Stochastic Calculus in Infinite Dimensions and SPDEs

These notes aim to take the reader from an elementary understanding of functional analysis and probability theory to a robust construction of the stochastic integral in Hilbert Spaces. We consider integrals driven at first by real valued martingales and later by Cylindrical Brownian Motion, introducing this concept and expanding into a basic set up for Stochastic Partial Differential Equations (SPDEs). The framework that we establish facilitates an exceedingly broad class of SPDEs and noise structures, in which we build upon standard SDE theory and rigorously deduce a conversion between the Stratonovich and It\^{o} Forms. The study of Stratonovich equations is largely motivated by the stochastic variational principle of SALT [Holm,2015] for fluid dynamics, and we discuss an application of the framework to a Navier-Stokes Equation with Stochastic Lie Transport as seen in [Goodair,2022]. Moreover we prove a fundamental existence and uniqueness result (which to the best of our knowledge, is not present in the literature) for SPDEs evolving in a finite dimensional Hilbert Space driven by Cylindrical Brownian Motion, assuming the analogy to the Lipschitz and linear growth conditions for the standard existence and uniqueness theory for SDEs