2. Statement of results

2.1. The model

Consider the sequence $\{X_k^N\}_{k\in \mathbb{N}_0}$ of $\Delta_N$ -valued random variables defined as

(1) \begin{equation}\begin{aligned}X_{k+1}^N &= X_k^N + \frac{1}{N} \eta^N_{k+1}\left(X_k^N\right), \qquad k \in \mathbb{N}_0,\\X_0^N &= x^N,\end{aligned}\end{equation}

where for each $x \in \Delta_N$ , $\eta^N_k(x)$ is a $\mathbb{Z}^d$ -valued random variable with distribution $\theta^N(\!\cdot\!| x)$ such that $\mbox{supp}(\theta^N(\cdot | x)) \subset \prod_{i=1}^d [-Nx_i, \infty)$ .

We will denote by $\mathbb{P}^N_{\nu}$ the probability measure under which the Markov chain $\big\{X_k^N\big\}$ has the initial distribution $\nu$ , namely $\mathbb{P}^N_{\nu}\big(X_0^N \in A\big) = \nu(A)$ . If $\nu = \delta_x$ , we write $\mathbb{P}^N_{\nu}$ as simply $\mathbb{P}^N_x$ .

Definition 1. A probability measure $\mu_N$ on $\Delta_N^o$ is said to be a quasi-stationary distribution (QSD) for the Markov chain $\{X_k^N\}$ if for every $n \in \mathbb{N}$

\begin{equation*}\mathbb{P}_{\mu_N}\big[X_k^N =j \big| X_k^N \in \Delta_N^o\big] = \mu_N(j), \; \mbox{ for all } j \in \Delta_N^o \mbox{ and } k \in \mathbb{N}.\end{equation*}

2.2. Definitions and assumptions

Consider the continuous-time process $\hat X^N$ obtained from a linear interpolation of $X^N$ , given as

(2) \begin{align}\hat{X}^N(t) = X^N_n + \big[X^N_{n+1} - X^N_{n}\big](Nt - n), \qquad t \in [n/N, (n+1)/N], \; n \in \mathbb{N}_0,\end{align}

The following assumption on the law-of-large-numbers behavior of $\hat X^N$ will play a central role in our study of asymptotic properties of QSD of $X^N$ .

Assumption 1. There is a Lipschitz function $G: \Delta \to \mathbb{R}^d$ such that for any sequence $x_N \to x$ with $x_N \in \Delta_N$ for every $N \in \mathbb{N}$ ,

(3) \begin{equation}\mathbb{P}_{x_N}\left(\sup_{0\le t \le T}\|\hat X^N(t) - \varphi_t(x)\| > \varepsilon\right) \to 0, \quad \mbox{ as } N \to \infty, \mbox{ for every } T\in [0,\infty) \mbox{ and } \varepsilon >0, \end{equation}

where $\{\varphi_t(x)\}_{t\ge 0}$ is the solution of the ODE

(4) \begin{equation}\dot{\varphi}(t) = G(\varphi(t)), \quad \varphi(0) = x.\end{equation}

We now introduce the notion of absorption-preserving pseudo-orbits for the flow associated with the ODE (4). Discrete-time analogues of these were introduced in [Reference Faure and Schreiber4].

Definition 2. Given $\delta, T > 0$ , consider a family of points $\xi = (\xi_0 = x,\dots, \xi_n = y) \in \Delta^{n+1}$ and a collection of times $T \leq T_1, \dots, T_{n-1}$ such that

• $\|\xi_0 - \xi_1\| < \delta$ ;

• whenever $\xi_i \in \partial \Delta$ , $\xi_{i+1} \in \partial \Delta$ ;

• $\|\xi_{i+1} - \varphi_{T_i}(\xi_i)\| < \delta$ for $1 \leq i \leq n-1$ .

The piecewise continuous path

\begin{equation*}\left(x, \{\varphi_t(\xi_1) : t \in [0,T_1]\}, \{\varphi_t(\xi_2) : t \in[0,T_2]\}, \dots, \{\varphi_t(\xi_{n-1}) : t \in [0,T_{n-1}]\}, y\right)\end{equation*}

is said to be a $(\delta,T)$ absorption-preserving pseudo-orbit (ap–pseudo-orbit) from x to y. Occasionally, we will also refer to the sequence $\{\xi_i\}_{i=0}^n$ as a $(\delta, T)$ ap–pseudo-orbit from x to y.

Definition 3. For two points $x,y \in \Delta$ , we say that $x <_{\small \rm{AP}} y$ if for all $\delta, T > 0$ there is a $(\delta,T)$ ap–pseudo-orbit from x to y. If $x <_{\small \rm{AP}} y$ and $y <_{\small \rm{AP}} x$ , we write $x \sim_{\small \rm{AP}} y$ . If $x \sim_{\small \rm{AP}} x$ , then x is said to be an ap–chain recurrent point. Let $\mathcal{R}_{\small \rm{AP}}$ denote the set of ap–chain recurrent points, and note that $\sim_{\small \rm{AP}}$ is an equivalence relation on $\mathcal{R}_{\small \rm{AP}}$ . For $x \in \mathcal{R}_{\small \rm{AP}}$ , the equivalence class $[x]_{\small \rm{AP}}$ of all $y \in \mathcal{R}_{{\small \rm{AP}}}$ such that $y \sim_{{\small \rm{AP}}} x$ is said to be an ap–basic class. Such a class is called maximal if, whenever for some $y\in \mathcal{R}_{{\small \rm{AP}}}$ , $x <_{\small \rm{AP}} y$ , we have $y \in [x]_{{\small \rm{AP}}}$ . A maximal ap–basic class is called an ap–quasiattractor. We let $\mathcal{R}_{{\small \rm{AP}}}^* \doteq \mathcal{R}_{{\small \rm{AP}}}\cap \Delta^o$ .

The following will be our main assumptions on the dynamical system $\{\varphi_t(x)\}$ . Parts (c) and (d) say that the velocity fields decay as the boundaries are approached but not at too fast a rate. Part (e) is our main stability assumption on the dynamics. Parts (a) and (b) are requirements on recurrence classes for the flow that are satisfied quite broadly.

Assumption 2.

1. (a) There are a finite number of ap–basic classes contained in $\Delta^o$ , which are denoted by $\{K_i\}_{i=1}^{v}$ . Each $K_i$ is a closed set. Additionally, for some $l < v$ , $\{K_i\}_{i=1}^l$ are ap–quasiattractors and $\{K_i\}_{i=l+1}^v$ are non-ap–quasiattractors.

2. (b) For each $i = 1, \ldots, v$ there is an $x_i\in K_i$ such that, for every $T>0$ , $\{\varphi_t(x_i): t\ge T\}$ is dense in $K_i$ .

3. (c) There exist $\varepsilon>0$ and $m>0$ such that for every $i= 1, \ldots, d$ , $G_i(x)> mx_i$ whenever $x \in \Delta^o$ and $x_i \le \varepsilon$ .

4. (d) For every $i=1, \ldots, d$ , as $\delta \to 0$ , $\sup\limits_{x \in \Delta: x_i \le \delta}G_i(x) \to 0$ .

5. (e) For some $\kappa \in (0,\infty)$ and $M \in (1,\infty)$ , $\langle x, G(x)\rangle \le -\kappa \|x\|^2$ for all $x \in \Delta$ with $\|x\|\ge M$ .

We will need certain assumptions on the moment generating functions of $\theta^N(\cdot | x)$ .

Assumption 3.

1. (a) For every $N\in \mathbb{N}$ , $\zeta \in \mathbb{R}^d$ , and $x \in \Delta_N^o$

   \begin{equation*}H^N(x,\zeta) \doteq \log \int_{\mathbb{R}^d} \exp\{\langle \zeta, y\rangle\} \theta^N(dy | x) <\infty.\end{equation*}

2. (b) There exists a stochastic kernel $\theta(dy | x)$ on $\mathbb{R}^d$ given $\Delta^o$ such that the following hold:

3. (i) For every $x\in \Delta^o$ , the convex hull of $\mbox{supp}(\theta(\cdot | x))$ equals $\mathbb{R}^d$ .

4. (ii) The map $x\mapsto \theta(\cdot | x)$ is a continuous map from $\Delta^o$ to $\mathcal{P}(\mathbb{R}^d)$ .

5. (iii) For every $\zeta \in \mathbb{R}^d$ and $K \in \mathcal{K}$ , $\sup_{x\in K} H(x,\zeta) <\infty$ , where

   \begin{equation*}H(x,\zeta) \doteq \log \int_{\mathbb{R}^d} \exp\{\langle \zeta, y\rangle\} \theta(dy | x).\end{equation*}
   Furthermore, as $N\to \infty$ ,
   \begin{equation*}\sup_{x\in K\cap \Delta_N} \big|H^N(x,\zeta) - H(x,\zeta)\big| \to 0.\end{equation*}

We introduce one final assumption to provide a lower bound on the probability that $X^N$ is absorbed when its initial state is sufficiently close to $\partial \Delta$ .

Assumption 4.

1. i. For each $N\in \mathbb{N}$ and $x,y \in \Delta_N^o$ , there is a $k \in \mathbb{N}$ such that $\mathbb{P}_y^N(X^N_k=x)>0$ .

2. ii. For every $\gamma \in (0,\infty)$ and $T\in \mathbb{N}$ , there is an open neighborhood $U_{\gamma}$ of $\partial \Delta$ in $\Delta$ such that

   \begin{equation*}\liminf_{N\to \infty} \inf_{x \in U_{\gamma}\cap \Delta_N} \frac{1}{N} \log \mathbb{P}_x\big(\hat X^{N}(T) \in \partial \Delta\big) \ge -\gamma.\end{equation*}

We now present our main results.

2.3. Main results

It is easy to see that under Assumption 2, for all $x\in \Delta$ and $t\ge 0$ , $\varphi_t(x) \in \Delta$ . In particular, $\varphi_t$ is a measurable map from $\Delta$ to itself for every $t\ge 0$ . We recall the definition of an invariant measure for the flow $\{\varphi_t\}$ .

Definition 4. A probability measure $\mu$ on $\Delta$ is $\{\varphi_t\}$ -invariant if $\mu(\varphi_t^{-1}(A)) = \mu(A)$ for every measurable $A \subseteq \Delta$ and $t > 0$ .

Theorem 1. Suppose that for every $N\in \mathbb{N}$ , there exists a QSD $\mu_N$ for $\{X^N_n\}_{n\in \mathbb{N}_0}$ , and the sequence $\{\mu_N\}$ is relatively compact as a sequence of probability measures on $\Delta^o$ . Suppose that Assumptions 1, 2, 3, and 4 are satisfied. Then any weak limit point $\mu$ of this sequence is $\{\varphi_t\}$ -invariant and is supported on $\cup_{i=1}^l K_i$ . Moreover, letting

(5) \begin{equation}\lambda_N \doteq \left[\mathbb{P}_{\mu_N}\left(X_1^N \in \Delta^o\right)\right]^N,\end{equation}

there is a $c>0$ and $N_0 \in \mathbb{N}$ such that $\lambda_N \ge 1 - e^{-cN}$ for all $N \ge N_0$ .

We now introduce a basic family of Markov chains, which we refer to as the binomial-Poisson models, for which Theorem 1 can be applied.

Consider a population with d types of particles evolving in discrete time, in which, at each time step, any given particle dies with probability $1/N$ , and given that the population size at previous time step was $Nx = (Nx_i)_{i=1}^d$ , the number of particles of type i that are produced at the next time step follows a Poisson distribution with mean $F_i(x)$ distribution for some $F:\Delta \to \mathbb{R}_+^d$ . Denote the total number of particles of type i at time k by $NX^{N,i}_k$ . The evolution of $X^N_k = (X^{N,1}_k, \ldots, X^{N,d}_k)$ is then given by (1), where, for each N, $\theta^N(dy|x) \equiv \theta^{N,*}(dy|x)$ is the distribution of $U-V$ , where $U = (U_i)_{i=1}^d$ , $V = (V_i)_{i=1}^d$ , $\{U_i, V_j, i,j=1, \ldots, d\}$ are mutually independent, $U_i \sim \mbox{Poi}(F_i(x))$ (namely, a Poisson random variable with mean $F_i(x)$ ), and $V_i \sim \mbox{Bin}(Nx_i, \frac{1}{N})$ (namely a binomial random variable with $Nx_i$ trials and probability of success $1/N$ ).

