Stochastic parabolic equations are widely used to model many random phenomena in natural sciences, such as the temperature distribution in a noisy medium, the dynamics of a chemical reaction in a noisy environment, or the evolution of the density of bacteria population. In many cases, the equation may involve an unknown moving boundary which could represent a change of phase, a reaction front, or an unknown population. In this paper, we focus on an inverse problem with the goal is to determine an unknown moving boundary based on data observed in a specific interior subdomain for the stochastic parabolic equation. The uniqueness of the solution of this problem is proved, and furthermore a stability estimate of log type is derived. This allows us, theoretically, to track and to monitor the behavior of the unknown boundary from observation in an arbitrary interior domain. The primary tool is a new Carleman estimate for stochastic parabolic equations. As a byproduct, we obtain a quantitative unique continuation property for stochastic parabolic equations.

中文翻译：

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确定未知时变边界的逆随机抛物线问题的稳定性估计*

随机抛物线方程广泛用于模拟自然科学中的许多随机现象，例如噪声介质中的温度分布、噪声环境中化学反应的动力学或细菌种群密度的演化。在许多情况下，方程可能涉及未知的移动边界，它可以代表相变、反应前沿或未知的总体。在本文中，我们关注一个反问题，其目标是根据在随机抛物线方程的特定内部子域中观察到的数据来确定未知的移动边界。证明了该问题解的唯一性，并推导了对数型的稳定性估计。从理论上讲，这使我们能够通过在任意内部域中的观察来跟踪和监控未知边界的行为。主要工具是随机抛物线方程的新卡尔曼估计。作为副产品，我们获得了随机抛物线方程的定量唯一连续性质。