Modern Stochastics: Theory and Applications

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.

We investigate the fractional Riemann-Liouville integral of a general continuous Gaussian process, focusing first on the functional weak convergence of suitably rescaled processes. Building on these results and recent theoretical advances, we deduce functional large deviation principles. For small times, the asymptotic behavior depends solely on the covariance of the underlying process at zero, while for large times, appropriate rescaling of the covariance is required and additional regularity assumptions must be imposed. A central aspect of our work is the explicit characterization of the reproducing kernel Hilbert spaces of the fractionally integrated processes, including concrete formulas for the related norms. As a notable special case, when the underlying Gaussian process is Gauss-Markov, the reproducing kernel Hilbert spaces and the covariance structure can be described explicitly, providing precise insights into the corresponding rate functions.