Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Intermittency for the stochastic heat and wave equations with generalized fractional noise

Ruxiao Qian

TL;DR

The dissertation investigates intermittency for stochastic heat and wave equations driven by generalized fractional noise, first showing that the Lyapunov exponents for the equation with Dobrić-Ojeda noise coincide with those for standard fractional noise. It then introduces a generalized fractional-noise framework that interpolates between the standard fractional and the DO noise, proving that the Lyapunov exponents depend only on the asymptotic variance order and are insensitive to the noise’s covariance structure in this setting. Methodologically, the work combines Walsh integration for the DO noise, Poisson-process moment representations, and Malliavin-calculus-based analysis for the generalized noise, yielding explicit moment formulas and sharp upper and lower bounds. The results suggest a form of universality: intermittency is governed by the time-variance order, not the precise temporal covariance, with consequences for robustness of intermittent behavior across a broad class of Gaussian noises and potential extensions to related SPDEs.

Abstract

We are looking at the stochastic heat and wave equations with different types of fractional noise. We are interested in the intermittency property and Lyapunov exponent for the solution. First we look at the equation driven by the Dobric-Ojeda noise and we show that the Lyapunov exponent matches that of the equation driven by standard fractional noise as obtained by Hu, Huang, Nualart, Tindel (2015) and Balan, Conus (2016). In the second part, we introduce a generalized fractional noise that includes both standard fractional noise and Dobric-Ojeda noise. It shows that, in this specific situation, the correlation structure of the noise does not change the Lyapunov exponent. We conjecture that this result would hold more generally

Intermittency for the stochastic heat and wave equations with generalized fractional noise

TL;DR

The dissertation investigates intermittency for stochastic heat and wave equations driven by generalized fractional noise, first showing that the Lyapunov exponents for the equation with Dobrić-Ojeda noise coincide with those for standard fractional noise. It then introduces a generalized fractional-noise framework that interpolates between the standard fractional and the DO noise, proving that the Lyapunov exponents depend only on the asymptotic variance order and are insensitive to the noise’s covariance structure in this setting. Methodologically, the work combines Walsh integration for the DO noise, Poisson-process moment representations, and Malliavin-calculus-based analysis for the generalized noise, yielding explicit moment formulas and sharp upper and lower bounds. The results suggest a form of universality: intermittency is governed by the time-variance order, not the precise temporal covariance, with consequences for robustness of intermittent behavior across a broad class of Gaussian noises and potential extensions to related SPDEs.

Abstract

We are looking at the stochastic heat and wave equations with different types of fractional noise. We are interested in the intermittency property and Lyapunov exponent for the solution. First we look at the equation driven by the Dobric-Ojeda noise and we show that the Lyapunov exponent matches that of the equation driven by standard fractional noise as obtained by Hu, Huang, Nualart, Tindel (2015) and Balan, Conus (2016). In the second part, we introduce a generalized fractional noise that includes both standard fractional noise and Dobric-Ojeda noise. It shows that, in this specific situation, the correlation structure of the noise does not change the Lyapunov exponent. We conjecture that this result would hold more generally

Paper Structure

This paper contains 31 sections, 34 theorems, 336 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Heat and Wave equations
  3. Stochastic heat and wave equations
  4. Intermittency
  5. An alternative fractional noise: the DO noise
  6. Main results and outline
  7. Existence and Uniqueness of solution for Dobrić-Ojeda noise
  8. Basic settings
  9. Existence and Uniqueness of the solution
  10. Moment formula for the DO noise case
  11. Preliminaries
  12. Second moment formula.
  13. $n$-th moment formula
  14. Wave equation formula
  15. Intermittency result for the DO noise case
  16. ...and 16 more sections

Key Result

Theorem 2.2.1

Let $W$ be the noise introduced in Definition DO noise def. Let $H\in (a/4,1)$, where $a$ is defined in eqn:a_param. The stochastic heat equation eqn:DO process has an almost-sure unique solution $u$ that satisfies where $p_{h}$ is the heat kernel introduced in heatkernel and $w(t,x)$ is the solution to the deterministic heat equation given in heatsolution. Also, $u$ satisfies

Figures (1)

  • Figure 1: Illustration of the process $X_t$ starting from $x=0$.

Theorems & Definitions (98)

  • Definition 1.1.1: Heat kernel
  • Definition 1.1.2: Wave kernel
  • Definition 1.2.2: Fractional Brownian motion
  • Remark 1.2.3
  • Definition 1.3.1: Lyapunov exponent
  • Definition 1.3.2: Full intermittency
  • Definition 1.3.3: Weak intermittency
  • Remark 1.3.4
  • Example 1.3.5
  • Example 1.3.6
  • ...and 88 more