Define

(6) \begin{equation}\tau^N_{\partial} \doteq \inf\big\{k \in \mathbb{N}_0: X_k^N \in \partial \Delta_N\big\}.\end{equation}

For a bounded and measurable $f : \Delta_N \rightarrow \mathbb{R}$ ,

(7) \begin{equation}P_n^N f (x) \doteq \mathbb{E}_x\left[ f\left({X}^N_n\right);\; \tau_{\partial}^N > n\right].\end{equation}

Theorem 2. Suppose that, for each N, $X^N$ is given by (1) with $\theta^N \equiv \theta^{N,*}$ . Further suppose that F is a bounded Lipschitz map and Parts (a)–(d) of Assumption 2.2 are satisfied with $G(x)=F(x)-x$ . Then there is a $\mu_N \in \mathcal{P}(\Delta^o_N)$ such that for every $N \in \mathbb{N}$ and $x_N \in \Delta^o_N$ ,

\begin{equation*} \frac{\delta_{x_N} P_n^N}{\delta_{x_N} P_n^N(1_{\Delta_N^o})}\end{equation*}

converges to $\mu_N$ in the total variation distance as $n \rightarrow \infty$ . The measure $\mu_N$ is a QSD for $\big\{X^N\big\}$ . The sequence $\{\mu_N\}_{N\in \mathbb{N}}$ is relatively compact as a sequence of probability measures on $\Delta$ , and any weak limit point $\mu$ of this sequence is $\{\varphi_t\}$ -invariant and is supported by $\cup_{i=1}^l K_i$ . Finally, letting $\lambda_N \doteq [\mathbb{P}_{\mu_N}(X_1^N \in \Delta^o)]^N$ , there is a $c>0$ and $N_0 \in \mathbb{N}$ such that $\lambda_N \ge 1 - e^{-cN}$ for all $N \ge N_0$ .

Theorem 1 is proved in Section 7 while Theorem 2 is established in Section 8.

3. Absorption-preserving pseudo-orbits

In this section we present some basic facts on absorption-preserving pseudo-orbits that will be used to prove Theorem 1. Throughout the section we will take Assumptions 1 and 2 to hold.

The proofs of many of these results are similar to those found in [Reference Faure and Schreiber4] for discrete-time flows, but we provide the details for completeness. Recall that the solution of the ODE (4) with initial value $\varphi(0)=x$ is denoted by $\{\varphi_t(x)\}_{t\ge 0}$ . The following lemma is a consequence of the stability condition in Assumption 2(e).

Proof. For fixed $x \in \Delta$ , $\|\varphi_t(x)\|^2$ solves the ODE

\begin{equation*}\frac{d}{dt} \|\varphi_t(x)\|^2 = 2 \langle G(\varphi_t(x)), \varphi_t(x)\rangle.\end{equation*}

From Assumption 2(e), when $\|x\| \ge M$

\begin{align*}2 \langle G(x), x\rangle\le - 2\kappa\|x\|^2.\end{align*}

This implies the following two facts:

1. (a) If for any $R\ge M$ , $x \in B_{R}\doteq \{z: \|z\| \le R\}$ , then $\varphi_t(x) \in B_{R}$ for every $t\ge 0$ .

2. (b) Given $T>0$ , define $\delta_0= \delta_0(T) \doteq \frac{\kappa T}{2}\wedge 1$ . Then for any $\delta \le \delta_0$ , and any $R\ge M$ , whenever $x \in B_{R+\delta}$ , we have that $\varphi_t(x) \in B_R$ for all $t \ge T$ .

Now fix $T>0$ and a compact $A \subset \mathbb{R}_+^d$ . Without loss of generality assume that there is an $R\ge M$ such that $A \subset B_R$ . Let $\delta_0 = \delta_0(T)$ be as defined above and consider a $(\delta_0,T)$ ap–pseudo-orbit $\{\xi_i\}_{i=0}^{n+1}$ with $\xi_0 \in A$ . Then the above two facts imply that $\xi_i \in B_{R+1}$ for all $i= 0, 1, \ldots, n+1$ . The result follows on taking $A_1 = B_{R+1}$ .

As a consequence of Lemma 1 we get the following result on the boundedness of ap–basic classes.

Proof. Fix $x \in \mathcal{R}_{\small \rm{AP}}$ and $y \in [x]_{\small \rm{AP}}$ . Let $T> 0$ and $A= \{x\}$ . From Lemma 1, there is a $\delta_0>0$ and a compact $A_1$ in $\Delta$ such that for each $\delta \leq \delta_0$ , any $(\delta,T)$ ap–pseudo-orbit starting at x is contained in $A_1$ . Since $y \in [x]_{\small \rm{AP}}$ , there must exist a $(\delta, T)$ ap–pseudo-orbit from x to y, which means that y must lie in $A_1$ . The result follows.

For $x \in \Delta^o$ , we denote the forward orbit of $\varphi$ by $\gamma^+(x) \doteq \{ \varphi_t(x) | t \geq 0\}.$ From Assumption 2(b) and arguments as in Lemma 1 the following result is immediate.

The proof of the next lemma follows from the observation (a) in the proof of Lemma 1.

Proof. Let $\varepsilon_1 \doteq \mbox{dist}(A, \partial \Delta)$ and let $\varepsilon$ and m be as in Assumption 2(c). Let $\varepsilon_0 \doteq (\varepsilon\wedge \varepsilon_1)/4$ . Note that for any $x \in \Delta$ and $i=1, \ldots, d$ ,

(8) \begin{equation} \frac{d}{dt}([\varphi_t(x)]_i)^2 = 2[\varphi_t(x)]_i G_i(\varphi_t(x)) > 2m([\varphi_t(x)]_i)^2 \quad \mbox{ whenever } [\varphi_t(x)]_i\le \varepsilon. \end{equation}

Since $m>0$ , we can choose a $T>0$ such that for any $x\in \Delta$ and $i=1, \ldots , d$ with $x_i\ge \varepsilon_0$ , we have $[\varphi_t(x)]_i > 3\varepsilon_0$ for all $t\ge T$ . Fix $\delta \in (0, \varepsilon_0)$ . Consider a $(\delta, T)$ ap–pseudo-orbit $\{\xi_k\}_{k=0}^n$ with $\xi_0\in A$ and associated time instants $T \leq T_1, \dots, T_{n-1}$ . Clearly $\xi_0 \not\in E_{2\varepsilon_0}$ , and by (8), $\varphi_t(\xi_0) \not\in E_{2\varepsilon_0}$ for all $t \in [0, T_1]$ . Also, by our choice of T, $\varphi_{T_1}(\xi_0) \not \in E_{3\varepsilon_0}$ and consequently $\xi_1 \not \in E_{2\varepsilon_0}$ . A recursive argument now shows that the pseudo-orbit has no intersection with $E_{\varepsilon_0}$ . The result follows.

We now recall a definition from the theory of dynamical systems.

Definition 5. The $\omega$ -limit set of $B \subset \Delta$ is

\begin{equation*}\omega(B) \stackrel{\cdot}{=} \left\{ x \in \Delta : \text{there is a sequence }t_n \uparrow \infty\text{ and a sequence }x_n \in B \text{ such that } \varphi_{t_n}(x_n) \rightarrow x\right\},\end{equation*}

so for $x \in \Delta$ ,

\begin{equation*}\omega(x) \doteq \left\{ y \in \Delta :\text{ there is a sequence }t_n \uparrow \infty\text{ such that }\varphi_{t_n}(x) \rightarrow y\right\}.\end{equation*}

The result below follows from classical arguments and on observing that under Assumption 2(b), if $x \in \Delta^o$ , then $\omega(x)\subset \Delta^o$ . For a proof of the lemma in the discrete-time setting see [Reference Faure and Schreiber4]. The proof for the continuous-time setting considered here is similar, so we omit details.

The following lemma gives a useful property of an ap–quasiattractor.

Proof. Suppose that $x <_{\small \rm{AP}} z$ . In order to show that $z \in [x]_{\small \rm{AP}}$ , it is enough to show that $z <_{\small \rm{AP}} x$ . Note that $\omega\left(z\right)$ is nonempty. Let $z' \in \omega\left(z\right)$ . From Lemma 6, $z' \in \mathcal{R}_{\small \rm{AP}}$ . We now show that $z <_{\small \rm{AP}} z'$ . Since $z' \in \omega\left(z\right)$ , there is a sequence $T_i \uparrow \infty$ such that $\varphi_{T_i}\left(z\right) \rightarrow z'$ . Fix $\delta, T > 0$ . Then we can find $T' > T$ such that $\| \varphi_{T'}\left(z\right) - z' \| < \delta$ . This shows that (z, z, z ^′, z ^′) is a $(\delta,T)$ ap–pseudo-orbit from z to z ^′. Since $\delta, T>0$ are arbitrary, we have $z <_{\small \rm{AP}} z'$ . Combining this with $x <_{\small \rm{AP}} z$ , we now see that $x <_{\small \rm{AP}} z'$ . Since $z' \in \mathcal{R}_{\small \rm{AP}}$ and $[x]_{\small \rm{AP}}$ is maximal, we must have $z' \in [x]_{\small \rm{AP}}$ , and therefore $z <_{\small \rm{AP}} x$ . This completes the proof of the lemma.

The following lemma provides an important invariance property of ap–classes under the flow $\{\varphi_t\}$ .

Proof. Let $y \in [x]_{\small \rm{AP}}$ . To begin, fix $t, \delta, T > 0$ , and let $T' > T + t$ . We can find some $\delta_0 \doteq \delta_0(y) < \delta$ such that if $\| y - x_0\| < \delta_0$ , then $\| \varphi_t(y) - \varphi_t(x_0)\| < \delta$ . Since $y \in \mathcal{R}_{\small \rm{AP}}$ , there is a $(\delta_0, T')$ ap–pseudo-orbit from y to y, which we denote by $\xi = (y, \xi_1, \dots, \xi_{n-1}, y)$ , with corresponding time instants $(T_1, \dots,T_{n-1})$ . Then $\tilde{\xi} \doteq ( \varphi_t(y), \varphi_t(\xi_1), \xi_2, \dots, \xi_n = y)$ is a $(\delta, T)$ ap–pseudo-orbit from $\varphi_t(y)$ to y with corresponding time instants $(T_1 - t, T_2, \dots,T_{n-1})$ , since

\begin{equation*}\| \varphi_t(x) - \varphi_t(\xi_1)\| < \delta,\quad \mbox{ and }\| \varphi_{T_1-t}(\varphi_t(\xi_1)) - \xi_2\| = \| \varphi_{T_1}(\xi_1) - \xi_2\| < \delta.\end{equation*}

Thus $\varphi_t(y) <_{\small \rm{AP}} y$ .

Next, define $\tilde{\xi} \doteq (y,\xi_1,\dots,\xi_{n-1}, \varphi_t(y))$ , and note that $\tilde{\xi}$ is a $(\delta,T)$ ap–pseudo-orbit from y to $\varphi_t(y)$ with time instants $(T_1,\dots,T_{n-2}, T_{n-1} + t)$ , since

\begin{equation*}\| \varphi_{T_{n-1}}(\xi_{n-1}) - y\| < \delta_0,\end{equation*}

which ensures that

\begin{equation*}\| \varphi_{T_{n-1}+t}(\xi_{n-1}) - \varphi_t(y)\| = \| \varphi_t(\varphi_{T_{n-1}}(\xi_{n-1})) - \varphi_t(y)\| < \delta.\end{equation*}

We have shown that $\varphi_t(y) \sim_{\small \rm{AP}} y$ , and so $\varphi_t(y) \in [y]_{\small \rm{AP}} = [x]_{\small \rm{AP}}$ . Since $y \in [x]_{\small \rm{AP}}$ is arbitrary, $\varphi_t([x]_{\small \rm{AP}}) \subset [x]_{\small \rm{AP}}$ . This proves the first part of the lemma.

