Switching diffusivity selects Pareto tail exponent in random growth with redistribution

Maxence Arutkin; Alexandre Vallée

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arXiv · 2605.19464

Switching diffusivity selects Pareto tail exponent in random growth with redistribution

Maxence Arutkin·Alexandre Vallée

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Abstract

Random multiplicative growth with redistribution generates stationary Pareto wealth tails in the Bouchaud-Mézard 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 switching diffusivity and ask how these fluctuations select the Pareto stationary tail. For a geometric Brownian motion with switching diffusivity, the long-time Gaussian limit holds when the redraw law has finite mean and variance. The asymptotic variance retains a contribution from diffusivity persistence. With redistribution and a general redraw law, the stationary large-wealth problem is characterized by a spectral condition for admissible algebraic modes. 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-Mézard limit.

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Maxence Arutkin, Alexandre Vallée. 2026-09-18. Switching diffusivity selects Pareto tail exponent in random growth with redistribution. https://arxiv.org/abs/2605.19464

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