2. Preliminaries: large deviations over finite time durations

In this section, we present a large deviation principle (LDP) for the process $\mu_N$ over finite time durations. This result will be used later to study the large-time behaviour of $\mu_N$ and the rate of convergence of $\mu_N$ to its invariant measure.

Fix $T>0$ . We introduce some notation. Let $\mathbb{P}_{\nu_N, [0,T]}^{(N)}$ denote the solution to the $D([0,T], M_1(\mathcal{Z}))$ -valued martingale problem for $L^N$ , i.e., the law of the empirical measure process $(\mu_N(t), 0 \leq t \leq T)$ , and let $\mathbb{P}_{\nu_N,T}^{(N)}$ denote the law of the terminal-time empirical measure $\mu_N(T) \in M_1(\mathcal{Z})$ , with a deterministic initial condition $ \mu_N(0) = \nu_N $ . Let $\mathcal{AC}[0,T]$ denote the space of absolutely continuous $M_1(\mathcal{Z})$ -valued paths on [0, T] (in particular they are differentiable for almost all $t \in [0,T]$ ; see [Reference Léonard30, Definition 3.1]). Define

\begin{align*} \tau^*(u) \,:\!=\, \left\{ \begin{array}{l@{\quad}l@{\quad}l} \infty & \text{ if } u < -1, \\[3pt] 1 & \text{ if } u = -1, \\[3pt] (u+1) \log\! (u+1) - u& \text{ if } u > -1, \end{array} \right.\end{align*}

which is the Fenchel–Legendre transform of $\tau(u) = e^u-u-1$ , $u \in \mathbb{R}$ . Recall the definition of the family of rate matrices $(\Lambda_\xi, \xi \in M_1(\mathcal{Z}))$ from Section 1. We have the following LDP for the sequence $\left\{\mathbb{P}_{\nu_N, [0,T]}^{(N)}\right\}_{N \geq 1}$ on $D([0,T], M_1(\mathcal{Z}))$ (see [Reference Léonard30, Theorem 3.1], [Reference Borkar and Sundaresan12, Theorem 3.2]). See [Reference Dembo and Zeitouni17, Section 1.2] for the definition of an LDP and a good rate function.

Theorem 2.1. Suppose that the initial conditions $\nu_N \to \nu$ in $M_1(\mathcal{Z})$ . Then the sequence of probability measures $\left\{\mathbb{P}_{\nu_N, [0,T]}^{(N)}, N\geq 1 \right\}$ on the space $D([0,T], M_1(\mathcal{Z}))$ satisfies the LDP with good rate function $S_{[0,T]}({\cdot}|\nu)$ defined as follows. If $\mu(0) = \nu$ and $\mu \in \mathcal{AC}[0,T]$ , then

\begin{align*} S_{[0,T]}(\mu|\nu) & = \int_{[0,T]} \sup_{\alpha \in \mathbb{R}^{|\mathcal{Z}|}} \Biggr\{ \sum_{z \in \mathcal{Z}} \alpha(z)(\dot{\mu}_t(z) - \Lambda_{\mu_t}^*\mu_t(z)) \\ & \qquad - \sum_{(z,z^\prime) \in \mathcal{E}} \tau(\alpha(z^\prime) - \alpha(z)) \lambda_{z,z^\prime}(\mu_t) \mu_t(z) \Biggr\} dt, \end{align*}

and $S_{[0,T]}(\mu | \nu) = +\infty$ otherwise. Moreover, if $S_{[0,T]}(\mu|\nu) < \infty$ , then there exists a unique family of rate matrices $L(t) = \big(l_{z,z^\prime}(t), z,z^\prime \in \mathcal{Z}\big)$ , $0 \leq t \leq T$ , such that $t \mapsto L(t)$ is measurable, $\mu$ is the solution to

\begin{align*} \dot{\mu}(t) = L(t)^* \mu(t), \, 0 \leq t \leq T, \, \mu(0) = \nu, \end{align*}

and

\begin{align*} & S_{[0,T]}(\mu|\nu) = \int_{[0,T]} \sum_{(z,z^\prime)\in \mathcal{E}} \mu(t)(z) \lambda_{z,z^\prime}(\mu(t)) \tau^*\left( \frac{l_{z,z^\prime}(t)}{\lambda_{z,z^\prime}(\mu(t))} -1 \right) dt, \end{align*}

where $L(t)^*$ denotes the transpose of L(t), $t \in [0,T]$ .

We can interpret the rate function $S_{[0,T]}$ as follows. Starting at $\nu_N$ , the process $\mu_N$ is likely to be in the neighbourhood of the solution to the McKean–Vlasov equation (1.1) with initial condition $\nu$ (with high probability). In order for the process $\mu_N$ to be in the neighbourhood of some other path, we need to apply a control given by the rate matrix L; $S_{[0,T]}(\mu|\nu)$ is the cost of this control. In particular, since the solution to the McKean–Vlasov equation starting at $\nu$ has zero cost (i.e., $S_{[0,T]}(\mu_{\nu}|\nu) = 0$ , where $\mu_\nu$ denotes the solution to (1.1) starting at $\nu$ ), the limiting behaviour that $\mu_N({\cdot}) \xrightarrow{\mathbb{P}} \mu_\nu({\cdot})$ in $D([0,T],M_1(\mathcal{Z}))$ as $N \to \infty$ follows. See [Reference Djehiche and Kaj18] for some remarks about the form of the rate function and for another representation of the rate function in terms of a relative entropy.

Here is an outline of the proof of Theorem 2.1: one looks at a system of non-interacting particles where the transition rates of a particle do not depend on the empirical measure, and considers the corresponding empirical measure process over [0, T]. Since at most one particle can jump at a given point in time, the measure $\mathbb{P}_{\nu_N, [0,T]}^{(N)}$ is absolutely continuous with the measure corresponding to the above non-interacting system on $D([0,T],M_1(\mathcal{Z}))$ . One can then write the Radon–Nikodym derivative using the Girsanov formula and show continuity properties of the same. An application of an extension of Sanov’s theorem (see [Reference Dawson and Gärtner14, Theorem 3.5]) tells us that the non-interacting particle system obeys the LDP on $D([0,T],M_1(\mathcal{Z}))$ . The above theorem then follows by an application of Varadhan’s integral lemma (see [Reference Dembo and Zeitouni17, Theorem 4.3.1]). This approach has been carried out for a system of interacting diffusions in [Reference Dawson and Gärtner14] and for jump processes in [Reference Léonard30, Reference Borkar and Sundaresan12]. One can also prove various special cases of Theorem 2.1 via other simpler methods; for example, for fixed initial conditions, i.e., when $\nu_N = \delta_{z}$ for some $z \in \mathcal{Z}$ and for all $N \geq 1$ , one can use a modification of Varadhan’s lemma to obtain the LDP for $\mathbb{P}_{\delta_{z}, [0,T]}^{(N)}$ (see [Reference Del Moral and Zajic16]). However, it is crucial to let the initial condition be arbitrary, except for the constraint that $\nu_N \to \nu$ weakly, to obtain a uniform version of Theorem 2.1 (see Corollary 2.1), which is used to prove our main results.

We now recall a theorem that gives the LDP for the sequence $\left\{\mathbb{P}_{\nu_N,T}^{(N)}\right\}_{N \geq 1}$ on $M_1(\mathcal{Z})$ . This can be obtained from the above theorem by an application of the contraction principle to the coordinate projection map $D([0,T], M_1(\mathcal{Z})) \ni \mu \mapsto \mu(T)$ (see [Reference Dembo and Zeitouni17, Theorem 4.2.1], [Reference Borkar and Sundaresan12, Theorem 3.3]).

Theorem 2.2. Suppose that the initial conditions $\nu_N \to \nu$ in $M_1(\mathcal{Z})$ . Then the sequence of probability measures $\left\{\mathbb{P}_{\nu_N,T}^{(N)}\right\}_{N \geq 1}$ on the space $M_1(\mathcal{Z})$ satisfies the LDP with the good rate function

\begin{align*} S_T(\xi|\nu) \,:\!=\, \inf \{S_{[0,T]}(\mu|\nu)\,:\, & \mu(0) = \nu, \mu(T) = \xi, \mu \in \mathcal{AC}[0,T]\}. \end{align*}

Moreover, the above infimum is attained, i.e., there exists a path $\hat{\mu} \in \mathcal{AC}[0,T] $ such that $\hat{\mu} (0)=\nu,\, \hat{\mu} (T) = \xi$ , and $S_{[0,T]}(\hat{\mu}|\nu) = S_T(\xi|\nu)$ .

Here, $S_T(\xi|\nu)$ can be interpreted as the minimum cost of passage from the profile $\nu$ to the profile $\xi$ in time T, among all paths from $\nu$ to $\xi$ in time T. It can be shown that $S_T$ is continuous on $M_1(\mathcal{Z}) \times M_1(\mathcal{Z})$ by constructing piecewise constant velocity trajectories between points on $M_1(\mathcal{Z})$ (see [Reference Borkar and Sundaresan12, Lemma 3.3]).

We also have the following uniform LDP for the sequence $\left\{\mathbb{P}_{\nu_N, [0,T]}^{(N)}\right\}_{ N \geq 1}$ (see [Reference Borkar and Sundaresan12, Corollary 3.1]) when the initial condition is allowed to lie in a compact set.

Corollary 2.1. For any compact set $K \subset M_1(\mathcal{Z})$ , any closed set $F \subset D([0,T], M_1(\mathcal{Z}))$ , and any open set $G \subset D([0,T], M_1(\mathcal{Z}))$ , we have

(2.1) \begin{align} \limsup_{N \to \infty} \frac{1}{N} \log \sup_{\nu \in K \cap M_1^N(\mathcal{Z})} \mathbb{P}_{\nu, [0,T]}^{(N)} (\mu_N \in F) \leq - \inf_{\nu \in K } \inf_{\mu \in F} S_{[0,T]} (\mu|\nu), \end{align}

and

(2.2) \begin{align} \liminf_{N \to \infty} \frac{1}{N} \log \inf_{\nu \in K \cap M_1^N(\mathcal{Z})} \mathbb{P}_{\nu, [0,T]}^{(N)} (\mu_N \in G) \geq - \sup_{\nu \in K} \inf_{\mu \in G} S_{[0,T]} (\mu|\nu). \end{align}

For a proof of the above, see [Reference Dembo and Zeitouni17, Corollary 5.6.15]. Note that, since the space $M_1(\mathcal{Z})$ is compact, we may take $K = M_1(\mathcal{Z})$ in the above corollary.

3. Large-time behaviour

In the study of the large-time behaviour of $\mu_N$ , an important role is played by the Freidlin–Wentzell quasipotential $V \,:\, M_1(\mathcal{Z}) \times M_1(\mathcal{Z}) \to [0, \infty)$ , defined by

\begin{align*} V(\nu,\xi) \,:\!=\, \inf\! \big\{S_{[0,T]}(\mu|\nu)\,:\,\mu(T)= \xi, T > 0\big\};\end{align*}

i.e., $V(\nu, \xi)$ denotes the minimum cost of transport from $\nu$ to $\xi$ in an arbitrary but finite time.

We say that $\nu \sim \xi$ ( $\nu$ is equivalent to $\xi$ ) if $V(\nu, \xi) = 0$ and $V(\xi, \nu) = 0$ . It is easy to see that $\sim$ defines an equivalence relation on $M_1(\mathcal{Z})$ . To study the large-time behaviour of the process $\mu_N$ , we make the following assumptions on the McKean–Vlasov equation (1.1) (see [Reference Freidlin and Wentzell20, Chapter 6, Section 2, Condition A]):

1. (B1) There exist a finite number of compact sets $K_1, K_2, \ldots, K_l$ such that the following hold:

• For each $i = 1,2, \ldots, l$ , $\nu_1, \nu_2 \in K_i$ implies $\nu_1 \sim \nu_2$ .

• For each $i \neq j$ , $\nu_1 \in K_i$ and $\nu_2 \in K_j$ implies $\nu_1 \nsim \nu_2$ .

• Every $\omega$ -limit set of the dynamical system (1.1) lies completely in one of the compact sets $K_i$ .

Since $V(\nu_1, \nu_2) = 0$ whenever $\nu_1,\nu_2 \in K_i$ for any $1 \leq i \leq l$ , we can define

\begin{align*} V\big(K_i, K_j\big) \,:\!=\, \inf\!\big\{S_{[0,T]}(\mu|\nu)\,:\,\nu \in K_i, \mu(T) \in K_j, T > 0\big\},\end{align*}

which is interpreted as the minimum cost of going from $K_i$ to $K_j$ . We also define the minimum cost of going from $K_i$ to $K_j$ without touching the other compact sets $K_k$ , $k \neq i,j$ , by

(3.1) \begin{align} \tilde{V}\big(K_i, K_j\big) \,:\!=\, \inf\!\big\{S_{[0,T]}(\mu|\nu)\,:\ &\nu \in K_i, \ \mu(t) \notin \cup_{k\neq i, j} K_k \nonumber \\ & \text{for all } 0 \leq t \leq T, \ \mu(T) \in K_j, \ T>0\big\}.\end{align}

Using the definition of the rate function $S_T$ , note that

\begin{align*} V(\nu,\xi) = \inf_{T > 0} S_T(\xi|\nu) \quad \text{ and } \quad V\big(K_i, K_j\big) = \inf_{\nu \in K_i, \xi \in K_j} V(\nu, \xi). \end{align*}

For a model of malware propagation where a limit cycle and an unstable equilibrium arise, see Benaïm and Le Boudec [Reference Benaïm and Boudec6, Section 4.1].

3.1. Preliminary results

It turns out that, under Assumption (B1), the large-time behaviour of the process $\mu_N$ can be studied via a discrete-time Markov chain whose state space is the union of small neighbourhoods of the compact sets $K_i$ , $1 \leq i \leq l$ . To study this chain, we introduce some notation. Let $L = \{1,2,\ldots, l\}$ . Given $0 < \rho_1 < \rho_0$ , let $\gamma_i$ (resp. $\Gamma_i$ ) denote the $\rho_1$ -open neighbourhood (resp. $\rho_0$ -open neighbourhood) of $K_i$ . Let $\gamma = \cup_{i=1}^l \gamma_i$ , $\Gamma = \cup_{i=1}^l \Gamma_i$ , and $C =M_1(\mathcal{Z}) \setminus \overline{\Gamma}$ . For a set $A \subset M_1(\mathcal{Z})$ and $\delta>0$ , let $[A]_\delta$ denote the $\delta$ -open neighbourhood of A, and for a subset $W \subset L$ , abusing notation, let $[W]_\delta$ denote the $\delta$ -open neighbourhood of $\cup_{i \in W} K_i$ . For each $n \geq 1$ , we define the sequence of stopping times $\tau_0\,:\!=\, 0$ , $\sigma_n \,:\!=\, \inf\{t>\tau_{n-1}\,:\, \mu_N(t) \in C\}$ , $\tau_n \,:\!=\, \inf\{t > \sigma_n \,:\, \mu_N(t) \in \gamma\}$ ; and we define $Z^N_n \,:\!=\, \mu_N(\tau_n)$ . Since $\mu_N$ is strong Markov, $Z^N$ is a discrete-time Markov chain, and $Z_n^N \in \gamma \cap M^N_1(\mathcal{Z})$ for all $n \geq 1$ . For a measurable set $A \in M_1(\mathcal{Z})$ , we define the stopping time $\tau_A \,:\!=\, \inf\{t > 0\,:\, \mu_N(t) \notin A\}$ , which denotes the time of first exit from the set A. Finally, for a subset $W \subset L$ , we define the stopping times $\hat{\tau}_W \,:\!=\, \inf\{t >0 \,:\, \mu_N(t) \in \cup_{i \in W}\gamma_i\}$ and $\bar{\tau}_W \,:\!=\, \inf \{t > 0\,:\, \mu_N(t) \in \cup_{i \in L \setminus W} \gamma_i \}$ , which denote the time of entry into the $\rho_1$ -neighbourhood of W and the time of entry into the $\rho_1$ -neighbourhood of $L \setminus W$ , respectively.