For the second part, suppose now that $[x]_{\small \rm{AP}} \subset \Delta^o$ . In order to see that $[x]_{\small \rm{AP}} \subset \varphi_t([x]_{\small \rm{AP}})$ for each $t \geq 0$ , let $y \in [x]_{\small \rm{AP}}$ and fix $t > 0$ . We need to show that there is some $z \in [x]_{\small \rm{AP}}$ such that $\varphi_t(z) = y$ . Fix a sequence $(\delta_k, T_k)$ such that $\delta_k \downarrow 0$ and $T^k \uparrow \infty$ . Since $y \in \mathcal{R}_{\small \rm{AP}}$ , we can find a sequence of $(\delta_k,T^k)$ ap–pseudo-orbits with corresponding time instants $\{T_i^k\}_{i=0}^{n(k)-1}$ from y to y, which we denote by $\xi^k = (\xi^k_0,\dots,\xi^k_{n(k)})$ . We assume without loss of generality that $T^k > t$ for all k and let $\tilde{T}^k \doteq T^k_{n(k) -1} - t$ . From Lemma 1 there is a compact $\tilde K$ in $\Delta$ such that for all sufficiently large k, $\xi^k_i \in \tilde{K}$ for all $i \in \{0,\dots,n(k)\}$ . From Lemma 4 we then have that, for all such k, $\varphi_{\tilde{T}^k}(\xi^k_{n(k)-1})$ lies in some compact set $\tilde{K}'$ . Thus (passing to a subsequence) we may assume that $\varphi_{\tilde{T}^k}(\xi^k_{n(k)-1}) \rightarrow z \in \tilde{K}'$ . Since

\begin{equation*}\varphi_{t}\left( \varphi_{\tilde{T}^k}\left(\xi^k_{n(k)-1}\right)\right) = \varphi_{T^k_{n(k)-1}}\left(\xi^k_{n(k)-1}\right) \rightarrow y,\end{equation*}

the continuity of $\varphi_t$ ensures that $\varphi_t(z) = y.$ Now we show that $z \in [x]_{{\small \rm{AP}}}$ . Fix $\delta, T > 0$ , and let k be large enough so that $\delta_k < \delta$ , $\big\| \varphi_{\tilde{T}^k}\big(\xi^k_{n(k)-1}\big) - z\big\| < \delta_k$ , $T^k > T$ , and $\tilde{T}^k > T$ . Then $\big(\xi^k_0, \dots, \xi^k_{n(k)-1}, z\big)$ is a $(\delta, T)$ ap–pseudo-orbit from y to z with corresponding time instants $\big(T^k_1,\dots,T^k_{n(k)-2}, \tilde{T}^k\big)$ , so $y <_{\small \rm{AP}} z$ . Now, fix $\tilde{t} > \max\{t, T\}$ , and note that

\begin{equation*}\varphi_{\tilde{t}}(z) = \varphi_{\tilde{t} -t }( \varphi_t(z)) = \varphi_{\tilde{t} - t}(y).\end{equation*}

Since $y \in [x]_{\small \rm{AP}}$ , it follows from the positive $\varphi_t$ -invariance of $[x]_{\small \rm{AP}}$ that $\varphi_{\tilde{t}-t}(y) \in [x]_{\small \rm{AP}}$ , so there is a $(\delta,T)$ ap–pseudo-orbit from $\varphi_{\tilde{t}-t}(y)$ to y, which we denote by $(\xi_0,\dots,\xi_n)$ . Denote the corresponding time instants by $T_1,T_2,\dots,T_{n-1}$ . Then $\tilde{\xi} \doteq (z, z, \xi_1, \dots, \xi_n)$ is a $(\delta,T)$ ap–pseudo-orbit from z to y with time instants $(\tilde{t},T_1,\dots,T_{n-1})$ , so $z <_{\small \rm{AP}} y$ and $z \in [x]_{\small \rm{AP}}$ .

We now recall the definition of an attractor for the flow $\{\varphi_t\}$ .

Definition 6. A compact set A is an attractor for the flow $\{\varphi_t\}$ if $\varphi_t(A) =A$ for each $t \geq 0$ and there is some neighborhood U of A such that

\begin{equation*}\lim\limits_{t\rightarrow\infty}\sup\limits_{x\in U}\mbox{dist}(\varphi_t(x),A) = 0.\end{equation*}

The neighborhood U is referred to as a fundamental neighborhood for the attractor A.

The proof of Corollary 1 follows from the proof of [Reference Kifer12, Proposition 4.2].

Proof. Recall that $\mathcal{R}_{{\small \rm{AP}}}^*$ denotes the collection of all ap–chain recurrent points in $\Delta^o$ . Note that, from Assumption 2(a) and Lemma 2, for each $z \in \mathcal{R}_{{\small \rm{AP}}}^*$ , $[z]_{{\small \rm{AP}}}$ is a compact set. Choose $\delta>0$ such that $\mathcal{N}^{\delta}([x]_{{\small \rm{AP}}})$ is an isolating neighborhood of $[x]_{{\small \rm{AP}}}$ with closure contained in $\Delta^o$ . Then, from Lemma 4 and Assumption 2(b), there is a compact $K_0 \subset \Delta^o$ such that for all $z\in \mathcal{N}^{\delta}([x]_{{\small \rm{AP}}})$ , $\varphi_t(z) \in K_0$ for all $t\ge 0$ . Let

\begin{equation*}\varepsilon^* \doteq \inf\limits_{y \in \mathcal{R}_{\small \rm{AP}}\setminus [x]_{\small \rm{AP}}}\mbox{dist}([y]_{\small \rm{AP}},[x]_{\small \rm{AP}}).\end{equation*}

For $\varepsilon \le \varepsilon^*$ , let

\begin{equation*}K^{\varepsilon} \doteq K_0 \setminus \left( \bigcup\limits_{y\in\mathcal{R}^*_{\small \rm{AP}} \setminus [x]_{\small \rm{AP}}}\mathcal{N}^{\varepsilon}([y]_{\small \rm{AP}})\right).\end{equation*}

We claim that there exist $\varepsilon \le \varepsilon^*$ and $\delta_0 \le \delta$ such that

(9) \begin{equation}\mbox{ for all } z \in \mathcal{N}^{\delta_0}([x]_{{\small \rm{AP}}}), \qquad \varphi_t(z) \in K^{\varepsilon} \mbox{ for all } t \ge 0.\end{equation}

We argue via contradiction. Suppose the claim is false; then, since there are finitely many ap–basic classes in $\mathcal{R}_{{\small \rm{AP}}}^*$ , there exist $\delta_n\downarrow 0$ , $\varepsilon_n \downarrow 0$ , $z_n \in \mathcal{N}^{\delta_n}([x]_{{\small \rm{AP}}})$ , $t_n\ge 0$ , $y \in \mathcal{R}^*_{\small \rm{AP}} \setminus [x]_{\small \rm{AP}}$ , such that $\varphi_{t_n}(z_n) \in \mathcal{N}^{\varepsilon_n}([y]_{{\small \rm{AP}}})$ . Passing to a subsequence we may assume that $z_n \to z$ and $\varphi_{t_n}(z_n) \to u$ . Then $z \in [x]_{\small \rm{AP}}$ and $u \in [y]_{\small \rm{AP}}$ . We consider two cases: (I) along a further subsequence $t_n$ converges to some $t^* <\infty$ ; (II) $t_n\to \infty$ . In Case I, $u= \varphi_{t^*}(z)$ and so by Lemma 8 $u \in [x]_{{\small \rm{AP}}}$ . But this is a contradiction since $y\notin [x]_{{\small \rm{AP}}}$ . In Case II, for every $\delta, T>0$ , there is a $(\delta, T)$ ap–pseudo-orbit from z to u, which means that $z <_{{\small \rm{AP}}} u$ . Since $[x]_{\small \rm{AP}}$ is a quasiattractor, from Lemma 7, $u \in [x]_{\small \rm{AP}}$ , which is once more a contradiction to the fact that $y\notin [x]_{{\small \rm{AP}}}$ . Thus we have the claim. Now fix $\delta_0 \le \delta^*$ and $\varepsilon \le \varepsilon^*$ so that (9) holds.

We now argue that

(10) \begin{equation} \mbox{ for some } \delta_1 \in (0, \delta_0), \mbox{ whenever } y \in \mathcal{N}^{\delta_1}([x]_{\small \rm{AP}}), \mbox{ we have } \varphi_t(y) \in \mathcal{N}^{\delta_0}([x]_{\small \rm{AP}}) \mbox{ for all } t \ge 0.\end{equation}

Once more we proceed via contradiction. Suppose the statement is false. Then there exist $\delta_n \downarrow 0$ , $y_n \in \mathcal{N}^{\delta_n}([x]_{\small \rm{AP}})$ , $t_n\ge 0$ such that $\varphi_{t_n}(y_n) \in \left(\mathcal{N}^{\delta_0}([x]_{\small \rm{AP}})\right)^c$ . We can find a subsequence along which $y_n \to y$ and $\varphi_{t_n}(y_n) \to u$ . We must have $y \in [x]_{\small \rm{AP}}$ and $u \in \left(\mathcal{N}^{\delta_0}([x]_{\small \rm{AP}})\right)^c$ . Once again we consider two cases as above. In Case I, $u=\varphi_{t^*}(y) \in [x]_{{\small \rm{AP}}}$ , which contradicts the fact that $u \in \left(\mathcal{N}^{\delta_0}([x]_{\small \rm{AP}})\right)^c$ . In Case II, $y<_{{\small \rm{AP}}} u$ , and so as before, $u \in [x]_{{\small \rm{AP}}}$ . Once more this is a contradiction. Thus we have shown (10). Now fix $\delta_1 \in (0, \delta_0)$ such that (10) holds. Let $U_0 \doteq \mathcal{N}^{\delta_0}([x]_{\small \rm{AP}})$ and $ U_1 \doteq \mathcal{N}^{\delta_1}([x]_{\small \rm{AP}})$ .

We will now show that

(11) \begin{equation}\lim\limits_{t\rightarrow\infty}\sup\limits_{y\in U_1}\mbox{dist}(\varphi_t(y), [x]_{\small \rm{AP}}) = 0.\end{equation}

Together with Lemma 8 we will then have that $[x]_{\small \rm{AP}}$ is an attractor, completing the proof of the result. In order to show (11) we will show that for each open neighborhood O of $[x]_{\small \rm{AP}}$ , $O\subset U_1$ , there is some $t(O) <\infty$ such that $\varphi_t(U_1) \subset O$ for all $t \geq t(O)$ . For any such O, let $O_1 \subset \subset O$ be an open neighborhood of $[x]_{\small \rm{AP}}$ such that for all $y \in O_1$ , $\varphi_t(y) \in O$ for all $t\ge 0$ . Here, for open sets $G_1, G_2$ , we write $G_1 \subset \subset G_2$ if $\bar G_1 \subset G_2$ . Existence of such an $O_1$ is shown in a similar manner as (10). It suffices to show that

\begin{equation*}t(O) \doteq \inf\{t : \varphi_t(U_0) \subset O\} < \infty,\end{equation*}

since then for each $t \geq t(O)$ ,

\begin{equation*}\varphi_t(U_1) = \varphi_{t(O)} ( \varphi_{t - t(O)}(U_1)) \subset \varphi_{t(O)}(U_0) \subset O,\end{equation*}

which will complete the proof.

In order to see that $t(O) < \infty$ for each such O, we argue by contradiction. Suppose that there is some O (with the associated $O_1$ ) such that $t(O) = \infty$ . Then we can find sequences $\{z_n\} \subset U_0$ and $T_n \uparrow \infty$ such that $\varphi_{T_n}(z_n) \in O^c$ . From the definition of $O_1$ , this says that $\varphi_t(z_n) \in O^c_1$ for all $0\le t \le T_n$ . Suppose that $z_n \to z$ along a subsequence. Then $\varphi_t(z) \in O_1^c$ for all $t > 0$ . Also, since $z \in \mathcal{N}^{\delta_0}([x]_{\small \rm{AP}})$ , by (9), $\varphi_t(z) \in K^{\varepsilon}$ for all $t \ge 0$ . Thus we have $\omega(z) \subset K^{\varepsilon} \setminus O_1$ . The final statement of Lemma 1 implies that for each $x \in \Delta^o$ , $\omega(x) \neq \emptyset$ and therefore $\omega(z)$ is a nonempty subset of $\mathcal{R}_{{\small \rm{AP}}}^*$ . Thus we have that $(K^{\varepsilon}\setminus O_1)\cap \mathcal{R}_{{\small \rm{AP}}}^*$ is nonempty, which contradicts the definition of $K^{\varepsilon}$ and $O_1$ . Thus we have that $t(O)<\infty$ , and the result follows.

The following lemma shows that suitable ap–pseudo-orbits come arbitrarily close to ap–recurrence classes.

Proof. Consider first Part (a). Fix $\delta>0$ and a compact $A \in \Delta$ , and let $T=1$ . With this choice of A and T, let $\delta_0$ and $A_1$ be as given in Lemma 1. For $x \in \Delta$ , let $T^{\delta}(x) \stackrel{\cdot}{=}\inf\{ t\geq0: \varphi_t(x) \in N^{\delta}(\mathcal{R}_{\small \rm{AP}})\}$ . Since $\omega(x)$ is a nonempty subset of $\mathcal{R}_{\small \rm{AP}}$ , $T^{\delta}(x) <\infty$ for each $x \in \Delta$ . We now claim that $T^{\delta}$ is an upper semicontinuous function on $\Delta$ . For this it suffices to argue that for each $\alpha > 0$ , the level set $L_{\alpha} \stackrel{\cdot}{=} \{x \in \Delta : T^{\delta}(x) \geq \alpha\}$ is closed. Let $\{x_n\} \subset L_{\alpha}$ be a sequence converging to some $x \in \Delta$ , and note that for each $t \ge 0$ , $\lim\limits_{n\rightarrow\infty}\varphi_t(x_n) = \varphi_t(x)$ . For $t < \alpha$ , $\varphi_t(x_n) \in \left(N^{\delta}(\mathcal{R}_{\small \rm{AP}})\right)^c$ , which is closed, so $\varphi_t(x) \in \left(N^{\delta}(\mathcal{R}_{\small \rm{AP}})\right)^c$ . Since this holds for all $t < \alpha$ , we have that $x \in L_{\alpha}$ . This shows that the level sets of $T^{\delta}$ are closed and thus establishes the claim. Since an upper semicontinuous function achieves its supremum over any compact set, $T_1 = \sup_{x\in A_1}T^{\delta}(x)<\infty$ . Let $T_A \doteq T_1 \vee 1$ . Then, from Lemma 1, any $(\delta_0, T_{A})$ ap–pseudo-orbit given by a collection of points $\xi = (\xi_0 = x,\dots, \xi_n = y) \in \Delta^{n+1}$ and a collection of times $T_A \leq T_1, \dots, T_{n-1}$ , with $x\in A$ , must satisfy $\xi_i \in A_1$ for every $i \in \{0, \ldots, n\}$ . Also, by the definition of $T_A$ , we must have that for each $i \in \{1,\dots,n-1\}$ , there is a $t \in [0, T_i]$ such that $\varphi_t(\xi_i) \in N^{\delta}(\mathcal{R}_{\small \rm{AP}})$ . The result in Part (a) follows.

