A non-nonstandard proof of Reimers' existence result for heat SPDEs

In 1989, Reimers gave a nonstandard proof of the existence of a solution to heat stochastic partial differential equations (SPDEs), driven by space–time white noise, when the diffusion coefficient is continuous and satisfies a linear growth condition. Using the martingale problem approach, we give a non-nonstandard proof of this fact, and with the aid of Girsanov's theorem for continuous orthogonal martingale measures (proved in a separate paper by the author), the result is extended to the case of a measurable drift.