This article investigates the role of the regularity of the test function when considering the weak error for standard spatial and temporal discretizations of SPDEs of the form $dX\left(t\right)=AX\left(t\right)dt+dW\left(t\right)$, driven by space–time white noise. In previous results, test functions are assumed (at least) of class ${\mathcal{C}}^2$ with bounded derivatives, and the weak order is twice the strong order.

We prove that to quantify the speed of convergence, it is crucial to control some derivatives of the test functions, even if the noise is non-degenerate. First, the supremum of the weak error over all bounded continuous functions, which are bounded by 1, does not converge to 0 as the discretization parameter vanishes. Second, when considering bounded Lipschitz test functions, the weak order of convergence is divided by 2, i.e. it is not better than the strong order.

This is in contrast with the finite dimensional case, where the Euler–Maruyama discretization of elliptic SDEs $dY\left(t\right)=f\left(Y\left(t\right)\right)dt+d{B}_t$ has weak order of convergence 1 even for bounded continuous functions.

本文探讨了当考虑形式为SPDE的标准时空离散的弱误差时测试函数的正则性的作用 $dX（Ť）=\mathrm{一种}X（Ť）dŤ+d\mathrm{w\; ^}（Ť）$，由时空白噪声驱动。在以前的结果中，假设测试功能（至少）是类别${\mathcal{C}}^2$ 有界导数，弱阶是强阶的两倍。

我们证明，要量化收敛速度，即使噪声没有退化，控制测试功能的某些导数也至关重要。首先，随着离散化参数的消失，所有有界的连续函数（以1为界）的微分误差的最大值不会收敛到0。其次，当考虑有界Lipschitz检验函数时，收敛的弱阶除以2，即它并不好于强阶。

这与有限维情况相反，在有限维情况下，椭圆形SDE的Euler–Maruyama离散化 $dÿ（Ť）=F（ÿ（Ť））dŤ+d{乙}_{Ť}$ 即使对于有界连续函数，也具有弱收敛阶数1。