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Abstract
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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.
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Keywords
nonreversible overdamped Langevin dynamics, metastability,
spectral theory, semiclassical analysis, Eyring–Kramers
formulas
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Mathematical Subject Classification 2010
Primary: 35P15, 35Q82, 60J60, 81Q12, 81Q20
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Milestones
Received: 26 September 2019
Revised: 24 August 2020
Accepted: 15 September 2020
Published: 16 November 2020
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