Stochastic Calculus in Infinite Dimensions and SPDEs

Abstract

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.

Existence and Uniqueness of Maximal Solutions to SPDEs with Applications to Viscous Fluid Equations

“…where we require that ½ A k is F 0 −measurable which is owing to the F 0 −measurability of Ψ 0 , and then apply Proposition 1.6.14 of [26]. Therefore the identity (3.15) holds È − a.s. on every A k hence È − a.s. on the whole of Ω.…”

“…We take W to be a cylindrical Brownian Motion over some Hilbert Space U with orthonormal basis (e i ). Recall ( [26], Subsection 1.4) that W admits the representation W t = ∞ i=1 e i W i t as a limit in L 2 (Ω; U ′ ) whereby the (W i ) are a collection of i.i.d. standard real valued Brownian Motions and U ′ is an enlargement of the Hilbert Space U such that the embedding J : U → U ′ is Hilbert-Schmidt and W is a JJ * −cylindrical Brownian Motion over U ′ .…”

“…1 A complete, direct construction of this integral, a treatment of its properties and the fundamentals of stochastic calculus in infinite dimensions can be found in [26] Section 1. Properties that we shall make frequent use of are the Burkholder-Davis-Gundy type inequality Theorem 1.6.8 and the passage of a bounded linear operator through the stochastic integral, Theorem 1.6.9.…”

“…A complete discussion of how B is then defined on U is given in [26] Subsection 2.2. We embed the divergence-free and zero-average conditions into the relevant function spaces and simply define our solutions as belonging to these spaces.…”

“…A detailed justification that the terms in this definition are well defined is given in[26] Subsections 2.2 and 2.4, referring to (3.5).…”