Fluid-Spatiotemporal Stochastic Geometry: Information Flow in Non-Stationary Fields

Abstract

The fundamental limits of information flow in spatial networks are usually characterized under stationary spatial point processes, but this assumption cannot capture non-stationary regimes where the node intensity field evolves continuously in space and time. This paper develops Fluid-Spatiotemporal Stochastic Geometry (F-STSG), treating dynamic network topology as a hydrodynamic limit of the discrete node constellation. We formulate the identification of latent network dynamics as an inverse boundary value problem and, using the minimum kinetic energy principle from optimal transport, establish the existence and uniqueness of a scalar potential field governing the compressive evolution of network load. The resulting field-theoretic formulation couples continuous Lagrangian transport with discrete Eulerian interference geometry. Based on this model, we derive the information flux vector as a sufficient statistic for macroscopic advection and the material derivative as a kinematic predictor of topological divergence. We further characterize non-stationary network limits through energy-density scaling and source-channel interpretation, showing how coordination overhead, topology deformation, and control signaling requirements are linked to the kinematic entropy of the evolving network topology.