We now state some results on the behaviour of the exit time from certain sets, which will be used in the paper subsequently. These results are known in the case of both Markov jump processes and diffusion processes; see [Reference Borkar and Sundaresan12, Appendix], and [Reference Freidlin and Wentzell20, Chapter 6, Section 2]. The main ingredients that are used in proving these results are (i) the strong Markov property of the $\mu_N$ process, (ii) Theorem 2.1 and Corollary 2.1 on the LDP for finite time durations, and (iii) the joint continuity of the terminal-time rate function $S_T({\cdot}|{\cdot})$ (see [Reference Borkar and Sundaresan12, Lemma 3.3]). Recall that $\mathbb{P}_\nu$ denotes the law of $(\mu_N(t), t \geq 0)$ with initial condition $\mu_N(0) = \nu$ and $\mathbb{E}_\nu$ denotes the corresponding expectation.

Lemma 3.1. ([Reference Borkar and Sundaresan12, Lemma A.3].) Let $K \subset M_1(\mathcal{Z})$ be a compact set such that all points in K are equivalent to each other (i.e., $\nu_1 \sim \nu_2$ for all $\nu_1, \nu_2 \in K$ ), and $K \neq M_1(\mathcal{Z})$ . Then, given $\varepsilon> 0$ , there exist $\delta > 0$ and $N_0 \geq 1$ such that for all $N \geq N_0$ and $\nu \in [K]_{\delta}\cap M^N_1(\mathcal{Z})$ ,

\begin{align*} \mathbb{E}_\nu \tau_{[K]_{\delta}} \leq \exp\{N\varepsilon\}. \end{align*}

Lemma 3.2. ([Reference Borkar and Sundaresan12, Lemma A.4].) Let $K \subset M_1(\mathcal{Z})$ be a compact set and let G be a neighbourhood of K. Then, given $\varepsilon > 0$ , there exist $\delta>0$ and $N_0 \geq 1$ such that for all $\nu \in \overline{[K]_{\delta}} \cap M_1^N(\mathcal{Z})$ and $N \geq N_0$ ,

\begin{align*} \mathbb{E}_\nu\!\left( \int_0^{\tau_G} 1_{\{\mu_N(t) \in \overline{[K]_\delta}\}} dt \right) \geq \exp\{{-}N\varepsilon\}. \end{align*}

Lemma 3.3. ([Reference Borkar and Sundaresan12, Lemma A.5].) Let $K \subset M_1(\mathcal{Z})$ be a compact set that does not contain any $\omega$ -limit set of (1.1) entirely. Then, there exist positive constants $c, T_0$ and $N_0 \geq 1$ such that for all $T \geq T_0$ , $N \geq N_0$ and any $\nu \in K \cap M_1^N(\mathcal{Z})$ , we have

\begin{align*} \mathbb{P}_{\nu}( \tau_K \geq T ) \leq \exp\{{-}Nc(T-T_0)\}. \end{align*}

Corollary 3.1. Under the conditions of Lemma 3.3, there exist $ C>0$ and $N_0 \geq 1$ such that for all $\nu \in K \cap M_1^N(\mathcal{Z})$ and $N \geq N_0$ ,

\begin{align*} \mathbb{E}_\nu \tau_K \leq C. \end{align*}

Recall the definition of the discrete-time Markov chain $Z^N$ on $\gamma \cap M_1^N(\mathcal{Z})$ . The next lemma gives upper and lower bounds on the one-step transition probabilities of the chain $Z^N$ . These estimates play an important role in the study of the large-time behaviour of the process $\mu_N$ , as we shall see in the sequel.

Lemma 3.4. ([Reference Borkar and Sundaresan12, Lemma A.6].) Given $\varepsilon >0$ , there exist $\rho_0 >0$ and $N_0 \geq 1$ such that, for any $\rho_2 < \rho_0$ , there exists $\rho_1 < \rho_2$ such that for any $\nu \in [K_i]_{\rho_2} \cap M_1^N(\mathcal{Z})$ and $N \geq N_0$ , the one-step transition probability of the chain $Z^N$ satisfies

(3.2) \begin{align} \exp\!\big\{{-}N\big(\tilde{V}\big(K_i, K_j\big)+\varepsilon\big)\big\} \leq \mathbb{P}\big(\nu, \gamma_j\big) \leq \exp\!\big\{{-}N\big(\tilde{V}\big(K_i, K_j\big)-\varepsilon\big)\big\}. \end{align}

Remark 3.1. In the above statement, $\mathbb{P}\big(\nu, \gamma_j\big)$ is defined as $\mathbb{P}\big(\nu, \gamma_j\big) \,:\!=\, \mathbb{P}_\nu\big(Z^N_1 \in \gamma_j\big) = \mathbb{P}_\nu\big(\mu_N(\tau_1) \in \gamma_j\big)$ .

The key ingredient in the proof of the above lemma is Corollary 2.1 on the uniform LDP on bounded sets. For the lower bound, one constructs a specific trajectory from $\nu$ to $K_j$ and examines its cost. For the upper bound, one uses the strong Markov property at the hitting time of $[L]_{\rho_1}$ and the uniform LDP. For details, the reader is referred to the proof of [Reference Borkar and Sundaresan12, Lemma A.6] for the case of Markov jump processes, and the proof of [Reference Freidlin and Wentzell20, Lemma 2.1, p. 152] for the case of small-noise diffusions.

3.2. Behaviour near attractors indexed by subsets of L

We now recall some results on the behaviour of the process $\mu_N$ near a small neighbourhood of attractors indexed by a given subset of L. Let $W \subset L$ , $W \neq \emptyset$ . A W-graph is a directed graph on L such that (i) each element of $L \setminus W$ has exactly one outgoing arrow and (ii) there are no closed cycles in the graph. We denote the set of W-graphs by G(W). For each $i \in L$ , we denote $G(\{i\})$ by G(i). For a W-graph g, define

(3.3) \begin{align} \tilde{V}(g) = \sum_{(m\to n) \in g} \tilde{V}(K_m, K_n).\end{align}

If g does not have any edges (e.g., when L is a singleton), we use the convention $\tilde{V}(g) = 0$ . Note that, using the estimate (3.2), $\tilde{V}$ can be used to estimate the probability that the process $\mu_N$ traverses a sequence of neighbourhoods in the order specified by the graph g.

For $i \in L\setminus W$ and $j \in W$ , let $G_{i,j}(W)$ denote the set of W-graphs in which there is a sequence of arrows leading from i to j. Define

\begin{align*} I_{i,j}(W) \,:\!=\, \min\{\tilde{V}(g)\,:\,g \in G_{i,j}(W)\} - \min\{\tilde{V}(g)\,:\, g \in G(W)\}.\end{align*}

We recall the following result on the probability that the first entry of $\mu_N$ into a neighbourhood of a set $W \subset L$ takes place via a given compact set $K_j$ , starting from a neighbourhood of $K_i$ .

Lemma 3.5. Let $W \subset L$ , and let $i \in L \setminus W$ and $j \in W$ . Given $\varepsilon > 0$ , there exist $\rho > 0$ and $N_0 \geq 1 $ such that for any $\rho_1 \leq \rho$ , $\nu \in \gamma_i \cap M^N_1(\mathcal{Z})$ , and $N \geq N_0$ , we have

\begin{align*} \exp\!\big\{{-}N\big(I_{i,j}(W)+\varepsilon\big)\big\} \leq \mathbb{P}_\nu\big(\mu_N\big(\hat{\tau}_W\big) \in \gamma_j\big) \leq \exp\!\big\{{-}N\big(I_{i,j}(W)-\varepsilon\big)\big\}. \end{align*}

Proof. The proof of [Reference Freidlin and Wentzell20, Lemma 3.3, p. 159] holds verbatim, by making use of the estimates in Lemma 3.4.

Remark 3.2. While the above lemma provides an estimate of the probability $\mathbb{P}_\nu\big(\mu_N\big(\hat{\tau}_W\big) \in \gamma_j\big)$ , it does not provide any information about the sequence of states in L visited by the process $\mu_N$ while traversing from i to j. The latter can be understood via studying the minimisations in the definition of $I_{i,j}$ ; see [Reference Gan and Cameron21].

Our next step is to understand the mean entry time $\mathbb{E}_\nu \hat{\tau}_W$ . For this, we need the following estimate on the stopping time $\tau_1$ ; see [Reference Hwang and Sheu25, Lemma 1.3, Part I] for a similar estimate for small-noise diffusion processes.

Lemma 3.6. Given $\varepsilon >0$ , there exist $\rho_1 >0$ and $N_0 \geq 1$ such that, for any $\nu \in \gamma \cap M_1^N(\mathcal{Z})$ and $N \geq N_0$ , we have

\begin{align*} \mathbb{E}_\nu \tau_1 \leq \exp\{N\varepsilon\}. \end{align*}

Proof. With a sufficiently small $\rho_1 > 0$ to be chosen later, let $\rho_0 = 2 \rho_1$ so that $[K_i]_{\rho_0}$ does not intersect with $[K_j]_{\rho_0}$ for all $j \neq i$ . Note that, for any $\nu \in \gamma$ ,

\begin{align*} \mathbb{E}_{\nu} \tau_1 = \mathbb{E}_\nu \sigma_1 +\mathbb{E}_\nu (\tau_1 - \sigma_1). \end{align*}

Consider the first term. By Lemma 3.1, there exist $\rho > 0$ and $N_0 \geq 1$ such that for all $\rho_1 \leq \rho$ , $\nu \in \gamma \cap M_1^N(\mathcal{Z})$ , and $N \geq N_0$ , we have

\begin{align*} \mathbb{E}_\nu \sigma_1 \leq \exp\{N\varepsilon/2\}. \end{align*}

Let $F = M_1(\mathcal{Z}) \setminus \gamma$ . By the strong Markov property, the second term is

\begin{align*} \mathbb{E}_\nu (\tau_1 - \sigma_1) = \mathbb{E}_{\mu_N(\sigma_1)} (\tau_{F}). \end{align*}

Therefore, it suffices to estimate $\mathbb{E}_{\nu^\prime} \tau_F$ for $\nu^\prime \in F$ . Since the compact set F does not contain any $\omega$ -limit set, by Corollary 3.1, there exist a constant $C >0$ and $N_1 \geq N_0$ such that for any $\nu^\prime \in F \cap M_1^N(\mathcal{Z})$

\begin{align*} \mathbb{E}_{\nu^\prime} \tau_{F} \leq C. \end{align*}

This completes the proof of the lemma.

Define

\begin{align*} I_i(W) \,:\!=\, &\min\{\tilde{V}(g)\,:\, g \in G(W)\} \\ & - \min\{\tilde{V}(g)\,:\, g \in G( W \cup \{i\}) \text{ or } g \in G_{i,j}(W \cup \{j\}), i \neq j, j \in L\setminus W\}.\end{align*}

The next lemma is about the mean entry time into a neighbourhood of a given set $W \subset L$ starting from a neighbourhood of $K_i$ ; see [Reference Hwang and Sheu25, Lemma 1.6, Part I] for a similar estimate on small-noise diffusion processes.

Lemma 3.7. Let $W \subset L$ , and let $i \in L \setminus W$ . Given $\varepsilon > 0$ , there exist $\rho > 0$ and $N_0 \geq 1 $ such that for any $\rho_1 \leq \rho$ , $\nu \in \gamma_i \cap M^N_1(\mathcal{Z})$ , and $N \geq N_0$ , we have

\begin{align*} \exp\{N(I_i(W)-\varepsilon)\} \leq \mathbb{E}_\nu \hat{\tau}_W \leq \exp\{N(I_i(W)+\varepsilon)\}. \end{align*}

Proof. We first prove the upper bound. Note that, by the strong Markov property, we have

\begin{align*} \mathbb{E}_\nu \hat{\tau}_W = \mathbb{E}_\nu \tau_v \leq \sum_{m=1}^\infty \mathbb{E}_\nu\!\left(1_{\{v = m\}} \times m \sup_{\nu^\prime \in \gamma} \mathbb{E}_{\nu^\prime} \tau_1 \right), \end{align*}

where v is the hitting time of the chain $Z_n^N$ on the set W. Using Lemma 3.6 and the upper bound on $\mathbb{E}_\nu v$ derived in [Reference Freidlin and Wentzell20, Lemma 3.4, p. 162], for sufficiently small $\rho_1$ and sufficiently large N, we have that

\begin{align*} \mathbb{E}_\nu \hat{\tau}_W \leq \exp\{N(I_i(W) +\varepsilon)\} \end{align*}

holds for all $\nu \in \gamma_i \cap M_1^N(\mathcal{Z})$ . For the lower bound, Lemma 3.2 implies that, for all sufficiently small $\rho_1$ and sufficiently large N, we have that

\begin{align*} \mathbb{E}_\nu \tau_1 \geq \exp\{{-}N\varepsilon\} \end{align*}

holds for all $\nu \in \gamma$ . Also,

\begin{align*} \mathbb{E}_\nu \hat{\tau}_W = \mathbb{E}_\nu \tau_v \geq \sum_{m=1}^\infty \mathbb{E}_\nu\!\left(1_{\{v = m\}} \times m \inf_{\nu^\prime \in \gamma} \mathbb{E}_{\nu^\prime} \tau_1 \right); \end{align*}

hence, using the lower bound on $\mathbb{E}_\nu v$ derived in [Reference Freidlin and Wentzell20, Lemma 3.4, p. 162], we get

\begin{align*} \mathbb{E}_\nu \hat{\tau}_W \geq \exp\{N(I_i(W) - \varepsilon)\} \end{align*}

for all $\nu \in \gamma_i \cap M_1^N(\mathcal{Z})$ and sufficiently large N.

