Random Surfaces and Higher Algebra

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 $\rho$-H\"older surfaces, where $\rho > \frac12$, 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.