Inverse random source problems for semilinear stochastic biharmonic wave equations

Abstract This paper concerns the direct and inverse source scattering problems for the stochastic biharmonic wave equation with a Lipschitz-type nonlinearity. The driven source is assumed to be a white noise. Given the rough random source, the direct problem is shown to be well-posed in the sense of distributions. For the inverse problem, it is shown that the strength of the random source can be uniquely determined by the expectation of the high frequency limit of the correlated far-field pattern. Moreover, we extend the results to the more general case where the driven source is a generalized Gaussian random function whose covariance operator is a classical pseudodifferential operator. For this case, we demonstrate by ergodicity that the strength of the random source can be uniquely determined by a single realization of the far-field pattern almost surely. This paper presents the first uniqueness result on inverse source scattering problems for nonlinear wave equations.