3.3. Cycles

We now define the notion of cycles, which helps us to describe the most probable way in which the process $\mu_N$ , for large N, traverses neighbourhoods of various compact sets $K_i$ , and the time required to go from one to another. Recall the definition of $\tilde{V}$ from (3.1). We interpret $\tilde{V}\big(K_i, K_j\big)$ as the ‘communication cost’ from i to j. Define $\tilde{V}(K_i) \,:\!=\, \min_{j \neq i} \tilde{V}\big(K_i, K_j\big)$ . We say that $i \to j$ if $\tilde{V}(K_i) = \tilde{V}\big(K_i,K_j\big)$ . For $j \neq i$ , the probability that $\mu_N$ hits a small neighbourhood of $K_j$ upon exit from a small neighbourhood of $K_i$ is of the form $\exp\!\big\{{-}N(\tilde{V}\big(K_i, K_j\big) - \tilde{V}(K_i))\big\}$ , and the mean exit time from a small neighbourhood of $K_i$ is of the form $\exp\!\big\{N\tilde{V}(K_i)\big\}$ [Reference Freidlin and Wentzell20, Chapter 6, Section 5]. In particular, the indices that attain the minimum above are the sets most likely to be visited by the process $\mu_N$ , for large enough N, starting from a neighbourhood of $K_i$ . For $i, j \in L$ , we say that $i \Rightarrow j$ if there exists a sequence of arrows leading from i to j, i.e., if there exist $i_1, i_2 , \ldots, i_n$ in L such that $i \to i_1 \to i_2 \to \cdots \to i_n \to j$ . Again, the above sequence of arrows from i to j is one among the locally most likely sequences for the process to traverse from a neighbourhood of $K_i$ to a neighbourhood of $K_j$ , for large N.

Definition 3.1. A 1-cycle $\pi$ is a directed graph on a subset of elements of L satisfying the following:

1. 1. $i \in \pi$ and $i \Rightarrow j$ implies $j \in \pi$ .

2. 2. For any $i \neq j$ in $\pi$ , we have $i \Rightarrow j $ and $j \Rightarrow i$ .

That is, a 1-cycle is a subset of the elements of L along with a certain assignment of arrows among them according to the numbers $\tilde{V}(\cdot, \cdot)$ . For example, if $L = \{1,2,3\}$ , and $1 \to 2$ , $2 \to 1$ , and $3 \to 1$ are the only possible arrows (i.e., $\tilde{V}(K_1) = \tilde{V}(K_1, K_2) < \tilde{V}(K_1, K_3)$ , $\tilde{V}(K_2) = \tilde{V}(K_2, K_1) < \tilde{V}(K_2, K_3)$ , and $\tilde{V}(K_3) = \tilde{V}(K_3, K_1) < \tilde{V}(K_3, K_2)$ ), then the graph on $\{1,2\}$ consisting of the arrows $1 \to 2$ and $2 \to 1$ is a 1-cycle. The set $\{3\}$ is not part of a 1-cycle. It can be shown that a 1-cycle always exists for all L (see the proof of [Reference Hwang and Sheu25, Lemma 1.9, Part I]).

We now define cycles of 1-cycles. Let $L_0 = L$ . Define

\begin{align*} L_1 \,:\!=\, \{\pi\,:\, \pi \text{ is a 1-cycle in } L\} \cup \{i \in L\,:\, i \text{ is not in any 1-cycle}\}.\end{align*}

That is, the elements of $L_1$ are either 1-cycles in L or elements of L that do not belong to any 1-cycle. In the previous example, $L_1$ is the set $\{\{1,2\}\} \cup \{\{3\}\}$ . Ultimately, we view the elements of $L_1$ as subsets of L. If $\pi \in L_1$ , we write $K \in \pi$ to indicate that the index of the compact set K in $\{1,2,\ldots, l\}$ is an element of $\pi$ . We now proceed to define the ‘communication cost’ between the elements of $L_1$ . For $\pi_1, \pi_2 \in L_1$ , $\pi_1 \neq \pi_2$ , define

\begin{align*} \hat{V}(\pi_1) \,:\!=\, \max\! \big\{\tilde{V}(K)\,:\, K \in \pi_1\big\},\end{align*}

\begin{align*} \tilde{V}(\pi_1, \pi_2) \,:\!=\, \hat{V}(\pi_1) + \min\!\big\{\tilde{V}(K_1, K_2)-\tilde{V}(K_1)\,:\, K_1 \in \pi_1, K_2 \in \pi_2\big\},\end{align*}

and

\begin{align*} \tilde{V}(\pi_1) \,:\!=\, \min\!\big\{\tilde{V}(\pi_1, \pi_2)\,:\, \pi_2 \in L_1, \pi_2 \neq \pi_1\big\}.\end{align*}

That is, $\tilde{V}(\pi_1, \pi_2)$ is the communication cost from $\pi_1$ to $\pi_2$ , and it generalises the quantity $\tilde{V}\big(K_i, K_j\big)$ to 1-cycles. Similarly, $\tilde{V}(\pi_1)$ generalises the quantity $\tilde{V}(K_i)$ to 1-cycles. If $\pi_1, \pi_2$ are 1-cycles, $\pi_1 \neq \pi_2$ , then upon exit from a small neighbourhood of the elements of $\pi_1$ , the probability that the process $\mu_N$ enters a small neighbourhood of the elements of $\pi_2$ is of the form $\exp\!\big\{{-}N\big(\tilde{V}(\pi_1, \pi_2) - \tilde{V}(\pi_1)\big)\big\}$ , and the mean exit time from a small neighbourhood of the elements of $\pi_1$ is of the form $\exp\!\big\{N\tilde{V}(\pi_1)\big\}$ . We say that $\pi_1 \to \pi_2$ if $ \tilde{V}(\pi_1) = \tilde{V}(\pi_1, \pi_2)$ , and we say that $\pi_1 \Rightarrow \pi_2$ if there is a sequence of arrows leading from $\pi_1$ to $\pi_2$ . This gives a cycle of 1-cycles, which we call a 2-cycle.

Let us now define the hierarchy of cycles. Having defined $(m-1)$ -cycles and the sets $L_0, L_1, \ldots, L_{m-2}$ , we define m-cycles as follows. Define

\begin{align*} L_{m-1} = \, & \big\{\pi^{m-1}\,:\, \pi^{m-1} \text{ is an } (m-1)\text{-cycle}\big\} \\ & \cup \big\{\pi^{m-2} \in L_{m-2} \,:\, \pi^{m-2} \text{ is not in any } (m-1)\text{-cycle}\big\}.\end{align*}

That is, the elements of $L_{m-1}$ are either $(m-1)$ -cycles or elements of $L_{m-2}$ that are not part of any $(m-1)$ -cycle; in both cases, they are ultimately viewed as subsets of L. Given $\pi^{m-1} \in L_{m-1}$ and an $(m-2)$ -cycle $\pi^{m-2}$ , we write $\pi^{m-2} \in \pi^{m-1}$ if the elements of $\pi^{m-2}$ (when it is viewed as a subset of L) are part of $\pi^{m-1}$ . For $\pi^{m-1} \in L_{m-1}$ , define

\begin{align*} \hat{V}\big(\pi^{m-1}\big) \,:\!=\, \max\! \big\{\tilde{V}\big(\pi^{m-2}\big)\,:\, \pi^{m-2} \in \pi^{m-1}\big\},\end{align*}

\begin{align*} \tilde{V}\big(\pi_1^{m-1}, \pi_2^{m-1}\big) \,:\!=\, \, & \hat{V}\big(\pi_1^{m-1}\big) \\ & + \min\!\big\{\tilde{V}\big(\pi_1^{m-2}, \pi_2^{m-2}\big)-\tilde{V}\big(\pi_1^{m-2}\big) \,:\, \pi_1^{m-2} \in \pi_1^{m-1}, \pi_2^{m-2} \in \pi_2^{m-1}\big\},\end{align*}

and

\begin{align*} \tilde{V}\big(\pi_1^{m-1}\big) \,:\!=\, \min\!\big\{\tilde{V}\big(\pi_1^{m-1}, \pi_2^{m-1}\big)\,:\, \pi_2^{m-1} \in L_{m-1}, \pi_2^{m-1} \neq \pi_1^{m-1}\big\}.\end{align*}

We say that $\pi_1^{m-1} \to \pi_2^{m-1}$ if $\tilde{V}\big(\pi_1^{m-1}\big) = \tilde{V}\big(\pi_1^{m-1}, \pi_2^{m-1}\big) $ . We then have the following definition.

Definition 3.2. An m-cycle $\pi^m$ is a directed graph on a subset of elements of $L_{m-1}$ satisfying the following:

1. 1. For $\pi_1^{m-1}, \pi_2^{m-1} \in L_{m-1}$ , $\pi_1^{m-1} \in \pi^m$ and $\pi_1^{m-1} \Rightarrow \pi_2^{m-1}$ implies $\pi_2^{m-1} \in \pi^m$ .

2. 2. For any $\pi_1^{m-1}, \pi_2^{m-1} \in \pi^m$ , we have $\pi_1^{m-1} \Rightarrow \pi_2^{m-1}$ and $\pi_2^{m-1}\Rightarrow \pi_1^{m-1}$ .

If we continue this way, for some $m \geq 1$ , the set $L_m$ will eventually be a singleton, at which point we stop. See [Reference Trouvé46] for a numerical example that consists of three 1-cycles and a 2-cycle when L has 9 elements.

We now state some results on the mean exit time from a cycle and the most probable cycle for the process $\mu_N$ to visit upon exit from a given cycle. For convenience, the set of elements of L constituting a k-cycle $\pi^k$ (through the hierarchy of cycles) is also denoted by $\pi^k$ . Also, for $W \subset L$ , we define $\gamma_W = \cup_{i \in W} \gamma_i$ .

Corollary 3.2. Let $\pi^k$ be a k-cycle and $K_i \in \pi^k$ . Let $W = L \setminus \pi^k$ . Given $\varepsilon > 0$ , there exist $\rho >0$ and $N_0 \geq 1$ such that for all $\rho_1 \leq \rho$ , $\nu \in \gamma_i \cap M_1^N(\mathcal{Z})$ , and $N \geq N_0$ , we have

\begin{align*} \exp\! \big\{N\big(\tilde{V}\big(\pi^k\big) - \varepsilon\big)\big\} \leq \mathbb{E}_\nu \hat{\tau}_W \leq \exp\! \big\{N\big(\tilde{V}\big(\pi^k\big) + \varepsilon\big)\big\}. \end{align*}

Corollary 3.3. Let $\pi_1^k, \pi_2^k$ be k-cycles, $\pi_1^k \neq \pi_2^k$ , and $K_i \in \pi_1^k$ . Let $W = L \setminus \pi_1^k$ . Given $\varepsilon > 0$ , there exist $\rho >0$ and $N_0 \geq 1$ such that for all $\rho_1 \leq \rho$ , $\nu \in \gamma_i \cap M_1^N(\mathcal{Z})$ , and $N \geq N_0$ , we have

\begin{align*} \exp\! \big\{{-}N\big(\tilde{V}\big(\pi_1^k, \pi_2^k\big) - \tilde{V}\big(\pi_1^k\big) + \varepsilon\big)\big\} & \leq \mathbb{P}_\nu \big(\mu_N(\hat{\tau}_W) \in \gamma_{\pi_2^k}\big) \\ &\leq \exp\! \big\{{-}N\big(\tilde{V}\big(\pi_1^k, \pi_2^k\big) - \tilde{V}\big(\pi_1^k\big) - \varepsilon\big)\big\}. \end{align*}

Remark 3.3. Note that Corollary 3.2 follows from Lemma 3.7 and the fact that $I_i(W) = \tilde{V}\big(\pi^k\big)$ (which is shown in [Reference Hwang and Sheu25, Corollary A.4, Appendix]). Corollary 3.3 is a consequence of Lemma 3.5 along with the fact that $\min\!\big\{I_{i,j}(W) \,:\, i \in \hat{\pi}^k\big\} = \tilde{V}\big(\pi^k, \hat{\pi}^k\big) - \tilde{V}\big(\pi^k\big)$ (see [Reference Hwang and Sheu25, Corollary A.6, Appendix]). Similar estimates as in Corollaries 3.2 and 3.3 in the case of small-noise diffusion processes have been shown in [Reference Hwang and Sheu25, Corollary 1.10, Part I] and [Reference Hwang and Sheu25, Corollary 1.11, Part I], respectively.

We also need the following lemmas that provide estimates on the probabilities of exit within certain times from given cycles.

Lemma 3.8. Let $\pi_1^k, \pi_2^k$ be k-cycles and let $\pi_1^k \to \pi_2^k$ . Then, given $\varepsilon >0$ , there exist $\delta >0$ , $\rho >0$ , and $N_0 \geq 1$ such that for all $\rho_1 \leq \rho$ , $\nu \in \gamma_{\pi_1^k} \cap M_1^N(\mathcal{Z})$ , and $N \geq N_0$ , we have

\begin{align*} \mathbb{P}_\nu\!\left(\bar{\tau}_{\pi_1^k} \leq \exp\!\big\{N\big(\tilde{V}\big(\pi_1^k\big)-\delta\big)\big\}, \mu_N\big(\bar{\tau}_{\pi_1^k}\big) \in \gamma_{\pi_2^k} \right) \geq \exp\{{-}N\varepsilon\}. \end{align*}

Lemma 3.9. Let $\pi^k$ be a k-cycle. Then, given $\varepsilon > 0$ , there exists $\rho > 0$ such that for all $\rho_1 \leq \rho$ , we have

\begin{align*} \lim_{N \to \infty} \sup_{\nu \in \gamma_{\pi^k} \cap M_1^N(\mathcal{Z})} \mathbb{P}_\nu\Big(\!\exp\!\big\{N\big(\tilde{V}\big(\pi^k\big) - \varepsilon\big)\big\} \leq \bar{\tau}_{\pi^k} \leq \exp\!\big\{N\big(\tilde{V}\big(\pi^k\big) + \varepsilon\big)\big\} \Big) =1. \end{align*}

Furthermore, given $\varepsilon > 0$ , there exist $\delta >0$ , $\rho > 0$ , and $N_0 \geq 1$ such that for all $\rho_1 \leq \rho$ , $N \geq N_0$ , and $\nu \in \gamma_{\pi^k} \cap M^N_1(\mathcal{Z})$ , we have

\begin{align*} \mathbb{P}_\nu\Big( \bar{\tau}_{\pi^k} < \exp\!\big\{N\big(\tilde{V}\big(\pi^k\big) - \delta\big)\big\} \Big) \leq \exp\{{-}N\varepsilon\},\ and \\ \mathbb{P}_\nu\Big( \bar{\tau}_{\pi^k} > \exp\!\big\{N\big(\tilde{V}\big(\pi^k\big) + \delta\big)\big\} \Big) \leq \exp\{{-}N\varepsilon\}. \end{align*}

Remark 3.4. Lemma 3.9 can be proved as follows. From Corollary 3.2, we have that the mean exit time from a small neighbourhood of the elements of $\pi_1^k$ is of the form $\exp\!\big\{N\tilde{V}\big(\pi_1^k\big)\big\}$ . From Corollary 3.3, we have that, upon exit from a small neighbourhood of the elements of $\pi_1^k$ , the probability that the process $\mu_N$ enters a small neighbourhood of the elements of $\pi_2^k$ is of the form $\exp\!\big\{{-}N\big(\tilde{V}\big(\pi_1^k, \pi_2^k\big) - \tilde{V}\big(\pi_1^k\big)\big\}$ . Using these facts, we can proceed via the proof of [Reference Freidlin and Wentzell20, Chapter 4, Theorem 4.2] to transfer the estimate on the mean of $\bar{\tau}_{\pi_1^k}$ to the estimates on the probability for $\bar{\tau}_{\pi_1^k}$ to lie between $\exp\!\big\{N\big(\tilde{V}\big(\pi_1^k\big) - \delta\big)\big\}$ and $\exp\!\big\{N\big(\tilde{V}\big(\pi_1^k\big) + \delta\big)\big\}$ . To prove Lemma 3.8, in addition to the above facts, we note that with high probability, the process $\mu_N$ enters a small neighbourhood of the elements of $\pi_2^k$ upon exit from a small neighbourhood of the elements of $\pi_1^k$ when $\pi_1^k \to \pi_2^k$ . Similar estimates as in Lemmas 3.8 and 3.9 in the case of small-noise diffusion processes have been shown in [Reference Hwang and Sheu25, Lemma 2.1, Part I] and [Reference Hwang and Sheu25, Lemma 2.2, Part I], respectively.

