Hi


There's Markov random field (MRF) which, by my Wikipedia-based knowledge, is an extension of Markov chain. I'd like to think of it as going from 1D to higher dimensional spaces. Inherent in its definition though, the space a MRF lives on (i.e., the index set of the stochastic process) is a discrete graph. So it's actually a lattice and not the continuum of a Euclidean space (or some manifold for that matter). I'm wondering if there exists a MRF of the latter form, in other words, an extension of the Markov process (as opposed to the Markov chain) to higher dimensions.


I know that a similar extension exists for the Poisson process, namely the spatial Poisson process. But for MRF, I'm not even sure how I'd define the Markov property.


Any references or remarks are appreciated.



Markov random field with continuous index set
Essentially the Markov property in higher dimensions means that for any index set $D$ with nice boundary, the conditional distribution of the restriction of the field to indices in $D$ conditioned on the realization outside of $D$ coincides with the same thing conditioned on the realization on the boundary of $D$.


From Rozanov's book [http://www.amazon.com/Markov-Random-Fields-Applications-Mathematics/dp/0387907084/ref=sr\_1\_1?ie=UTF8&s=books&qid=1300042943&sr=8-1](http://rads.stackoverflow.com/amzn/click/0387907084)
that I looked into about 10 years ago just for fun, I vaguely remember that this property sometimes has to be altered a little, but maybe this is only because Rozanov wanted to consider generalized Markov fields ("generalized" means in Sobolev-Schwartz sense).


