Dynamic Spatial Treatment Effects as Continuous Functionals: Theory and Evidence from Healthcare Access

Abstract

I develop a continuous functional framework for spatial treatment effects grounded in Navier-Stokes partial differential equations. Rather than discrete treatment parameters, the framework characterizes treatment intensity as continuous functions $\tau(\mathbf{x}, t)$ over space-time, enabling rigorous analysis of boundary evolution, spatial gradients, and cumulative exposure. Empirical validation using 32,520 U.S. ZIP codes demonstrates exponential spatial decay for healthcare access ($\kappa = 0.002837$ per km, $R^2 = 0.0129$) with detectable boundaries at 37.1 km. The framework successfully diagnoses when scope conditions hold: positive decay parameters validate diffusion assumptions near hospitals, while negative parameters correctly signal urban confounding effects. Heterogeneity analysis reveals 2-13 $\times$ stronger distance effects for elderly populations and substantial education gradients. Model selection strongly favors logarithmic decay over exponential ($\Delta \text{AIC} > 10,000$), representing a middle ground between exponential and power-law decay. Applications span environmental economics, banking, and healthcare policy. The continuous functional framework provides predictive capability ($d^*(t) = \xi^* \sqrt{t}$), parameter sensitivity ($\partial d^*/\partial \nu$), and diagnostic tests unavailable in traditional difference-in-differences approaches.