From: israel@math.ubc.ca (Robert Israel)
Subject: Re: Markov measures and specification
Date: 17 Jan 2000 18:40:08 GMT
Newsgroups: sci.math.research
Summary: [missing]

In article <85r5vl$i2d$1@nnrp1.deja.com>,
 v_i_smirnov@my-deja.com writes:
> Could someone give a precise definition of "Markov specification" for
> measures and explain why this indeed gives rise to a Markov measure?

You didn't specify the context, and there are various levels of generality
that can be used, so for convenience I'll assume we're talking about a 
system on a graph S (in physics often referred to as a "lattice"); for each
i in S there is a random variable X_i which takes values in a finite set W.
A configuration x is a function from S to W; for any subset A of W we let
x_A be the restriction of x to A.  Let B be the sigma-field generated by
all the X_i, and for each subset A of S let B_A be the sigma-field generated
by the X_i for i in A.
A specification is a system of functions p_A(x|y) on W^A x W^(S\A) for 
finite subsets A of S satisfying

1) p_A(x|y) >= 0 for all x,y
2) sum_{x in W^A} p_A(x|y) = 1 for all y
3) p_A(x|.) is measurable in B_{S\A} for each x
4) For any subset B of A, x in W^A and y in W^(S\A),
    p_A(x|y) = p_B(x_B|x_{A\B}, y) sum_{z in W^B} p_A((z,x_{A\B})| y)

A Markov specification is one where each p_A(x|.) is measurable in B_{b(A)}
where b(A) consists of all s in S\A which have neighbours in A (i.e. there
is an edge (s,t) of the graph with t in A).  A probability measure P on 
(W^S,B) is a Gibbs distribution for the specification p if 
P(X_A = x_A | B_{S\A}) = p_A(x_A, .) P-a.s.  

It follows directly from the definitions that a Gibbs distribution for a 
Markov specification has the Markov property P(E | B_{S\A}) = P(E | B_{b(A)})
for A finite and E in B_A.  If this is the case for infinite A as well,
we say that P has the global Markov property.  This is not always the case:
see e.g. my paper "Some Examples Concerning the Global Markov Property",
Commun. math. Phys. 105 (1986), 669-673

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2