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

TL;DR

This paper establishes multiscaling limit theorems for solutions of stochastic partial differential equations with cyclic long-range dependent initial conditions, providing explicit spectral representations and discussing limitations for higher Hermite ranks.

Contribution

It introduces new multiscaling limit theorems for SPDEs with cyclic long-range dependence and derives spectral and covariance representations of the limit fields.

Findings

Limit theorems for cyclic long-range dependence initial conditions

Spectral and covariance representations of limit fields

Numerical examples illustrating theoretical results

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.