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 $$d{X}_t=-U_h(X_t)dt+\sqrt{2h}d{B}_t$$ in the low temperature regime $h\to 0$. In this work, we study the spectrum of the associated diffusion $L=-h\mathrm{\Delta }+U_h\cdot \mathrm{\nabla }$ under the assumptions that $U_h=U_0+h\nu$, where the vector fields $U_{0}:\mathbb R^{d}\to \mathbb R^{d}$and$\nu:\mathbb R^{d}\to \mathbb R^{d}$areindependentof$h\in(0,1]$, and that the dynamics admits $e^{-\frac{V}{h}}$ as an invariant measure for some smooth function $V:{\mathbb{R}}^d\to \mathbb{R}$. Assuming additionally that $V$ is a Morse function admitting $n_0$ local minima, we prove that there exists $ϵ>0$ such that in the limit $h\to 0$, $L$ admits exactly $n_0$ eigenvalues in the strip $\{0\le \mathrm{Re}(z)<ϵ\}$, 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.