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→ 0 for the mean exit time from a bounded domain for the non-reversible process dXₜ=b(Xₜ)dt + √ h \, dBₜ under a generic orthogonal decomposition of b and when the boundary of $Ω$ is assumed to be non characteristic. The main contribution of this work lies in the fact that we do not assume that the process (Xₜ,t≥ 0) is Gibbsian. In this case, a new correction term characterizing the 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.