The proof of Part (b) can be completed in a similar manner on observing that from Lemma 3, for every $x \in A$ , the forward orbit $\gamma^+(x)$ is contained in a compact subset of $\Delta^o$ . We omit the details.

The following lemma gives key properties of pseudo-orbits in relation to their visits to neighborhoods of ap–quasiattractors and non-quasiattractors.

Proof. For the first statement in the lemma we will argue via contradiction. By Lemma 5 we can choose $\bar \delta>0, \bar T>0$ , and $\tilde K \in \mathcal{K}$ such that any $(\bar \delta, \bar T)$ ap–pseudo-orbit starting from $N^{\bar\delta}(K_i)$ lies in $\tilde K$ for every $i = 1, \ldots, v$ . Henceforth we only consider $(\delta,T)$ ap–pseudo-orbits with $\delta <\bar \delta$ and $T > \bar T$ . Fix $\theta>0$ and suppose that there is a sequence $\theta>\delta_k \downarrow 0$ and $T_k \uparrow \infty$ , such that for every k there is a $(\delta_k,T_k)$ ap–pseudo-orbit $\xi^k \doteq (\xi^k_0,\dots, \xi^k_{n(k)})$ that satisfies (12) (with $\xi,\delta, n$ replaced by $\xi_k,\delta_k, n(k)$ ), with $i=i'$ . Let $j(k) \in \{1,\dots, n(k) - 1\}$ be such that $\xi^k_{j(k)} \in (N^{\theta}(K_i))^c$ . By passing to a subsequence if necessary, we can find $x, y \in K_i$ and $z \in (N^{\theta}(K_i))^c\cap \tilde K$ such that $\xi^k_0 \rightarrow x$ , $\xi^k_{n(k)} \rightarrow y$ , and $\xi^k_{j(k)} \rightarrow z$ .

In order to see that $x \leq_{\small \rm{AP}} z$ , fix $\delta,T > 0$ and let k be large enough so that

\begin{equation*}\delta_k < \frac{\delta}{2}, \qquad T_k > T, \qquad \big\|x - \xi^k_0\big\| < \frac{\delta}{2}, \qquad \text{and } \big\| \xi^k_{j(k)} - z\big\| < \frac{\delta}{2}.\end{equation*}

Then $\| x - \xi^k_1\| \leq \| x - \xi^k_0\| + \|\xi^k_0 - \xi^k_1\| < \delta$ , and

\begin{equation*}\Big\| \varphi_{T^k_{j(k)-1}}\left(\xi^k_{j(k)-1}\right) - z\Big\| \leq \Big\| \varphi_{T^k_{j(k)-1}}\left(\xi^k_{j(k)-1}\right) - \xi^k_{j(k)}\Big\| + \Big\| \xi^k_{j(k)} - z\Big\| < \delta,\end{equation*}

and so $\tilde{\xi} \doteq \big(x, \xi^k_1, \dots, \xi^k_{j(k)-1}, z\big)$ is a $(\delta,T)$ ap–pseudo-orbit from x to z. Thus $x <_{\small \rm{AP}} z$ . Similarly, $z <_{\small \rm{AP}} y$ , which shows that $z \in K_i$ . However, since $z \in (N^{\theta}(K_i))^c$ , this is a contradiction. This proves (a).

Now consider Part (b). Fix $i,i' \in \{1, \ldots, v\}$ and suppose that for each $\delta, T > 0$ there is some $(\delta,T)$ ap–pseudo-orbit $\xi \doteq (\xi_0,\dots,\xi_n)$ such that $\xi_0 \in N^{\delta}(K_i)$ and $\xi_n \in N^{\delta}(K_{i'})$ . Let $\delta_k \downarrow 0$ and $T_k \uparrow \infty$ and let $\xi^k \doteq \big(\xi^k_0,\dots, \xi^k_{n(k)}\big)$ be a $(\delta_k,T_k)$ ap–pseudo-orbit such that $\xi^k_0 \in N^{\delta_k}(K_i)$ and $\xi^k_n \in N^{\delta_k}\big(K_{i'}\big)$ . Passing to subsequences if necessary, we can find $x \in K_i$ and $y \in K_{i'}$ such that $\xi^k_0 \rightarrow x$ and $\xi^k_{n(k)}\rightarrow y$ . Thus, for any fixed $\delta, T > 0$ , when k is sufficiently large, $\tilde{\xi} \doteq \big(x, \xi^k_1,\dots, \xi^k_{n(k)-1}, y\big)$ is a $(\delta,T)$ ap–pseudo-orbit from $K_i$ to $K_{i'}$ , showing that $K_{i} \leq_{\small \rm{AP}} K_{i'}$ . So if for some i, i ^′, $K_{i} \leq_{\small \rm{AP}} K_{i'}$ does not hold, there must exist $\bar \delta =\delta(i,i')>0$ and $\bar T = T(i,i')<\infty$ such that there is no $(\bar \delta, \bar T)$ ap–pseudo-orbit $\xi \doteq (\xi_0,\dots,\xi_n)$ with the property that $\xi_0 \in N^{\bar\delta}(K_i)$ and $\xi_n \in N^{\delta}(\bar K_{i'})$ . Define $\delta = \min_{(i,i')} \delta(i,i')$ and $T= \max_{(i,i')} T(i,i')$ . Clearly, the statement in Part (b) holds with this choice of $(\delta,T)$ .

The final result of this section is a consequence of Lemma 9 and Lemma 10. It summarizes key properties of ap–pseudo-orbits in relation to ap–recurrent classes. This result will be used in Section 7 in the proof of Theorem 1.

Proof. Fix $\delta, T \in (0,\infty)$ and a compact $A \in \Delta^o$ . Since $K_i$ is an attractor for each $i \in \{1,\dots,l\}$ , there is a bounded open neighborhood ${O}_i$ of $K_i$ , with $\bar O_i \subset N^{\delta}(K_i)\cap\Delta^o$ , such that

(13) \begin{equation}\lim\limits_{t\rightarrow\infty}\sup\limits_{x\in {O}_i}\mbox{dist}(\varphi_t(x),K_i) = 0.\end{equation}

For each $i \in \{l+1,\dots,v\}$ , let ${O}_i$ be an arbitrary bounded, open, and isolating neighborhood of $K_i$ such that $\bar O_i \subset N^{\delta}(K_i)\cap\Delta^o$ . Denote the $(\delta, T)$ given by Lemma 10(b) by $(\delta^*_1, T^*_1)$ , and denote the $(\delta_0, T_A)$ given by Lemma 9(a) by $(\delta^*_2, T^*_2)$ . Let $\theta>0$ be small enough so that $\overline{N^{\theta}(K_i)} \subset {O}_i$ for each $i \in \{1,\dots, v\}$ . From Lemma 10 we can find $\delta_1 < \min\{\theta, \delta^*_1, \delta^*_2\} $ and $T_1 > \max\{T, T^*_1, T^*_2\}$ such that if $\xi \doteq (\xi_0,\dots,\xi_n)$ is a $(\delta_1,T_1)$ ap–pseudo-orbit with $\xi_0 \in N^{\delta_1}(K_i)$ and $\xi_n \in N^{\delta_1}(K_j)$ such that $\xi_m \in (N^{\theta}(K_i))^c$ for some $m \in \{1,\dots, n-1\}$ , then $i \neq j$ and $K_{i} \leq_{\small \rm{AP}} K_j$ .

Now, let $V_i \doteq N^{\theta + \varepsilon}(K_i)$ , where $\varepsilon > 0$ is small enough so that $\overline{V_i} \subset {O}_i$ for all $i \in \{1,\dots,v\}$ , and let $\delta_2 < \delta_1$ be small enough so that $\overline{N^{\delta_2}(V_i)} \subset {O}_i$ . Thus, for every $i \in \{1,\dots,v\}$

\begin{equation*}K_i \subset N^{\delta_2}(K_i)\subset \subset N^{\theta}(K_i)\subset \subset V_i \subset \subset N^{\delta_2}(V_i)\subset \subset O_i,\end{equation*}

where, as before, for open sets $G_1, G_2$ , we write $G_1 \subset \subset G_2$ if $\bar G_1 \subset G_2$ .

From (13), there is some $T_2 > T_1$ such that if $t \geq T_2$ , then for each $i \in \{1,\dots,l\}$ ,

\begin{equation*}\sup\limits_{u \in {O}_i}\mbox{dist}(\varphi_t(u), K_i) < \delta_2.\end{equation*}

Then Parts 1 and 2 hold when $\delta_0 \doteq \delta_2$ and $T_0 \doteq T_2$ . Additionally, Part 3 holds from the property of $(\delta_1, T_1)$ ap–pseudo-orbits noted above, since $V_i^c \subset (N^{\theta}(K_i))^c$ for each $i \in \{1,\dots,v\}$ . Finally, since $T_0\ge T_2^*$ and $\delta_0 \le \delta_2^*$ , from Lemma 9, Part 4 holds as well.

4. Large deviation estimates

Throughout this section we will assume that Assumption 3 is satisfied. We will give some key uniform large deviation bounds that will be used in Sections 5, 6, and 7.

For $\alpha \in (0,1)$ let $\mathcal{V}_{\alpha} \doteq \{x \in \Delta^o: \mbox{dist}(x,\partial \Delta) >\alpha\}$ . For each compact $K \in \mathcal{K}$ , let $\mathcal{V}_{\alpha, K} \doteq\mathcal{V}_{\alpha} \cap K$ , and let $\pi_{\alpha, K}$ denote the projection map from $\mathbb{R}^d$ to $\bar{\mathcal{V}}_{\alpha, K}$ , defined as

\begin{equation*}\pi_{\alpha, K}(x) \doteq \mbox{arg min}_y \{\|y-x\|: y \in \bar{\mathcal{V}}_{\alpha, K}\}.\end{equation*}

Similarly, denote by $\pi^N_{\alpha, K}$ the projection map from $\mathbb{R}^d$ to $\bar{\mathcal{V}}_{\alpha, K}\cap \Delta_N$ . Let $\theta^{N,\alpha, K}$ be a transition probability kernel on $\mathbb{R}^d$ defined by

\begin{equation*}\theta^{N,\alpha, K}(\cdot | x) \doteq \theta^N\left(\cdot | \pi^N_{\alpha, K}(x)\right).\end{equation*}

Let $\big\{X^{N,\alpha,K}_n\big\}$ be an $\mathbb{R}^d$ -valued chain defined as in (1) but with $\theta^N$ replaced with $\theta^{N,\alpha,K}$ . We consider continuous-time processes $\hat X^{N,\alpha,K}$ associated with $\{X^{N,\alpha,K}_n\}$ as

\begin{equation*}\hat{X}^{N,\alpha,K}(t) = X^{N,\alpha,K}_n + \left[X^{N,\alpha,K}_{n+1} - X^{N,\alpha,K}_{n}\right](Nt - n), \qquad t \in [n/N, (n+1)/N], \; n \in \mathbb{N}_0.\end{equation*}

We now present a basic large deviation result for $\hat X^{N,\alpha,K}$ . Recall the stochastic kernel $\theta(dy| x)$ from Assumption 3(b). For $x, \zeta \in \mathbb{R}^d$ , define

\begin{equation*}H_{\alpha, K}(x, \zeta) \doteq \log \int_{\mathbb{R}^d} \exp\{ \langle \zeta , y\rangle\} \theta(dy| \pi_{\alpha, K}(x)),\end{equation*}

and let

\begin{equation*}L_{\alpha, K}(x,\beta) \doteq \sup_{\zeta \in \mathbb{R}^d} \{ \langle \zeta , \beta \rangle - H_{\alpha, K}(x, \zeta)\}.\end{equation*}

