Interface Fluctuations for the D = 1 Stochastic Ginzburg–Landau Equation with Nonsymmetric Reaction Term

Front Fluctuations in One Dimensional Stochastic Phase Field Equations

“…(1.1) with λ = 0 but with γ > 0. This is well studied in the literature, [9,10,13], even though in slightly different contexts. The scaling procedure is the same as in Theorem 1.1 and it leads to the same limit law but without drift, i.e.…”

“…The term η n (8). In order to show η n (8) is negligible (as specified at the beginning of the present proof) we cannot use directly the a priori bounds, but we use equation (5.6) to get…”

“…(see [8] for the last inequality). It is easy to see that the estimates (B.3) and (B.4) also hold for z (n,⊥) .…”

“…In practice, however, it is better to work with a more "geometrical" definition. Following [10,9], we define a center ξ of m as a real number which minimizes the L 2 -norm of m −m x (as a function of x). Then ξ is such that…”

“…Then, if m has center ξ, the deviation v = m −m ξ by (2.2) has no component alongm ξ , hence the linearized evolution starting from v decays exponentially fast and correspondingly m converges tom ξ , so that the center of m is also the center of the instanton to which the linearized AC evolution converges. The pair {ξ, m −m ξ } is known as the Fermi coordinates of m. This notion of a center of a function plays an important role also in our proofs, so we will spend a few more words, recalling Proposition 3.2 in [9], which gives a sufficient condition for m = m(x) to have a center. …”