Dynamic Spatial Treatment Effect Boundaries

• Dynamic Spatial Treatment Effect Boundaries are continuously evolving space-time frontiers where treatment effects diminish, defined and modeled using PDE frameworks.
• They employ advection–diffusion equations and self-similar solutions with special functions to rigorously quantify and interpret spatial causal impacts.
• Empirical validations, including satellite data and Monte Carlo simulations, demonstrate their robustness and relevance for policy-making in environmental and financial contexts.

Dynamic spatial treatment effect boundaries describe the evolving, often non-sharp frontiers in space and time at which the causal effect of an intervention transitions from non-negligible to negligible under realistic propagation mechanisms. The concept is grounded in the recognition that treatments—whether policy interventions, environmental shocks, or service facility placements—propagate over both space and time according to physics-based or stochastic diffusion processes. Unlike discrete or manually selected cutoff approaches, continuous functional frameworks, especially those derived from partial differential equations such as the Navier–Stokes system, yield a mathematically rigorous, empirically valid, and diagnostic platform for both estimating and interpreting the extent of treatment influence in interconnected systems.

1. Theoretical Framework

This framework conceptualizes the treatment effect not as a discrete difference but as a continuous concentration field $\tau(\mathbf{x}, t)$ defined over space-time, $\mathbf{x} \in \mathbb{R}^d$, $t \geq 0$. The evolution of this field is governed by advection–diffusion equations, often simplified (in the absence of drift) to

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

where $\nu$ is the diffusion coefficient and $S(\mathbf{x}, t)$ encodes the source/sink structure (e.g., instantaneous emissions, policy pulses). The treatment effect $\tau(\mathbf{x}, t)$ thus assimilates and mathematically parallels concentration fields in fluid and transport theory.

Self-similar and exact solutions to this PDE are central. For point-source diffusion in an isotropic, homogeneous domain, self-similarity requires that

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

with exponents determined by the spatial dimension $d$—for standard Brownian motion diffusion, $\alpha = d/2$, $\beta = 1/2$. Special functions such as the modified Bessel function $K_0$ and the Kummer confluent hypergeometric function $M(a, b, z)$ (linked to $I_\nu(z)$ via transformation identities) are employed for analytic tractability in cylindrical or more complex symmetries.

The dynamic spatial boundary—where treatment becomes policy-irrelevant—is defined implicitly by a fixed-threshold criterion: $\tau(d^*(t), t) = \tau_{\min}$ For standard scaling exponents, this gives $d^*(t) \propto t^\beta$; i.e., the boundary advances as a predictable function of time under pure diffusion.

2. Scaling Laws and Analytical Solution Structure

The scaling law

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

implies that for diffusive propagation in $d$ dimensions, the treatment effect profile collapses to a universal shape when properly rescaled. Particular solutions include:

• Instantaneous Point Source, $d=3$:

$$\tau(r, t) = \frac{Q}{(4\pi \nu t)^{3/2}} \exp\left(-\frac{r^2}{4\nu t}\right)$$

with $Q$ conserved total effect, $r = |\mathbf{x}|$.

• Cylindrical Symmetry:

$$\tau(r, t) = \frac{A}{t} K_0\left(\frac{r}{2\sqrt{\nu t}}\right)$$

The spatial boundary evolves as

$d^*(t) = \xi^* t^\beta$

where $\xi^*$ depends on detection threshold and profile function $f$.

The relevant policy implication is that the velocity of boundary expansion,

$\frac{d}{dt} d^*(t) = \beta \xi^* t^{\beta-1}$

slows over time ($\beta < 1$), so the influence “front” decelerates after an initial rapid propagation phase.

3. Empirical Validation

Empirical analysis using 42 million TROPOMI satellite observations of NO$_2$ concentrations from U.S. coal-fired power plants (2019–2021) confirms the theoretical scaling forms. Specifically:

• NO$_2$ concentrations exhibit strong exponential spatial decay, captured by the relationship

$\tau(d) \propto \exp(-\kappa_s d)$

with an estimated decay parameter $\kappa_s = 0.004028$ per km (SE = 0.000016, $p < 0.001$), implying a $0.4\%$ reduction per km.

