Multiscaling limit theorems for stochastic FPDE with cyclic long-range dependence

Abstract

The paper studies solutions of stochastic partial differential equations with random initial conditions. First, it overviews some of the known results on scaled solutions of such equations and provides several explicit motivating examples. Then, it proves multiscaling limit theorems for renormalized solutions for the case of initial conditions subordinated to random processes with cyclic long-range dependence. Two cases of stochastic partial differential equations are examined. The spectral and covariance representations for the corresponding limit random fields are derived. Additionally, it is discussed why analogous results are not valid for subordinated cases with Hermite ranks greater than 1. Numerical examples that illustrate the obtained theoretical results are presented.

22 pages, 4 figures

References in corpus (9)

• A Caputo fractional derivative of a function with respect to another function
• Caputo derivatives of fractional variable order: numerical approximations
• On some properties of the Mittag-Leffler function $E_{\alpha }(-t^{\alpha })$, completely monotone for $t>0$ with $0<\alpha <1$
• Limit theorems for weighted nonlinear transformations of Gaussian stationary processes with singular spectra
• Statistical challenges in the analysis of Cosmic Microwave Background radiation
• Random spherical hyperbolic diffusion
• Limit theorems for weighted functionals of cyclical long-range dependent random fields
• Spherically Restricted Random Hyperbolic Diffusion
• Analysis of Spherical Monofractal and Multifractal Random Fields