Continuous Framework for Spatial Treatment Effects

• The paper introduces a continuous framework that models treatment intensity as a function over space and time using PDEs.
• It leverages physical laws and nonparametric techniques to quantify spatial spillovers, decay rates, and dynamic boundary evolution.
• Empirical validation with environmental and health data demonstrates the model's capability to optimize policy evaluation and exposure assessment.

A continuous functional framework for spatial treatment effects formalizes the treatment intensity as a real-valued function over space, and often also over time, allowing rigorous analysis of propagation, decay, heterogeneity, and boundary formation in response to interventions. Contemporary developments in this area integrate mathematical physics, spatial econometrics, deep learning, and nonparametric estimation to address the complex realities of spatial spillovers, heterogeneous diffusion, and dynamic boundary evolution in diverse fields such as environmental science, health geography, transportation, and policy analysis.

1. Mathematical and Physical Foundations

The central theoretical innovation is the specification of the treatment effect not as a binary or discrete event, but as a continuous function—$\tau(\mathbf{x}, t)$—mapping spatial (and potentially temporal) coordinates to effect intensities. This “treatment field” is often modeled as the solution to partial differential equations (PDEs) derived from physical laws, notably the Navier–Stokes and the advection–diffusion equations:

$$\frac{\partial\tau}{\partial t} + \mathbf{v}(\mathbf{x},t)\cdot \nabla \tau = \nu \nabla^2 \tau + S(\mathbf{x}, t)$$

where $\nu$ is the diffusion coefficient, $\mathbf{v}$ the advection (velocity) field, and $S$ the source term for the intervention (Kikuchi, 17 Oct 2025, Kikuchi, 16 Oct 2025). This formulation grounds spillover estimation in first principles of spatial dispersion, so that effects arising from a point source (e.g., emissions from a plant, access from a hospital, or propagation from a bank branch) diffuse according to the physics governing the medium (air, population, economic network).

Solutions to these PDEs admit exact self-similar forms, including:

• Gaussian kernels (for isotropic pure diffusion)
• Modified Bessel functions ($K_0$) for cylindrical or radially symmetric systems (Kikuchi, 16 Oct 2025, Kikuchi, 1 Oct 2025)
• Kummer confluent hypergeometric functions for more general advection–diffusion or nonlinear settings

The scaling law

$$\tau(d, t) = t^{-\alpha} f\left(\frac{d}{t^\beta}\right)$$

with domain-specific scaling exponents $(\alpha, \beta)$, implies that treatment boundaries and intensities evolve predictably with distance and time.

2. Spatial and Temporal Decay, and Boundary Definition

Under pure diffusion (low Reynolds and Péclet numbers), analytic solutions exhibit exponential spatial decay:

$\tau(d) = Q\, \exp(-\kappa\, d)$

where $\kappa$ is the spatial decay parameter and $Q$ is the local effect at the source (Kikuchi, 16 Oct 2025, Kikuchi, 13 Oct 2025, Kikuchi, 17 Oct 2025). The detectable spatial boundary $d^*$ for a threshold $\epsilon$ (e.g., $10\%$ of initial effect) is given by

$$d^* = \frac{1}{\kappa} \ln\!\left(\frac{1}{\epsilon}\right)$$

Temporal effects may exhibit analogous relationships, with boundary times $\tau^*$ derived as

$$\tau^* = \frac{1}{\delta} \ln\!\left(\frac{\kappa}{\delta\,\epsilon}\right)$$

where $\delta$ is the temporal decay or depreciation rate (Kikuchi, 1 Oct 2025).

For time-dependent treatments, spatial domain expansion follows diffusive scaling:

$d^*(t) = \xi^* \sqrt{t}$

with $\xi^*$ determined by diffusion parameters (Kikuchi, 17 Oct 2025, Kikuchi, 16 Oct 2025). The theory also accommodates nonlinear regimes, with decay modulated by advection and chemical reaction terms, leading to region- or context-specific asymmetry or acceleration in boundary movement (Kikuchi, 13 Oct 2025).

3. Regimes, Scope Conditions, and Diagnostic Criteria

The framework delineates scope conditions via fluid dynamics dimensionless numbers (Kikuchi, 13 Oct 2025, Kikuchi, 16 Oct 2025):

• Péclet number ($\text{Pe} = UL/D$): ratio of advection to diffusion; $\text{Pe} \ll 1$ signals diffusion dominance.
• Reynolds number ($\text{Re} = UL/\nu$): turbulence indicator; $\text{Re} < 2000$ ensures applicability of laminar/diffusive models.
• Damköhler number ($\text{Da} = \lambda L^2/D$): reaction-diffusion balance.

Positive, statistically significant decay parameters ($\kappa > 0$) validate the scope for exponential decay. Negative or flat decay ($\kappa \le 0$) is interpreted as evidence that alternate sources (urban contributors, advection, or network effects) dominate, signaling that the continuous diffusion framework is inapplicable (Kikuchi, 13 Oct 2025, Kikuchi, 16 Oct 2025, Kikuchi, 17 Oct 2025).

