Tsallis statistics and Langevin equation with multiplicative noise in different orders of prescription

TL;DR

This paper explores the relationship between Tsallis statistics and Langevin equations with multiplicative noise, analyzing different discretization prescriptions and their impact on stationary solutions, with applications to population growth models.

Contribution

It establishes a general connection between Tsallis distributions and Langevin equations with multiplicative noise across various discretization schemes.

Findings

Tsallis distribution relates to Langevin equations with multiplicative noise.

The Tsallis index and prescription parameter depend on drift and diffusion coefficients.

Application to population growth models demonstrates the distribution's descriptive power.

Abstract

Usually discussions on the question of interpretation in the Langevin equation with multiplicative white noise are limited to the Ito and Stratonovich prescriptions. In this work, a Langevin equation with multiplicative white noise and its Fokker-Planck equation are considered. From this Fokker-Planck equation a connection between the stationary solution and the Tsallis distribution is obtained for different orders of prescription in discretization rule for the stochastic integrals; the Tsallis index $q$ and the prescription parameter (\lambda) are determined with the drift and diffusion coefficients. The result is quite general. For application, one shows that the Tsallis distribution can be described by a class of population growth models subject to the linear multiplicative white noise.