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In this paper, we study the null controllability for some linear andsemi-linear parabolic SPDEs involving both the state and the gradient of thestate. To start with, an improved global Carleman estimate for linear forward(resp. backward) parabolic SPDEs with general random coefficients and$L^2$-valued source terms is derived. Based on this, we further develop a newglobal Carleman estimate for linear forward (resp. backward) parabolic SPDEswith $H^{-1}$-valued source terms, which enables us to deal with the globalnull controllability for linear backward (resp. forward) parabolic SPDEs withgradient terms. As byproduct, a special energy-type estimate for the controlledsystem that explicitly depends on the parameters $\lambda,\mu$ and the weightedfunction $\theta$ is obtained. Furthermore, by employing a fixed-pointargument, we extend the previous linear controllability results to somesemi-linear backward (resp. forward) parabolic SPDEs.

New global Carleman estimates and null controllability for forward/backward semi-linear parabolic SPDEs