# Sharp spectral asymptotics for non-reversible metastable diffusion processes

## TL;DR

This paper analyzes the spectral properties of non-reversible diffusion processes in low temperature regimes, revealing the number and behavior of small eigenvalues related to metastable states and providing Eyring-Kramers type formulas.

## Contribution

It establishes the existence and asymptotic behavior of small eigenvalues of the diffusion operator for non-reversible processes with Morse potential barriers.

## Key findings

- Exactly $n_0$ eigenvalues in the low-temperature limit
- Eigenvalues have exponentially small moduli
- Asymptotic behavior described by Eyring-Kramers formulas

## Abstract

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 $\{0\leq \operatorname{Re}(z)< \epsilon\}$, 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.