4. Nonparametric Identification, Estimation, and Inference

Theoretical and empirical work advances nonparametric boundary identification robust to failures of physical idealizations such as steady winds and homogeneous terrain (Kikuchi, 14 Oct 2025). The mean outcome is modeled as a flexible function of distance:

$m(d) = \mathbb{E}[Y_i \mid D_i = d]$

Boundaries are estimated as the minimum $d$ satisfying $m(d) \le \epsilon m(0)$. Local polynomial estimators (e.g., local linear regression with kernel smoothing) allow for nonparametric recovery:

$$\hat{m}(d_0) = \arg\min_{\beta_0, \beta_1} \sum_i [Y_i - \beta_0 - \beta_1 (D_i - d_0)]^2 K_h(D_i - d_0)$$

Bandwidth choice is optimized for boundary recovery accuracy, often starting from rules-of-thumb and iteratively refined (Kikuchi, 14 Oct 2025). The derived estimator for $d^*$ is shown to admit asymptotic normality:

$$\sqrt{n h_n}\, [\hat{d}^* - d^* - B_n] \to_d N(0, V)$$

with explicit bias ($B_n$) and variance ($V$) terms, providing valid inference in very large spatial datasets (e.g., >40 million observations).

5. Empirical Validation and Regional Heterogeneity

Large-scale empirical analyses support the continuous functional framework:

• NO₂ satellite data (42 million observations, TROPOMI): Estimated $\kappa_s = 0.004$ per km, $R^2 = 0.35$, boundary at $d^* \approx 572$ km. Nonparametric estimation outperforms discrete parametric models in RMSE and boundary detection (Kikuchi, 16 Oct 2025, Kikuchi, 14 Oct 2025).
• PM₂.₅ monitors (ground network, >500,000 obs): Decay rates and boundaries consistent with physical transport parameters (Kikuchi, 13 Oct 2025).
• Healthcare access (32,520 ZIP codes): Exponential decay in lack of insurance with distance to nearest hospital, $\kappa = 0.002837$ per km with boundary $d^* \approx 37$ km; heterogeneity detected by age (elderly up to 13× effect) and education (5–13× gradient) (Kikuchi, 17 Oct 2025).

Critical diagnostic capability is provided by region-specific analysis: positive (valid) and negative (invalid) decay regimes are detected based on local source dominance and confounding, respectively. Monte Carlo evidence confirms that the functional approach has low false positive rates and high accuracy in complex, realistic scenarios (Kikuchi, 16 Oct 2025, Kikuchi, 14 Oct 2025).

6. Policy Applications and Practical Implications

The continuous functional framework underpins a mathematically principled approach to boundary and exposure quantification in policy evaluation:

• Explicit, data-driven boundary computation informs regulatory decisions (e.g., pollution buffer zones, healthcare or banking service deserts) (Kikuchi, 13 Oct 2025, Kikuchi, 16 Oct 2025, Kikuchi, 17 Oct 2025).
• Exposure functions ($\int_0^T \tau(\mathbf{x}, t)dt$) enable rigorous health and cost–benefit analyses in environmental policy.
• Predictive and sensitivity analysis: The framework yields explicit formulas for boundary evolution (e.g., $d^*(t) = \xi^* \sqrt{t}$) and allows calculation of effect sizes under counterfactual changes in diffusion parameters, supporting intervention optimization (Kikuchi, 17 Oct 2025).

The systematic model selection (via information criteria such as AIC) can reveal when decay is best captured by logarithmic or power-law rather than exponential forms, reflecting diminishing marginal effects with distance and guiding the appropriate functional specification (Kikuchi, 17 Oct 2025).

7. Limitations and Contemporary Directions

While the continuous functional paradigm offers robust theoretical and empirical advances, several limitations and research avenues remain (Kikuchi, 13 Oct 2025, Kikuchi, 14 Oct 2025, Kikuchi, 17 Oct 2025):

• Assumption violations: Steady-state, pure diffusion assumptions may be invalid in the presence of strong advection, turbulence, or non-point sources.
• Scope diagnosis: Users are urged to perform ex ante diagnostic checks (e.g., $\kappa_s$ sign and significance, $R^2$) before applying functional estimators.
• Model flexibility: Nonparametric approaches are preferred when physical idealizations fail; ongoing work combines physically-informed priors with machine learning to capture heterogeneous or dynamically evolving patterns.
• Network generalizations: Future directions include incorporating non-Euclidean (network) distance, dynamic boundary propagation, and joint estimation with general equilibrium spillovers.

Summary Table: Key Components of the Continuous Functional Framework

Component	Mathematical Structure	Empirical Implementation
Treatment intensity	$\tau(\mathbf{x}, t)$, PDE solutions	Regression of outcomes vs. distance/time
Spatial decay	$\exp(-\kappa d)$, scaling laws	Log-linear regression, nonparametric m(d)
Boundary estimation	$d^* = (1/\kappa)\log(1/\epsilon)$	Plug-in estimator, kernel methods
Diagnostic test	Sign and significance of $\kappa_s$	Regional regression, information criteria
Predictive evolution	$d^*(t) = \xi^* \sqrt{t}$	Boundary trajectory estimation

This continuous functional framework, drawing on mathematical physics and advanced nonparametric statistics, has significantly expanded the methodological and empirical toolkit for spatial treatment effect analysis, enabling physically and econometrically rigorous policy evaluation and boundary detection across multiple domains (Kikuchi, 13 Oct 2025, Kikuchi, 16 Oct 2025, Kikuchi, 17 Oct 2025, Kikuchi, 14 Oct 2025).