down the speed of outward excursions and enables us to overcome the issue described in the counterexamples. To highlight this, consider a modified example of Section 1.1.2 where $\lambda_{z,z+1}=\lambda_{f}/(z+1)$, $z\in\mathcal{Z}$; the rest of the description remains the same. Let $\tilde{\xi}_{W}\in\mathcal{M}_{1}(\mathcal{Z})$ denote the invariant probability measure associated with one particle. It can be checked that $\tilde{\xi}_{W}(z)$ is of the order of $\exp\{-\vartheta(z)\}$, unlike $\xi^{*}_{W}$ which has geometric decay. As a consequence, $I(\xi\|\tilde{\xi}_{W})$ is finite if and only if the $\vartheta$-moment of $\xi$ is finite. Hence, by imposing (A2), we have ensured that the barriers in $\mathcal{M}_{1}(\mathcal{Z})$ that are insurmountable for finite time duration trajectories continue to remain insurmountable in the stationary regime; this is the key property that enables us to prove the main result of this paper. Assumption (A3) is a uniform Lipschitz continuity property for the transition rates which is required for the process-level LDP for $\mu^{N}_{\nu_{N}}$ to hold and for the the McKean-Vlasov equation (1.2) to be well-posed.

Our second set of assumptions is on the McKean-Vlasov equation (1.2). Let $\mu_{\nu}$, $\nu\in\mathcal{M}_{1}(\mathcal{Z})$, denote the solution to the limiting dynamics (1.2) with initial condition $\nu\in\mathcal{M}_{1}(\mathcal{Z})$. Recall the function $\vartheta$. Define $\mathscr{K}_{M}\coloneqq\{\xi\in\mathcal{M}_{1}(\mathcal{Z}):\langle\xi,\vartheta\rangle\leq M\}$, $M>0$.

1. (B1)

   There exists a unique globally asymptotically stable equilibrium $\xi^{*}$ for the McKean-Vlasov equation (1.2).

2. (B2)

   $\langle\xi^{*},\vartheta\rangle<\infty$ and $\lim_{t\to\infty}\sup_{\nu\in\mathscr{K}_{M}}\langle\mu_{\nu}(t),\vartheta\rangle=\langle\xi^{*},\vartheta\rangle$ for each $M>0$.

The first assumption above asserts that all the trajectories of (1.2) converge to $\xi^{*}$ as time becomes large. The proof of the LDP upper and lower bounds for the family $\{\wp^{N},N\geq 1\}$ involves construction of trajectories that start at suitable compact sets, reach the stable equilibrium $\xi^{*}$ using arbitrarily small cost, and then terminate at a desired point in $\mathcal{M}_{1}(\mathcal{Z})$ starting from $\xi^{*}$. All these are enabled by assumption (B1) (see more remarks about this assumption in Section 1.3). The second assumption asserts that the $\vartheta$-moment of the solution to the limiting dynamics converges uniformly over initial conditions lying in sets of bounded $\vartheta$-moment. In the case of a non-interacting system that satisfies (A1) but with constant forward transition rates (for example, see $L^{W}$ in Section 1.1.2), the analogue of this assumption can easily be verified: the first moment of the solution to the limiting dynamics converges uniformly over initial conditions lying in sets of bounded first moment. In fact, one can explicitly write down the first moment of the solution to the limiting dynamics in this case and verify this assumption easily. Assumption (B2) is the analogous statement for our mean-field system that satisfies the $1/z$-decay of the forward transition rates in assumption (A2).

We now state the main result of this paper, namely the LDP for the family of invariant measures $\{\wp^{N},N\geq 1\}$ under the assumptions (A1)–(A3) and (B1)–(B2).

We first assert the existence and uniqueness of the invariant measure $\wp^{N}$ for $\mathscr{L}^{N}$ for each $N\geq 1$, and the exponential tightness of the family $\{\wp^{N},N\geq 1\}$.

Recall the quasipotential $V$ defined in (1.3). We now state the main result of this paper.

