1.2. HEURISTICS OF OUR MATHEMATICAL APPROACH 11
Figure 1.11. Solution trajectory of the diffusion XtT
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(
), 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(
). Accordingly, the transition time from 1 to −1 in
U1 is given by exp(
), as long as T = T (ε) exp(
). 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].
ε ln T (ε) V.
−1, t ∈ [k, k +
1, t ∈ [k +
, k + 1), k ∈ N0.
Then the Lebesgue measure of the set
(1.11) t ∈ [0, 1]: |Xt
− 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