Michael Eckhoff
Precise Asymptotics of Small Eigenvalues of Reversible Diffusions
in the Metastable Regime
(442K, postscript)
ABSTRACT. We investigate the connection between metastability and the
spectrum near zero corresponding to the elliptic, second-order,
differential operator
$L_\e\equiv -\e\D+\nabla F\cdot\nabla$, $\e>0$,
with $F:\R^d\to\R$. For generic $F$ to each local minimum of $F$
there corresponds an eigenvalue of $L_\e$ which converges to
zero exponentially fast as $\e\downarrow 0$. Modulo errors
of exponentially small order in $\e$ this eigenvalue is
given as the inverse of the expected metastable relaxation
time. The corresponding eigenstate, which may be
viewed as a metastable state, is highly concentrated in the
basin of attraction of this trap.