the proof.

### Perfect Phase Approximation of Probability Density Function of Escape Times {#chap_approx_pdf}

The PDF for the escape times derived in Theorem \[chap\_4\_pdf\_thm\] had to differentiate between left and right escapes and are conditioned on the time $u$ of entrance into the well. Suppose now that $t$ is the escape time from any well, which does not differentiate between left and right escape. Note that $t$ is the actual time it takes to escape from a well and is not a time coordinate. The PDF for $t$ is given by $$\begin{aligned}
p_{tot}(t)=
\frac{1}{2}
\int_{0}^{T}p_-(t+u,u)m_-(u)+p_+(t+u,u)m_+(u)\,du\end{aligned}$$ This is because after a long time has elapsed we would expect that many transitions would have occurred between left and right. The number of transitions escaping from the left and right should be roughly the same. The $m_-(u)$ is a PDF for the time of entrance into the left well and the $m_+(u)$ is a PDF for the time of entrance into the right well. We may not have explicit expressions for $m_-(u)$ and $m_+(u)$. We derive an approximate expression for $p_{tot}$ without an explicit expressions for $m_-(u)$ and $m_+(u)$. Let $m_-(u)$ and $m_+(u)$ be approximated by $$\begin{aligned}
m_-(u)\approx\delta\left(u-T/2\right)
\quad \text{where} \quad 
m_+(u)\approx
\frac{1}{2}\delta\left(u\right)
