1.2. HEURISTICS OF OUR MATHEMATICAL APPROACH 11 Figure 1.11. Solution trajectory of the diffusion tT (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]: |Xε t 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
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