In this paper, we develop and analyze a time-domain perfectly matched layer (PML) method for the stochastic acoustic wave equation driven by spatially white additive Gaussian noise. We begin by establishing the well-posedness and stability of the direct problem through a rigorous analysis of the associated time-harmonic stochastic Helmholtz equation and the application of an abstract Laplace transform inversion theorem. To address the low regularity of the random source, we employ scattering theory to investigate the meromorphic continuation of the Helmholtz resolvent defined on rough fields. Based on a piecewise constant approximation of the white noise, we construct an approximate wave solution and formulate a time-domain PML method. The convergence of the PML method is established, with explicit dependence on the PML layer's thickness and medium properties, as well as the piecewise constant approximation of the white noise. In addition, we propose a frequency-domain approach for solving the inverse random source problem using time-domain boundary measurements. A logarithmic stability estimate is derived, highlighting the ill-posedness of the inverse problem and offering guidance for the design of effective numerical schemes.

翻译：本文针对空间白加性高斯噪声驱动的随机声波方程，发展并分析了一种时域完美匹配层（PML）方法。我们首先通过对关联的时谐随机亥姆霍兹方程的严格分析，并结合抽象拉普拉斯变换反演定理的应用，建立了正定问题的适定性与稳定性。为处理随机源的低正则性，我们利用散射理论研究了定义于粗糙场上的亥姆霍兹预解式的亚纯延拓。基于对白噪声的分段常数逼近，我们构造了近似波解并构建了时域PML方法。文中建立了PML方法的收敛性，明确给出了收敛性与PML层厚度、介质属性以及白噪声分段常数逼近的依赖关系。此外，我们提出了一种利用时域边界测量求解逆随机源问题的频域方法。推导了对数型稳定性估计，该结果揭示了逆问题的不适定性，并为设计有效的数值格式提供了指导。