12 1. HEURISTICS
Figure 1.12. Solution trajectory of the diffusion XtT
ε
(ε)
in the
double-well periodic step potential.
have to explain the asymptotics of the minimal time a Brownian particle needs to
exit from the (frozen) starting well, say the left one. Freezing the potential at some
time s, the asymptotics of its exit time is derived from the classical large deviation
theory of randomly perturbed dynamical systems, see Freidlin and Wentzell [40].
Let us assume that U is locally Lipschitz continuous. We recall that for any t 0
the potential U(t, ·) has its minima at −1 and 1, separated by the saddle point
0. The law of the first exit time τ1
ε
= inf{t 0: Xt
ε
0} is described by some
particular functional related to large deviation. For t 0, we introduce the action
functional on the space of real valued continuous functions C([0,t], R) on [0,t] by
St
s(ϕ)
=

⎨1

2
t
0
˙ ϕ
u
+

∂x
U(s, ϕu)
2
du if ϕ is absolutely continuous,
+∞ otherwise,
which is non-negative and vanishes on the set of solutions of the ordinary differential
equation ˙ ϕ =

∂x
U(s, ϕ). Let x and y be real numbers. With respect to the
(frozen) action functional, we define the (frozen) quasipotential
Vs(x, y) = inf{St
s(ϕ):
ϕ C([0,t], R), ϕ0 = x, ϕt = y, t 0}
which represents the minimal work the diffusion with a potential frozen at time
s and starting in x has to do in order to reach y. To switch wells, the Brownian
particle starting in the left well’s bottom −1 has to overcome the barrier. So we let
V
s
= Vs(−1, 0).
This minimal work needed to exit from the left well can be computed explicitly,
and is equal to twice its depth at time s. The asymptotic behavior of the exit time
is expressed by
lim
ε→0
ε ln Exτ1
ε
= V
s
or in generalization of (1.6)
lim
ε→0
Px e
V s−δ
ε
τ1
ε
e
V s+δ
ε
= 1 for any δ 0 and x 0.
Let us now assume that the left well is the deeper one at time s. If the Brownian
particle has enough time to cross the barrier, i.e. if T (ε) e
V
s
ε
, then, generalizing
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