Exactly solvable nonlinear model with two multiplicative Gaussian colored noises

TL;DR

This paper presents an exactly solvable nonlinear model with two multiplicative Gaussian colored noises, revealing phenomena like anomalous diffusion and stochastic localization through analytical solutions of probability distributions.

Contribution

It introduces a novel exactly solvable nonlinear stochastic model with specific noise relations, providing explicit solutions for probability distributions and moments.

Findings

Demonstrates anomalous diffusion behavior

Shows stochastic localization of particles

Provides exact analytical expressions for distributions

Abstract

An overdamped system with a linear restoring force and two multiplicative colored noises is considered. Noise amplitudes depend on the system state $x$ as $x$ and $|x|^{\alpha}$. An exactly soluble model of a system is constructed due to consideration of a specific relation between noises. Exact expressions for the time-dependent univariate probability distribution function and the fractional moments are derived. Their long-time asymptotic behavior is investigated analytically. It is shown that anomalous diffusion and stochastic localization of particles, not subjected to a restoring force, can occur.