Towards Abstract Wiener Model Spaces

TL;DR

The paper develops a unifying framework, AWMS, that extends abstract Wiener space theory to enhanced Gaussian objects arising in rough path theory and regularity structures. By organizing both Top-Down and Bottom-Up constructions, it provides a robust route to obtain large deviations, Fernique estimates, and Cameron–Martin translations directly at the enhanced level, while remaining anchored to the classical AWS. The main contributions include a precise definition of AWMS, two complementary construction methods, and general LDP and Fernique results for AWMS, with a comprehensive suite of examples (Gaussian rough paths, Itô Brownian motion, rough volatility structures, and Phi4_d/PAM-type models). These results unify previous enhanced Gaussian large deviations results and extend tail and quasi-invariance properties to the augmented spaces relevant for singular SPDEs. The framework has potential impact on stochastic analysis of rough path/regularity structure models and on Laplace- and tail-type asymptotics for stochastic PDEs driven by Gaussian noises.

Abstract

Wiener spaces are in many ways the decisive setting for fundamental results on Gaussian measures: large deviations (Schilder), quasi-invariance (Cameron--Martin), differential calculus (Malliavin), support description (Stroock--Varadhan), concentration of measure (Fernique), etc. Analogues of these classical results have been derived in the "enhanced" context of Gaussian rough paths and, more recently, regularity structures equipped with Gaussian models. The aim of this article is to propose a similar notion directly on this enhanced level - an abstract Wiener model space - that encompasses the aforementioned. More specifically, we focus here on enhanced Schilder type results, Cameron--Martin shifts and Fernique estimates, offering a somewhat unified view on results of Friz--Victoir and Hairer--Weber.

Figures (5)

• Figure 1: Diagram of abstract and concrete setting.
• Figure 2: Diagram of the definition of AWMS.
• Figure 3: Diagram of the definition of AWMS. By $\texttt{CM}(E, \mu)$ we denote the Cameron--Martin space associated to $(E, \mu)$. The lower level corresponds to the data which belongs to the classical theory, while the upper level corresponds to data in the enhanced setting. The two lifts $\mathop{\mathrm{\mathfrak{L}}}\nolimits$ and $\hat{\mathop{\mathrm{\mathfrak{L}}}\nolimits}$ provide a connection between the two. Black symbols represent data which needs to be chosen in the definition an AWMS, while blue symbols can be defined from that choice. Arrows of the form $\twoheadrightarrow$ represent projections, while arrows of the form $\hookrightarrow$ represent inclusions.
• Figure 4: Summary of the Top-Down construction. Symbols in black are assumed (Top-Down Data) while symbols in blue are constructed.
• Figure 5: Summary of the Bottom-Up construction. Symbols in black are assumed (Bottom-Up Data) while symbols in blue are constructed.

Theorems & Definitions (87)

• Remark : Para-controlled calculus and Polchinski flow
• Definition : Abstract Wiener Model Space
• Definition 2.1: Abstract Wiener Space, e.g. deuschelLargeDeviations1989
• Remark
• Example 2.2: Running Example
• Definition 2.3: Ambient Space
• Example 2.4: Running Example continued
• Definition 2.5: Lift
• Remark : Aida--Kusuoka--Stroock
• Remark : Renormalized Bottom-Up