Dynamic roughening of cities driven by multiplicative noise.

The evolution of urban landscapes is rapidly altering the surface of our planet. Yet our understanding of the urbanization phenomenon remains far from complete. A fundamental challenge is to describe spatiotemporal changes in the built environment, both vertical and horizontal. In this work, we model global building-height dynamics as a zero-dimensional geometric Brownian motion (GBM): multiplicative noise produces stochastic fluctuations around a drift linked to economic growth. To account for intra-city correlations, we extend the GBM with spatial coupling, revealing how local interactions effectively mitigate noise-driven fluctuations and shape urban morphology. In the continuum limit, the spatial model reduces to the Kardar-Parisi-Zhang (KPZ) equation, and roughness exponents estimated from data fall within the KPZ range for most cities. Our results indicate that multiplicative noise, moderated by local coupling, governs the evolution of urban roughness, placing city growth within a well-established statistical-physics framework.