Abstract

Let $U_h:{ℝ}^d\to {ℝ}^d$ be a smooth vector field and consider the associated overdamped Langevin equation

in the low temperature regime $h\to 0$. 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

are independent of $h\in \left(0,1\right]$, and that the dynamics admits $e^{-\frac{V}{h}}dx$ as an invariant measure for some smooth function $V:{ℝ}^d\to ℝ$. 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 $\left\{0\le Re\left(z\right)<𝜖\right\}$, 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.

Keywords

Mathematical Subject Classification 2010

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