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Multiscaling asymptotic behavior of solutions to random high-order heat equations

Maha Mosaad A Alghamdi, Nikolai Leonenko, Andriy Olenko

TL;DR

This work analyzes high-order heat-type PDEs with random initial data exhibiting long-memory and cyclic long-range dependence due to spectral singularities at zero and nonzero frequencies. Using spectral methods and scaling, it proves that suitably rescaled solutions converge to Gaussian random fields, with even-order equations yielding limits stationary in space but not in time, and odd-order equations requiring kernel smoothing to obtain stationary Gaussian limits. The limiting fields are characterized by explicit spectral representations and covariances, including Fox–Wright function components, and depend critically on parity and the presence of zero-frequency singularities. Numerical experiments illustrate the theoretical limits and reveal how the multiscaling behavior evolves with the equation order and spectral structure, underscoring the approach's relevance for modeling systems with both long-range and cyclic dependence.

Abstract

This paper studies high-order partial differential equations with random initial conditions that have both long-memory and cyclic behavior. The cases of random initial conditions with the spectral singularities, both at zero (representing classical long-range dependence) and at non-zero frequencies (representing cyclic long-range dependence), are investigated. Using spectral methods and scaling techniques, it is proved that, after proper rescaling and normalization, the solutions converge to Gaussian random fields. For each type of equation, spectral representations and covariance functions of limit fields are given. For odd-order equations, we apply the kernel averaging of solutions to obtain nonexplosive and nondegenerate limits. It is shown that the different limit fields are determined by the even or odd orders of the equations and by the presence or absence of a spectral singularity at zero. Several numeric examples illustrate the obtained theoretical results.

Multiscaling asymptotic behavior of solutions to random high-order heat equations

TL;DR

This work analyzes high-order heat-type PDEs with random initial data exhibiting long-memory and cyclic long-range dependence due to spectral singularities at zero and nonzero frequencies. Using spectral methods and scaling, it proves that suitably rescaled solutions converge to Gaussian random fields, with even-order equations yielding limits stationary in space but not in time, and odd-order equations requiring kernel smoothing to obtain stationary Gaussian limits. The limiting fields are characterized by explicit spectral representations and covariances, including Fox–Wright function components, and depend critically on parity and the presence of zero-frequency singularities. Numerical experiments illustrate the theoretical limits and reveal how the multiscaling behavior evolves with the equation order and spectral structure, underscoring the approach's relevance for modeling systems with both long-range and cyclic dependence.

Abstract

This paper studies high-order partial differential equations with random initial conditions that have both long-memory and cyclic behavior. The cases of random initial conditions with the spectral singularities, both at zero (representing classical long-range dependence) and at non-zero frequencies (representing cyclic long-range dependence), are investigated. Using spectral methods and scaling techniques, it is proved that, after proper rescaling and normalization, the solutions converge to Gaussian random fields. For each type of equation, spectral representations and covariance functions of limit fields are given. For odd-order equations, we apply the kernel averaging of solutions to obtain nonexplosive and nondegenerate limits. It is shown that the different limit fields are determined by the even or odd orders of the equations and by the presence or absence of a spectral singularity at zero. Several numeric examples illustrate the obtained theoretical results.
Paper Structure (5 sections, 6 theorems, 86 equations, 9 figures)

This paper contains 5 sections, 6 theorems, 86 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The case of even-order equations
  4. Limit theorems for odd order equations
  5. Conclusion

Key Result

Theorem 1

alghamdi2024 Consider the random field $u(t,x),$$t>0,$$x\in \mathbb R,$ that is the solutions of the initial value problem (classical heat equation) and (initial Airy) with $M(x)\equiv 0.$ If Assumption AssA holds true, then, for $\varepsilon \to 0,$ the finite-dimensional distributions of the rand converge weakly to the finite-dimensional distributions of the zero-mean Gaussian random field wit

Figures (9)

  • Figure 1: Covariance and spectral structure of the initial condition $\eta(x)$
  • Figure 2: The case of the 4th-order heat equation with $A_{0}=0$
  • Figure 3: Covariance structures of $U_{0}(t,x)$ for $m$th-order heat equations with $A_{0}=0$
  • Figure 4: The case of the 4th-order heat equation with $A_{0}\neq0$
  • Figure 5: Covariance structures of $U_{0}(t,x)$ for $m$th-order heat equations with $A_{0}\neq0$
  • ...and 4 more figures

Theorems & Definitions (22)

  • Theorem 1
  • Example 1
  • Lemma 1
  • Remark 1
  • Theorem 2
  • proof
  • Remark 2
  • Example 2
  • Theorem 3
  • proof
  • ...and 12 more