Let
${U}_{h}:{\mathbb{R}}^{d}\to {\mathbb{R}}^{d}$
be a smooth vector field and consider the associated overdamped Langevin
equation
$$d{X}_{t}={U}_{h}\left({X}_{t}\right)dt+\sqrt{2h}d{B}_{t}$$
in the low temperature regime
$h\to 0$.
In this work, we study the spectrum of the associated diffusion
$L=h\Delta +{U}_{h}\cdot \nabla $ under the
assumptions that
${U}_{h}={U}_{0}+h\nu $,
where the vector fields
$${U}_{0}:{\mathbb{R}}^{d}\to {\mathbb{R}}^{d}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\nu :{\mathbb{R}}^{d}\to {\mathbb{R}}^{d}$$
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
$V:{\mathbb{R}}^{d}\to \mathbb{R}$. Assuming additionally
that
$V$ is a Morse function
admitting
${n}_{0}$ local minima,
we prove that there exists
$\mathit{\epsilon}>0$
such that in the limit
$h\to 0$,
$L$ admits exactly
${n}_{0}$ eigenvalues in the
strip
$\left\{0\le Re\left(z\right)<\mathit{\epsilon}\right\}$, 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.
