# Markov Random Topological Spaces

1. Aug 21, 2011

### alexfloo

The Markov chain, as you know, is a sequence of random variables with the property that any two terms of the sequence X and Y are conditionally independent given any other random variable Z that is between them. This sequence (which is in fact a family, indexed by the naturals) can and has been generalized to continuous-time Markov processes, which are families indexed by the real numbers, with the similar property that:

for all
• for all a < b < c, Xa and Xc are conditionally independent given Xb.

This condition (I believe) is equivalent to the more common formulation that the Markov random process is conditionally independent of its history given the present.

I may be failing to understand this but I believe that condition can be generalized to say that every point on the interior of an interval I is conditionally independent of every point on the exterior of that interval given the boundary of the interval.

My question is this: can that reformulation be used (or, more interestingly, has it been used, and to what end?) to generalize this notion to families of random variables indexed by an arbitrary topological space (X,$\theta$) such that, for any subset S of X, each point in the interior is independent of each point on the exterior given the values on the boundary?

(Or, put another way, random variables in the topological space interact only along continuous paths?)

This appears to be a natural (if not necessarily useful) generalization, but I haven't found anything about it anywhere.

2. Aug 21, 2011

### alexfloo

Just after posting this, I just stumbled across the idea of a general random field, and it does appear to be a common practice to index them by topological spaces. I'm not certain, however, whether the implications of the Markov property in arbitrary spaces have been explored.

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