On asymptotic behavior of solutions to random fractional Riesz-Bessel equations with cyclic long memory initial conditions

TL;DR

This paper studies the long-term behavior of solutions to fractional Riesz-Bessel equations with random initial conditions exhibiting long-range and cyclic dependence, showing convergence to specific Gaussian random fields.

Contribution

It introduces asymptotic analysis of solutions with cyclic long memory initial conditions, revealing their convergence to spatio-temporal Gaussian fields with detailed spectral properties.

Findings

Rescaled solutions converge to Gaussian random fields

Limit fields are stationary in space, non-stationary in time

Multiscaling limit theorems are established for regularly varying cases

Abstract

This paper investigates fractional Riesz-Bessel equations with random initial conditions. The spectra of these random initial conditions exhibit singularities both at zero frequency and at non-zero frequencies, which correspond to the cases of classical long-range dependence and cyclic long-range dependence, respectively. Using spectral methods and asymptotic theory, it is shown that the rescaled solutions of the equations converge to spatio-temporal Gaussian random fields. The limit fields are stationary in space and non-stationary in time. The covariance and spectral structures of the resulting asymptotic random fields are provided. The paper further establishes multiscaling limit theorems for the case of regularly varying asymptotics. A numerical example illustrating the theoretical results is also presented.