  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
where the goal is to determine an unknown moving boundary based on data
observed in a specific interior subdomain for the stochastic parabolic equation
and prove that the unknown boundary depends logarithmically on the interior
measurement. This allows us, theoretically, to track and to monitor the
behavior of unknown boundary from observation in an arbitrary interior domain.
The stability estimate is based on a new Carleman estimate for stochastic
parabolic equations. As a byproduct, we obtain a quantitative unique
continuation property for stochastic parabolic equations.