The proof of this result is carried out in Sections 4–7. We begin with the process-level uniform LDP for $\mu^{N}_{\nu_{N}}$ over compact subsets of $\mathcal{M}_{1}(\mathcal{Z})$; this uniform LDP gives us the large deviation estimates for the process $\mu^{N}_{\nu_{N}}$ uniformly over the initial conditions $\nu_{N}$ lying in a given compact set (see Definition 2.2 and Theorem 2.1). We prove the LDP for the family $\{\wp^{N},N\geq 1\}$ by transferring this process-level uniform LDP for $\mu^{N}_{\nu_{N}}$ over compact subsets of $\mathcal{M}_{1}(\mathcal{Z})$ to the stationary regime. The proof of the LDP lower bound (in Section 4) considers specific trajectories and lower bounds the probability of small neighbourhoods of points in $\mathcal{M}_{1}(\mathcal{Z})$ under $\wp^{N}$ using the probability that the process $\mu^{N}_{\nu_{N}}$ remains close to these trajectories. For the proof of the upper bound, we require certain regularity properties of the quasipotential. These properties are established in Section 5. We first show a controllability⁵⁵5This terminology is from Cerrai and Röckner [11]. property for $V$: $V(\xi)$ is finite if and only if $\langle\xi,\vartheta\rangle<\infty$. Using the lower bound proved in Section 4, we then show that the level sets of $V$ are compact subsets of $\mathcal{M}_{1}(\mathcal{Z})$. Since $\mathcal{M}_{1}(\mathcal{Z})$ is not locally compact and $V$ has compact lower level sets, we do not expect $V$ to be continuous on $\mathcal{M}_{1}(\mathcal{Z})$. Indeed, if $\xi\in\mathcal{M}_{1}(\mathcal{Z})$ is such that $V$ is continuous at $\xi$ and $V(\xi)<\infty$, given $\varepsilon>0$ there exists a $\delta>0$ such that $d(\xi^{\prime},\xi)<\delta$ implies that $|V(\xi^{\prime})-V(\xi)|<\varepsilon$. In particular, $\{\xi^{\prime}\in\mathcal{M}_{1}(\mathcal{Z}):V(\xi^{\prime})\leq V(\xi)+\varepsilon\}\supset B(\xi,\delta)$. Since $\{\xi^{\prime}\in\mathcal{M}_{1}(\mathcal{Z}):V(\xi^{\prime})\leq V(\xi)+\varepsilon\}$ is compact in $\mathcal{M}_{1}(\mathcal{Z})$, this shows that $\xi$ has a relatively compact neighborhood in $\mathcal{M}_{1}(\mathcal{Z})$, which is a contradiction. This shows that, for any $\xi\in\mathcal{M}_{1}(\mathcal{Z})$ such that $V(\xi)<\infty$, $V$ is discontinuous at $\xi$. However we show the following small cost connection property: whenever $\xi_{n}\to\xi^{*}$ in $\mathcal{M}_{1}(\mathcal{Z})$ and $\langle\xi_{n},\vartheta\rangle\to\langle\xi^{*},\vartheta\rangle$ as $n\to\infty$, we have $\lim_{n\to\infty}V(\xi_{n})=V(\xi^{*})=0$. These properties of the quasipotential are then used to transfer the process-level uniform LDP upper bound for $\mu^{N}_{\nu_{N}}$ (uniform over compact subsets of $\mathcal{M}_{1}(\mathcal{Z})$) to the LDP upper bound for the family of invariant measures. The proof of the upper bound is carried out in Section 6. Finally, we complete the proof of the theorem in Section 7.

While the proofs of our lower and upper bounds follow the general methodology of Sowers [33], there are significant model-specific difficulties that arise in our context. The main novelty in the proof of Theorem 1.1 is to establish the small cost connection property of the quasipotential $V$ under assumptions (A1)–(A3) and (B1)–(B2). That is, we can find trajectories of small cost that start at $\xi^{*}$ and end at points in $\mathcal{M}_{1}(\mathcal{Z})$ whose $\vartheta$-moment is not very far from that of $\xi^{*}$. In the work of Sowers [33], this has been carried out by considering the “straight-line” trajectory that connects the attractor to the nearby point under consideration. Such a trajectory may not have small cost in our case since the mass transfer is restricted to the edges in $\mathcal{E}$. We overcome this difficulty by considering a piecewise constant velocity mass transfer via the edges in $\mathcal{E}$. We then carefully estimate the cost of this trajectory and prove the necessary small cost connection property. We also simplify the proof of the compactness of the lower level sets of $V$; while Sowers [34, Proposition 7] studies the minimisation of the costs of trajectories over the infinite-horizon, we arrive at it by using the LDP lower bound and the exponential tightness of the family $\{\wp^{N},N\geq 1\}$. We also remark that the methodology of Sowers [33] has been used by Cerrai and Röckner [11] in the context of stochastic reaction diffusion equations and by Cerrai and Paskal [9] in the context of two-dimensional stochastic Navier-Stokes equations.

