1.4. DIFFUSIONS WITH CONTINUOUSLY VARYING POTENTIALS 19

in higher dimensions, as well as to potential diffusions themselves. This will be done

in detail in Chapter 3. However, it will turn out that diffusions and their reduced

dynamics Markov chains are not as similar as expected. Indeed, in a reasonably

large time window around the resonance point for Y

ε,

the tuning picture of the

spectral power amplification for the diffusion is different. Under weak regularity

conditions on the potential, it exhibits strict monotonicity in the window. Hence

optimal tuning points for diffusion and Markov chain differ essentially. In other

words, the diffusion’s SPA tuning behavior is not robust for passage to the reduced

model (see Chapter 3, subsection 3.4.4). This strange deficiency is diﬃcult to

explain. The main reason of this subtle effect appears to be that the diffusive

nature of the Brownian particle is neglected in the reduced model. In order to

point out this feature, we may compute the SPA coeﬃcient of g(Xε) where g is

a particular function designed to cut out the small fluctuations of the diffusion in

the neighborhood of the bottoms of the wells, by identifying all states there. So

g(x) = −1 (resp. 1) in some neighborhood of −1 (resp. 1) and otherwise g is the

identity. This results in

˜X

η (ε, T ) =

1

0

Eν g(XsT

ε

)

e2πis

ds

2

, ε 0, T ≥ 0.

In the small noise limit this quality function admits a local maximum close to the

resonance point of the reduced model: the growth rate of Topt(ε) is also given by

the arithmetic mean of the wells’ depths. So the lack of robustness seems to be due

to the small fluctuations of the particle in the wells’ bottoms.

In any case, this clearly calls for other quality measures to be used to transfer

properties of the reduced model to the original one. Our discussion indicates that

due to their emphasis on the pure transition dynamics, a second more probabilis-

tic family of quality measures should be used. This will be made mathematically

rigorous in Chapter 4. The family is composed of quality measures based on tran-

sition times between the domains of attraction of the local minima, residence times

distributions measuring the time spent in one well between two transitions, or inter-

spike times. To explain its main features there is no need to restrict to landscapes

frozen in time independent potential states on half period intervals. So from now

on the potential U(t, x) is a continuous function in (t, x). For simplicity — remain-

ing in the one-dimensional case — we further suppose that its local minima are

given by ±1, and its only saddle point by 0, independently of time. So the only

meta-stable states on the time axis are ±1. Let us denote by

v−(t)

2

(resp.

v+(t)

2

the

depth of the left (resp. right) well. These function are continuous and 1-periodic.

We shall assume that they are strictly monotonous between their global extrema.

Let us now consider the motion of the Brownian particle in this landscape. As

in the preceding case, according to Freidlin’s law of quasi-deterministic motion its

trajectory gets close to the global minimum, if the period is large enough. The

exponential rate of the period should be large enough to permit transitions: if

T (ε) =

eμ/ε

with μ ≥ maxi=± supt≥0 vi(t) meaning that μ is larger than the max-

imal work needed to cross the barrier, then the particle often switches between

the two wells and should stay close to the deepest position in the landscape. By

defining φ(t) = 2I{v+(t)v−(t)} − 1, in the small noise limit the Lebesgue measure

of the set

{t ∈ [0, 1]: |XtT − φ(t)| δ}