12 1. HEURISTICS Figure 1.12. Solution trajectory of the diffusion tT (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 Ss(ϕ) t = ⎨1 2 t 0 ˙ 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 ˙ = ∂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{Ss(ϕ): t 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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