Multiscaling Asymptotic Behaviour of Random Fields Driven by SPDEs

Abstract

This thesis studies the asymptotic behaviour of random fields which are solutions of stochastic partial differential equations with random initial conditions exhibiting long-range and cyclic long-range dependence. The main focus is on investigating multiscaling limits and their spectral and covariance structures. The cases of solutions to fractional, high-order, and Riesz–Bessel type equations are considered. The results are obtained for the cases of arbitrary multiple spectral singularities, which have not been considered before. First, the thesis investigates multiscaling limits for stochastic fractional equations with random initial conditions possessing cyclic long-memory. The corresponding rescaled solutions are shown to converge to Gaussian limit fields. Their explicit spectral and covariance representations are provided. Next, the thesis studies high-order heat equations with random initial conditions whose spectra have singularities at zero and at non-zero frequencies. Using spectral and scaling techniques, it is proved that the normalised solutions converge to Gaussian random fields determined by the presence or absence of a zerofrequency singularity. Kernel averaging is introduced for odd-order equations to obtain non-degenerate limits. Finally, the thesis analyses fractional Riesz–Bessel equations with initial conditions exhibiting both classical and cyclic long-range dependence. It is proved that the rescaled solutions converge to spatio-temporal Gaussian random fields that are stationary in space and non-stationary in time. In addition, multiscaling limit theorems are derived for the case of regularly varying asymptotics.