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

TL;DR

This paper derives bounds on the errors caused by perturbing the noise in the stochastic Burgers equation with additive trace-class noise, showing how approximation affects solution accuracy.

Contribution

It provides the first explicit bounds for weak and strong errors due to noise covariance operator perturbations in the stochastic Burgers equation.

Findings

Weak error is of order $ig\| (-A)^{-1^{-}} (Q_1-Q_2) ig\|_{ ext{trace}}$

Strong error is of order $ig\| (-A)^{-1/2^{-}} |Q_1^{1/2} -Q_2^{1/2}| ig\|_{ ext{Hilbert-Schmidt}}$

Error bounds guide finite-dimensional noise approximations

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.