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Random Surfaces and Higher Algebra

Darrick Lee, Harald Oberhauser

TL;DR

This work extends the path-development framework to random surfaces by embedding surface data into matrix double groups via surface holonomy. It constructs a surface development theory using matrix 2-connections and crossed modules, and proves existence and continuity in the Young regime ($\rho>\tfrac12$), enabling a characteristic-function viewpoint for laws of random surfaces. The authors establish universality and characteristicness for both unparametrized and parametrized surfaces, and introduce a metric on laws that metrizes weak convergence on compact sets; they also illustrate the framework with examples such as fractional Brownian sheets. Collectively, the results provide a computable, geometry-aware description of random surfaces and a natural statistical distance between their laws, with potential applications to stochastic analysis and higher-dimensional data.

Abstract

We introduce a characteristic function for laws of random surfaces $\mathbf{X}: [0,s] \times [0,t] \to \mathbb{R}^d$, in the spirit of expected path developments for one-dimensional stochastic processes into matrix groups. A key property is that path development is structure preserving: path concatenation becomes matrix multiplication. The main challenge is to account for two distinct concatenation operations for surfaces: horizontal and vertical. To address this, we use the notion of surface holonomy from higher geometry to define surface developments, and study this in a stochastic context. We generalize surface developments to the Young setting of $ρ$-Hölder surfaces, where $ρ> \frac12$, show that such developments characterize parametrized surfaces. Our main result shows that the resulting expected surface development provides a computable and structured description of laws of random surfaces and leads to a natural metric on the space of probability measures on surfaces.

Random Surfaces and Higher Algebra

TL;DR

This work extends the path-development framework to random surfaces by embedding surface data into matrix double groups via surface holonomy. It constructs a surface development theory using matrix 2-connections and crossed modules, and proves existence and continuity in the Young regime (), enabling a characteristic-function viewpoint for laws of random surfaces. The authors establish universality and characteristicness for both unparametrized and parametrized surfaces, and introduce a metric on laws that metrizes weak convergence on compact sets; they also illustrate the framework with examples such as fractional Brownian sheets. Collectively, the results provide a computable, geometry-aware description of random surfaces and a natural statistical distance between their laws, with potential applications to stochastic analysis and higher-dimensional data.

Abstract

We introduce a characteristic function for laws of random surfaces , in the spirit of expected path developments for one-dimensional stochastic processes into matrix groups. A key property is that path development is structure preserving: path concatenation becomes matrix multiplication. The main challenge is to account for two distinct concatenation operations for surfaces: horizontal and vertical. To address this, we use the notion of surface holonomy from higher geometry to define surface developments, and study this in a stochastic context. We generalize surface developments to the Young setting of -Hölder surfaces, where , show that such developments characterize parametrized surfaces. Our main result shows that the resulting expected surface development provides a computable and structured description of laws of random surfaces and leads to a natural metric on the space of probability measures on surfaces.
Paper Structure (30 sections, 41 theorems, 189 equations, 6 figures)

This paper contains 30 sections, 41 theorems, 189 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Path Development
  3. Algebraic Structure of Surfaces
  4. Matrix Crossed Modules
  5. From Groups to Double Groups.
  6. Crossed Modules of Groups
  7. Crossed Modules of Matrix Lie Groups
  8. Crossed Modules of Matrix Lie Algebras
  9. Matrix Surface Development
  10. Matrix Surface Development
  11. Young Surface Development
  12. Existence and Continuity
  13. Young Double Groups
  14. Parametrized Surface Development
  15. Characterizing the Boundary
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $\rho > \frac{1}{2}$ and let $\mathcal{P}^\rho$ denote the space of probability measures valued in the $\rho$-Hölder surfaces $C^\rho([0,1]^2, \mathbb{R}^d)$. The surface development characteristic function (SDCF) separates probability measures in $\mathcal{P}^\rho$. Furthermore, there exists a metric $d: \mathcal{P}^\rho \times \mathcal{P}^\rho \to \mathbb{R}$ defined by the SDCF which metriz

Figures (6)

  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • ...and 1 more figures

Theorems & Definitions (92)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Theorem 2.5
  • Remark 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4: faria_martins_fundamental_2011
  • ...and 82 more