Applying Dynkin's isomorphism: An alternative approach to understand the Markov property of the de Wijs process

TL;DR

This paper extends Dynkin's isomorphism to the de Wijs process, a Gaussian Markov random field in spatial statistics, revealing new insights into its connection with planar Brownian motion and kriging.

Contribution

It introduces an extension of Dynkin's isomorphism to the de Wijs process, linking it with planar Brownian motion and enhancing understanding of spatial Markov models.

Findings

Extended Dynkin's isomorphism to the de Wijs process.

Connected the de Wijs process with recurrent planar Brownian motion.

Provided new insights into Matheron's kriging formula.

Abstract

Dynkin's (Bull. Amer. Math. Soc. 3 (1980) 975-999) seminal work associates a multidimensional transient symmetric Markov process with a multidimensional Gaussian random field. This association, known as Dynkin's isomorphism, has profoundly influenced the studies of Markov properties of generalized Gaussian random fields. Extending Dykin's isomorphism, we study here a particular generalized Gaussian Markov random field, namely, the de Wijs process that originated in Georges Matheron's pioneering work on mining geostatistics and, following McCullagh (Ann. Statist. 30 (2002) 1225-1310), is now receiving renewed attention in spatial statistics. This extension of Dynkin's theory associates the de Wijs process with the (recurrent) Brownian motion on the two dimensional plane, grants us further insight into Matheron's kriging formula for the de Wijs process and highlights previously unexplored relationships of the central Markov models in spatial statistics with Markov processes on the plane.