A weighted identity for stochastic partial differential operators and its applications

In this paper, a weighted identity for some stochastic partial differential operators (with complex principal parts) is established. This identity presents a unified approach in establishing Carleman-type estimates for some deterministic/stochastic partial differential equations. Based on this identity, one can deduce some known global Carleman estimates for stochastic parabolic equations, stochastic Schrödinger equations, stochastic transport equations and their deterministic counterparts. Meanwhile, as its applications, we derive two different Carleman estimates for linear forward stochastic complex Ginzburg-Landau equations. They can be used to study the controllability/observability and inverse problems for some stochastic complex Ginzburg-Landau equations, respectively.