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 Itô Forms. The study of Stratonovich equations is largely motivated by the stochastic variational principle of SALT [4] for fluid dynamics, and we discuss an application of the framework to a Navier-Stokes Equation with Stochastic Lie Transport as seen in [1]. 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.

We present here a criterion to conclude that an abstract SPDE posseses a unique maximal strong solution, which we apply to a Stochastic Navier-Stokes Equation on a boundeded domain of R 3 . Inspired by the work of Kato and Lai [3] in the deterministic setting, we provide a comparable result here in the stochastic case whilst facilitating a variety of noise structures such as additive, multiplicative and transport. In particular our criterion is designed to fit viscous fluid dynamics models with Stochastic Advection by Lie Transport (SALT) as introduced in [4]. Our application to the Incompressible Navier-Stokes equation on a bounded domain in R 3 matches the deterministic theory and represents the first well-posedness result for an SPDE with SALT noise on a bounded domain. This short work summarises the results and announces two papers [1], [2] which give the full details for the abstract well-posedness arguments and application to the Navier-Stokes Equation respectively.

We present two criteria to conclude that a stochastic partial differential equation (SPDE) posseses a unique maximal strong solution. This paper provides the full details of the abstract well-posedness results first given in [25], and partners the paper [27] which rigorously addresses applications to the 3D SALT (Stochastic Advection by Lie Transport, [30]) Navier-Stokes Equation in velocity and vorticity form, on the torus and the bounded domain respectively. Each criterion has its corresponding set of assumptions and can be applied to viscous fluid equations with additive, multiplicative or a general transport type noise.

No abstract