Space-Time Infill Asymptotics

• Space-time infill asymptotics is a rigorous framework that studies the behavior of estimators and solutions when spatial and temporal observations become arbitrarily dense in a fixed domain.
• It examines how estimator variance, consistency, and spatial confounding evolve as increased data density leads to strengthened correlations among observations.
• This framework applies to stochastic PDEs, geometric analysis, and general relativity, offering theoretical insights for model calibration, adaptive numerics, and resolving boundary ambiguities.

Space-time infill asymptotics is a rigorous framework and collection of methods for the statistical and geometric analysis of stochastic processes, partial differential equations, and geometric structures as the observational resolution increases within a fixed spatiotemporal domain. This topic traverses probability theory, nonparametric statistics, spacetime geometry, stochastic analysis of PDEs, model selection, and the global structure of general relativity. Central concerns include the asymptotic properties of estimators and solutions as the grid in space and time grows increasingly dense (asymptotic mesh size tending to zero), the impact of this densification on variance and consistency, and the means by which geometric, analytic, and probabilistic structures interpolate ("infill") between spatial and temporal regimes.

1. Foundations and Regimes of Infill Asymptotics

Space-time infill asymptotics addresses statistical behavior and geometric structure as the observation mesh of a fixed domain is refined indefinitely in both spatial and temporal directions. In contrast to increasing-domain asymptotics—where data are collected over ever-larger domains—the infill regime holds the underlying region constant and analyzes what happens when sample points become arbitrarily dense.

A prototypical situation involves observing a process over $D \subset \mathbb{R}^d \times [0,T]$, with the number of spatial and temporal points increasing while $D$ remains unchanged (see (Natoli et al., 8 Mar 2024, Natoli et al., 9 Mar 2024, Lieshout et al., 2022, Petersson et al., 28 Aug 2025)). In stochastic PDEs, this regime ensures that consistency and rate results must account for growing correlations among observations since the effective number of independent increments does not grow without bound as in classical settings.

Within geometric analysis and general relativity, "space-time infill" may refer to how the local, or "near-infinity," asymptotic structures (e.g., at null, spatial, or timelike infinity) interpolate and link global conserved quantities, symmetries, and boundary structures (Figueroa-O'Farrill et al., 2021, Ashtekar et al., 2023). Here, manifold boundaries at infinity and their degeneracies are resolved or "infilling" is achieved by blowing up points into richer boundary geometries (e.g., grassmannians).

2. Statistical Inference and Effects of Infill Sampling

Infill asymptotic analysis of spatial, temporal, or spatiotemporal models produces several distinctive statistical phenomena:

• Estimator Variance: With infill sampling, variances of parameter estimators generally do not vanish as sample size increases; instead, they converge to strictly positive limits. This outcome is a direct consequence of the increasing dependence among observations as spacing decreases but the domain remains fixed (Natoli et al., 8 Mar 2024, Natoli et al., 9 Mar 2024).
• Loss of Consistency: Even unbiased estimators (BLUE, UMVUE) such as generalized least squares estimates for regression coefficients become inconsistent: their asymptotic variance does not shrink to zero, and estimation error remains non-negligible no matter how finely the domain is sampled (Natoli et al., 8 Mar 2024).
• Sample Size Determination: There exists a point of diminishing returns for adding samples: additional data beyond a particular density contribute negligible incremental information (see percent-of-total-variation analyses in (Natoli et al., 9 Mar 2024)).
• Covariate Smoothness and Spatial Confounding: The consistency and even qualitative behavior (convergence to zero or divergence to infinity) of regression coefficients are sensitive to the smoothness of covariates versus the kernel of the spatial random field (Bolin et al., 27 Mar 2024). If the covariate is "too rough" compared to the underlying spatial process, the effect estimate may vanish entirely under infill asymptotics, despite high raw correlations—a phenomenon formalized using Cameron–Martin spaces and measure equivalence considerations.

Regime	Estimator Variance	Consistency	Remarks
Increasing domain	$\to 0$	Yes	Classical consistency
Infill (fixed domain)	$\not\to 0$	Generally No	Diminishing return, see (Natoli et al., 8 Mar 2024)

3. Advanced Applications: SPDEs, Covariance Testing, and High-Resolution Models

Space-time infill asymptotics provides theoretical groundwork for inference in high-dimensional SPDE settings and goodness-of-fit testing for covariance structures:

• Nonparametric Covariance Estimation: In parabolic SPDEs, realized covariation statistics over grid spacings $(\Delta, h) \to 0$ consistently estimate the noise covariance kernel $Q$ under mild assumptions, with CLT-type results in the Hilbert–Schmidt norm (Petersson et al., 28 Aug 2025).
• Error Dominance and Coupling: The asymptotic variance and bias achieve their "infill" limits under explicit coupling rates; for example, spatial discretization error remains asymptotically negligible as long as $h = o(\Delta^{1/2})$ relative to the temporal mesh, even when spatial resolution is much coarser than temporal (Petersson et al., 28 Aug 2025).
• Omnibus Goodness-of-Fit Procedures: Limiting distributions of empirical covariance tests are derived as generalized $\chi^2$ mixtures, permitting hypothesis tests for $Q$ that are independent of the evolution operator $A$ of the SPDE (Petersson et al., 28 Aug 2025).

