Diffusing diffusivity selects Pareto tail exponent in random growth with redistribution

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.