In this paper, we study two inverse problems for stochastic parabolic equations with additive noise. One is to determinate the history of a stochastic heat process and the random heat source simultaneously by the observation at the final time 𝑇. For this inverse problem, we obtain a conditional stability result. The other one is an inverse source problem to determine two kinds of sources simultaneously by the observation at the final time and on the lateral boundary. The main tool for solving the inverse problems is a new global Carleman estimate for the stochastic parabolic equation.

中文翻译：

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带有附加噪声的随机抛物方程的反问题

在本文中，我们研究了带有加性噪声的随机抛物方程的两个反问题。一种是通过最后时刻observation的观测同时确定随机热过程和随机热源的历史。对于这个反问题，我们获得了条件稳定性结果。另一个是逆源问题，通过最终时间和横向边界上的观测来同时确定两种源。解决逆问题的主要工具是对随机抛物方程进行新的全局Carleman估计。