Conditional formulation of Gaussian systems on spatial networks with irregularly distributed points

In this talk I develop a mathematical and statistical framework for analyzing spatial systems on irregularly distributed point locations. Using conditional formulations, this framework constructs Gaussian systems on spatial networks with symmetric weights for both stationary and non-stationary point processes. The framework associates these systems with reversible Markov chains and, through Kipnis–Varadhan invariance principles, establishes geostatistical scaling limits that recover the de Wijs process or Gaussian free field under stationary sampling and generalize to more complex limits under inhomogeneous sampling. The framework further studies convergence through Beurling–Deny decomposition, the effects of non-uniform sampling on limiting fields, and normalization procedures based on Hungarian embeddings and Sinkhorn–Knopp scaling for restoring canonical limits. It also enables scalable matrix-free computation, REML estimation, conditional simulation, spatial prediction, and sampling-invariant inference on large irregular spatial networks, for which there is currently little parallel in the geostatistical literature. The talk concludes with applications to mapping soil organic carbon in Tanzania, reconstructing global seawater oxygen isotope fields, and analyzing leukemia survival data from northwest England.

This is joint work with postdoctoral scholar Subhrajyoty Roy.