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Quantifying the effect of noise perturbation for the stochastic Burgers equation with additive trace-class noise

Sonja Cox, Matas Urbonas

Abstract

We establish upper bounds for the weak and strong error resulting from a perturbation of the noise driving the stochastic Burgers equation, where we assume the noise to be additive and of trace class and the initial value to be sufficiently regular. More specifically, replacing the covariance operator of the driving noise $Q_1 \in \mathcal{L}_1(L^2)$ in the Burgers equation by a covariance operator $Q_2 \in \mathcal{L}_1(L^2)$ results in a weak error of $\mathcal{O}\big(\| (-A)^{-1^{-} } (Q_1-Q_2) \|_{\mathcal{L}_1(L^2)}\big)$ and a strong error of $\mathcal{O}\big(\big\| (-A)^{-1/2^{-}}\big|Q_1^{1/2} -Q_2^{1/2}\big| \big\|_{\mathcal{L}_2(L^2)}\big)$. Here $\|\cdot \|_{\mathcal{L}_1}$ is the trace class norm, $\|\cdot \|_{\mathcal{L}_2}$ is the Hilbert-Schmidt norm, and $A$ is the one-dimensional Dirichlet Laplacian that represents the leading term in the Burgers equation. In particular, our results provide upper bounds for the weak and strong error arising when approximating the trace class noise by finite-dimensional noise; the rates we obtain reflect the general philosophy that the weak convergence rate should be twice the strong rate.

Quantifying the effect of noise perturbation for the stochastic Burgers equation with additive trace-class noise

Abstract

We establish upper bounds for the weak and strong error resulting from a perturbation of the noise driving the stochastic Burgers equation, where we assume the noise to be additive and of trace class and the initial value to be sufficiently regular. More specifically, replacing the covariance operator of the driving noise in the Burgers equation by a covariance operator results in a weak error of and a strong error of . Here is the trace class norm, is the Hilbert-Schmidt norm, and is the one-dimensional Dirichlet Laplacian that represents the leading term in the Burgers equation. In particular, our results provide upper bounds for the weak and strong error arising when approximating the trace class noise by finite-dimensional noise; the rates we obtain reflect the general philosophy that the weak convergence rate should be twice the strong rate.
Paper Structure (12 sections, 21 theorems, 143 equations)

This paper contains 12 sections, 21 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. The setting for the 1D stochastic Burgers equation
  5. Properties of the operators $A$ and $B$
  6. Estimates for stochastic convolutions
  7. Moment bounds
  8. The Kolmogorov equation
  9. Estimates for the solution of the Kolmogorov equation and related bounds
  10. Weak convergence
  11. Strong convergence
  12. Proofs of Lemmas \ref{['lem:Y exp bound']}--\ref{['lem:sup of psi']}

Key Result

Theorem \ifnumequal100.11.1

Assume that there exist $\gamma_0>0$ and $p>32$ such that $\mathbb{E}\left[ \exp(\gamma_0\|X_0\|_{L^2}^2) \right] < \infty$ and $\mathbb{E}\left[ \|(-A)^{\frac{1}{2}}X_0\|_{L^2}^p \right] < \infty$. Let $K_{\max} \in (0,\infty)$ and let $Q_1,Q_2 \in \mathcal{L}_1(L^2)$ be positive and self-adjoint, and for all $\epsilon >0$, $r\in [1,\frac{p}{4})$ there exists $C_{\epsilon,\gamma_0,p,r,T}(X_0,K_{

Theorems & Definitions (39)

  • Theorem \ifnumequal100.11.1
  • Lemma \ifnumequal200.12.1
  • Proposition \ifnumequal200.22.2
  • proof
  • Proposition \ifnumequal200.32.3
  • proof
  • Lemma \ifnumequal200.42.4
  • proof
  • Lemma \ifnumequal200.52.5
  • Lemma \ifnumequal200.62.6
  • ...and 29 more