\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: $\tau_0\coloneqq 0$, $\sigma_n \coloneqq \inf\{t>\tau_{n-1}: \mu_N(t) \in C\}$, $\tau_n \coloneqq \inf\{t > \sigma_n : \mu_N(t) \in \gamma\}$, and define $Z^N_n \coloneqq \mu_N(\tau_n)$, which is a discrete time Markov chain on the state space $\gamma \cap M^N_1(\mathcal{Z})$. For a measurable set $A \in M_1(\mathcal{Z})$, we define the stopping time $\tau_A \coloneqq \inf\{t > 0: \mu_N(t) \notin A\}$, which denotes the time exit from the set $A$. Finally, for a subset $W \subset L$, we define the stopping time $\hat{\tau}_W \coloneqq \inf\{t >0 : \mu_N(t) \in \cup_{i \in W}\gamma_i\}$, and $\bar{\tau}_W \coloneqq \inf \{t > 0: \mu_N(t) \in \cup_{i \in L \setminus W} \gamma_i \}$, 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 