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This work concerns the direct and inverse potential problems for thestochastic diffusion equation driven by a multiplicative time-dependent whitenoise. The direct problem is to examine the well-posedness of the stochasticdiffusion equation for a given potential, while the inverse problem is todetermine the potential from the expectation of the solution at a fixedobservation point inside the spatial domain. The direct problem is shown toadmit a unique and positive mild solution if the initial value is nonnegative.Moreover, an explicit formula is deduced to reconstruct the square of thepotential, which leads to the uniqueness of the inverse problem for nonnegativepotential functions. Two regularization methods are utilized to overcome theinstability of the numerical differentiation in the reconstruction formula.Numerical results show that the methods are effective to reconstruct bothsmooth and nonsmooth potential functions.

An inverse potential problem for the stochastic diffusion equation with a multiplicative white noise