Stochastic Maps, Continuous Approximation and Stable Distribution

TL;DR

This paper develops a continuous approximation framework for non-linear stochastic maps with Gaussian noise, deriving Langevin equations and explicit formulas for stable distributions, validated by numerical simulations.

Contribution

It introduces a novel continuum approximation for stochastic maps, deriving Langevin equations with multiplicative noise and explicit stable distribution formulas.

Findings

Langevin equations derived from stochastic maps with Gaussian noise

Explicit formulas for stable distributions of stochastic maps

Good agreement between theoretical results and numerical simulations

Abstract

A continuous approximation framework for non-linear stochastic as well as deterministic discrete maps is developed. For the stochastic map with uncorelated Gaussian noise, by successively applying the It\^o lemma, we obtain a Langevin type of equation. Specifically, we show how non-linear maps give rise to a Langevin description that involves multiplicative noise. The multiplicative nature of the noise induces an additional effective force, not present in the absence of noise. We further exploit the continuum description and provide an explicit formula for the stable distribution of the stochastic map and conditions for its existence. Our results are in good agreement with numerical simulations of several maps.