Let $D([0,T],\mathcal{S})$ denote the space of $\mathcal{S}$-valued functions on $[0,T]$ that are right continuous with left limits. It is equipped with the Skorohod topology which makes it a complete and separable metric space (see, for example, Ethier and Kurtz [16, Chapter 3]). Let $\rho$ denote a metric on $D([0,T],\mathcal{S})$ that generates the Skorohod topology. An element of $D([0,T],\mathcal{S})$ is called a “trajectory”, and we shall refer to the process-level large deviations rate function evaluated on a trajectory as the “cost” associated with that trajectory. For a trajectory $\varphi$, let both $\varphi_{t}$ and $\varphi(t)$ denote the evaluation of $\varphi$ at time $t$. For $N\geq 1$ and $\nu_{N}\in\mathcal{M}_{1}^{N}(\mathcal{Z})$, let $\mathbb{P}^{N}_{\nu_{N}}$ denote the solution to the $D([0,T],\mathcal{M}_{1}^{N}(\mathcal{Z}))$-valued martingale problem for $\mathscr{L}^{N}$ with initial condition $\nu_{N}\in\mathcal{M}_{1}^{N}(\mathcal{Z})$ (whenever the martingale problem for $\mathscr{L}^{N}$ is well-posed). Let $\mu^{N}_{\nu_{N}}$ denote the random element of $D([0,T],\mathcal{M}_{1}^{N}(\mathcal{Z}))$ whose law is $\mathbb{P}^{N}_{\nu_{N}}$. For each $\xi\in\mathcal{M}_{1}(\mathcal{Z})$, let $L_{\xi}$ denote the generator acting on functions $f$ on $\mathcal{Z}$ by

$\displaystyle f\mapsto L_{\xi}(z)\coloneqq\sum_{z^{\prime}:(z,z^{\prime})\in\mathcal{E}}\lambda_{z,z^{\prime}}(\xi)(f(z^{\prime})-f(z)),\,z\in\mathcal{Z},$

i.e., the generator of the single particle evolving on $\mathcal{Z}$ under the static mean-field $\xi$.

Let $C_{0}^{1}([0,T]\times\mathcal{Z})$ denote the space of real-valued functions on $[0,T]\times\mathcal{Z}$ with compact support that are continuously differentiable in the first argument. Given a trajectory $\varphi\in D([0,T],\mathcal{M}_{1}(\mathcal{Z}))$ such that the mapping $[0,T]\ni t\mapsto\varphi_{t}\in\mathcal{M}_{1}(\mathcal{Z})$ is absolutely continuous (see Dawson and Gärtner [12, Section 4.1]), one can define $\dot{\varphi}_{t}\in\mathbb{R}^{\infty}$ for almost all $t\in[0,T]$ such that

$\displaystyle\langle\varphi_{t},f_{t}\rangle=\langle\varphi_{0},f_{0}\rangle+\int_{[0,t]}\langle\dot{\varphi}_{u},f_{u}\rangle du+\int_{[0,t]}\langle\varphi_{u},\partial_{u}f_{u}\rangle du$

holds for each $f\in C_{0}^{1}([0,T]\times\mathcal{Z})$ and $t\in[0,T]$.

Finally, let $\mathcal{M}_{1}(D([0,T],\mathcal{Z}))$ denote the space of probability measures on $D([0,T],\mathcal{Z})$ equipped with the usual weak topology. Also, let $\mathcal{M}_{1}(\mathcal{M}_{1}(D([0,T],\mathcal{Z})))$ denote the space of probability measures on $\mathcal{M}_{1}(D([0,T],\mathcal{Z}))$ equipped with the weak topology.