We note that for every $\beta, \zeta \in \mathbb{R}^d$ , $H_{\alpha, K}(x, \zeta) = H_{\alpha', K'}(x, \zeta)$ and $L_{\alpha,K}(x,\beta) = L_{\alpha',K'}(x,\beta)$ whenever $\pi_{\alpha, K}(x)= x= \pi_{\alpha', K'}(x)$ . For $x \in \Delta^o$ and $\beta, \zeta \in \mathbb{R}^d$ , define

\begin{equation*}L(x,\beta) \doteq L_{\alpha, K}(x,\beta), \; H(x,\zeta) \doteq H_{\alpha, K}(x,\zeta) \mbox{ if } x \in \mathcal{V}_{\alpha, K}.\end{equation*}

For $\alpha > 0$ , $x \in \mathbb{R}^d$ , $K \in \mathcal{K}$ , $T \in (0,\infty)$ , and $\phi \in C([0,T]:\mathbb{R}^d)$ , define

\begin{equation*}S_{\alpha, K}(x,T, \phi) \doteq \left\{\begin{array}[c]{cc}\int_0^T L_{\alpha, K}(\phi(t),\dot{\phi}(t)) dt\;\;\;\;\; & \text{if } \phi \mbox{ is absolutely continuous,}\\\infty & \text{otherwise.}\end{array}\right.\end{equation*}

Note that if, for $\alpha, \alpha' >0$ and $K,K'\in \mathcal{K}$ , $\phi \in C([0,T]:\bar{\mathcal{V}}_{\alpha, K})\cap C([0,T]:\bar{\mathcal{V}}_{\alpha', K'})$ , then $S_{\alpha, K}(\phi(0),T, \phi) = S_{\alpha', K'}(\phi(0),T, \phi)$ . Thus, for $\phi \in C([0,T]:\mathbb{R}^d)$ that satisfies $\phi(0) = x$ and $\phi(t) \in \Delta^o$ for all $t \in [0,T]$ , we define

(14) \begin{equation}S(x, T, \phi) = S_{\alpha, K}(x,T, \phi) \quad \mbox{ if } \phi \in C([0,T]:\bar{\mathcal{V}}_{\alpha, K}) \mbox{ for some } \alpha >0 \mbox{ and }K \in \mathcal{K}.\end{equation}

The following uniform large deviation principle will be used several times in this work.

Theorem 3. Suppose Assumption 3 is satisfied. Fix $T\in (0,\infty)$ , $\alpha>0$ , and $K, K' \in \mathcal{K}$ . For each $a \in (0,\infty)$ , let

\begin{equation*}\Phi_{x,\alpha,K',T}(a) \doteq \Big\{\phi \in C\big([0,T]: \mathbb{R}^d\big) : S_{\alpha, K'}(x, T, \phi) \leq a\Big\}.\end{equation*}

1. (a) (Compact level sets.) For every $a \in (0, \infty)$ , the set $\bigcup\limits_{x\in K}\Phi_{x,\alpha,K',T}(a)$ is compact.

2. (b) (Upper bound.) Given $\delta, \gamma \in (0,1)$ and $L \in (0,\infty)$ , there is some $N < \infty$ such that

   \begin{equation*}\mathbb{P}_x\Big(\Big\|\hat{X}^{n,\alpha, K'} - \phi\Big\|_{*,T} < \delta\Big) \geq \exp\big(-n\big(S_{\alpha,K'}(x,T,\phi) + \gamma\big)\big)\end{equation*}
   for all $n \geq N$ , $x \in K \cap \Delta_N$ , and $\phi \in \Phi_{x,\alpha,K',T}(L)$ .

3. (c) (Lower bound.) Given $\delta, \gamma \in (0,1)$ and $L \in (0,\infty)$ , there is some $N < \infty$ such that

   \begin{equation*}\mathbb{P}_x\left( d\left(\hat{X}^{n,\alpha,K'}, \Phi_{x,\alpha,K',T}(l)\right) \geq \delta\right) \leq \exp( -n(l - \gamma))\end{equation*}
   for all $n \geq N$ , $x \in K \cap \Delta_N$ , and $l \in [0,L]$ .

Proof. We will apply [Reference Dupuis and Ellis3, Theorem 6.7.5]. For $x, \zeta \in \mathbb{R}^d$ , let

\begin{equation*}H_{\alpha, K}^N(x, \zeta) \doteq \log \int_{\mathbb{R}^d} \exp\{ \langle \zeta , y\rangle\} \theta^N\left(dy| \pi^N_{\alpha, K}(x)\right).\end{equation*}

By Assumption 3(b)(iii), for each compact $A \subset \mathbb{R}^d$ and $\zeta \in \mathbb{R}^d$ ,

(15) \begin{equation}\sup_{N \in \mathbb{N}} \sup_{x\in \mathbb{R}^d} H^N_{\alpha,K}(x,\zeta)<\infty ,\qquad \sup_{x\in \mathbb{R}^d} H_{\alpha,K}(x,\zeta)<\infty,\end{equation}

and

(16) \begin{equation}\sup_{x\in A} \Big|H^N_{\alpha,K}(x,\zeta) - H_{\alpha,K}(x,\zeta)\Big| \to 0 \quad \mbox{ as } N\to \infty. \end{equation}

Furthermore, from Assumption 3(b)(ii), $x \mapsto \theta (dy| \pi_{\alpha, K}(x))$ is a continuous map from $\mathbb{R}^d$ to $\mathcal{P}\big(\mathbb{R}^d\big)$ . Thus, Conditions 6.2.1 and 6.7.2 of [Reference Dupuis and Ellis3] are satisfied. Next, since from Assumption 3(b)(i) the convex hull of the support of $\theta (dy| \pi_{\alpha, K}(x))$ is all of $\mathbb{R}^d$ , [Reference Dupuis and Ellis3, Condition 6.7.4] is satisfied as well. Thus, from [Reference Dupuis and Ellis3, Theorem 6.7.5] we have that, for every $T \in (0,\infty)$ , $\{\hat{X}^{N,\alpha,K}\}_{N\in \mathbb{N}}$ satisfies a Laplace principle, uniformly on compact subsets of $\mathbb{R}^d$ , in the sense of [Reference Dupuis and Ellis3, Definition 1.2.6], with rate function $S_{\alpha,K}(x, T, \cdot)$ . It is shown in [Reference Salins, Budhiraja and Dupuis17, Theorem 4.3] that a uniform Laplace principle of the form given in [Reference Dupuis and Ellis3, Theorem 6.7.5] implies a uniform large deviation principle in the sense of Freidlin and Wentzell [Reference Freidlin and Wentzell6], which means that Parts (a)–(c) of the theorem hold. The result follows.

Proof. The proof follows from [Reference Dupuis and Ellis3, Lemma 6.5.2] on noting that, by Assumption 3(b) for every $x \in \mathbb{R}^d$ , the convex hull of the support of $\theta(dy| \pi_{\alpha, K}(x))$ is $\mathbb{R}^d$ , and $\sup_{x \in \mathbb{R}^d} H_{\alpha,K}(x, \zeta)<\infty$ for every $\zeta \in \mathbb{R}^d$ .

An important consequence of the above uniform large deviation principle is the following uniform upper bound for closed sets F in $C([0,T]:\mathbb{R}^d)$ .

Theorem 4. Fix $T\in (0,\infty)$ , $\alpha>0$ , and $K,K' \in \mathcal{K}$ . Then, for every closed set F in $C([0,T]: \mathbb{R}^d)$ ,

\begin{equation*} \limsup_{N\to \infty} \frac{1}{N} \log \sup_{x\in K\cap \Delta_N}\mathbb{P}_x\Big(\hat X^{N,\alpha,K'} \in F\Big) \le -\inf_{x \in K} \inf_{\phi \in F} S_{\alpha,K'}(x,T, \phi).\end{equation*}

Proof. Fix $T, \alpha, K, K'$ as in the statement of the theorem. We begin by showing that for each $s \geq 0$ and $\delta > 0$ there is some $\varepsilon \doteq \varepsilon(\delta) \in (0,1)$ such that for all $x,y \in K$ with $\|x-y\|\le \varepsilon$ ,

(17) \begin{align}\Big\{ \phi \in C\big([0,T]: \mathbb{R}^d\big) : \ &d\Big(\phi, \Phi_{x,\alpha,K',T}(s)\Big) \leq\delta\Big\}\nonumber \\ &\supseteq \left\{\phi \in C\big([0,T]: \mathbb{R}^d\big): d\left(\phi, \Phi_{y,\alpha,K',T}\left(s - \frac{\delta}{4}\right)\right) \leq \frac{\delta}{2}\right\}.\end{align}

Let $\kappa_0 \doteq 1+\sup_{x\in K, \|\beta\|\le 1} L(x,\beta)$ . From Lemma 12, $\kappa_0 <\infty$ . Since

\begin{equation*}\cup_{y \in K} \Phi_{y,\alpha,K',T}\left(s - \frac{\delta}{4}\right)\end{equation*}

is a compact set, we can find $\varepsilon \in \Big(0, \frac{\delta}{8\kappa_0}\Big)$ such that for all

\begin{equation*}\psi \in \cup_{y \in K} \Phi_{y,\alpha,K',T}\left(s - \frac{\delta}{4}\right)\end{equation*}

and $0 \le t_1\le t_2 \le T$ with $|t_1-t_2|\le \varepsilon$ , we have $\|\psi(t_2)- \psi(t_1)\| \le \frac{\delta}{8}$ .

Fix $y \in K$ and $\phi$ in the set on the right side of (17). Then there is a $\psi_1 \in \Phi_{y,\alpha,K',T}\left(s - \frac{\delta}{4}\right)$ such that

\begin{equation*}\|\phi - \psi_1\|_{*,T} \le \frac{\delta}{2} + \frac{\delta}{8} = \frac{5\delta}{8}.\end{equation*}

Note in particular that $\psi_1(0)=y$ . Fix $x \in K$ such that $\|y-x\| \le \varepsilon$ . Let $t_0 \doteq \|x-y\|$ and define the function $\eta_{x,y} : [0,t_0] \rightarrow \mathbb{R}^d$ as

(18) \begin{equation}\eta_{x,y}(t) \doteq x + \frac{(y-x)}{\|y-x\|}t.\end{equation}

Define $\psi_2:[0,T]\to \mathbb{R}^d$ as

\begin{equation*}\psi_2(s) \doteq \eta_{x,y}(s) 1_{[0, t_0]}(s) + \psi_1(s-t_0)1_{(t_0, T]}(t).\end{equation*}

Note that $\psi_2(0)=x$ and

\begin{align*}S_{\alpha,K'}(x,T,\psi_2) &= \int_0^{t_0} L_{\alpha, K'}(\psi_2(t),\dot{\psi}_2(t)) dt+ \int_{t_0}^T L_{\alpha, K'}(\psi_2(t),\dot{\psi}_2(t)) dt\\& \le \varepsilon \kappa_0 + s - \frac{\delta}{4} = \frac{\delta}{8} +s - \frac{\delta}{4} \le s.\end{align*}

Thus $\psi_2 \in \Phi_{x,\alpha,K',T}\left(s \right)$ . Furthermore,

\begin{equation*}\|\phi - \psi_2\|_{*,T} \le \|\phi - \psi_1\|_{*,T} + \|\psi_1 - \psi_2\|_{*,T} \le \frac{5\delta}{8} + \|\psi_1 - \psi_2\|_{*,T}.\end{equation*}

Also, for $t \in (t_0, T]$ ,

\begin{equation*}\|\psi_1(t) - \psi_2(t)\| = \|\psi_1(t) - \psi_1(t- t_0)\| \le \frac{\delta}{8},\end{equation*}

and for $t \in [0, t_0]$ ,

\begin{equation*}\|\psi_1(t) - \psi_2(t)\| \le \|\psi_2(t)-y\| + \|\psi_1(t) - \psi_1(0)\| \le \varepsilon + \frac{\delta}{8} \le \frac{\delta}{8} + \frac{\delta}{8} = \frac{\delta}{4}.\end{equation*}

Thus

\begin{equation*}\|\phi - \psi_2\|_{*,T} \le \frac{5\delta}{8} + \frac{\delta}{4} \le \delta.\end{equation*}

Since $\psi_2 \in \Phi_{x,\alpha,K',T}\left(s \right)$ , we have $d\Big(\phi, \Phi_{x,\alpha,K',T}\left(s \right)\Big) \le \delta$ , and thus $\phi$ is in the set on the left side of (17). This proves the inclusion in (17).

