From Weakly Chaotic Dynamics to Deterministic Subdiffusion via Copula Modeling

TL;DR

This paper introduces a copula-based approach to connect weakly chaotic dynamics with deterministic subdiffusion, revealing that their joint distribution involves an extreme value copula with Mittag-Leffler marginals, supported by theoretical and numerical evidence.

Contribution

It develops a novel copula modeling framework linking chaotic dynamics and subdiffusion distributions, including methods for exact distribution calculation and validation through simulations.

Findings

Subdiffusion tail distribution is an extreme value copula with Mittag-Leffler marginals.

Decomposition of jumps distribution into bivariate form using ergodic theory and inference.

Numerical simulations confirm the theoretical copula model's accuracy.

Abstract

Copula modeling consists in finding a probabilistic distribution, called copula, whereby its coupling with the marginal distributions of a set of random variables produces their joint distribution. The present work aims to use this technique to connect the statistical distributions of weakly chaotic dynamics and deterministic subdiffusion. More precisely, we decompose the jumps distribution of Geisel-Thomae map into a bivariate one and determine the marginal and copula distributions respectively by infinite ergodic theory and statistical inference techniques. We verify therefore that the characteristic tail distribution of subdiffusion is an extreme value copula coupling Mittag-Leffler distributions. We also present a method to calculate the exact copula and joint distributions in the case where weakly chaotic dynamics and deterministic subdiffusion statistical distributions are already known. Numerical simulations and consistency with the dynamical aspects of the map support our results.