A weighted identity for stochastic partial differential operators and its applications

In this paper, a pointwise weighted identity for some stochastic partial differential operators (with complex principal parts) is established. This identity presents a unified approach in studying the controllability, observability and inverse problems for some deterministic/stochastic partial differential equations. Based on this identity, one can deduce all the known Carleman estimates and observability results, for some deterministic partial differential equations, stochastic heat equations, stochastic Schr\"odinger equations and stochastic transport equations. Meanwhile, as its new application, we study an inverse problem for linear stochastic complex Ginzburg-Landau equations.