A Toolbox for Nonlinear SPDEs

We present a range of results toward the analysis of nonlinear SPDEs. First of all we prove the well-posedness of an SPDE evolving in a finite dimensional Hilbert space but driven by a Cylindrical Brownian Motion, under traditional Lipschitz and linear growth constraints. Criteria to deduce the existence of a limit from a select sequence of processes are also established, through both tightness and Cauchy arguments, with particular novelty in the Cauchy approach. An energy equality for SPDEs under loose assumptions is also proven.