White Noise Driven Parabolic SPDEs with Measurable Drift

We prove existence and uniqueness of the solution of a white noise driven parabolic SPDE, in case the drift is measurable and satisfies a "one sided linear growth condition," and the diffusion coefficient is nondegenerate, has a locally Lipschitz derivative, and satisfies a linear growth condition. The proof combines arguments similar to those of Gyِngy and Pardoux together with an estimate of the density of the solution of the equation without drift, which is obtained with the help of the Malliavin calculus.