  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.
