Exponentially distributed noise—its correlation function and its effect on nonlinear dynamics

Abstract

We propose a simple Langevin equation as a generator for a noise process with Laplace-distributed values (pure exponential decays for both positive and negative values of the noise). We calculate explicit expressions for the correlation function, the noise intensity, and the correlation time of this noise process and formulate a scaled version of the generating Langevin equation such that correlation time and variance or correlation time and noise intensity for the desired noise process can be exactly prescribed. We then test the effect of the noise distribution on a classical escape problem: the Kramers rate of an overdamped particle out of the minimum of a cubic potential. We study the problem both for constant variance and constant intensity scalings and compare to an Ornstein–Uhlenbeck process with the same noise parameters. We demonstrate that specifically at weak fluctuations, the Laplace noise induces more frequent escapes than its Gaussian counterpart while at stronger noise the opposite effect is observed.