Multiscale asymptotic analysis of kernel-smoothed solutions to fractional Riesz-Bessel equations with random initial conditions
Pub. online: 18 August 2026
Type: Research Article
Received
28 May 2026
28 May 2026
Revised
7 August 2026
7 August 2026
Accepted
7 August 2026
7 August 2026
Published
18 August 2026
18 August 2026
Notes
The paper is dedicated to the 75th anniversary of Professor Nikolai Leonenko.
Abstract
This paper investigates fractional Riesz–Bessel equations with random initial conditions that exhibit either classical or cyclic long-range dependence. It studies zoom-in asymptotics for the corresponding solutions and establishes multiscaling limit theorems. It is known that for similar problems, non-degenerate multiscaling limits may not exist in general. The paper develops a kernel-smoothing approach for these equations and obtains non-degenerate limit fields under suitable normalisation and rescaling. It proves that the kernel-smoothed solutions converge weakly to Gaussian random fields, which are non-stationary in both time and space. Their stochastic integral representations and covariance functions are derived. The paper also analyses the regularity and dependence structure of the limit fields. In particular, under appropriate general assumptions on the smoothing kernel, the limits exhibit long-range dependence in time and short-range dependence in space. Numerical examples for the case of Matérn-type kernels are provided to illustrate the theoretical results.
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