Now fix a closed set F in $C([0,T]: \mathbb{R}^d)$ . If

\begin{equation*}\inf\limits_{x\in K}\inf\limits_{\phi \in F}S_{\alpha,K'}(x,T,\phi) = 0,\end{equation*}

then the result clearly holds, so we assume that

\begin{equation*}\bar S \doteq \inf\limits_{x\in K}\inf\limits_{\phi \in F}S_{\alpha,K'}(x,T,\phi) > 0.\end{equation*}

Fix $s \in \left(0, \bar S \right)$ , and let $\{x_n\} \subset K$ and $\varepsilon \downarrow 0$ . Since K is compact, we may pass to a subsequence and assume that $x_n \rightarrow \tilde{x}$ for some $\tilde{x} \in K$ . Since

\begin{equation*}\inf\limits_{\phi\in F}S_{\alpha,K'}(\tilde{x}, T, \phi) > s,\end{equation*}

$F \cap \Phi_{\tilde{x},\alpha,K',T}(s) = \emptyset$ . This, along with the facts that $\Phi_{\tilde{x},\alpha,K',T}(s)$ is compact and F is closed, ensures that there is some $\delta \in (0,s)$ such that

\begin{equation*}F \subset \Big\{\phi \in C\Big([0,T]:\mathbb{R}^d\Big): d\Big(\phi,\Phi_{\tilde{x},\alpha,K'}(s)\Big) > \delta \Big\}.\end{equation*}

Let $\varepsilon = \varepsilon(\delta) > 0$ be chosen as above (17). Without loss of generality we assume that $\|\tilde{x} - x_n\| \le \varepsilon$ for all n. Then, for every $n\in \mathbb{N}$ ,

\begin{align*}F \ &\subset \Big\{ \phi \in C\bigg([0,T]: \mathbb{R}^d : d\Big(\phi, \Phi_{\tilde x,\alpha,K',T}(s)\Big) > \delta\Big\}\\&\subset \left\{\phi \in C\bigg([0,T]: \mathbb{R}^d : d\left(\phi, \Phi_{x_n,\alpha,K',T}\left(s - \frac{\delta}{4}\right)\right) > \frac{\delta}{2}\right\}.\end{align*}

From the upper bound in Theorem 3(c) we see that

\begin{equation*}\begin{split}\limsup\limits_{N\rightarrow\infty}\!\frac{1}{N}\!\log \mathbb{P}_{x_N}\!\left(\hat{X}^{N, \alpha,K'} \!\in F\right) &\leq \limsup\limits_{N\rightarrow\infty}\frac{1}{N} \log \mathbb{P}_{x_N}\left(d\left(\hat{X}^{N,\alpha,K'}, \Phi_{x_N,\alpha,K',T}\!\left(s - \frac{\delta}{4}\right)\right) > \frac{\delta}{2}\right)\\&\leq - \left(s - \frac{\delta}{4}\right).\\\end{split}\end{equation*}

The result follows from letting $\delta \to 0$ and $s \to \bar S$ .

5. Asymptotic behavior of QSD

In this section we assume that Assumptions 1, 2, 3, and 4 are satisfied. Using these assumptions we will provide several exponential probability estimates and use them to deduce some asymptotic properties of the QSD $\{\mu_N\}$ (when they exist). For $N\in \mathbb{N}$ and $T\in (0,\infty)$ , let

\begin{equation*}D^N_T \stackrel{\cdot}{=} \sup\limits_{0\leq t \leq T}\Big\|\hat{X}^N(t) - \varphi_t\left(X_0^N\right)\Big\| = \Big\|\hat X^N - \varphi_{\cdot}\left(X^N_0\right)\Big\|_{*,T}.\end{equation*}

The estimates obtained in Lemma 13 and Lemma 14 are the key steps in the proof of Theorem 5, which gives the asymptotics of $\lambda_N \doteq \Big[\mathbb{P}_{\mu_N}\Big(X^N_1 \in \Delta^o\Big)\Big]^N$ , where $\mu_N$ is a QSD for $\{X^N\}.$ Recall the definition of $\mathcal{V}_{\alpha}$ from Section 4.

Lemma 13. For each $ \alpha > 0$ , compact set $K \subset \mathcal{V}_{\alpha}$ , $\varepsilon > 0$ , and $T > 0$ , there is a $c\in (0,\infty)$ and $N_0 \in \mathbb{N}$ such that for every $N\ge N_0$ ,

\begin{equation*}\sup\limits_{x \in K \cap \Delta_N} \mathbb{P}_x\left[ D^N_T \geq \varepsilon \right] \leq \exp(-Nc).\end{equation*}

Proof. Let $\alpha > 0$ and let $K \subset V_{\alpha}$ be compact. For each $\varepsilon \in (0,\alpha)$ , let

\begin{equation*}F_{\varepsilon} = \left\{ \psi \in C\big([0,T] : \mathbb{R}^d\big) : \sup\limits_{0\leq t \leq T}\| \psi(t)- \varphi_t(\psi(0))\| \geq \varepsilon \right\}.\end{equation*}

Using Lemma 3 we can (and will) assume without loss of generality that $\varepsilon$ is small enough so that the compact set

\begin{equation*}K' \doteq \overline{N^{\varepsilon}\left( \varphi_{[0,\infty)}(K)\right)} \subset \Delta^o.\end{equation*}

Note that

\begin{equation*}\sup\limits_{x \in K \cap \Delta_N}\mathbb{P}_x\left[ D^N_T \geq \varepsilon\right] = \sup\limits_{x \in K \cap \Delta_N}\mathbb{P}_x\left[ \sup\limits_{0\leq t \leq T}\Big\| \hat{X}^{N, \alpha, K'}(t) - \varphi_t\Big(\hat{X}^{N, \alpha, K'}(0)\Big)\Big\| \geq \varepsilon\right].\end{equation*}

Since $F_{\varepsilon}$ is closed, Theorem 4 says that for each $\delta>0$ , there is an $N_{\delta}\in \mathbb{N}$ such that for all $N\ge N_{\delta}$ ,

\begin{equation*}\begin{split}\log \sup\limits_{x \in K \cap \Delta_N}\mathbb{P}_x\left[ \sup\limits_{0\leq t \leq T} \| \hat{X}^{N,\alpha,K'}(t) - \varphi_t(x)\| \geq \varepsilon\right]&= \log \sup\limits_{x\in K \cap \Delta_N} \mathbb{P}_x\Big(\hat{X}^{N, \alpha,K'} \in F_{\varepsilon}\Big)\\&\leq - N \left[\inf\limits_{x \in K }\inf\limits_{\psi \in F_{\varepsilon}} S_{\alpha,K'}(x,T,\psi) - \delta\right].\end{split}\end{equation*}

To prove the result, it suffices to show that

\begin{equation*}\inf\limits_{x \in K }\inf\limits_{\psi \in F_{\varepsilon}} S_{\alpha,K'}(x,T,\psi) >0.\end{equation*}

Arguing by contradiction, suppose that this infimum is 0. Then there are sequences $\{x_n\} \subset K$ and $\{\psi_n\} \subset C([0,T]: \mathbb{R}^d)$ such that $\psi_n \in F_{\varepsilon}$ for each n and

\begin{equation*}\lim\limits_{n\rightarrow\infty} S_{\alpha,K'}(x_n,T,\psi_n) = 0.\end{equation*}

Since $S_{\alpha,K'}(x, T, \phi) < \infty$ if and only if $\phi(0) = x$ , we can assume without loss of generality that $x_n = \psi_n(0)$ for every n. For each $\varepsilon' > 0$ ,

\begin{equation*}\psi_n \in \Big\{\phi \in C\big([0,T]: \mathbb{R}^d\big): S_{\alpha,K'}(y,T,\phi) \le \varepsilon' \mbox{ for some } y \in K\Big\}\end{equation*}

whenever n is sufficiently large. Since K is compact, Theorem 3 ensures that $\{\psi_n\}$ is precompact in $C([0,T]: \mathbb{R}^d)$ , and so there is a convergent subsequence of $\{\psi_n\}$ . Denoting this subsequence by $\{\psi_{n_k}\}$ and its limit by $\psi$ , we have that

\begin{equation*} \lim\limits_{k\rightarrow\infty}(\psi_{n_{k}}, x_{n_{k}}) = \lim\limits_{k\rightarrow\infty}(\psi_{n_{k}}, \psi_{n_{k}}(0)) = (\psi, \psi(0)). \end{equation*}

Since $\phi \mapsto S_{\alpha,K'}(\phi(0), T, \phi)$ is lower semicontinuous, it follows that

\begin{equation*}S_{\alpha,K'}(\psi(0), T, \psi) \leq \lim\limits_{k\rightarrow\infty} S_{\alpha,K'}\big(\psi_{n_k}(0), T, \psi_{n_{k}}\big) = 0,\end{equation*}

which says that $\psi(t) = \varphi_t(\psi(0))$ . However, this is a contradiction, since $\psi \in F_{\varepsilon}$ . The result follows.

Lemma 14. Let U be a fundamental neighborhood of an attractor $A \subset \Delta^o$ such that $\bar U \subset \Delta^o$ . Then for every $T_0 \in (0,\infty)$ , there are $c_0 \in (0,\infty)$ , $T \in (T_0,\infty)$ , and $N_0 \in \mathbb{N}$ such that

\begin{equation*}\sup\limits_{x \in U \cap \Delta_N}\mathbb{P}_x\left( X_{\lfloor NT \rfloor}^N \in U^c\right) \leq \exp\left(-c_0N\right)\end{equation*}

for all $N \ge N_0$ .

Proof. Let $\alpha \doteq \mbox{dist}(A, U^c)$ . Since U is a fundamental neighborhood of the attractor A, we can find $T > T_0$ such that

\begin{equation*}\sup\limits_{t \geq T}\sup\limits_{x \in U}\mbox{dist}(\varphi_t(x), A) < \alpha/2.\end{equation*}

Let $K \in \mathcal{K}$ be a compact set containing U. From Lemma 3 there exists a $\sigma \in (0, \alpha/4)$ and a $K' \in \mathcal{K}$ such that $\overline{N^{\sigma}(\gamma^+(U))} \subset K'$ , where $\gamma^+(U) = \cup_{x\in U}\gamma^+(x)$ . Then for each $x \in U \cap \Delta_N$ , we have

(19) \begin{equation}\begin{split}\mathbb{P}_x\left( X_{\lfloor NT \rfloor}^N \in U^c\right) &\leq \mathbb{P}_x\left(\mbox{dist}\left(X_{\lfloor NT \rfloor}^N, A\right) > \alpha\right)\leq \mathbb{P}_x\left( \Big\|X_{\lfloor NT\rfloor}^N - \varphi_T(x)\Big\| > \alpha/2\right)\\&\leq \mathbb{P}_x\left(\Big\|X_{\lfloor NT \rfloor}^N -\hat{X}^N(T)\Big\| + \Big\|\hat{X}^N(T)- \varphi_T(x)\Big\| > \alpha/2\right)\\&\leq \mathbb{P}_x\Big(D_T^N > \sigma \Big)+ \mathbb{P}_x\left( \Big\|X_{\lfloor NT \rfloor}^N- \hat{X}^N(T)\Big\| > \alpha/4, D_T^N \le \sigma\right).\\\end{split}\end{equation}

Using the Markov property, we have

\begin{equation*}\begin{split}\mathbb{P}_x\left( \Big\|X_{\lfloor NT \rfloor}^N- \hat{X}^N(T)\Big\| > \alpha/4, D_T^N \le \sigma\right)&\le \mathbb{P}_x\left( \Big\|X_{\lfloor NT \rfloor}^N- X_{\lfloor NT \rfloor + 1}^N\Big\| > \alpha/4, D_T^N \le \sigma\right)\\&\le \sup\limits_{x\in K'\cap \Delta_N}\mathbb{P}_x\Big( \Big\|X^N_1-x\Big\| > \alpha/4\Big).\end{split}\end{equation*}

From Assumption 3 we have that for every $\lambda>0$ ,

\begin{equation*}C(\lambda) \doteq \sup_{N \in \mathbb{N}} \sup_{x \in K' \cap \Delta_N} \mathbb{E}_x\Big(e^{\lambda N \big\|X^N_1-x\big\| }\Big) < \infty.\end{equation*}

Thus, for any $\lambda >0$ ,

\begin{equation*}\sup\limits_{x\in K'\cap \Delta_N}\mathbb{P}_x\Big( \Big\|X^N_1-x\Big\| > \alpha/4\Big) \le c(\lambda)e^{-\lambda N \alpha/4}.\end{equation*}

The result follows on using the above estimate and Lemma 13 in (19).

The following lemma says that for every open $U \subset \Delta^o$ , the support of $\mu_N$ (when it exists) has a nonempty intersection with U when N is sufficiently large.