4. Space-Time Infill Asymptotics in Geometric and Relativistic Analysis

The notion of infill asymptotics parallels and extends to geometric and general relativistic settings as follows:

• Asymptotic Boundaries as Manifolds: The "blow-up" of points at spatial and timelike infinity leads to boundary manifolds (Spi, Ti, Ni) carrying intrinsic geometric structures (carrollian, pseudo-carrollian, doubly-carrollian) (Figueroa-O'Farrill et al., 2021, Ashtekar et al., 2023). These boundaries may be characterized as grassmannians of affine Lorentzian, spacelike, or null hyperplanes, with implications for holography and asymptotic charges.
• Symmetry Algebra Unification: The symmetry algebra of these boundary geometries—most notably, the BMS algebra on conformal null infinity—coincides with that induced by the "doubly-carrollian" structure of Ni, indicating a deep link between geometric infill and asymptotic symmetries (Figueroa-O'Farrill et al., 2021).
• Gluing Procedures and Consistent Conserved Charges: By strengthening the regularity and directionality properties of the conformal metric (e.g., $C^{>0}$ smoothness) at the interfaces between spatial and null infinity, space-time infill asymptotics "glue" the two descriptions and guarantee that physically meaningful charges (energy, momentum, angular momentum) are unambiguous and consistent across the entire asymptotic region (Ashtekar et al., 2023). This eliminates the so-called supertranslation ambiguity by selecting a canonical Poincaré subgroup.

5. Quantum and Stochastic Dynamical Phenomena under Space-Time Infill

Space-time infill asymptotics also emerges as a core concept in quantum semiclassical analysis, stochastic PDEs, and wave propagation:

• Quantization and Kernel Asymptotics: In Berezin–Toeplitz quantization, simultaneous infill scaling in time and phase space reveals local kernel structures and produces global trace asymptotics—obtained as fine-scale infill limits—that unify fixed-point formulas and spectral geometry (Paoletti, 2013).
• Intermittency and Extremes in Anderson Models: For stochastic parabolic PDEs (e.g., parabolic Anderson models), infill asymptotics characterize the spatial growth rate of extremes (e.g., maxima of solutions over expanding windows), with precise exponents and constants depending on the interplay of spatial and temporal noise structure (Chen, 2016, Huang et al., 2016). These phenomena link temporal intermittency and spatial "spikes" via shared variational formulas.
• Adaptive and High-Order Numerics: In computational mathematics, adaptive isogeometric analysis (IgA) methods with local stabilization parameters naturally realize space-time infill by concentrating mesh refinement—and thus error reduction—where dynamic phenomena or gradients occur, demonstrating optimal convergence rates under local mesh shrinking (Langer et al., 2018).

6. Key Methodological Considerations and Extensions

Several salient methodological consequences and extensions of space-time infill asymptotics include:

• Design and Power Calculations: Sample size planning in infill domains is fundamentally different: practitioners must account for the diminishing returns due to estimator variance limits and can use nonparametric cubic regression links between covariance tuning parameters (e.g., from an Ornstein–Uhlenbeck process) and variance ratios to determine efficient stopping rules (Natoli et al., 9 Mar 2024).
• Handling Spatial Confounding: Statistically consistent coefficient estimation in spatial regression—especially under infill regimes—requires harmonizing the smoothness of covariates and errors, either via pre-smoothing or regularization; failure to do so can result in systematic bias or even divergence in the estimation process (Bolin et al., 27 Mar 2024).
• Nonparametric Goodness-of-Fit and Model Adequacy: The ability to test and validate covariance structures nonparametrically, even when the evolution operator is unknown or inapplicable, is critical for robust modeling in high-dimensional, high-frequency regimes (Petersson et al., 28 Aug 2025).

7. Connections and Contrasts Across Domains

Space-time infill asymptotics bridges techniques and conceptual frameworks across diverse domains:

• In statistical analysis, it represents a shift from the classical paradigm and requires new convergence and sampling theories tailored to high-dependence, high-density regimes.
• In geometric and gravitational analysis, it provides a unifying language for "filling in" the structure at infinity, encoding both symmetry reduction (to canonical Poincaré or BMS subgroups) and the global structure of conserved quantities.
• In numerical and stochastic PDEs, it delivers theoretical guarantees and tools for fine-scale model calibration, parameter estimation, and adaptive computation in the presence of space-time correlated structure.

This unified and rigorous framework continues to play a central role where models and data are increasingly high-resolution, multi-scale, and demand the seamless integration of space and time in both theory and application.