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.
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BibTeX
@misc{belardinelli2026driftdiffusion,
title = {{A Drift-Diffusion Approach for Pearson Type IV Heavy-Tail Distributions}},
author = {Daniele Belardinelli and Simone Benella and Mirko Stumpo and Giuseppe Consolini},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06749},
eprint = {2611.06749},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06749},
}
RIS
TY - UNPB
TI - A Drift-Diffusion Approach for Pearson Type IV Heavy-Tail Distributions
AU - Daniele Belardinelli
AU - Simone Benella
AU - Mirko Stumpo
AU - Giuseppe Consolini
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06749
UR - https://arxiv.org/abs/2611.06749
AB - 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.
ER -