where $\left|\nabla^2(x)\right|$ denotes the determinant of the Hessian of the potential at the well $x$, $\left\Vert \nabla^2V(z_i) \right\Vert$ denotes the modulus of the determinant of the potential at the saddle $z_i$ and $\left|\lambda(z_i)\right|$ denotes the minimum eigenvalue of the Hessian of the potential at the saddle $z_i$. This gives the escape rate in the next order of approximation to be $$\begin{aligned}
R&=\sum_{i=1}^nR_i
=\sum_{i=1}^n k_ie^{-2\Delta V_i /\epsilon^2}\end{aligned}$$ which is rewritten as $$\begin{aligned}
R=
\frac{\sqrt{\left|\nabla^2V(x)\right|}}{2\pi}
\sum_{i=1}^n
\frac{\left|\lambda(z_i)\right|}{\sqrt{\left\Vert \nabla^2V(z_i) \right\Vert}}
\exp\left\{\frac{-2\left(V(z_i)-V(x)\right)}{\epsilon^2}\right\}\end{aligned}$$ The last order of approximation for higher noise $\epsilon$ is done by bounding the error on Kramers coefficient. This is $$\begin{aligned}
k_i=\frac{\sqrt{\left|\nabla^2V(x)\right|}}{2\pi}
\frac{\left|\lambda(z_i)\right|}{\sqrt{\left\Vert \nabla^2V(z_i) \right\Vert}}
\left(
\frac{1}{1+\mathcal{O}\left(\frac{\epsilon^2}{2}\ln\frac{\epsilon^2}{2}\right)}
\right)\end{aligned}$$ which is rewritten as $$\begin{aligned}
\label{chap_2_krammers}
\frac{1}{k_i}
=
\frac{2\pi}{\sqrt{\left|\nabla^2V(x)\right|}}
\frac{\sqrt{\left\Vert \nabla^2V(z_i) \right\Vert}}{\left|\lambda(z_i)\right|}
\left[
1+\mathcal{O}\left(\frac{\epsilon^2}{2}\ln\frac{\epsilon^2}{2}\right)
\right]\end{aligned}$$ The derivation of Kramers formula was done in [@Bovier02metastabilityin].

Oscillatory Potential {#chapter_oscil_times}
=====================

Let $V:\mathbb{R}^r\rightarrow\mathbb{R}$ be a potential with two wells. This potential is subjected to a periodic forcing $F\in\mathbb{R}^r$ with frequency $\Omega$ and perturbed by noise $\epsilon$, which is described by the SDE. $$\begin{aligned}
\dot{X^\epsilon_t}=-\nabla V+F\cos(\Omega t)+\epsilon \dot{W_t}
\label{chap_4:markov}\end{aligned}$$ where $W_t$ is a Wiener process in $\mathbb{R}^r$ and $F\in\mathbb{R}^r$. We call $X^\epsilon_t$ the diffusion case. In this Section we study Equation \[chap\_4:markov\] by considering the escape times between 