Lemma 3.10. Let $\pi^k$ be a k-cycle and assume that $\tilde{V}\big(\pi^k\big) > 0$ . Given $\varepsilon > 0$ , there exist $\delta>0$ , $\rho >0$ , and $N_0 \geq 1$ such that for all $\rho_1 \leq \rho$ , $\nu \in M_1^N(\mathcal{Z})$ , and $N \geq N_0$ , we have

\begin{align*} \mathbb{P}_{\nu}\big(\bar{\tau}_{\pi^k} \leq \exp\!\big\{N\big(\hat{V}\big(\pi^k\big)+\delta\big\}\big) \leq \exp\!\big\{{-}N\big(\tilde{V}\big(\pi^k\big) - \hat{V}\big(\pi^k\big) - \varepsilon\big)\big\}. \end{align*}

Proof. We proceed via the steps in the proof of [Reference Hwang and Sheu25, Lemma 2.1, Part III]. Let $\pi^{k-1} \in \pi^k$ be a $(k-1)$ -cycle such that $\tilde{V}\big(\pi^{k-1}\big) = \hat{V}\big(\pi^k\big)$ . With $\rho_1>0$ to be chosen later, for each $n \geq 1$ , define the minimum of $\bar{\tau}_{\pi^k}$ and successive entry and exit times from a $\rho_1$ -neighbourhood of $\pi^{k-1}$ as follows:

\begin{align*} \hat{\theta}_0 & \,:\!=\, \inf\!\big\{ t >0 \,:\, \mu_N(t) \in \big[\pi^{k-1}\big]_{\rho_1}\big\} \wedge \bar{\tau}_{\pi^k}, \\ \bar{\theta}_n & \,:\!=\, \inf\!\big\{t > \hat{\theta}_{n-1}\,:\, \mu_N(t) \in \big[L \setminus \pi^{k-1}\big]_{\rho_1} \big\} \wedge \bar{\tau}_{\pi^k} , \\ \hat{\theta}_{n+1}& \,:\!=\, \inf\!\big\{t > \bar{\theta}_n \,:\, \mu_N(t) \in \big[\pi^{k-1}\big]_{\rho_1} \big\} \wedge \bar{\tau}_{\pi^k}. \end{align*}

With $\delta > 0$ to be chosen later, using the strong Markov property, for any $\nu \in \big[\pi^k\big]_{\rho_1} \cap M_1^N(\mathcal{Z})$ , we have

(3.4) \begin{align} \mathbb{P}_{\nu}&\big(\bar{\tau}_{\pi^k} \leq \exp\!\big\{N\big(\hat{V}\big(\pi^k\big)+\delta\big)\big\}\big) \nonumber \\ & = \mathbb{P}_\nu \big(\hat{\theta}_0 = \bar{\tau}_{\pi^k}, \bar{\tau}_{\pi^k} \leq \exp\!\big\{N\big(\hat{V}\big(\pi^k\big)+\delta\big)\big\}\big) \nonumber \\ & \qquad + \mathbb{P}_\nu\Bigg(\hat{\theta}_0 < \bar{\tau}_{\pi^k}, \bigcup_{n \geq 1} \left\{\bar{\tau}_{\pi^k} = \bar{\theta}_n, \bar{\tau}_{\pi^k} \leq \exp\!\big\{N\big(\hat{V}\big(\pi^k\big)+\delta\big)\big\}, \bar{\tau}_{\pi^k} \geq \hat{\theta}_{n-1} \right\} \Bigg) \nonumber \\ & \qquad + \mathbb{P}_\nu\Bigg(\hat{\theta}_0 < \bar{\tau}_{\pi^k}, \bigcup_{n\geq 1} \left\{\bar{\tau}_{\pi^k} = \hat{\theta}_n, \bar{\tau}_{\pi^k} \leq \exp\!\big\{N\big(\hat{V}\big(\pi^k\big)+\delta\big)\big\}, \bar{\tau}_{\pi^k} \geq \bar{\theta}_n \right\} \Bigg). \end{align}

We now upper-bound each of the terms in (3.4). Consider the first term. It can be shown using Corollary 3.3 and [Reference Hwang and Sheu25, Corollary A.6, Appendix] that there exist $\rho_1>0$ and $\delta > 0$ such that for any $\nu \in \big[\pi^k\big]_{\rho_1}$ and sufficiently large N, we have

\begin{align*} \mathbb{P}_\nu\big(\hat{\theta}_0 = \bar{\tau}_{\pi^k}\big) \leq \exp\!\big\{{-}N\big(\tilde{V}\big(\pi^k\big) - \hat{V}\big(\pi^k\big) - \varepsilon\big)\big\}. \end{align*}

Consider the second term in (3.4). For any $\nu_1 \in \big[\pi^{k-1}\big]_{\rho_1} \cap M_1^N(\mathcal{Z})$ , the probability of the unionised event can be bounded above by

\begin{align*} & \mathbb{P}_{\nu_1}\! \left(\bigcup_{n \geq 1}\left\{\bar{\tau}_{\pi^k} = \bar{\theta}_n, \bar{\tau}_{\pi^k} \leq \exp\!\big\{N\big(\hat{V}\big(\pi^k\big)+\delta\big)\big\}, \bar{\tau}_{\pi^k} \geq \hat{\theta}_{n-1} \right\} \right)\\ & \quad\leq \mathbb{P}_{\nu_1}\! \left(\bigcup_{n=1}^M \left\{\bar{\tau}_{\pi^k} = \bar{\theta}_n, \bar{\tau}_{\pi^k} \leq \exp\!\big\{N\big(\hat{V}\big(\pi^k\big)+\delta\big)\big\}, \bar{\tau}_{\pi^k} \geq \hat{\theta}_{n-1} \right\} \right) \\ & \qquad\! + \mathbb{P}_{\nu_1}\! \left( \bigcup_{n \geq M+1} \left\{\bar{\tau}_{\pi^k} = \bar{\theta}_n, \bar{\tau}_{\pi^k} \leq \exp\!\big\{N\big(\hat{V}\big(\pi^k\big)+\delta\big)\big\}, \bar{\tau}_{\pi^k} \geq \hat{\theta}_{n-1} \right\} \right) \\ & \quad\leq \mathbb{P}_{\nu_1} \big(\bar{\tau}_{\pi^k} = \bar{\theta}_n \text{ and } \bar{\tau}_{\pi^k} \geq \hat{\theta}_{n-1} \text{ for some } n \leq M\big) \\ & \qquad\! + \mathbb{P}_{\nu_1} \big(\hat{\theta}_M \leq \exp\!\big\{N\big(\hat{V}\big(\pi^k\big)+\delta\big)\big\} \text{ and } \hat{\theta}_M \leq \bar{\tau}_{\pi^k}\big)\\ & \quad\leq \mathbb{P}_{\nu_1}\big(\hat{\theta}_M = \bar{\tau}_{\pi^k}\big) + \mathbb{P}_{\nu_1} \big(\hat{\theta}_M \leq \exp\!\big\{N\big(\hat{V}\big(\pi^k\big)+\delta\big)\big\} \text{ and } \hat{\theta}_M \leq \bar{\tau}_{\pi^k}\big). \end{align*}

Again, the first term above can be bounded by

\begin{align*} \mathbb{P}_{\nu_1}\big(\hat{\theta}_M \leq \bar{\tau}_{\pi^k}\big) \leq \exp\!\big\{{-}N\big(\tilde{V}\big(\pi^k\big) - \hat{V}\big(\pi^k\big) - \varepsilon\big)\big\}, \end{align*}

for all $\nu_1 \in \big[\pi^{k-1}\big]_{\rho_1} \cap M_1^N(\mathcal{Z})$ and sufficiently large N. The second term can be bounded by $\exp\{{-}NM\}$ for large enough M, by the same argument used in the proof of [Reference Hwang and Sheu25, Lemma 1.7, Part I]. If we choose M sufficiently large, the above implies that the second term in (3.4) is bounded by $\exp\!\big\{{-}N\big(\tilde{V}\big(\pi^k\big) - \hat{V}\big(\pi^k\big) - \varepsilon\big)\big\}$ . A similar argument gives the same bound for the third term in (3.4).

3.4. LDP for the invariant measure

Using the estimates (3.2) of the transition probabilities of the discrete-time Markov chain $Z^N$ , we can study large deviations for the process $\mu_N$ in the stationary regime. Recall that $\wp_N$ denotes the unique invariant probability measure of the process $\mu_N$ . Also recall that G(i) is the set of all directed graphs g on L such that (a) every node other than i has exactly one outgoing arrow and (b) there are no closed cycles in g. We state the following result.

Theorem 3.1. ([Reference Borkar and Sundaresan12, Theorem 2.2].) Assume (A1), (A2), and (B1). Then the sequence of invariant measures $\{\wp_N\}_{N \geq 1}$ satisfies the LDP on $M_1(\mathcal{Z})$ with good rate functions given by

(3.5) \begin{align} s(\xi) = \min_{1 \leq i \leq l} \{W(i) + V(K_i, \xi)\} - \min_{1 \leq j \leq l} W(j), \end{align}

where

\begin{align*} W(i) = \min_{g \in G(i)} \sum_{(m,n) \in g} \tilde{V}(m,n). \end{align*}

The form of the rate function s in Theorem 3.1 is also related to the form of the invariant measure in the context of Markov chains on finite state spaces whose transition kernels are of the form (3.2); see, for example, [Reference Del Moral and Miclo15, Section 1.1]. Also, see [Reference Bodineau and Giacomin9] for an analogous result in a boundary-driven symmetric simple exclusion process, which involves the study of the LDP for the invariant measure in an infinite-dimensional setting. However, our focus is on sharp estimates on the rate of convergence to the invariant measure, which is the subject of the next section.

3.5. Convergence to the invariant measure

In this section, we prove our first main result on the time required for the convergence of $\mu_N$ to its invariant measure.

Let $i_0 \in L$ be such that $ \min\!\big\{\tilde{V}(g)\,:\, g \in G(i_0)\big\} = \min\!\big\{\tilde{V}(g)\,:\, g \in G(i), i \in L \big\} $ . We anticipate that $K_{i_0}$ is one of the most stable $\omega$ -limit sets (possibly among others) for the dynamics (1.1). This is because Theorem 3.1 tells us that the rate function that governs the LDP for $\{\wp_N\}_{N \geq 1}$ vanishes on $K_{i_0}$ . Hence, for a large but fixed N, over large time intervals, one expects that there is positive probability (in the exponential scale) for the process $\mu_N$ to be in a small neighbourhood of $K_{i_0}$ .

Define

(3.6) \begin{align} \Lambda \,:\!=\, \left\{ \begin{array}{l@{\quad}l} \min\!\big\{\tilde{V}(g)\,:\, g \in G( \{ i \} ), i \in L \big\} - \min\!\big\{\tilde{V}(g)\,:\, g \in G(\{ i, j\}), i, j \in L, i \neq j\big\} & \text{ if } |L|\geq 2,\\[4pt] 0 &\text{ if } |L| = 1. \end{array} \right. \end{align}

Since we are interested in the case when (1.1) has multiple $\omega$ -limit sets, we assume throughout that $\Lambda > 0$ . The motivation to define this constant $\Lambda$ is the following. Since the process $\mu_N$ spends most of the time near one of the compact sets $K_i$ , we expect that it mixes well when the discrete-time Markov chain $Z^N$ , with transition probabilities of the form $\exp\!\big\{{-}N\tilde{V}\big(K_i, K_j\big)\big\}$ given in (3.2), mixes well. The mixing time of $Z^N$ is determined by the real part of $\big(1-\hat{\lambda}^N_2\big)$ , where $\hat{\lambda}^N_2$ is the second-largest (in absolute value) eigenvalue of an $l \times l$ transition probability matrix whose (i, j)th entry, for $i \neq j$ , is given by $\exp\!\big\{{-}N\tilde{V}\big(K_i, K_j\big)\big\}$ ; it turns out that this scales as $\exp\{{-}N\Lambda\}$ [Reference Freidlin and Wentzell20, Chapter 6, Theorem 7.3]. Thus, we expect that, when time is of the order $\exp\{N\Lambda\}$ , the process $\mu_N$ mixes well.

Let $\mathbb{P}_T(\nu, \cdot ) = \mathbb{P}_{\nu}(\mu_N(T) \in \cdot )$ denote the transition probability kernel associated with the process $\mu_N$ . Note that we suppress the dependence on N for greater readability. We first show a lower bound for the transition probability $\mathbb{P}_T\big(\nu_1, K_{i_0} \big) $ of reaching a small neighbourhood of $K_{i_0}$ when T is of the order $\exp\{N(\Lambda-\delta_0)\}$ for some $\delta_0>0$ .

