Stationary states in 2D systems driven by bi-variate L\'evy noises

TL;DR

This paper explores the complex behavior of 2D systems driven by bi-variate $\alpha$-stable noises, revealing how spectral measures and potential steepness influence stationary states, including multimodality and local minima.

Contribution

It demonstrates that 2D systems driven by bi-variate $\alpha$-stable noises exhibit unique stationary states influenced by spectral measures and potential steepness, extending understanding beyond 1D cases.

Findings

Stationary states can be multimodal and have local minima at the origin.

Potential wells must be steep enough to produce bounded states.

Stationary state shapes reflect the symmetry of the noise's spectral measure.

Abstract

Systems driven by $\alpha$-stable noises could be very different from their Gaussian counterparts. Stationary states in single-well potentials can be multimodal. Moreover, a potential well needs to be steep enough in order to produce stationary states. Here, it is demonstrated that 2D systems driven by bi-variate $\alpha$-stable noises are even more surprising than their 1D analogs. In 2D systems, intriguing properties of stationary states originate not only due to heavy tails of noise pulses, which are distributed according to $\alpha$-stable densities, but also because of properties of spectral measures. Consequently, 2D systems are described by a whole family of Langevin and fractional diffusion equations. Solutions of these equations bear some common properties but also can be very different. It is demonstrated that also for 2D systems potential wells need to be steep enough in order to produce bounded states. Moreover, stationary states can have local minima at the origin. The shape of stationary states reflects symmetries of the underlying noise, i.e. its spectral measure. Finally, marginal densities in power-law potentials also have power-law asymptotics.