Proof. Let $N_0$ be large enough so that $U \cap \Delta^o_N$ is nonempty for all $N\ge N_0$ . Fix $N \geq N_0$ , $x \in U \cap \Delta^o_N$ , and $w \in \Delta^o_N$ with $\mu_N(w)>0$ . From Assumption 4(a), there is a $k \in \mathbb{N}$ such that $\mathbb{P}_w(X^N_k=x)>0$ . Then

\begin{equation*}\begin{split}\mu_N(U) &\geq \mu_N(x) = \frac{\sum\limits_{y \in \Delta_N^o} \mu_N(y)\mathbb{P}_y\left(X_k^N = x\right)}{ \sum\limits_{z\in \Delta^o_N}\sum\limits_{y \in \Delta_N^o} \mu_N(y)\mathbb{P}_y\left( X_k^N = z\right)} \geq \frac{ \mu_N(w)\mathbb{P}_w\left(X_k^N = x\right)}{ \sum\limits_{z\in \Delta^o_N}\sum\limits_{y \in \Delta_N^o} \mu(y)\mathbb{P}_y\left(X_k^N = z\right)} > 0.\end{split}\end{equation*}

The following lemma quantifies the asymptotic behavior of the sequence $\{\lambda_N\}$ introduced in (5).

Theorem 5. Suppose that for each $N \in \mathbb{N}$ , $X^N$ has a QSD $\mu_N$ . Then there exist $c, c' \in (0,\infty)$ such that for all $N\in \mathbb{N}$ ,

\begin{equation*}0 \leq 1 - \lambda_N \leq c' \exp(-cN).\end{equation*}

Proof. From Assumption 2 and Corollary 1 there exists an attractor A in $\Delta^o$ . Let $U \subset \Delta^o$ be a fundamental neighborhood of A. From Lemma 14 there are $c_0 \in (0,\infty)$ and $T,N_1 \in \mathbb{N}$ such that for all $N\ge N_1$ ,

(20) \begin{equation}\sup\limits_{x \in U \cap \Delta_N}\mathbb{P}_x\left( X_{NT}^N \in U^c\right) \leq \exp\left(-c_0N\right).\end{equation}

From Lemma 15 there is an $N_2 \in \mathbb{N}$ such that for all $N\ge N_2$ , $\mu_N(U) > 0$ . Fixing $N \geq N_1\vee N_2$ , we have

\begin{equation*}\begin{split}\lambda^{T}_N \mu_N(U) &= \sum\limits_{x \in \Delta^o_N}\mathbb{P}_{x}\left(X^N_{NT} \in U\right)\mu_N(x)\\&\geq \sum\limits_{x \in U \cap \Delta^o_N}\mathbb{P}_x\left(X_{NT}^N \in U\right)\mu_N(x)\\&\geq \inf\limits_{x\in U \cap \Delta^o_N}\mathbb{P}_x\left(X_{NT}^N \in U\right) \sum\limits_{x \in U \cap \Delta^o_N}\mu_N(x)\\&=\inf\limits_{x\in U \cap \Delta^o_N}\mathbb{P}_x\left(X_{NT}^N \in U\right)\mu_N(U).\end{split}\end{equation*}

Thus, for all $N\ge N_1\vee N_2$ ,

\begin{equation*}\lambda_N\ge \lambda^{T}_N \geq \inf\limits_{x\in U \cap \Delta^o_N}\mathbb{P}_x\left(X_{NT}^N \in U\right) = 1 - \sup\limits_{x\in U \cap \Delta^o_N}\mathbb{P}_x\left(X_{NT}^N \in U^c\right)\ge 1-\exp\left(-c_0N\right),\end{equation*}

where the last inequality uses (20). The result follows.

For $\delta>0$ , $T \in \mathbb{N}$ , and $K\in \mathcal{K}$ , let

(21) \begin{equation}\beta_{\delta, K}^N(T) \doteq \sup_{x \in \Delta_N\cap K} \mathbb{P}_x\big[\big\|\hat X^N - \varphi_{\cdot}(x)\big\|_{*,T} \ge \delta\big].\end{equation}

The following lemma gives a different lower bound on $\lambda_N$ . This bound will be needed in the proof of Theorem 6 below.

Proof. Since A is an attractor, there is a fundamental neighborhood U of A contained in $\tilde U$ . Thus we can find a $\delta>0$ and $T \in \mathbb{N}$ such that $N^{\delta}(\varphi_T(\overline{U})) \subset U$ . From Lemma 15 we can find an $N_0 \in \mathbb{N}$ such that $\mu_N(U)>0$ for all $N \ge N_0$ . Following the proof of Theorem 5, we see that

\begin{equation*}\begin{split}\lambda_N^T \mu_N(U) \geq \left(1 - \sup\limits_{x\in U \cap \Delta^o_N}\mathbb{P}_x\left(X^N_{NT} \in U^c\right)\right)\mu_N(U).\end{split}\end{equation*}

From our choice of U and $\delta$ it now follows that

\begin{equation*}\begin{split}\lambda_N^T &\geq 1 - \sup\limits_{x\in U \cap \Delta^o_N}\mathbb{P}_x\Big(X^N_{NT} \in \big(N^{\delta}\big(\varphi_T(\overline{U})\big)\big)^c\Big) \geq 1 - \beta^N_{\delta,K}(T).\\\end{split}\end{equation*}

A key consequence of the following theorem is that the support of any weak limit point of $\mu_N$ is contained in $\Delta^o$ . This, along with a further characterization of the support of such weak limit points given in Corollary 2, is a key element in the proof of Theorem 1.

Theorem 6. Suppose that for each $N \in \mathbb{N}$ , $X^N$ has a QSD $\mu_N$ . Then, for every $\delta>0$ , $T\in \mathbb{N}$ , and $K \in \mathcal{K}$ , there exists an open neighborhood $V_K$ of $\partial \Delta$ in $\Delta$ such that

(22) \begin{equation}\limsup_{N\to \infty} \mu_N(V_K) \le \limsup_{N\to \infty} \frac{\beta_{\delta, K}^N(T) }{\inf_{x \in V_K \cap \Delta_N} \mathbb{P}_x\big[\hat X^N(T) \in \partial \Delta\big]}=0.\end{equation}

Suppose in addition that $\mu_N$ converges along some subsequence to some probability measure $\mu$ on $\Delta$ . Then there is an open neighborhood $V_0$ of $\partial \Delta$ in $\Delta$ such that $\mu(V_0) = 0$ .

Proof. Fix $\delta, T,K$ as in the statement of the theorem. Let

\begin{equation*}\delta_0 \doteq \frac{1}{2} \inf\limits_{t \in [0,T]} \inf\limits_{x\in K}\mbox{dist}(\varphi_t(x), \partial\Delta)\end{equation*}

and let $K' \doteq \overline{ N^{\delta_0}( \varphi_{[0,T]}(K))}$ , and consider the closed set

\begin{equation*}F \doteq \left\{\phi \in C\big([0,T]:\mathbb{R}^d\big) : \|\phi -\varphi_{\cdot}(\phi(0))\|_{*,T} \geq \delta_0\right\}.\end{equation*}

Fix $\alpha \in (0,\delta_0)$ and $K_1 \in \mathcal{K}$ that contains some open neighborhood of K ^′. Then from Theorem 4,

\begin{equation*}\begin{split}\limsup\limits_{N\rightarrow\infty} \frac{1}{N} \log\sup_{x \in K \cap \Delta_N} \mathbb{P}_x \big(\hat{X}^{N} \in F\big) &= \limsup\limits_{N\rightarrow\infty} \frac{1}{N} \log \sup_{x \in K \cap \Delta_N} \mathbb{P}_x \big(\hat{X}^{N,\alpha,K_1} \in F\big)\\&\leq - \inf_{x \in K} \inf_{\phi \in F} S_{\alpha,K_1}(x,T,\phi) \doteq -c(K).\end{split}\end{equation*}

Clearly $c(K)>0$ . From Assumption 4(b) we can find an open neighborhood $V_K$ of $\partial \Delta$ such that

\begin{equation*}\liminf\limits_{N\rightarrow\infty}\inf_{x\in V_K\cap \Delta_N}\frac{1}{N} \log \mathbb{P}_x \big( \hat{X}^N(T) \in \partial \Delta\big) \geq - c(K)/4.\end{equation*}

Combining last two displays, we can find an $N_1 \in \mathbb{N}$ such that for all $N \ge N_1$ ,

\begin{equation*}\begin{split}\frac{\beta_{\delta, K}^N(T) }{\inf_{x \in V_K \cap \Delta_N} \mathbb{P}_x\big[\hat X^N(T) \in \partial \Delta\big]} =\frac{\sup\limits_{x \in \Delta_N\cap K} \mathbb{P}_x\left(\big\| \hat X^N - \varphi_{\cdot}(x)\big\|_{*,T} \ge \delta\right)}{\inf\limits_{x \in V_K \cap \Delta_N} \mathbb{P}_x\big[\hat X^N(T) \in \partial \Delta\big]} \leq \exp(-Nc(K)/2),\end{split}\end{equation*}

which converges to 0 as $N \rightarrow\infty$ . This proves the last equality in (22).

Next, from Assumption 2 and Corollary 1 there exists an attractor A in $\Delta^o$ . Let $\tilde U \in \Delta^o$ be an open set containing A, and let $K \in \mathcal{K}$ be such that $\tilde U\subset K$ . Then from Lemma 16 there exist $\delta > 0$ and $T, N_0 \in \mathbb{N}$ such that

\begin{equation*}\lambda_N^{T} \geq 1 - \beta_{\delta,K}^N(T) \quad \mbox{ for each } N \ge N_0.\end{equation*}

Since $\mu_N(\Delta^o_N) = 1$ , we have, with $V_K$ given as in the first part of the theorem,

\begin{equation*}\begin{split}1 - \beta_{\delta,K}^N(T) &\leq \lambda_N^T \mu_N(\Delta^o_N)\\&= \sum\limits_{x\in \Delta^o_N}\left(1 - \mathbb{P}_x\left( X_{NT}^N \in \partial \Delta\right)\right)\mu_N(x)\\&= \sum\limits_{x \in V_K\cap \Delta^o_N}\left(1 - \mathbb{P}_x\big( \hat{X}^N(T) \in \partial \Delta\big)\right)\mu_N(x) + \sum\limits_{x \in \Delta^o_N \setminus V_K}\left(1 - \mathbb{P}_x\big( \hat{X}^N(T) \in \partial \Delta\big)\right)\mu_N(x)\\ & \leq \left(1 - \inf\limits_{x\in V_K \cap \Delta^o_N}\mathbb{P}_x\big(\hat{X}^N(T)\in \partial \Delta\big) \right)\mu_N(V_K) +\mu_N(\Delta^o_N \setminus V_K)\\&= 1 - \inf\limits_{x\in V_K \cap \Delta^o_N}\mathbb{P}_x\big(\hat{X}^N(T) \in \partial \Delta\big)\mu_N(V_K). \end{split}\end{equation*}

Rearranging the previous inequality, we obtain

\begin{equation*}\mu_N(V_K) \leq \frac{\beta^N_{\delta,K}(T)}{\inf\limits_{x\in V_K \cap \Delta^o_N}\mathbb{P}_x\big(\hat{X}^N(T) \in \partial \Delta\big)}.\end{equation*}

This proves the first inequality in (22).

Finally, let $V_0$ be an open neighborhood of $\partial \Delta$ such that $\bar V_0 \subset V_K$ . From the first part of the theorem, taking the limit along the convergent subsequence,

\begin{equation*}\mu(V_0) \le \liminf_{N\to \infty} \mu_N( V_0) \le \liminf_{N\to \infty} \mu_N(V_K) = 0.\end{equation*}

The result follows.

The following theorem proves the invariance of $\mu$ under the flow $\{\varphi_t\}$ .