Theorem 3.2. Given $\varepsilon >0$ , there exist $\delta_0 > 0$ , $\rho > 0$ , and $N_0 \geq 1$ such that for all $\rho_1 \leq \rho$ , $N \geq N_0$ , $\nu \in M_1^N(\mathcal{Z})$ , we have

(3.7) \begin{align} \mathbb{P}_{T_0}(\nu, \gamma_{i_0}) \geq \exp\{{-}N\varepsilon\}, \end{align}

where $T_0 =\exp\{N(\Lambda - \delta_0)\}$ . Furthermore, there exist $\nu_0 \in M_1(\mathcal{Z})$ and $\beta >0$ such that for all $N \geq N_0$ and $\nu \in [\nu_0]_{\rho_1} \cap M_1^N(\mathcal{Z})$ ,

(3.8) \begin{align} \mathbb{P}_{T_0}(\nu, \gamma_{i_0}) \leq \exp\{{-}N\beta\}. \end{align}

Proof. We follow the steps in Hwang and Sheu [Reference Hwang and Sheu25, Part I, Theorem 2.3]. With $\rho > 0$ to be chosen later, we first show that (3.7) holds for all $\nu \in \gamma \cap M_1^N(\mathcal{Z})$ . Towards this, let m be the smallest integer such that $L_{m+1}$ is a singleton. For $0 \leq k \leq m$ , let $\pi_0^k \in L_k$ be the k-cycle containing $i_0$ . Let $V_k = \max\! \big\{\tilde{V}\big(\pi^k\big) \,:\, \pi^k \subset \pi_0^{k+1}, \pi^k \neq \pi_0^k\big\}$ . Using [Reference Hwang and Sheu25, Lemma A.10, Appendix], we have $\Lambda = \max\{V_k\,:\, 0 \leq k \leq m\}$ .

Fix $j \in L$ and consider $\nu \in [K_j]_{\rho}$ . Let $\pi_1^m \in L_m$ be such that $K_j \in \pi_1^m$ . If $\pi_1^m \neq \pi_0^m$ , then we have $\pi_1^m \Rightarrow \pi_0^m$ ; that is, there exist $\pi_2^m, \pi_3^m, \ldots, \pi_n^m =\pi_0^m$ , $n \leq l$ , such that $\pi_1^m \to \pi_2^m \to \pi_3^m \to \cdots \to \pi_n^m = \pi_0^m$ . Therefore, with $\delta$ to be chosen later, by the strong Markov property, we have

\begin{align*} \mathbb{P}_\nu \big(\hat{\tau}_{\pi^m_0} & \leq n \exp\!\big\{N(V_m - \delta)\big\}\big) \\ & \geq \mathbb{E}_\nu \bigg(1_{\big\{\bar{\tau}_{\pi^m_1} \leq \exp\!\big\{N(V_m -\delta )\big\}\big\}}\cdot 1_{\big\{ \mu_N\big(\bar{\tau}_{\pi^m_1}\big) \in \pi_2^m\big\}} \\ & \qquad \times \mathbb{E}_{\mu_N\big(\bar{\tau}_{\pi^m_1}\big)}\bigg(1_{\big\{\bar{\tau}_{\pi_2^m} \leq \exp\!\big\{N(V_m -\delta )\big\}\big\}} \cdot 1_{\big\{\mu_N\big(\bar{\tau}_{\pi^m_2}\big) \in \pi_3^m\big\}} \\ & \qquad \cdots \times \mathbb{E}_{\mu_N\big(\bar{\tau}_{\pi^m_{n-2}}\big)}\bigg( 1_{\big\{\bar{\tau}_{\pi^m_{n-1}} \leq \exp\!\big\{N(V_m -\delta )\big\}\big\}} \cdot 1_{\big\{\mu_N\big(\bar{\tau}_{\pi^m_{n-1}}\big) \in \pi_0^m\big\}}\bigg)\\ & \qquad \cdots \bigg)\bigg). \end{align*}

Since $V\big(\pi_i^m\big) \leq V_m$ for all $1 \leq i \leq n$ , the above becomes

\begin{align*} \mathbb{P}_\nu \Big(\hat{\tau}_{\pi^m_0} & \leq n \exp\{N(V_m - \delta)\}\Big) \\ & \geq \mathbb{E}_\nu \bigg(1_{\big\{\bar{\tau}_{\pi^m_1} \leq \exp\!\big\{N\big(\tilde{V}(\pi_1^m) -\delta \big)\big\}\big\}} \cdot 1_{\big\{ \mu_N\big(\bar{\tau}_{\pi^m_1}\big) \in \pi_2^m\big\}} \\ & \qquad \times \mathbb{E}_{\mu_N\big(\bar{\tau}_{\pi^m_1}\big)}\bigg(1_{\big\{\bar{\tau}_{\pi_2^m} \leq \exp\!\big\{N\big(\tilde{V}\big(\pi_2^m\big)-\delta \big)\big\}\big\}}\cdot 1_{\big\{\mu_N\big(\bar{\tau}_{\pi^m_2}\big) \in \pi_3^m\big\}} \\ & \qquad \cdots \times \mathbb{E}_{\mu_N\Big(\bar{\tau}_{\pi^m_{n-2}}\Big)}\bigg(1_{\big\{\bar{\tau}_{\pi^m_{n-1}} \leq \exp\!\big\{N\big(\tilde{V}\big(\pi_{n-1}^m\big) -\delta \big)\big\}\big\}} \cdot 1_{\big\{\mu_N\big(\bar{\tau}_{\pi^m_{n-1}}\big) \in \pi_0^m\big\}}\bigg)\\ & \qquad \cdots \bigg)\bigg). \end{align*}

By Lemma 3.8, there exist $\rho >0$ , $\delta > 0$ , and $N_0 \geq 1 $ such that each of the above probabilities is at least $\exp\{{-}N\varepsilon/l\}$ for sufficiently large N, i.e., we have

\begin{align*} \mathbb{P}_\nu \Big(\hat{\tau}_{\pi^m_0} \leq n \exp\!\big\{N(V_m - \delta))\big\}\Big) \geq \exp\{{-}N n \varepsilon/l\} \geq \exp\{{-}N\varepsilon\}, \end{align*}

On the other hand, if $K_j$ is such that $K_j \in \pi_0^m$ , the above holds trivially. Therefore, there exist $\delta_1 > 0$ and $N_1 \geq 1$ such that for all $\nu \in \gamma \cap M_1^N(\mathcal{Z})$ and $N \geq N_1$ , we have

\begin{align*} \mathbb{P}_\nu \Big(\hat{\tau}_{\pi^m_0} \leq \exp\{N(V_m - \delta_1)\}\Big) \geq \exp\{{-}N \varepsilon\}. \end{align*}

We now use the above bound to show (3.7). Let $T = \exp\{N(\Lambda - \delta_1)\}$ , $T_m = \exp\{N(V_m - \delta_1)\}$ , and $T_{m-1} =\exp\{N(V_{m-1}-\delta_1)\}$ . Then, for any $\nu \in \gamma \cap M_1^N(\mathcal{Z})$ and $N \geq N_1$ , we have

(3.9) \begin{align} \mathbb{P}_\nu \big(\mu_N(T) \in \gamma_{i_0}\big)& \geq \mathbb{E}_\nu \bigg(1_{\big\{\hat{\tau}_{\pi^m_0} \leq T_m\big\}} \cdot 1_{\big\{\mu_N\big(T-\hat{\tau}_{\pi^m_0}\big) \in \gamma_{i_0}\big\}}\bigg) \nonumber\\ & = \mathbb{E}_\nu \bigg(1_{\big\{\hat{\tau}_{\pi^m_0} \leq T_m\big\}} \cdot \mathbb{E}_{\mu_N\big(\hat{\tau}_{\pi_0^m}\big)} \bigg(1_{\big\{\mu_N\big(T-\hat{\tau}_{\pi^m_0}\big) \in \gamma_{i_0}\big\}}\bigg)\bigg) \nonumber \\ & \geq \inf_{\substack{\nu^\prime \in \big[\pi_0^m\big]_{\rho}\cap M_1^N\big(\mathcal{Z}\big) \\ T-T_m \leq t \leq T}} \mathbb{P}_{\nu^\prime} \big(\mu_N(t) \in \gamma_{i_0}\big) \mathbb{P}_\nu \big(\hat{\tau}_{\pi^m_0}\leq T_m\big) \nonumber \\ & \geq \inf_{\substack{ \nu^\prime \in \big[\pi_0^m\big]_{\rho}\cap M_1^N(\mathcal{Z}) \\ T-T_m \leq t \leq T}} \mathbb{P}_{\nu^\prime} \big(\mu_N(t) \in \gamma_{i_0}\big) \exp\!\big\{{-}N\varepsilon\big\}, \end{align}

where the second equality follows from the strong Markov property. To get a lower bound for the above infimum, fix $\nu \in \big[\pi_0^m\big]_\rho \cap M_1^N(\mathcal{Z})$ and $T-T_{m} \leq t \leq T$ . Define the stopping time $\theta \,:\!=\, \inf\!\big\{s> t-T_{m-1} \,:\, \mu_N(s) \in \big[\pi_0^m\big]_{\rho}\big\}$ . Then, for a large $T^*$ (not depending on N) to be chosen later, we have

(3.10) \begin{align} \mathbb{P}_\nu &\big(\mu_N(t) \in \gamma_{i_0}\big) \nonumber \\ &\geq \mathbb{E}_\nu \bigg(1_{\big\{\theta \leq t-T_{m-1}+T^* , \bar{\tau}_{\pi_0^m} > T\big\}} \cdot \mathbb{E}_{\mu_N(\theta)}\bigg( 1_{\big\{\mu_N(t-\theta) \in \gamma_{i_0}\big\}}\bigg) \nonumber \\ & \geq \mathbb{P}_\nu\Big(\theta \leq t-T_{m-1}+T^* , \bar{\tau}_{\pi_0^m} > T\Big) \inf_{\substack{\nu^\prime \in \big[\pi_0^m\big]_{\rho}\cap M_1^N\big(\mathcal{Z}\big) \\ T_{m-1}-T^*\leq t \leq T_{m-1}}} \mathbb{P}_{\nu^\prime} \big(\mu_N(t) \in \gamma_{i_0}\big). \end{align}

Note that

\begin{align*} \mathbb{P}_\nu\big(\theta \leq t-T_{m-1}+T^*, \bar{\tau}_{\pi_0^m} > T\big)= \mathbb{P}_\nu\Big(\bar{\tau}_{\pi^m_0} > T\Big) - \mathbb{P}_\nu\big(\theta > t-T_{m-1}+T^* , \bar{\tau}_{\pi_0^m} > T\big). \end{align*}

By Lemma 3.9, since $\Lambda \leq \tilde{V}\big(\pi^m_0\big)$ , we have

\begin{align*} \mathbb{P}_\nu\Big(\bar{\tau}_{\pi^m_0} > T\Big) \geq \mathbb{P}_\nu\big(\bar{\tau}_{\pi^m_0} > \exp\!\big\{N\big(\tilde{V}\big(\pi^m_0\big) - \delta\big)\big\}\big) \to 1 \end{align*}

as $N \to \infty$ . For the second term, note that

\begin{align*} \mathbb{P}_\nu& \big(\theta > t-T_{m-1}+T^* , \bar{\tau}_{\pi_0^m} > T\big) & \\ & = \mathbb{P}_\nu \big(\mu_N(s) \notin \big[\pi_0^m\big]_\rho \text{ for all } t-T_{m-1}\leq s \leq t-T_{m-1}+T^*, \bar{\tau}_{\pi_0^m} > T\big) \\ & = \mathbb{P}_\nu \big(\mu_N(s) \notin \gamma \text{ for all } t-T_{m-1}\leq s \leq t-T_{m-1}+T^*, \bar{\tau}_{\pi_0^m} > T\big) \\ & \leq \mathbb{P}_\nu \big(\mu_N(s) \notin \gamma \text{ for all } t-T_{m-1}\leq s \leq t-T_{m-1}+T^*\big). \end{align*}

The second equality follows since $\mu_N(s) \notin \big[\pi_0^m\big]_\rho$ and $\bar{\tau}_{\pi_0^m} > T$ implies that we have exited $\big[\pi_0^m\big]_\rho$ and we have not yet entered a neighbourhood of any other attractor, which is the same as saying $\mu_N(t) \notin \gamma$ and $\bar{\tau}_{\pi_0^m} >T$ . By the Markov property, the above probability equals

\begin{align*} \mathbb{E}_\nu \! \left( \mathbb{E}_{\mu_N\big(t-T_{m-1}\big)}\bigg(1_{\big\{\mu_N(s) \notin \gamma \text{ for all } s \in \big[t-T_{m-1}, t-T_{m-1}+T^*\big]\big\}}\bigg) \right) \leq \sup_{\nu^\prime \in F} \mathbb{P}_{\nu^\prime} \big(\tau_F \geq T^*\big), \end{align*}

where $F = M_1(\mathcal{Z}) \setminus \gamma$ . By Lemma 3.3, $T^*$ can be chosen large enough (not depending on N) that the above probability is at most $1/2$ . Therefore, (3.10) becomes

\begin{align*} \inf_{\substack{\nu \in \big[\pi_0^m\big]_\rho \cap M_1^N(\mathcal{Z}) \\ T-T_m \leq t \leq T}} \mathbb{P}_\nu\big(\mu_N(t) \in \gamma_{i_0}\big) \geq \frac{1}{2} \inf_{\substack{\nu^\prime \in \big[\pi_0^m\big]_{\rho} \cap M_1^N(\mathcal{Z}) \\ T_{m-1}-T^* \leq t \leq T_{m-1}}} \mathbb{P}_{\nu^\prime} \big(\mu_N(t) \in \gamma_{i_0}\big), \end{align*}

and (3.9) becomes

\begin{align*} \mathbb{P}_\nu \big(\mu_N(T) \in \gamma_{i_0}\big) \geq \frac{1}{2} \exp\{{-}N\varepsilon\} \inf_{\substack{\nu^\prime \in \big[\pi_0^m\big]_{\rho} \cap M_1^N(\mathcal{Z}) \\ T_{m-1}-T^* \leq t \leq T_{m-1}}} \mathbb{P}_{\nu^\prime} \big(\mu_N(t) \in \gamma_{i_0}\big), \end{align*}

for sufficiently large N and $\nu \in \gamma \cap M_1^N(\mathcal{Z})$ . Repeating the above argument m times, we see that there exists $N_2 \geq 1$ such that for all $\nu \in \gamma$ and $N \geq N_2$ , we have

\begin{align*} \mathbb{P}_\nu\big(\mu_N(T) \in \gamma_{i_0}\big) & \geq \left(\frac{1}{2}\right)^m\exp\{{-}Nm\varepsilon\} \inf_{\substack{\nu^\prime \in \big[\pi_0^1\big]_{\rho}\cap M_1^N(\mathcal{Z}) \\ T_0 -T^*\leq t \leq T_0}} \mathbb{P}_{\nu^\prime} \big(\mu_N(t) \in \gamma_{i_0}\big) \\ & \geq \left(\frac{1}{2}\right)^{m}\exp\{{-}N(m+1)\varepsilon\} \inf_{\substack{\nu^\prime \in [K_0]_{\rho} \cap M_1^N(\mathcal{Z})\\ T_0 -T^*\leq t \leq T_0}} \mathbb{P}_{\nu^\prime} \big(\mu_N(t) \in \gamma_{i_0}\big) \\ & \geq \left(\frac{1}{2}\right)^{m+1}\exp\{{-}N(m+1)\varepsilon\}, \end{align*}

where $T_0 = \exp\{N(V_0 - m\delta)\}$ . Thus, we conclude that there exist $N_3 \geq 1$ , $\delta_3>0$ , and $\rho > 0$ such that for all $\nu \in \gamma \cap M_1^N(\mathcal{Z})$ and $N \geq N_3$ , we have

\begin{align*} \mathbb{P}_\nu\big(\mu_N(T) \in \gamma_{i_0}\big) \geq \exp\{{-}N(m+3)\varepsilon\}, \end{align*}

where $T = \exp\{N(\Lambda-\delta_3)\}$ . This establishes (3.7) for all $\nu \in \gamma \cap M_1^N(\mathcal{Z})$ . For any $\nu \in M_1^N(\mathcal{Z})\setminus \gamma$ , from Lemma 3.3, there exist $T^\prime$ large enough and $N_4 \geq N_3$ such that $\mathbb{P}_\nu\big(\tau_{M_1(\mathcal{Z}) \setminus \gamma} \leq T^\prime\big) \geq \frac{1}{2}$ for all $N \geq N_4$ . Therefore, we have

\begin{align*} \mathbb{P}_\nu\big(\mu_N(T) \in \gamma_{i_0}\big) & \geq \mathbb{E}_\nu \bigg( 1_{\big\{\tau_{M_1(\mathcal{Z}) \setminus \gamma} \leq T^\prime\big\}} \cdot \mathbb{E}_{\mu_N(\tau_F)} \bigg(1_{\big\{\mu_N(T-T^\prime) \in \gamma_{i_0}\big\}}\bigg)\bigg) \\ & \geq \frac{1}{2} \inf_{\nu^\prime \in \gamma} \mathbb{P}_{\nu^\prime}\big(\mu_N(T-T^\prime) \in \gamma_{i_0}\big) \\ & \geq \frac{1}{2}\exp\{{-}N(m+3)\varepsilon\}. \end{align*}

Thus, we have established (3.7) for any $\nu \in M_1^N(\mathcal{Z})$ .

