the two wells, modelling Equation \[chap\_4:markov\] by a continuous time Markov Chain, considering six measures of stochastic resonance and a new statistical test called the conditional Kolmogorov-Smirnov Test.

Markov Chain Reduction
----------------------

Stochastic resonance usually involves studying transitions between two stable states. If one solely concentrate on the transitions times one can reduce the model to a continuous time Markov chain $Y_t^\epsilon$ with state space $Y^\epsilon_t=\pm 1$ symbolizing the two stable states. $$\begin{aligned}
X^\epsilon_t
\longrightarrow
Y^\epsilon_t\end{aligned}$$ Let $w_{l}(t)$ denote the position of the left well at time $t$ and $w_{r}(t)$ the position of the right well at time $t$. Note that $w_l(t)$ and $w_r(t)$ are also continuous in time. Let $R\in\mathbb{R}$ be constant. The reduction from the $X^\epsilon_t$ to the Markov Chain $Y^\epsilon_t$ is $$\begin{aligned}
Y^\epsilon_t&=
\left\{
\begin{array}{lll}
-1 & \text{if} &
\left|X^\epsilon_t-w_l(t)\right|\leq R\\[0.5em]
+1 & \text{if} &
\left|X^\epsilon_t-w_r(t)\right|\leq R\\[0.5em]
Y^\epsilon_s & \text{if} & \left|X^\epsilon_t-w_l(t)\right|> R \quad \text{and} \quad \left|X^\epsilon_t-w_r(t)\right|> R
\end{array}
\right.\end{aligned}$$ where $s$ is given by $$\begin{aligned}
s&=\max\left\{s_1,s_2\right\}\\
\text{where}\quad s_1&=\max_{s<t}\left\{s:\left|X^\epsilon_s-w_l(s)\right|\leq R\right\}\\
\text{where}\quad s_2&=\max_{s<t}\left\{s:\left|X^\epsilon_s-w_r(s)\right|\leq R\right\}\end{aligned}$$ When $Y^\epsilon_t=-1$ we say the particle is in the left well and when $Y^\epsilon_t=+1$ we say the particle is in the right well. Hence we keep $Y^\epsilon_t$ constant even when the particle is in 