Carleman estimate for stochastic degenerate wave equation with drift and its application

Abstract This paper is devoted to establishing the Carleman estimate for the stochastic degenerate wave equation with drift in the weakly degenerate case. Due to the degeneracy, we first study an approximate nondegenerate system. In order to overcome the difficulty arising from the drift term, we introduce an auxiliary function by which the diffusion and the drift term are transformed into a complex whole. After complicated and detailed estimates, we then derive the Carleman estimate for the stochastic degenerate wave equation. As an application for our Carleman estimate, we solve an inverse problem of determining three unknowns simultaneously by some suitable observations for the stochastic degenerate wave equation with drift. More precisely, we obtain a global uniqueness result for our inverse problem.