We now turn to (3.8). Since $\Lambda = \max\{V_k, 0 \leq k \leq m\}$ , there exists a k such that $V_k = \Lambda$ . From the definition of $V_k$ , we see that there exists $\pi^k \in L_k$ such that

\begin{align*} \tilde{V}\big(\pi^k\big) = \Lambda, \qquad \pi^k \subset \pi_0^{k+1}, \qquad \text{and }\pi^k \neq \pi^k_0, \end{align*}

where $\pi_0^{k+1}$ is the $(k+1)$ -cycle that contains $K_{i_0}$ . Therefore, Lemma 3.9 implies that, for some $\beta > 0$ , for some $\delta_4 < \delta_3$ and an appropriately chosen $\rho > 0$ , with $T = \exp\{N(\Lambda-\delta_3)\} = \exp\!\big\{N\big(\tilde{V}\big(\pi^k\big) - \delta_3\big)\big\}$ , we have

\begin{align*} \mathbb{P}_\nu\big(\mu_N(T) \in \gamma_{i_0}\big) \leq \mathbb{P}_\nu\big(\bar{\tau}_{\pi^k} \leq T\big) \leq \exp\{{-}N\beta\}, \end{align*}

for any $\nu \in \big[\pi^k\big]_{\rho} \cap M_1^N(\mathcal{Z})$ and sufficiently large N. This completes the proof of the theorem.

The above theorem immediately gives a lower bound on $\mathbb{P}_T(\nu, \xi)$ for any $\xi$ in a small neighbourhood of $K_{i_0}$ , over time durations of order $\exp\{N(\Lambda-\delta)\}$ for some $\delta > 0$ . Let us make this precise.

Corollary 3.4. Under the conditions of Theorem 3.2, for all $\nu \in M_1^N(\mathcal{Z})$ , $\xi \in \gamma_{i_0} \cap M_1^N(\mathcal{Z})$ , and N sufficiently large, we have

\begin{align*} \mathbb{P}_{T_0}(\nu, \xi) \geq \exp\{{-}2N\varepsilon\}. \end{align*}

Proof. Given $\varepsilon > 0$ , let $\rho$ , $N_0$ , and $T_0$ be as in the statement of Theorem 3.2. Choose t large enough (not depending on N) and $\rho^\prime < \rho$ so that for all $\rho_1 \leq \rho^\prime$ we have $S_t(\nu_1|\nu_2) \leq \varepsilon/2$ for all $\nu_1, \nu_2 \in \gamma_{i_0}$ . This is possible by the joint continuity of the rate function $S_t({\cdot}|{\cdot})$ and the fact that $V(\nu_1, \nu_2) = 0$ whenever $\nu_1, \nu_2 \in K_{i_0}$ . Therefore, using the large deviation lower bound, there exists $N_2 \geq N_1$ such that

\begin{align*} \mathbb{P}_t(\nu_1, \nu_2) \geq \exp\{{-}N(S_t(\nu_2 | \nu_1) + \varepsilon/2)\} \geq \exp\{{-}N\varepsilon\}, \end{align*}

for all $\nu_1, \nu_2 \in \gamma_{i_0} \cap M_1^N(\mathcal{Z})$ and $N \geq N_2$ . Therefore, by Theorem 3.2, for $\nu \in M_1^N(\mathcal{Z})$ , $\xi \in \gamma_{i_0} \cap M_1^N(\mathcal{Z})$ , and $N \geq N_2$ , we have

\begin{align*} \mathbb{P}_{T_0}(\nu, \xi) & = \sum_{\nu_2 \in \gamma_{i_0} \cap M_1^N(\mathcal{Z})} \mathbb{P}_{T_0-t}(\nu_1, \nu_2) \mathbb{P}_t\big(\nu_2, \xi\big) \\ & \geq \mathbb{P}_{T_0-t}\big(\nu_1, \gamma_{i_0}\big) \inf_{\nu_2 \in \gamma_{i_0} \cap M_1^N(\mathcal{Z})} \mathbb{P}_t\big(\nu_2, \xi\big) \\ & \geq \exp\{{-}2N \varepsilon\}. \end{align*}

3.5.1. Proofs of Theorem 1.1 and Theorem 1.2

We now prove our first main result (Theorem 1.1) on the convergence of $\mu_N$ to the invariant measure and its converse Theorem 1.2. Theorem 1.1 together with Theorem 1.2 shows that the constant $\Lambda$ is sharp (in the exponential scale) for the time required for $\mu_N$ to equilibrate.

Define $\tilde{L}_0 \,:\!=\, \{i \in L \,:\, s(K_i) = 0\}$ ; i.e, $\tilde{L}_0$ denotes the set of minimisers of the rate function s (see (3.5)). Let $B(M_1(\mathcal{Z}))$ denote the space of bounded Borel-measurable functions on $M_1(\mathcal{Z})$ .

Proof of Theorem 1.1.We follow the steps in Hwang and Sheu [Reference Hwang and Sheu25, Part I, Theorem 2.5]. Let $\varepsilon > 0$ , and let $T_0$ , $\delta_0$ , $\rho$ , $\rho_1$ , and $N_0 \geq1$ be as in the statement of Theorem 3.2. Note that, for any $\nu \in M_1^N(\mathcal{Z})$ , $\xi \notin \big[\tilde{L}_0\big]_{\rho_1}$ and for some fixed $t > 0$ ,

\begin{align*} \mathbb{P}_{T_0}(\nu,\xi)& = \sum_{\nu^\prime \in \big[K_{i_0}\big]} \mathbb{P}_{T_0-t}(\nu, \nu^\prime) \mathbb{P}_t(\nu^\prime, \xi)\\ & \geq \exp\{{-}2N\varepsilon\} \inf_{\nu^\prime \in \big[K_{i_0}\big]} \mathbb{P}_t(\nu^\prime, \xi) \\ & \geq \exp\{{-}2N\varepsilon\} \exp\!\Bigg\{{-}N\sup_{\nu^\prime \in \big[K_{i_0}\big]} S_t\big(\xi|\nu^\prime\big)\Bigg\}, \end{align*}

where the first inequality follows from Corollary 3.4 and the second from the uniform LDP (Corollary 2.1). Hence, we can find a function $U \,:\, M_1(\mathcal{Z}) \to [0, \infty)$ such that $U(\xi) = 0$ for $\xi \in \big[\tilde{L}_0\big]_{\rho_1}$ and

(3.11) \begin{align} \mathbb{P}_{T_0}(\nu,\xi) \geq c_N \exp\{{-}NU(\xi)\} \end{align}

holds for all $\nu \in M_1^N(\mathcal{Z})$ , $\xi \notin \big[\tilde{L}_0\big]_{\rho_1}$ and sufficiently large N; here $c_N$ is such that

\begin{align*} \pi_N(\xi) = c_N\exp\{{-}NU(\xi)\} \end{align*}

is a probability measure on $M_1^N(\mathcal{Z})$ . Define $Q_{T_0}(\nu ,\cdot) \,:\!=\, \mathbb{P}_{T_0}(\nu, \cdot)/ \pi_{N}({\cdot})$ . Note that $c_N \to 0$ exponentially fast as $N \to \infty$ . Indeed, since $U(\xi) = 0$ for all $\xi \in \big[\tilde{L}_0\big]_{\rho_1}$ , each of these points yield $\pi_N(\xi) = c_N$ . Since the number of points in $\big[\tilde{L}_0\big]_{\rho_1} \cap M_1^N(\mathcal{Z})$ is exponential in N and since $\pi_N$ is a probability measure, we see that $c_N$ must decay exponentially fast in N. We have, for any $\nu_1, \nu_2 \in M_1^N(\mathcal{Z})$ and sufficiently large N,

\begin{align*} \mathbb{E}_{\nu_1}&(\,f(\mu_N(T_0))) - \mathbb{E}_{\nu_2}(\,f(\mu_N(T_0))) \\ & = \sum_{\xi \in M_1^N(\mathcal{Z})}\mathbb{P}_{T_0}(\nu_1,\xi) f(\xi) - \sum_{\xi \in M_1^N(\mathcal{Z})}\mathbb{P}_{T_0}(\nu_2,\xi) f(\xi) \\ & = \sum_{\xi \in M_1^N(\mathcal{Z})}Q_{T_0}(\nu_1,\xi) f(\xi) \pi_N(\xi)- \sum_{\xi \in M_1^N(\mathcal{Z})}Q_{T_0}(\nu_2,\xi) f(\xi) \pi_N(\xi) \\ & = \sum_{\xi \in M_1^N(\mathcal{Z})}(Q_{T_0}(\nu_1,\xi)- \exp\{{-}2N\varepsilon\})f(\xi) \pi_N(\xi) \\ & \qquad -\sum_{\xi \in M_1^N(\mathcal{Z})}(Q_{T_0}(\nu_2,\xi) - \exp\{{-}2N\varepsilon\}) f(\xi) \pi_N(\xi) \\ & \leq (1-\exp\{{-}2N\varepsilon\}) \bigg(\sup_\xi f(\xi) - \inf_\xi f(\xi)\bigg), \end{align*}

where the last inequality follows from (3.11) and the fact that $Q_{T_0}(\cdot, \cdot) \geq 1$ . Therefore, we have that

\begin{align*} \sup_{\nu_1, \nu_2} | \mathbb{E}_{\nu_1}&(\,f(\mu_N(T_0))) - \mathbb{E}_{\nu_2}(\,f(\mu_N(T_0))) | \leq (1-\exp\{{-}2N\varepsilon\}) \| f\|_\infty. \end{align*}

Continuing this procedure k times, and by using the Markov property, we get

\begin{align*} \sup_{\nu_1, \nu_2} |\mathbb{E}_{\nu_1}&(\,f(\mu_N(kT_0))) - \mathbb{E}_{\nu_2}(\,f(\mu_N(kT_0))) | \leq (1-\exp\{{-}2N\varepsilon\})^k \| f\|_\infty, \end{align*}

and hence we have

\begin{align*} \sup_{\nu} | \mathbb{E}_{\nu}&(\,f(\mu_N(kT_0))) - \langle\, f, \wp_N \rangle | \leq (1-\exp\{{-}2N\varepsilon\})^k \| f\|_\infty. \end{align*}

Choose $k = \exp\{N(\delta_0+\delta)\}$ ; then we have $kT_0 = \exp\{N(\Lambda + \delta)\}$ , and the above becomes

\begin{align*} \sup_{\nu} | \mathbb{E}_{\nu}&(\,f(\mu_N(kT_0))) - \langle\, f, \wp_N \rangle | \leq \exp\{{-}\exp\!(N({-}2\varepsilon+\delta_0 + \delta))\}. \end{align*}

We can choose $\varepsilon$ small enough so that the quantity $-2\varepsilon+ \delta > 0$ , and hence for some $\varepsilon^\prime > 0$ , we have

\begin{align*} \sup_{\nu} | \mathbb{E}_{\nu}&(\,f(\mu_N(T))) - \langle\, f, \wp_N \rangle | \leq \exp\{{-}\exp\!(N\varepsilon^\prime)\}, \end{align*}

for sufficiently large N, where $T = \exp\{N(\Lambda+\delta)\}$ . This establishes the result.

Proof of Theorem 1.2.This is a direct consequence of (3.8), established in Theorem 3.2.

4. Asymptotics of the second-largest eigenvalue for reversible processes

In this section, our goal is to understand the rate of convergence of $\mu_N$ to its invariant measure for a fixed N. For this purpose, we shall assume that the Markov process $\mu_N$ is reversible. That is, the operator $L^N$ is self-adjoint in $L^2(\wp_N)$ and it admits a spectral expansion; let $0 =\lambda_1^N > -\lambda_2^N \geq -\lambda_3^N \ldots$ denote its eigenvalues in decreasing order, and let $u_1^N \equiv 1, u_2^N, u_3^N,\ldots$ denote their corresponding eigenfunctions. Without loss of generality, we assume that the eigenfunctions are orthonormal, i.e., $\big(u_i^N, u_i^N\big) = 1$ for each i and $\big(u_i^N, u_j^N\big) = 0$ for each $i \neq j$ , where $( \cdot, \cdot )$ denotes the inner product in $L^2(\wp_N)$ . The spectral expansion [Reference Bakry, Gentil and Ledoux5, Section 1.7.2] enables us to write, for any $f \in B(M_1(\mathcal{Z}))$ ,

(4.1) \begin{align} \mathbb{E}_\nu f(\mu_N(t)) = \langle\, f, \wp_N\rangle + \sum_{k \geq 2} e^{-t\lambda_k^N} \big( f, u_k^N \big) u_k^N(\nu). \end{align}

Therefore, the rate of convergence of $\mathbb{E}_\nu f(\mu_N(t))$ to its stationary value $\langle\, f, \wp_N \rangle$ is determined by the leading term in the above sum, which is the second-largest eigenvalue $-\lambda_2^N$ . Hence, to understand the convergence of $\mu_N$ to its invariant measure, we study the asymptotics of $\lambda_2^N$ .

