Markov random field with continuous index set

Question

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.

Answer 1

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 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).

Answer 2

There is the continuum 2-dimensional Gaussian free field, which is a higher dimensional generalization of Brownian motion. The continuum GFF satisfies the so-called domain Markov property (meaning if you condition on the value of the GFF on a subset of $\mathbb{R}^2$, its value outside that subset depends only on the value on the boundary of that subset), and can be viewed as weak limit of GFF defined on 2-dimensional infinite lattices. The discrete GFF is basically a probability distribution on $\mathbb{R}^{\mathbb{Z}^2}$, i.e., the space of real-valued functions on $\mathbb{Z}^2$. The distribution is given in terms of a Hamiltonian, $$\displaystyle H(x) = \sum_{(i,j) \in E} x_i x_j + \sum_{i \in \mathbb{Z}^2} x_i^2 $$ In other words, the probability density is proportional to $\exp(-H(x))$. Thus it's the most natural Gaussian measure on $\mathbb{R}^{\mathbb{Z}^2}$ that takes into account the underlying graph structure.

As an analogy, the Brownian motion at discrete time points, say $\mathbb{Z}$ is the Gaussian free field on $\mathbb{Z}$. Another perspective is to view Gaussian free field as the standard Gaussian random variable on the $\mathbb{Z}^2$ or $\mathbb{Z}$ but with the Dirichlet inner product instead of the usual Euclidean inner product; since Dirichlet product is basically the $\mathcal{L}^2$ inner product of the gradient, one needs to impose boundary condition or define certain equivalence classes of functions in order for the inner product to be nondegenerate.

When we take finer and finer grid in $\mathbb{Z}$ or $\mathbb{Z}^2$, we obtain in the weak limit the Brownian motion and the continuum 2-dimensional Gaussian free field. The only thing weird about the continuum GFF in 2 dimensions is that it's no longer a probability law on a function space, but rather on the space of distributions, or generalized functions on $\mathbb{R}^2$; Brownian motion on the other hand can still be viewed as a distribution on continuous functions on $\mathbb{R}$. For more details on how to define such a limiting object, see the lecture notes by Scott Sheffield: Gaussian free fields for mathematicians.

Answer 3

In general, the simplest extension of (real-valued) one-parameter processes is to (real-valued) multi-parameter processes, where the index set is $\mathbb{R}^n_+$ (usually $n=2$). This includes, for example, the $n$-parameter Brownian sheet, and more generally, $n$-parameter L\'evy sheets (or processes). However, such processes are much simpler than general set-indexed processes.

You'll probably find the following paper very useful (especially the introduction, since the set-indexed framework might be too heavy): A Markov Property For Set-Indexed Processes.