Upon the recent development of the quasi-reversibility method for terminal value parabolic problems in Nguyen et al. (2019), it is imperative to investigate the convergence analysis of this regularization method in the stochastic setting. In this paper, we positively unravel this open question by focusing on a coupled system of Dirichlet reaction–diffusion equations with additive white Gaussian noise on the terminal data. In this regard, the approximate problem is designed by adding the so-called perturbing operator to the original problem and by exploiting the Fourier reconstructed terminal data. By this way, Gevrey-type source conditions are included, while we successfully maintain the logarithmic stability estimate of the corresponding stabilized operator, which is necessary for the error analysis. As the main theme of this work, we prove the error bounds for the concentrations and for the concentration gradients, driven by a large amount of weighted energy-like controls involving the expectation operator. Compared to the classical error bounds in $L^2$ and $H^1$ that we obtained in the previous studies, our analysis here needs a higher smoothness of the true terminal data to ensure their reconstructions from the stochastic fashion. Two numerical examples are provided to corroborate the theoretical results.

在Nguyen等人的终端价值抛物线问题的准可逆性方法的最新发展中。（2019），必须在随机环境中研究这种正则化方法的收敛性分析。在本文中，我们将重点放在末端数据上具有加性高斯白噪声的Dirichlet反应扩散方程的耦合系统中，从而积极地解决这个悬而未决的问题。在这方面，通过将所谓的扰动算子添加到原始问题并利用傅立叶重构的终端数据来设计近似问题。这样，就包括了Gevrey型源条件，同时我们成功地维护了相应稳定算子的对数稳定性估计，这对于误差分析是必需的。作为这项工作的主题，我们证明了浓度和浓度梯度的误差范围是由涉及期望算子的大量加权能量样控制所驱动的。与经典误差范围相比${\mathrm{大号}}^2$ 和 $H^{\mathrm{1个}}$从先前的研究中获得的结果来看，这里的分析需要真实终端数据的更高平滑度，以确保从随机方式重建它们。提供了两个数值示例，以证实理论结果。