We first need the following lemma, which estimates the probability that the process $\mu_N$ is outside a small neighbourhood of the set $\cup_{i=1}^l K_i$ .

Lemma 4.1. Fix $\rho_1 >0$ and let B be the $\rho_1$ -neighbourhood of $\cup_{i \in L} K_i$ . Given $\varepsilon > 0$ , there exist $\delta >0$ and $N_0 \geq 1$ such that for each $\nu \in M_1^N(\mathcal{Z})$ and $N \geq N_0$ , we have

\begin{align*} \mathbb{P}_\nu\!\left(\mu_N(T) \in M_1^N(\mathcal{Z}) \setminus B\right) \leq \exp\{{-}N\delta\}, \end{align*}

where $T = \exp\{N(\Lambda+ \varepsilon)\}$ .

This result can be proved using Theorem 1.1, which deals with the convergence to the invariant measure, and Theorem 3.4, which addresses large deviations of the invariant measure $\{\wp_N\}_{N \geq 1}$ . Indeed, from Theorem 3.1, since the function s in (3.5) is strictly positive outside $\cup_{i=1}^l K_i$ , the probability of the complement of a small neighbourhood of $\cup_{i=1}^l K_i$ under $\wp_N$ decays exponentially in N. But when T is of the order $\exp\{N(\Lambda+\varepsilon)\}$ , from Theorem 1.1, the law of the random variable $\mu_N(T)$ is not far from $\wp_N$ . Therefore, the probability that $\mu_N(T)$ lies outside a small neighbourhood of $\cup_{i=1}^l K_i$ decays exponentially in N.

We are now ready to prove our next main result (Theorem 1.3) on the asymptotics of the modulus of the second-largest eigenvalue $\big($ i.e., $\lambda_2^N\big)$ .

Proof of Theorem 1.3.Lower bound: Suppose that there exists a subsequence $\{N_k\}_{k \geq 1}$ such that

(4.2) \begin{align} \log \lambda_2^{N_k} < -N_k(\Lambda + \varepsilon) \end{align}

for some $\varepsilon > 0$ . We will show that this contradicts $\int\!\Big(u_2^{N_k}(\nu)\Big)^2 \wp_N(d\nu) =1 $ for sufficiently large k. Fix $\rho >0 $ and define $B \,:\!=\, \cup_{i=1}^l [K_i]_{\rho}$ . Then, using the lower semicontinuity of the rate function $S_t({\cdot}|{\cdot})$ and Corollary 2.1 on uniform LDP, we see that for sufficiently large t, there exists $\delta_1 >0$ such that $\inf\{S_t(\xi|\nu)\,:\,\xi, \nu \in B^c\} = \delta_1 > 0$ . Therefore, for any $\nu \in B^c \cap M_1^N(\mathcal{Z})$ and any $\delta_2>0$ , there exists $N_0 \geq 1$ such that for all $N \geq N_0$ ,

\begin{align*} \mathbb{P}_\nu(\mu_N(t) = \nu) \leq \exp\{{-}N(S_t(\nu|\nu) - \delta_2)\} \leq \exp\{{-}N(\delta_1 + \delta_2)\}. \end{align*}

On the other hand, (4.1) implies that

\begin{align*} \mathbb{P}_\nu(\mu_N(t) = \nu) & = \mathbb{E}_\nu(1_{\{\nu\}}(\mu_N(t))) \\ &\geq e^{-\lambda_2^Nt} \big(u_2^N(\nu)\big)^2 \wp_N(\nu), \end{align*}

so that

(4.3) \begin{align} \int_{B^c} \big|u_2^N\big|^2 \wp_N (d \nu) \leq \exp\{{-}N(\delta_1 + \delta_2)\} \end{align}

for all $N \geq N_0$ . To bound the integral over B, by Theorem 1.1, with $T = \exp\{N(\Lambda+\varepsilon/2)\}$ , there exist $\delta_3>0$ and $N_1 \geq N_0$ such that for all $N \geq N_1$ ,

\begin{align*} \left|\mathbb{E}_\nu f(\mu_N(T)) - \langle\, f, \wp_N \rangle \right| \leq \|\,f\|_\infty \exp\{{-}\exp\!(N\delta_3)\}, \end{align*}

for any $f \in B(M_1(\mathcal{Z}))$ . On the other hand, from (4.1), for any $\nu \in B \cap M_1^N(\mathcal{Z})$ , with $f = 1_{\{\nu\}}$ , we have

\begin{align*} \left| \mathbb{E}_\nu f(\mu_N(T)) - \langle\, f, \wp_N \rangle \right| & = \biggr| \sum_{i \geq 2} \exp\!\big\{{-}\lambda_i^NT\big\} \big( f, u_i^N(\nu)\big) u_i^{N}(\nu) \biggr| \\ & \geq \exp\big\{{-}\lambda_2^{N}T\big\} \big(u_2^{N}(\nu)\big)^2 \wp_N(\nu), \end{align*}

so that, by our assumption (4.2), there exists a $k_0 \geq 1$ such that

\begin{align*} u_2^{N_k}(\nu))^2 \wp_{N_k}(\nu) & \leq \exp\!\big\{\lambda_2^{N_k}T\big\} \exp\{{-}\exp\!(N_k\delta_3)\} \\ & \leq \exp\{2\exp\!({-}N_k(\Lambda+\varepsilon)) \exp\!(N_k(\Lambda+\varepsilon/2))\} \exp\{{-}N_k\delta_3\} \end{align*}

for all $k \geq k_0$ . Since $\big|M_1^{N_k}(\mathcal{Z})\big| \leq (N_k+1)^{|\mathcal{Z}|}$ for all k, the above implies that, for some $\delta_4 > 0$ ,

(4.4) \begin{align} \int_B \Big(u_2^{N_k}(\nu)\Big)^2 \wp_{N_k}(d\nu) \leq \exp\{{-}N_k\delta_4\} \end{align}

for all $k \geq k_0$ . Therefore, (4.3) and (4.4) imply that, for some $\delta > 0$ ,

\begin{align*} \int_{M_1(\mathcal{Z})} \Big(u_2^{N_k}(\nu)\Big)^2 \wp_{N_k}(d\nu) \leq \exp\{{-}N_k\delta\} \end{align*}

for all sufficiently large k, which is a contradiction to $\int\!\Big(u_2^{N_k}(\nu)\Big)^2 \wp_{N_k} (d\nu) =1$ for all sufficiently large k.

Upper bound: Suppose that there exists a subsequence $\{N_k\}_{k \geq 1}$ such that $\log \lambda_2^N > $ $N_k({-}\Lambda + \varepsilon)$ for some $\varepsilon>0$ . Let $\nu_0$ , $\delta_0<\varepsilon/2$ , $\rho$ , $N_0$ be as in Theorem 3.2. Then, with $f(\nu) =1_{\big\{\big[K_{i_0}\big]_{\rho/2}\big\}}(\nu)$ and $T = \exp\{N(\Lambda-\delta_0/2)\}$ , (3.8) implies that

\begin{align*} \mathbb{E}_\nu f(\mu_N(T)) = \mathbb{P}_\nu\big(\mu_N(T) \in \big[K_{i_0}\big]_{\rho/2}\big) \leq \exp\{{-}N\beta\} \end{align*}

for all $N \geq N_0$ and $\nu \in [\nu_0]_{\rho/2 } \cap M_1^N(\mathcal{Z})$ . Also, by Theorem 3.1, for any $\delta >0$ , there exists $N_1 \geq N_0$ such that for all $N \geq N_1$ , we have

\begin{align*} \langle\, f, \wp_N \rangle = \wp_N\big(\big[K_{i_0}\big]_{\rho/2}\big) \geq \exp\{{-}N\delta\}. \end{align*}

This is possible since $\inf_{\xi \in \big[K_{i_0}\big]_{\rho/2}}s(\xi) =0$ . Therefore, for all $N \geq N_1$ ,

\begin{align*} \int_{M_1(\mathcal{Z})} |\mathbb{E}_\nu(\,f(\mu_N(T))) & - \langle\, f, \wp_N \rangle |^2 \wp_N(d\nu) \\ &\geq \int_{[\nu_0]_{\rho/2}} \left|\mathbb{E}_\nu(\,f(\mu_N(T))) - \langle\, f, \wp_N \rangle \right|^2 \wp_N(d\nu) \\ & \geq \wp_N([\nu_0]_{\rho/2}) (\!\exp\{{-}N\beta\} - \exp\{{-}N\delta\}) \\ &\geq \wp_N([\nu_0]_{\rho/2}) \exp\{{-}N\delta_1\}, \text{ for some } \delta_1 > 0 \\ &\geq \exp\{{-}N\delta_2\}, \text{ for some } \delta_2 > 0, \end{align*}

where the last inequality follows by Theorem 3.1. On the other hand, for any function f with $\int\! |f|^2 d\wp_N \leq 1$ , we have

\begin{align*} \int_{M_1(\mathcal{Z})} |\mathbb{E}_\nu(\,f(\mu_N(T))) & - \langle\, f, \wp_N \rangle |^2 \wp_N(d\nu) \\ & = \int_{M_1(\mathcal{Z})} \sum_{k\geq 2} e^{-2\lambda_k^NT} \big( f, u_k^N\big)^2 u_k^N(\nu)^2 \wp_N(d\nu) \\ &\leq \exp\!\big\{{-}2\lambda_2^NT\big\} \int_{M_1(\mathcal{Z})} |f|^2 d\wp_N \\ &\leq \exp\!\big\{{-}2\lambda_2^NT\big\}. \end{align*}

Therefore, we have $ \exp\!\big\{{-}2\lambda_2^{N}T\big\} \geq \exp\{{-}N\delta_2\}$ whenever $N \geq N_1$ . By our assumption, we see that

\begin{align*} \exp\{{-}2\exp\!({-}N_k(\Lambda-\varepsilon)) \exp\!(N_k(\Lambda-\delta_0))\} \geq \exp\{{-}N_k\delta_1\} \end{align*}

for sufficiently large k, which is a contradiction since $\delta_0 < \varepsilon/2$ .

Using the above theorem, we see that if $\Lambda>0$ , then as N becomes large, it takes longer for the process $\mu_N$ to be close to its invariant measure. This means in particular that metastable states reduce the rates of convergence of $\mu_N$ to its invariant measure. On the other hand, if there is a unique global attractor of the limiting McKean–Vlasov equation (1.1), then we see that $\Lambda = 0$ , and the rate of convergence of $\mu_N$ to its invariant measure does not suffer from such a slowing-down phenomenon.

Note that the spectral expansion in (4.1) is crucial in the proof of Theorem 1.3, to enable the use of the results on the large-time behaviour of $\mu_N$ established in Section 3 to obtain the asymptotics of $\lambda^N_2$ . The main purpose of Theorem 1.3 is to demonstrate that, in the reversible case, the asymptotics of $\lambda^N_2$ can easily be obtained as an application of the study of the large-time behaviour of $\mu_N$ . Even in the non-reversible case, one can obtain asymptotics of the real part of $\lambda^N_2$ via other approaches; see, for example, [Reference Wentzel47], where the author obtains the asymptotics of the real part of the second-largest eigenvalue of the generator corresponding to a small-noise diffusion process by examining the eigenvalues of a discrete-time chain (with transition probabilities of the form appearing in (3.2)) and transferring them to the operator. Furthermore, the asymptotics of all the eigenvalues of an $l \times l$ transition probability matrix whose (i, j)th entry, for $i \neq j$ , is given by $\exp\!\big\{{-}N\tilde{V}\big(K_i, K_j\big)\big\}$ can also be obtained; the real part of $\big(1-\hat{\lambda}^N_k\big)$ (where $\hat{\lambda}^N_k$ is the kth eigenvalue of the matrix, $2 \leq k \leq l$ ) decays exponentially in N, where the exponent is given by a quantity analogous to $\Lambda$ in (3.6) in which the first minimum is taken over all graphs in G(W) with $|W| = k-1$ and the second minimum is taken over all graphs in G(W) with $|W| = k$ , see Freidlin and Wentzell [Reference Freidlin and Wentzell20, Chapter 6, Theorem 7.3]. However, it is not clear how to transfer these asymptotics to the eigenvalues of $L^N$ using the large-time behaviour of $\mu_N$ , a question that we leave for the future. This question is also related to the behaviour of $\mu_N$ over times of the order of the inverse of these eigenvalues. For reversible Markov chains with a small parameter, such questions have been studied by Miclo [Reference Miclo34, Reference Miclo35]. In particular, it would be interesting to investigate the asymptotics of $\lambda^N_3$ , since the rate of convergence of $\mu_N$ to $\wp_N$ depends on whether $\lambda^N_3$ decays as $\exp\{{-}N\Lambda\}$ or $\lambda^N_3 \gg \exp\{{-}N\Lambda\}$ .