Proof. From Corollary 1, for each $i \in \{1,\dots,l\}$ , $K_i$ is an attractor. Fix $1\le i \le l$ , $\delta > 0$ , and $K \in \mathcal{K}$ such that $N^{\gamma}(K_i ) \subset K$ for some $\gamma > 0$ . Let $\beta_{\delta,K}^N $ be as in (21). It suffices to show that for any continuous and bounded $f: \Delta \to \mathbb{R}$ and $t > 0$ , $\mu(f) = \mu(f\circ\varphi_t)$ . Fix f and t as above and let $\varepsilon>0$ be arbitrary. Using the fact that $\{\mu_N\}$ (considered along the convergent subsequence) is tight, we can assume that the K chosen above satisfies

\begin{equation*}\sup_{N\ge 1}\mu_N(K^c) \le \frac{\varepsilon}{2\|f\|_{\infty}}. \end{equation*}

Let $t_N = \lfloor Nt\rfloor/N$ . Note that $t_N \to t$ as $N \to \infty$ , and from Theorem 5, $\lambda_N^{t_N}\to 1$ as $N\to \infty$ . For a bounded $g: \Delta^o \to \mathbb{R}$ and $k \in \mathbb{N}$ , let

\begin{equation*}\mathcal{P}^N_k f(x) \doteq \mathbb{E}_x\left[f\left(X^N_k\right); \tau^N_{\partial} >k\right], \qquad x \in \Delta^o_N,\end{equation*}

and

\begin{equation*}\mathcal{P}^{N,*}_k f(x) \doteq \mathbb{E}_x\left[f\left(X^N_k\right)\right], \qquad x \in \Delta^o_N.\end{equation*}

Then

\begin{equation*}\mu_N(f) = \lambda_N^{-t_N} \mu_N\left(\mathcal{P}^N_{\lfloor N t\rfloor} f\right).\end{equation*}

In particular, as $N\to \infty$ ,

\begin{equation*}\Big|\mu_N(f)-\mu_N\left(\mathcal{P}^N_{\lfloor N t\rfloor} f\right)\Big| \le \|f\|_{\infty} \Big|1-\lambda_N^{t_N}\Big| \to 0.\end{equation*}

Also,

\begin{equation*} \left|\mu_N\left(\mathcal{P}^N_{\lfloor N t\rfloor} f\right)-\mu_N\big(f\circ\varphi_t\big)\right| \le \frac{\varepsilon}{2\|f\|_{\infty}} 2 \|f\|_{\infty} + \sup_{x\in K}\left|\mathcal{P}^N_{\lfloor N t\rfloor} f(x) - f\circ\varphi_t(x)\right|. \end{equation*}

For each $x \in K \cap \Delta_N$ ,

\begin{equation*} \begin{split} \left| \mathcal{P}^N_{\lfloor N t \rfloor} f(x) - f \circ \varphi_t(x)\right| &\leq \left| \mathcal{P}^{N,*}_{\lfloor N t \rfloor} f(x) - \mathcal{P}^{N}_{\lfloor N t \rfloor} f(x) \right| + \left| \mathcal{P}^{N,*}_{\lfloor N t \rfloor} f(x) - f \circ \varphi_t(x)\right|\\ &\leq ||f||_{\infty}\mathbb{P}_x\left( \tau^N_{\partial} \leq \lfloor N t \rfloor \right) + \left| \mathcal{P}^{N,*}_{\lfloor N t \rfloor} f(x) - f \circ \varphi_t(x)\right|, \end{split} \end{equation*}

and Assumption 1 ensures that as $N \rightarrow \infty$ ,

\begin{equation*} \sup\limits_{x\in K} \left| \mathcal{P}^{N,*}_{\lfloor N t \rfloor} f(x) - f \circ \varphi_t(x)\right| \rightarrow 0. \end{equation*}

Let $\tilde{\delta} \doteq \inf_{x\in K, 0 \leq s \leq t}\mbox{dist}(\varphi_{s}(x), \partial \Delta) > 0$ , and note that Assumption 1 ensures that as $N\rightarrow\infty$ ,

\begin{equation*} \begin{split} \sup\limits_{x \in K}\mathbb{P}_x\left(\tau_{\partial}^N \leq \lfloor N t\rfloor \right) \leq \sup\limits_{x\in K}\mathbb{P}_x\big( \big\|X^N - \varphi_{\cdot}(x)\big\|_{*,t} > \tilde{\delta}\big) \rightarrow 0. \end{split} \end{equation*}

Combining the two previous convergence properties, we see that as $N \rightarrow \infty$ ,

\begin{equation*} \left| \mathcal{P}^N_{\lfloor N t \rfloor} f(x) - f \circ \varphi_t(x)\right| \rightarrow 0, \end{equation*}

and therefore that

\begin{equation*}|\mu(f)- \mu(f \circ \varphi_t)| \le \limsup_{N\to \infty} |\mu_N(f)- \mu_N(f \circ \varphi_t)| \le \varepsilon.\end{equation*}

Since $\varepsilon>0$ is arbitrary, the result follows.

We now recall the definition of the Birkhoff center of $\{\varphi_t\}$ .

Definition 7. The Birkhoff center of $\{\varphi_t : t\geq 0\}$ is

\begin{equation*} BC(\varphi) \doteq \overline{ \{x \in \Delta : x \in \omega(x)\}}. \end{equation*}

Proof. Let $\delta, T > 0$ and suppose that $x \in \omega(x)$ . There is a sequence of time instants $t_i \uparrow \infty$ such that $\varphi_{t_i}(x) \rightarrow x$ , so if we let

\begin{equation*} j = \min\{ i : t_i > T \text{ and } \|\varphi_{t_i}(x) - x\| < \delta\}, \end{equation*}

then $(x, x, \phi_{t_j}(x), x, x)$ is a $(\delta,T)$ ap–pseudo-orbit from x to x. Since $\delta, T$ are arbitrary, $x \in \mathcal{R}_{{\small \rm{AP}}}$ . This proves the first part of the lemma. The second part is now immediate on using Assumption 2(a).

We will use the PoincarÉ recurrence theorem given below. For a proof see [Reference Hasselblatt and Katok10, Theorem 4.1.19].

Theorem 8. Let $\nu$ be a measure which is invariant under $\{\varphi_t\}$ . Then for each measurable $B \subset \Delta$ and $T > 0$ ,

\begin{equation*}\nu( \{ x \in B : \{\varphi_t(x)\}_{t\geq T} \subset \Delta \setminus B\}) = 0.\end{equation*}

The next result is a consequence of Lemma 17 and Theorem 8. It shows that the support of $\mu$ is contained in $\mathcal{R}_{{\small \rm{AP}}}^*$ .

Proof. From Theorem 7, $\mu$ is invariant under $\{\varphi_t\}$ . Enumerate the d-dimensional rationals in $\Delta$ as $\mathbb{Q}^d \doteq \{q_1,q_2,\dots\}$ , and for $m,n \in \mathbb{N}$ , denote the ball of radius $n^{-1}$ centered at $q_m$ by $B(q_m,n^{-1})$ . Then for each $m,n \in \mathbb{N}$ , Theorem 8 says that

\begin{equation*}\mu\big(\tilde{B}\big(q_m,n^{-1}\big)\big) = \mu\big(B\big(q_m,n^{-1}\big)\big),\end{equation*}

where

\begin{equation*}\tilde{B}\big(q_m,n^{-1}\big) \doteq \big\{ x \in B\big(q_m,n^{-1}\big) : \mbox{ there exist } t_k \uparrow \infty \mbox{ with } \varphi_{t_k}(x) \in B\big(q_m,n^{-1}\big) \text{ for all }k \in \mathbb{N} \big\}.\end{equation*}

Let $R \doteq \cap_{n=1}^{\infty}\cup_{m=1}^{\infty} \tilde{B}\big(q_m,n^{-1}\big)$ ; then

\begin{equation*}1 = \mu\left(\cap_{n=1}^{\infty}\cup_{m=1}^{\infty} B\left(q_m,n^{-1}\right)\right) = \mu\left(\cap_{n=1}^{\infty}\cup_{m=1}^{\infty} \tilde B\left(q_m,n^{-1}\right)\right) = \mu( R),\end{equation*}

which together with Theorem 6 implies that $\text{supp}(\mu) \subseteq \overline{R} \cap \Delta^o$ . Furthermore, if $x \in R$ , then $x \in \omega(x)$ , so $R \subseteq \text{BC}(\varphi)$ , and consequently $\bar{R} \cap \Delta^o \subset \text{BC}(\varphi) \cap \Delta^o$ . It now follows from Lemma 17 that

\begin{equation*}\text{supp}(\mu) \subseteq \overline{R}\cap \Delta^o \subseteq \text{BC}(\varphi)\cap \cap \Delta^o \subseteq \mathcal{R}_{{\small \rm{AP}}}^*.\end{equation*}

Combining the results of Corollary 2, Theorem 7, and Theorem 5, we have most of Theorem 1. In particular, we have the lower bound on probabilities of non-extinction given in Theorem 1, and we also have that the limit points $\mu$ of the QSD are invariant under the flow and are supported on the union of absorption-preserving recurrence classes in the interior. The final step is to show that the support in fact lies in the union of the interior attractors. For this we will introduce another notion of recurrence which is given in terms of the quasipotential associated with the rate functions in the underlying large deviation principles.

6. Quasipotential and chain recurrence

In this section we suppose that Assumptions 1, 2, and 3 are satisfied. Recall the rate function S introduced in (14). For $x,y \in \Delta^o$ , let $\mathcal{C}(x,y,T) \doteq \{\phi\in C([0,T]:\Delta^o):\phi(0)=x, \phi(T)=y\}$ , and define

(23) \begin{equation}V(x,y) \doteq \liminf_{T\to \infty} \inf_{\phi \in C(x,y,T)}S(x,T,\phi).\end{equation}

For $x,y \in \Delta^o$ , we say x leads to y in $ \Delta^o$ if $V(x,y) =0$ , and we write $x<_{V} y$ . We say $x \in \Delta^o$ is V-chain recurrent if $x <_{V} x$ . The collection of V-chain recurrent points is denoted by $\mathcal{R}_V$ . For $x,y \in \Delta^o$ we say $x\sim_{V} y$ if $x<_{V} y$ and $y<_{V} x$ . Equivalence classes under $\sim_{V}$ will be called V-basic classes, and the equivalence class associated with an $x \in \mathcal{R}_V$ will be denoted by $[x]_V$ . For $x, y \in \mathcal{R}_V$ we say $[x]_V \prec [y]_V$ if $x <_{V} y$ . A V-basic class $[x]_V$ is said to be maximal if, whenever $y \in \mathcal{R}_V$ satisfies $[x]_V \prec [y]_V$ , we have that $y \in [x]_V$ . A maximal V-basic class is a called a V-quasiattractor. The following is the main result of this section.

Before proving Theorem 9 we will establish some basic results regarding $V(\cdot,\cdot)$ and $\mathcal{R}_{V}$ . The following lemma is a consequence of the stability properties of the ODE (4) studied in Lemma 3 and the property that low-cost trajectories closely follow the solution of the ODE.

Proof. For $n\ge 1$ , let

\begin{equation*}\tau(\phi_n) \doteq \inf\{t \in [0,T_n]\;:\; \mbox{dist}(\phi_n(t), \partial \Delta) \le \alpha_1 \mbox{ or } \|\phi_n(t)\| \ge M_1\},\end{equation*}

where the infimum is taken to be $T_n$ if the above set is empty. Note that the result holds trivially if the above set is empty for all but finitely many n. Now, arguing by contradiction, suppose the set is nonempty for infinitely many n. Consider the subsequence along which the above sets are nonempty, and denote the subsequence once more by $\{n\}$ . Also assume without loss of generality that $\gamma_n\doteq S(\phi_n(0), T_n, \phi_n) \le 1$ for every n.

We claim that there is a $\delta>0$ and $k_0\in \mathbb{N}$ such that $\tau(\phi_n) \ge \delta$ for all $n \ge k_0$ . Indeed, otherwise, by passing to a further subsequence (once more denoted by $\{n\}$ ) we can find a sequence $\delta_n \to 0$ such that for every n,

\begin{equation*}\phi_n(0) \in \mathcal{V}_{\alpha, K},\; \phi_n(\delta_n) \in [\mathcal{V}_{\alpha_1, K_1}]^c.\end{equation*}

Since $S(\phi_n(0), T_n, \phi_n)\le \gamma_n\le 1$ , we must have from the compactness-of-level-sets property in Theorem 3 that $\phi_n(0)$ and $\phi_n(\delta_n)$ converge along a subsequence to the same limit, which is a contradiction.

Let $\delta > 0$ be such that $\tau(\phi_n) \geq \delta$ for all sufficiently large n. For each such n let $\hat \tau_n \doteq \tau(\phi_n)-\delta$ , and define $\phi_n^*: [0,\delta] \to \Delta^o$ as $\phi_n^*(t) = \phi_n(t+\hat \tau_n)$ , $t \in [0,\delta]$ . Then $\phi_n^* \in C([0,\delta]: \mathcal{V}_{\alpha_1, K_1})$ . Also,

(24) \begin{align}\int_0^{\delta} L_{\alpha_1, K_1}({\phi}_n^*(t), \dot{\phi}_n^*(t))dt &= \int_0^{\delta} L({\phi}_n^*(t), \dot{\phi}_n^*(t))dt\nonumber\\[4pt]&\leq \int_0^{T_n}L(\phi_n(t), \dot{\phi}_n(t))dt = \gamma_n \le 1.\end{align}

In particular,

\begin{equation*}\{\phi_n^*\} \subset \cup_{x\in K_1}\Phi_{x,\alpha_1,K_1,\delta}(1).\end{equation*}

From Theorem 3, the latter set is compact, and so, along some subsequence, $\phi_n^*$ converges to some $\phi^*$ in $C([0,\delta]: \mathcal{V}_{\alpha_1, K_1})$ . Using the compactness of level sets again, we have from (24) and the fact that $\gamma_n\to 0$ that