• The $R^2 = 0.35$ for the exponential regression evidences substantial explanatory power, especially in light of atmospheric complexities.
• The detected boundary (e.g., $d^* \approx 572$ km at a $10\%$ threshold) quantifies the operational spatial reach of coal plant emissions.
• Regional heterogeneity is validated by observing positive decay parameters within a 100 km radius (coal plant dominance), and negative parameter estimates beyond 100 km, consistent with local sources (urban transport) overtaking plant-generated effects.

These results allow for precise mapping of where policy interventions are effective and where alternative determinants dominate, fulfilling the diagnostic and analytic agenda of the framework.

4. Monte Carlo Simulation Evidence

A broad array of simulation data generating processes (DGPs)—exponential decay, hump-shaped, and flat zero-boundary cases—are used to validate boundary detection. Key findings include:

• The continuous functional (nonparametric) approach achieves 95\% interval coverage and low bias/RMSE in boundary estimation when the true boundary exists.
• When no boundary is present, the approach correctly rejects the boundary 94\% of the time; in contrast, discrete parametric methods falsely declare spurious boundaries in 73\% of replications under the null.
• Continuous approaches flexibly accommodate non-monotonic and near-flat patterns, making them robust against misspecification in real-world data, where boundary presence/absence cannot be postulated a priori.

This diagnostic power strengthens confidence in empirical applications involving pollution, financial access, or service diffusion.

5. Applications across Domains

The continuous functional framework enables unified treatment of dynamic spatial boundaries in various empirical settings:

• Environmental Economics: Quantification of pollution footprints (e.g., coal plant NO$_2$) for exposure and benefit–cost analysis, with direct integration of boundaries into health or regulatory assessments.
• Banking and Financial Services: Assessment of the spatial reach of bank branches on loan demand, revealing that "boundaries" may be weak or non-existent, and facilitating diagnoses of supply-demand asymmetry and misinterpretation of banking deserts.
• Healthcare: Application to spatial influence of health facilities on service uptake or outcomes, leveraging the boundary evolution for designing catchment areas and measuring cumulative exposures (using, for instance, $\Phi(\mathbf{x}) = \int_0^T \tau(\mathbf{x}, t) dt$).

For each context, parameterized boundary velocity and cumulative exposure enable more granular and theoretically defensible policy targeting than ad hoc cutoff methods.

6. Diagnostic and Computational Features

A distinctive feature is the framework’s self-diagnosis. Boundary presence or absence, and boundary evolution rate, provide a direct test for functional form misspecification or for alternative explanations of spatial patterns. The use of analytic special functions (Bessel, Kummer confluent hypergeometric) enables computationally efficient, closed-form solutions in many practically important settings. The framework also suggests the use of the calculus of variations and Hamilton–Jacobi–Bellman methods for optimizing intervention targeting given resource and boundary constraints.

7. Implications and Future Directions

The continuous functional perspective, rooted in Navier–Stokes PDEs and their scaling laws, advances both theory and practice in spatial causal inference:

• Policy evaluation gains rigor, as dynamic boundaries are quantitatively estimated rather than imposed.
• Credible counterfactual and cumulative exposure estimation is feasible in sensor-rich environments.
• The approach’s diagnostic properties help prevent misinterpretation of empirical patterns in environmental, financial, and public health data.
• Extensions to heterogeneous, networked, or multi-layer systems are possible, leveraging the flexibility of PDE-based methodologies and modern nonparametric regression schemes.

Overall, this paradigm unifies spatial econometric analysis and mathematical physics, supporting robust and interpretable boundary detection and effect exposure mapping for complex spatial and temporal interventions (Kikuchi, 16 Oct 2025).