Eyring-Kramers formula for the mean exit time of non-Gibbsian elliptic processes: the non characteristic boundary case

In this work, we derive a new sharp asymptotic equivalent in the small temperature regime $h\to 0$ for the mean exit time from a bounded domain for the non-reversible process $dX\mathrm{\_}t=b(X\mathrm{\_}t)dt+\sqrt{h}dB\mathrm{\_}t$ under a generic orthogonal decomposition of $b$ and when the boundary of $\mathrm{\Omega }$ is assumed to be $\mathit{\text{non}}\mathit{\text{characteristic}}$. The main contribution of this work lies in the fact that we do not assume that the process $(X\mathrm{\_}t,t\ge 0)$ is $\mathit{\text{Gibbsian}}$. In this case, a new correction term characterizing the $\mathit{\text{non-Gibbsianness}}$ of the process appears in the equivalent of the mean exit time. The proof is mainly based on tools from spectral and semi-classical analysis.