First-exit-time probability density tails for a local height of a non-equilibrium Gaussian interface
Abstract
We study the long-time behavior of the probability density Qt of the first exit time from a bounded interval [-L,L] for a stochastic non-Markovian process h(t) describing fluctuations at a given point of a two-dimensional, infinite in both directions Gaussian interface. We show that Qt decays when t ∞ as a power-law $-1 - α, where α is non-universal and proportional to the ratio of the thermal energy and the elastic energy of a fluctuation of size L. The fact that α appears to be dependent on L, which is rather unusual, implies that the number of existing moments of Qt depends on the size of the window [-L,L]. A moment of an arbitrary order n, as a function of L, exists for sufficiently small L, diverges when L approaches a certain threshold value Ln, and does not exist for L > Ln. For L > L1, the probability density Qt is normalizable but does not have moments.