Let
be a smooth vector field and consider the associated overdamped Langevin
equation
in the low temperature regime
.
In this work, we study the spectrum of the associated diffusion
under the
assumptions that
,
where the vector fields
are independent of
, and
that the dynamics admits
as an invariant measure for some smooth function
. Assuming additionally
that
is a Morse function
admitting
local minima,
we prove that there exists
such that in the limit
,
admits exactly
eigenvalues in the
strip
, which have
moreover exponentially small moduli. Under a generic assumption on the potential barriers of the
Morse function
,
we also prove that the asymptotic behaviors of these small eigenvalues are given by
Eyring–Kramers type formulas.