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In this paper, we consider a null controllability and an inverse sourceproblem for stochastic Grushin equation with boundary degeneracy andsingularity. We construct two special weight functions to establish twoCarleman estimates for the whole stochastic Grushin operator with singularpotential by a weighted identity method. One is for the backward stochasticGrushin equation with singular weight function. We then apply it to prove thenull controllability for stochastic Grushin equation for any $T$ and anydegeneracy $\gamma>0$, when our control domain touches the degeneracy line$\{x=0\}$. In order to study the inverse source problem of determining twokinds of sources simultaneously, we prove the other Carleman estimate, which isfor the forward stochastic Grushin equation with regular weight function. Basedon this Carleman estimate, we obtain the uniqueness of the inverse sourceproblem.

Null controllability and inverse source problem for stochastic Grushin equation with boundary degeneracy and singularity