think that in the case of synchronized saddle no stochastic resonance is possible. To study this in detail we need to have a more careful look at the escape time statistics. The problem we are facing is that the distribution of the escape times strictly speaking depends on the entrance time phase $u$. As we have no theoretical result for the distribution of $u$, we are developing here a technique to study the distribution of the escape conditioned on the entrance phase $u$, that is $p(t ,u)$.

### Kolmogorov-Smirnov Test

First we recall well known results about the Kolmogorov-Smirnov statistic and the Kolmogorov-Smirnov test [@ks_test_1]. Let $
\xi_1, \xi_2, \ldots, \xi_n
$ be $n$ independently and identically distributed real random variables, where each $\xi_i$ is distributed with CDF $F(\cdot)$. The empirical CDF $F_n(\cdot)$ is defined. This gives $$\begin{aligned}
P(\xi_i\leq x)=F(x)
\quad \text{and} \quad
F_n(x)=
\frac{1}{n}
\sum_{i=1}^n
\mathbf{1}_{(-\infty,x]}
(\xi_i)\end{aligned}$$ where $\mathbf{1}_{A}$ is the indicator function for a set $A$. Consider the following $$\begin{aligned}
D_n=
\left\Vert
F_n-F
\right\Vert_\infty
=\sup_{x\in\mathbb{R}}
\left|
\frac{1}{n}
\sum_{i=1}^n
\mathbf{1}_{(-\infty,x]}
(\xi_i)
-F(x)
\right|\end{aligned}$$ where $D_n$ is called the Kolmogorov-Smirnov statistic or KS statistic. We define what we mean by the null hypothesis.

Let $\xi_1,\xi_2,\ldots,\xi_n$ be $n$ real random variables. 