Eyring-Kramers Law for the Underdamped Langevin Process

Consider the underdamped Langevin process $(q(t),p(t){)}_{t\ge 0}$ in ${\mathbb{R}}^d\times {\mathbb{R}}^d$. We derive the low-temperature asymptotic of its mean-transition time between basins of attraction for a double-well potential. This asymptotic is called Eyring-Kramers law and often relies in the literature on Potential theory tools which are ill-defined for hypoelliptic processes like the underdamped Langevin process. In this work, we implement a novel approach which circumvents the use of these traditional methods.