Random Surfaces and Higher Algebra

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 $\ddot{I}$-Hölder surfaces, where $\ddot{I}>\frac{1}{2}$, 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.

38 pages, substantially shortened exposition and proofs, now use Hölder regularity instead of p-variation