$N$ is a discrete number of periods. This leads us to define the invariant measure as the state probabilities in the limit as $N\longrightarrow\infty$ $$\begin{aligned}
\overline{\nu}(n):=\lim_{N\longrightarrow\infty}\nu(NT+n)\end{aligned}$$ The rate of convergence to the invariant measure would depend on the value of $p$ and $q$ themselves. Define the relaxation time $T_{relax}$ as the first time $t=T_{relax}$ such that $$\begin{aligned}
\left|
\overline{\nu}\left(T_{relax}\right)
-\nu\left(T_{relax}\right)
\right|
\leq
e^{-1}\end{aligned}$$ which is a measure of the rate of convergence to the invariant measure.

### Continuous Time Markov Chain - Alternating Saddles $p\neq q$

Notice that $p$ may be interpreted as the probability of escape from the left well and $q$ as the probability of escape from the right well. If $p$ and $q$ are cyclic over $[0,T]$, then this can be interpreted as modelling a potential with periodic forcing in continuous time.

\[chap\_4\_thm:equal\_contin\] Let $p\neq q$ and $t\geq0$. The state probabilities are given by $$\begin{aligned}
\nu_-(t)&=\frac{\nu_-(0)+\int_0^tq(s)\exp\left\{\int_0^sp(u)+q(u)\,du\right\}\,ds}{\exp\left\{\int_0^tp(u)+q(u)\,du\right\}}\\[0.5em]
\nu_+(t)&=\frac{\nu_+(0)+\int_0^tp(s)\exp\left\{\int_0^sp(u)+q(u)\,du\right\}\,ds}{\exp\left\{\int_0^tp(u)+q(u)\,du\right\}}\end{aligned}$$

\[corollary\_invariant\_measure\_contin\] For the state probabilities in Theorem \[chap\_4\_thm:equal\_contin\] the invariant measures are $$\begin{aligned}
\overline{\nu}_-(t)
&=
\frac{\int^t_0p(s)g(s)\,ds}{g(t)}
+\frac{\int^T_0p(s)g(s)\,ds}{g(t)\left(g(T)-1\right)}\\[0.5em]
\overline{\nu}_+(t)
&=
\frac{\int^t_0q(s)g(s)\,ds}{g(t)}
+\frac{\int^T_0q(s)g(s)\,ds}{g(t)\left(g(T)-1\right)}\\[0.5em]
\text{where}\quad
g(t)&=\exp\left\{\int^t_0p(u)+q(u)\,du\right\}\end{aligned}$$

We derive the invariant measure for $\overline{\nu}_-(t)$. The case for $\overline{\nu}_+(t)$ is similar. Consider the fact that $p(\cdot)$ and $q(\cdot)$ are cyclic on $[0,T]$ and let $i$ 