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Hyperbolic Anderson equations with general time-independent Gaussian noise: Stratonovich regime

Xia Chen, Yaozhong Hu

Abstract

In this paper, we investigate the hyperbolic Anderson equation generated by a time-independent Gaussian noise with two objectives: The solvability and intermittency. First, we prove that Dalang's condition is necessary and sufficient for existence of the solution. Second, we establish the precise long time and high moment asymptotics for the solution under the usual homogeneity assumption of the covariance of the Gaussian noise. Our approach is fundamentally different from the ones existing in literature. The main contributions in our approach include the representation of Stratonovich moment under Laplace transform via the moments of the Brownian motions in Gaussian potentials and some large deviation skills developed in dealing effectively with the Stratonovich chaos expansion.

Hyperbolic Anderson equations with general time-independent Gaussian noise: Stratonovich regime

Abstract

In this paper, we investigate the hyperbolic Anderson equation generated by a time-independent Gaussian noise with two objectives: The solvability and intermittency. First, we prove that Dalang's condition is necessary and sufficient for existence of the solution. Second, we establish the precise long time and high moment asymptotics for the solution under the usual homogeneity assumption of the covariance of the Gaussian noise. Our approach is fundamentally different from the ones existing in literature. The main contributions in our approach include the representation of Stratonovich moment under Laplace transform via the moments of the Brownian motions in Gaussian potentials and some large deviation skills developed in dealing effectively with the Stratonovich chaos expansion.
Paper Structure (16 sections, 17 theorems, 291 equations)

This paper contains 16 sections, 17 theorems, 291 equations.

Table of Contents

  1. Introduction
  2. Stratonovich expansion and approximations
  3. Green's function
  4. Stratonovich integral
  5. Stratonovich moments
  6. Strantonovich moment representation
  7. Stratonovich integrability and Fubini's theorem
  8. Link to Dalang-Mueller-Tribe's work
  9. Proof of Theorem \ref{['th-1']}
  10. Proof of Theorem \ref{['th-2']}
  11. Upper bounds of (\ref{['intro-9']}) and (\ref{['intro-10']})
  12. Lower bound for (\ref{['intro-9']})
  13. Lower bounds for (\ref{['intro-10']})
  14. Appendix
  15. Moment bounds for Brownian intersection local times
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $d=1,2,3$ and assume that $u_0(x)=1$ and $u_1(x)=0$ in (intro-1).

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • Remark 3.2
  • ...and 27 more