Null controllability for stochastic heat equations with singular inverse-square potentials

This paper is devoted to the analysis of control properties for stochastic heat equations with a singular inverse-square potential arising in quantum mechanics and combustion theory. Using some results in functional analysis, we demonstrate that the null controllability of this system can be reduced to the observability of a backward stochastic system introduced here. Then, we prove the desired observability inequality by establishing a new Carleman inequality for the backward system. Our proof of Carleman inequality is based on Itô's formula, a splitting argument on the domain, two improved versions of the Hardy inequality.