Sharp spectral asymptotics for non-reversible metastable diffusion processes

Let $U_h:\\mathbb R^{d}\\to \\mathbb R^{d}$ be a smooth vector field and consider the associated overdamped Langevin equation $$dX_t=-U_h(X_t)\\,dt+\\sqrt{2h}\\,dB_t$$ in the low temperature regime $h\\rightarrow 0$. In this work, we study the spectrum of the associated diffusion $L=-h\\Delta+U_h\\cdot\\nabla$ under the assumptions that $U_h=U_{0}+h\\nu$, where the vector fields $U_{0}:\\mathbb R^{d}\\to \\mathbb R^{d}$ and $\\nu:\\mathbb R^{d}\\to \\mathbb R^{d}$ are independent of $h\\in(0,1]$, and that the dynamics admits $e^{-\\frac Vh}$ as an invariant measure for some smooth function $V:\\mathbb{R}^d\\rightarrow\\mathbb{R}$. Assuming additionally that $V$ is a Morse function admitting $n_0$ local minima, we prove that there exists $\\epsilon>0$ such that in the limit $h\\to 0$, $L$ admits exactly $n_0$ eigenvalues in the strip $\\{\\operatorname{Re}(z)< \\epsilon\\}$ which have moreover exponentially small moduli. Under a generic assumption on the potential barriers of the Morse function $V$, we also prove that the asymptotic behaviors of these small eigenvalues are given by Eyring-Kramers type formulas.