Example 4.1. We provide a simplified Curie–Weiss model of magnetisation for which $L^N$ is reversible with respect to $\wp_N$ for all $N \geq 1$ [Reference Comets13]. Let $\mathcal{Z} = \{{-}1, +1\}$ . The states represent the direction of magnetisation of the particles. For each $N \geq 1$ , consider the following probability measure on $\mathcal{Z}^N$ :

(4.5) \begin{align} \pi_N\big(\mathbf{z}^N\big) = \frac{1}{C_N} \exp\!\left\{N \left(\frac{1}{N}\sum_{n=1}^N z_n\right)^2\right\}, \, \mathbf{z}^N = (z_1,z_2, \ldots, z_N) \in \mathcal{Z}^N, \end{align}

where $C_N$ is a normalisation constant. Given $\xi \in M_1(\mathcal{Z})$ , define the total magnetisation by $\xi_{\text{tot}} = \xi({+}1) - \xi({-}1)$ . Define the transition rates

\begin{align*} \lambda_{z,({-}z)}(\xi) = \exp\{{-}2 z \xi_{\text{tot}}\}, \, z \in \mathcal{Z}, \, \xi \in M_1(\mathcal{Z}). \end{align*}

It is straightforward to verify that the Markov process $\big(X_1^N, \ldots, X_N^N\big)$ that describes the joint evolution of the states of all the particles is reversible with respect to its invariant measure $\pi_N$ in (4.5). That is, for every $\mathbf{z}^N, \tilde{\mathbf{z}}^N \in \mathcal{Z}^N$ that differ on the nth component, we have the following $\big($ recall that $\overline{\mathbf{z}^N} = \frac{1}{N}\sum_{n=1}^N \delta_{z_n}\big)$ :

(4.6) \begin{align} \pi_N\big(\mathbf{z}^N\big) \lambda_{z_n, ({-}z_n)}\big( \overline{\mathbf{z}^N}\big) = \pi_N\big(\tilde{\mathbf{z}}^N\big) \lambda_{({-}z_n), z_n}\big(\overline{\tilde{\mathbf{z}}^N}\big). \end{align}

From the reversibility of $\big(X_1^N, \ldots, X_N^N\big)$ , noting that $\wp_N(\xi)$ is the sum of $\pi_N\big(\mathbf{z}^N\big)$ over all $\mathbf{z}^N$ such that $\overline{\mathbf{z}^N} = \xi$ , it is straightforward to check that $\mu_N$ is reversible. For completeness, we show the reversibility of $\mu_N$ by checking the detailed balance condition. Towards this, we first note that for any $\xi, \tilde{\xi} \in M_1^N(\mathcal{Z})$ such that $\xi(z) = \tilde{\xi}(z)+1/N$ for some $z \in \mathcal{Z}$ (which ensures that $\xi(z) > 0$ , and hence $\tilde{\xi}({-}z) > 0$ ),

(4.7) \begin{align} &\xi(z) \times \big(\text{number of } \mathbf{z}^N \in \mathcal{Z}^N \text{ such that } \overline{\mathbf{z}^N} = \xi\big) \nonumber \\ & \qquad = \tilde{\xi}({-}z) \times \big(\text{number of } \mathbf{z}^N \in \mathcal{Z}^N \text{ such that } \overline{\mathbf{z}^N} = \tilde{\xi}\big). \end{align}

Let $\mathbf{z}^N_\xi \in \mathcal{Z}^N$ $\Big($ resp. $\tilde{\mathbf{z}}^N_{\tilde{\xi}} \in \mathcal{Z}^N\Big)$ be such that $\overline{\mathbf{z}^N} = \xi$ $\big($ resp. $\overline{\tilde{\mathbf{z}}^N} = \tilde{\xi}\big)$ . Noting that $\pi\big(\mathbf{z}^N\big)$ depends only on $\overline{\mathbf{z}^N}$ , for any $z \in \mathcal{Z}$ with $\xi(z) > 0$ , we have

\begin{align*} \wp_N(\xi) N \xi(z) \lambda_{z,({-}z)}(\xi) & = \pi_N\big(\mathbf{z}^N_\xi\big) \big(\text{number of } \mathbf{z}^N \in \mathcal{Z}^N \text{ such that } \overline{\mathbf{z}^N} = \xi\big) N \xi(z) \lambda_{z,({-}z)}(\xi)\\ &= \pi_N\big(\mathbf{z}^N_\xi\big) \big(\text{number of } \mathbf{z}^N \in \mathcal{Z}^N \text{ such that } \overline{\mathbf{z}^N} = \tilde{\xi}\big) N \tilde{\xi}({-}z) \lambda_{z,({-}z)}(\xi) \\ & = \pi_N\big(\mathbf{z}^N_{\tilde{\xi}}\big) \big(\text{number of } \mathbf{z}^N \in \mathcal{Z}^N \text{ such that } \overline{\mathbf{z}^N} = \tilde{\xi}\big) N \tilde{\xi}({-}z) \lambda_{({-}z),z}\big(\tilde{\xi}\big) \\ & = \wp_N\big(\tilde{\xi}\big) N \tilde{\xi}({-}z) \lambda_{({-}z),z}\big(\tilde{\xi}\big), \end{align*}

where we have used (4.7) in the second equality and (4.6) in the third equality. It follows that $\mu_N$ is reversible.

Remark 4.1. Another situation where $\mu_N$ is reversible with respect to $\wp_N$ is in the non-interacting case (i.e., when, for each $(z,z^\prime) \in \mathcal{E}$ , $\lambda_{z,z^\prime}({\cdot})$ is a constant function, which we denote by $\lambda_{z,z^\prime}$ ), where the Markov process on $\mathcal{Z}$ with generator

\begin{align*} f \mapsto \sum_{z^\prime:(z,z^\prime) \in \mathcal{E}} (\,f(z^\prime) - f(z)) \lambda_{z,z^\prime}, \qquad z \in \mathcal{Z}, \end{align*}

is reversible with respect to its invariant measure (i.e., when the Markov process corresponding to a single particle’s evolution on $\mathcal{Z}$ is reversible with respect to its invariant measure). This results in a reversible empirical measure process $\mu_N$ . However, the authors are not aware of a general condition (in terms of the transition rates $\lambda_{z,z^\prime}({\cdot}), (z,z^\prime) \in \mathcal{E}$ ) that characterises reversibility of $\mu_N$ .

5. Convergence to a global minimum via controlled addition of particles

In this section, our goal is to increase the number of particles N over time so as to obtain, with high probability, convergence of the empirical measure process to a global minimum of the rate function s that governs the LDP for the sequence of invariant measure $\{\wp_N\}_{N\geq 1}$ .

Fix $c>0$ . Let $N_0 = \min\{n \in \mathbb{N} \,:\, \exp\{nc\} - 2 \geq 0 \}$ , $t_{N_0} = 0$ , and for each $N > N_0$ , let $t_N = \exp\{Nc\}-2$ . For each $N \geq N_0$ define the generator $L^N_t$ acting on bounded measurable functions f on $M_1(\mathcal{Z})$ by

\begin{align*} L_t^Nf (\xi) \,:\!=\, \sum_{(z,z^\prime) \in \mathcal{E}} N_t \xi(z) \lambda _{z,z^\prime}(\xi) \left[ f\left(\xi+\frac{\delta_{z^\prime}}{N_t}- \frac{\delta_z}{N_t}\right) - f(\xi) \right], \qquad t\in [t_{N}, t_{N+1}),\end{align*}

where $N_t = N$ for $t \in [t_N, t_{N+1})$ . Let $z_0 \in \mathcal{Z}$ be a fixed state and let $\nu \in M_1^{N_0}(\mathcal{Z})$ . We say that a probability measure $\mathbb{P}_{0,\nu}$ on $D([0,\infty),M_1(\mathcal{Z}))$ is a solution to the martingale problem for $\big\{L^N\big\}_{N \geq N_0}$ with initial condition $\nu$ if $\mathbb{P}_{0,\nu}(\bar{\mu}\,:\, \bar{\mu}(0) = \nu)=1$ , for each $N \geq N_0$ , the restriction of $\mathbb{P}_{0,\nu}$ on $D([t_N, t_{N+1}), M_1^{N}(\mathcal{Z}))$ is a solution to the $D([t_N, t_{N+1}), M_1^{N}(\mathcal{Z}))$ -valued martingale problem for $L^N$ , and

\begin{align*} \mathbb{P}_{0,\nu}\!\left(\bar{\mu}\,:\, \bar{\mu}(t_{N+1}) = \frac{N}{1+N}\bar{\mu}\big(t_{N+1}^-\big) + \frac{1}{N+1} \delta_{z_0}\right) =1.\end{align*}

Again, by the boundedness assumption on transition rates (A2), for each $\nu \in M_1^{N_0}(\mathcal{Z})$ , there exists a unique probability measure $\mathbb{P}_{0,\nu}$ that solves the martingale problem for $\big\{L^N\big\}_{N \geq N_0}$ with initial condition $\nu$ . Let $\bar{\mu}$ be the process on $D([0,\infty),M_1(\mathcal{Z}))$ whose law is $\mathbb{P}_{0,\nu}$ . To describe the process in words, we start with $N_0$ particles and follow the mean-field interaction described in Section 1, except that at each time instant $t_N$ , $N > N_0$ , we add a new particle whose state is set to $z_0$ . Similarly, for $t > 0$ and $\nu \in \mathcal{M}_1^{\lfloor \log\! (t+2)\rfloor}$ , let $\mathbb{P}_{t,\nu}$ denote the law of the process $(\bar{\mu}(t^\prime), t^\prime \geq t)$ with $\bar{\mu}(t) = \nu$ .

We anticipate that if c is small, then $N_t$ is so large that the fluid limit kicks in too quickly over time and the process $\bar{\mu}$ converges (over time) to a local minimum of s with positive probability depending on the initial condition $\bar{\mu}(0)$ . When c is sufficiently large, we anticipate that there is enough time for exploration and therefore we will converge to a global minimum of s. Recall that the set of global minimisers of s is denoted by $\tilde{L}_0$ . Our interest in this section is in finding a constant $c^*$ such that for all $c > c^*$ and $\nu \in M_1^{N_0}(\mathcal{Z})$ , we have

(5.1) \begin{align} \mathbb{P}_{0,\nu} \big(\bar{\mu}(t) \text{ lies in a neighbourhood of }\tilde{L}_0\big) \to 1 \end{align}

as $t \to \infty$ .

We use the results in the previous sections to identify the constant $c^*$ . Since $N_t \to \infty$ as $t \to \infty$ , for a fixed $T > 0$ and large enough t, the large deviation properties of the process $(\bar{\mu}(s), t \leq s \leq t+T)$ from the limiting dynamics (1.1) starting at an arbitrary $\bar{\mu}(t)$ can be obtained similarly to the LDP of the process $\mu_N$ studied in Theorem 2.1 and Corollary 2.1. Therefore, the results in the previous sections on the large-time behaviour for the process $(\mu_N(t), t\geq 0)$ are also valid for $(\bar{\mu}(t) , t \geq 0)$ when the t is large enough; we make these precise now.

Lemma 5.1. (See Lemma 3.8.) Let $\pi_1^k$ and $\pi_2^k$ be k-cycles and suppose that $\pi_1^k \to \pi_2^k$ and $\tilde{V}\big(\pi_1^k\big)/c < 1$ . Then, given $\varepsilon > 0$ , there exist $\delta>0$ and $\rho>0$ such that for all $\rho_1 < \rho$ , there is $t^* >0$ such that

\begin{align*} \mathbb{P}_{t,\nu}\Big(\bar{\tau}_{\pi^k_1} \leq t + t^{(\tilde{V}\big(\pi_1^k\big) -\delta)/c}, \bar{\mu}\big(\bar{\tau}_{\pi_1^k}\big) \in \gamma_{\pi_2^k}\Big) \geq t^{-\varepsilon/c} \end{align*}

holds uniformly for all $\nu \in \big[\pi_1^k\big]_{\rho_1} \cap M_1^{N_{t}}(\mathcal{Z})$ and $t \geq t^*$ .

Lemma 5.2. (See Lemma 3.9.) Let $\pi^k$ be a k-cycle and suppose that $\tilde{V}\big(\pi^k\big)/c < 1$ . Then, given $\delta > 0$ such that $\big(\tilde{V}\big(\pi^k\big)+\delta\big)/c < 1$ , there exist $\varepsilon > 0$ and $\rho>0$ such that for all $\rho_1 < \rho$ , there is $t^* > 0$ such that

\begin{align*} \mathbb{P}_{t,\nu}\Big(\bar{\tau}_{\pi^k} & < t + t^{\big(\tilde{V}\big(\pi^k\big) -\delta\big)/c}\Big) \leq t^{-\varepsilon/c} \quad and \\ \mathbb{P}_{t,\nu}\Big(\bar{\tau}_{\pi^k}& > t + t^{\big(\tilde{V}\big(\pi^k\big) +\delta\big)/c}\Big) \leq t^{-\varepsilon/c} \end{align*}

hold uniformly for all $\nu \in \big[\pi^k\big]_{\rho_1} \cap M_1^{N_{t}}(\mathcal{Z})$ and $t \geq t^*$ .

Lemma 5.3. (See Lemma 3.10.) Let $\pi^k$ be a k-cycle and suppose that $\hat{V}\big(\pi^k\big)/c < 1$ . Given $\varepsilon>0$ , there exist $\delta \in \big(0, c-\hat{V}\big(\pi^k\big)\big)$ and $\rho>0$ such that for all $\rho_1 \leq \rho$ , there is $t^* >0$ such that

\begin{align*} \mathbb{P}_{t,\nu}\Big(\bar{\tau}_{\pi^k}\leq t + t^{\big(\hat{V}\big(\pi^k\big)+\delta\big)/c}\Big) \leq t^{-(\tilde{V}\big(\pi^k\big) - \hat{V}\big(\pi^k\big)-\varepsilon)/c} \end{align*}

holds uniformly for all $\nu \in \big[\pi^k\big]_{\rho_1} \cap M_1^{N_{t}}(\mathcal{Z})$ and $t \geq t^*$ .

Recall the definition of the sets L and C from Section 3.

Lemma 5.4. (See Lemma 3.3.) Given $\rho_0 > 0$ and $\rho_1 <\rho_0$ and their associated sets L and C, given $v>0$ , there exist $T^* > 0$ and $t^* >0 $ such that

\begin{align*} \mathbb{P}_{t,\nu}\big(\hat{\tau}_L \geq t + T^*\big) \leq t^{-v/c} \end{align*}

holds uniformly for all $\nu \in C \cap M_1^{N_{t}}(\mathcal{Z})$ and $t \geq t^*$ .

To answer the question on the convergence of $\bar{\mu}$ to a global minimum of s, we define the following quantities, analogously to what is done in Hwang and Sheu [Reference Hwang and Sheu25]. Let m be such that $L_{m+1}$ is a singleton $\big($ denote it by $\big\{\pi^{m+1}\big\}\big)$ . Define

\begin{align*} A_m \,:\!=\, \big\{\pi^m \in L_m\,:\,\tilde{V}(\pi^m) = \hat{V}\big(\pi^{m+1}\big)\big\}.\end{align*}

Inductively define, for each $\pi^{k+1} \in L_{k+1}$ ,

\begin{align*} A_{k}\big(\pi^{k+1}\big) \,:\!=\, \big\{\pi^k \in \pi^{k+1} \,:\, \tilde{V}\big(\pi^k\big) = \hat{V}\big(\pi^{k+1}\big)\big\},\end{align*}

and for each $k \geq 1$ , define

\begin{align*} A_k \,:\!=\, \bigcup_{\pi^{k+1} \in A_{k+1}} A_k\big(\pi^{k+1}\big).\end{align*}

Also, for each $\pi^k \in L_k$ , define

\begin{align*} c_{k-1}\big(\pi^k\big) \,:\!=\, \left\{\begin{array}{l@{\quad}l} 0 \qquad \text{ if } \big\{ \pi^{k-1} \in \pi^k \,:\, \pi^{k-1} \notin A_{k-1}\big(\pi^k\big) \big\} = \emptyset, \\[4pt] \max\!\big\{\tilde{V}\big(\pi^{k-1}\big)\,:\,\pi^{k-1} \notin A_{k-1}\big(\pi^k\big), \pi^{k-1} \in \pi^k\big\} \qquad \text{ otherwise}, \end{array} \right.\end{align*}

and for each $k \geq 1$ , define

\begin{align*} c_{k-1} \,:\!=\, \max\!\big\{c_{k-1}\big(\pi^k\big), \,:\,\pi^k \in A_k\big\}.\end{align*}

Finally, define

\begin{align*} c^* \,:\!=\, \max\{c_k, 0 \leq k \leq m\}.\end{align*}

Figure 1 illustrates these definitions. Similarly to [Reference Hwang and Sheu25, Lemma A.11, Appendix], we can show that $A_0 = \tilde{L}_0$ , the set of minimisers of the rate function s that governs the LDP for the invariant measure $\{\wp_N\}_{N\geq 1}$ . We now prove Theorem 1.4 on the convergence of $\bar{\mu}$ to the set of global minimisers.