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

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.

💡 Research Summary

This paper presents a rigorous asymptotic analysis of solutions to fractional Riesz-Bessel equations (FRBE) driven by random initial conditions exhibiting long-range dependence. The FRBE is a fractional partial differential equation of the form ∂^β u(t,x)/∂t^β = -μ(I-Δ)^{γ/2}(-Δ)^{α/2} u(t,x), where the operators represent fractional derivatives in time (order β ∈ (0,1]) and composite Riesz-Bessel operators in space (orders α≥0, γ>0). The initial condition u(0,x)=ξ(x) is a zero-mean Gaussian random field whose covariance function is specifically designed to have a spectral density with singularities not only at the zero frequency (classical long-range dependence) but also at non-zero frequencies ±w_j (cyclic or seasonal long-range dependence).

The core of the analysis lies in studying the limiting distribution of the rescaled solution field U_ε(t,x) = ε^{-β/(2α)} u(t/ε, x/ε^{β/α}) as the scaling parameter ε → 0. Using spectral representation techniques and properties of the Mittag-Leffler function (which naturally arises from solving the fractional differential equation), the authors prove that under the condition α > 1/2 and when the initial condition possesses only cyclic long memory (i.e., spectral singularity at non-zero frequencies), the finite-dimensional distributions of U_ε(t,x) converge weakly to those of a zero-mean Gaussian random field U_0(t,x). This limit field has an explicit spectral representation given in Theorem 3.1. A key finding is that the limiting field is stationary in the spatial variable x but non-stationary in the temporal variable t, a consequence of the memory effect inherent in the fractional-time derivative. The explicit covariance structure of the limit field is derived.

Furthermore, the paper establishes multiscaling limit theorems for the case of regularly varying asymptotics. This extends the results to more general spectral behaviors beyond pure power laws, allowing for a unified treatment of scaling limits under different spatial and temporal scaling regimes. This part addresses mixed scenarios of classical and cyclic long memory. The theoretical developments are complemented by a numerical example, demonstrating the practical implications and validity of the asymptotic results. The work bridges fractional calculus, random field theory, and limit theorems, providing new insights into the long-term behavior of complex spatio-temporal systems with memory and oscillatory correlation structures.