A note on stochastic semilinear equations and their associated Fokker–Planck equations

The main purpose of this paper is to prove existence and uniqueness of (probabilistically weak and strong) solutions to stochastic differential equations (SDE) on Hilbert spaces under a new approximation condition on the drift, recently proposed in [6] to solve Fokker–Planck equations (FPE), extended in this paper to a considerably larger class of drifts. As a consequence we prove existence of martingale solutions to the SDE (whose time marginals then solve the corresponding FPE). Applications include stochastic semilinear partial differential equations with white noise and a non-linear drift part which is the sum of a Burgers-type part and a reaction diffusion part. The main novelty is that the latter is no longer assumed to be of at most linear, but of at most polynomial growth. This case so far had not been covered by the existing literature. We also give a direct and more analytic proof for existence of solutions to the corresponding FPE, extending the technique from [6] to our more general framework, which in turn requires to work on a suitable Gelfand triple rather than just the Hilbert state space.