First-exit-time probability density tails for a local height of a non-equilibrium Gaussian interface

TL;DR

This paper investigates the long-time tail behavior of the first exit time probability density for a non-Markovian Gaussian interface, revealing non-universal power-law decay and size-dependent moments.

Contribution

It introduces a detailed analysis of the tail decay and moment existence for the first exit time density in a non-equilibrium Gaussian interface, highlighting non-universality and size dependence.

Findings

Q_t decays as a power-law with a non-universal exponent.

The existence of moments depends on the interval size L.

For large L, Q_t is normalizable but moments do not exist.

Abstract

We study the long-time behavior of the probability density Q_t 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 Q_t decays when t \to \infty as a power-law $^{-1 - \alpha}, where \alpha 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 \alpha appears to be dependent on L, which is rather unusual, implies that the number of existing moments of Q_t 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 L_n, and does not exist for L > L_n. For L > L_1, the probability density Q_t is normalizable but does not have moments.