A Drift-Diffusion Approach for Pearson Type IV Heavy-Tail Distributions

Abstract

Non-Gaussian stationary fluctuations with heavy tails and finite-order moment cutoffs are ubiquitous in non-equilibrium systems. Here, we derive the Pearson type IV distribution as the stationary solution of a standardized Fokker-Planck equation with linear drift and quadratic diffusion. We establish necessary and sufficient conditions for its emergence and show that the linear-quadratic structure, although sufficient, is not necessary. We also give explicit closed expressions for the cumulative distribution function, a moment recurrence relation, skewness, kurtosis, and mode. From a physical perspective, our results reveal that heavy tails and asymmetry originate from distinct features of state-dependent diffusion: its local nonlinear curvature and gradient, respectively. In particular, asymmetric stationary fluctuations can emerge even in the presence of a symmetric linear restoring drift. This framework provides a general stochastic description of asymmetric heavy-tailed fluctuations in non-equilibrium systems, including space plasma turbulence.