Diffusing diffusivity selects Pareto tail exponent in random growth with redistribution

TL;DR

This paper investigates how fluctuating diffusivity influences the Pareto tail exponent in models of wealth distribution with redistribution, revealing that high-diffusivity states dominate large-wealth events and affect the tail behavior.

Contribution

It introduces a model with diffusing diffusivity in wealth growth processes and derives how these fluctuations select the Pareto tail exponent, extending previous fixed-noise models.

Findings

High-diffusivity states dominate rare large-wealth events.

Exact tail analysis shows the Pareto exponent interpolates between limits.

Diffusing diffusivity leads to stationary Pareto tails influenced by volatility fluctuations.

Abstract

Random multiplicative growth with redistribution generates stationary Pareto wealth tails in the Bouchaud-M\'ezard model, but assumes a fixed multiplicative noise intensity. This is restrictive for physical and financial growth processes, where volatility (diffusivity) is often fluctuating. We replace the constant noise intensity by a diffusing diffusivity and ask how these fluctuations select the Pareto stationary tail. For a geometric Brownian motion with diffusing diffusivity, the effect is transient: log-returns show non-Gaussian short-time statistics but self-average to a Gaussian form at long times. With redistribution, the same persistence becomes stationary. Agents remaining in high-diffusivity states dominate rare large-wealth events, so the Pareto exponent is not obtained by replacing the diffusivity by its mean. For a two-state diffusivity, an exact tail analysis gives a Pareto exponent interpolating between the high-diffusivity slow-refresh limit and the mean-diffusivity fast-refresh Bouchaud-M\'ezard limit.