1.2. HEURISTICS OF OUR MATHEMATICAL APPROACH 11

Figure 1.11. Solution trajectory of the diffusion XtT

ε

(ε)

in the

time independent double-well potential U.

1, with V v (and U2 wells with respectively opposite roles). Let us briefly point

out the main features of the transition times for periodic step potentials described

in (1.6). According to (1.6) the exponential rate of the transition time from −1 to

1 in U1 in the small noise limit is asymptotically given by exp(

V

ε

), as long as the

time scale T of the diffusion allows no switching of the potential states before, i.e.

as long as T = T (ε) exp(

V

ε

). Accordingly, the transition time from 1 to −1 in

U1 is given by exp(

v

ε

), as long as T = T (ε) exp(

v

ε

). Similar statements hold for

transitions between states of U2. It is therefore also plausible that (1.8) generalizes

to the following statement due to Freidlin [39, Theorem 2].

Suppose

(1.10) lim

ε→0

ε ln T (ε) V.

Define

φ(t) =

−1, t ∈ [k, k +

1

2

),

1, t ∈ [k +

1

2

, k + 1), k ∈ N0.

Then the Lebesgue measure of the set

(1.11) t ∈ [0, 1]: |Xt

ε

T (ε)

− φ(t)| δ

tends to 0 as ε → 0 in Px-probability, for any δ 0, x ∈ R.

Again, this just means that the process Xε, run in a time scale T (ε) large

enough, will spend most of the time in the minimum of the deepest potential well

which is given by the time periodic function φ. Excursions to the other well are

exponentially negligible on this scale, as ε → 0. The picture is typically the one

depicted in Figure 1.12.

1.2.4. Periodic potentials and quasi-deterministic motion. Since the

function φ appearing in the previous theorem is already discontinuous, it is plausible

that the step function potential is in fact a reasonable approximation of the general

case of continuously and (slowly) periodically changing potential functions. It is

intuitively clear how the result has to be generalized to this situation. We just have

to replace the periodic step potentials by potentials frozen along a partition of the

period interval on the potential state taken at its starting point, and finally let the

mesh of the partition tend to 0. To continue the discussion in the spirit of the

previous section and with the idea of instantaneously frozen potential states, we