Measure - Theoretic Boundaries of Markov Chains , 0 - 2 Laws And

The classic Poisson formula giving an integral representation of bounded harmonic functions in the unit disk in terms of its boundary values has a long history (as it follows from its very name). Given a Markov operator P on a state space X one can easily deene harmonic functions as invariant functions of the operator P, but in order to speak about their boundary values one needs a boundary, because no boundary is normally attached to the state space of a Markov chain (as distinct from bounded Euclidean domains common for the classic potential theory). One way of getting rid of this nuisance is to try to nd a topological compactiication of the state space naturally connected with the Markov operator P. This problem was solved by Martin by constructing the famous Martin boundary which when applied to a Markov operator gives an integral representaion for all positive harmonic functions in terms of extreme harmonic functions (see 17] 46] for references). Nonetheless, the Martin approach has some drawbacks. In the rst place, one needs a topology on the state space to be able to speak about its compactiications, and the transition probabilities must be absolutely continuous to ensure that the cone of positive harmonic functions is a simplex 17]. The Martin boundary doesn't have good functorial properties (see 20], 40], 43] for a discussion of problems connected with the Martin boundary for a product of two chains, and 7] for an example when the Martin boundary for a polynomial P = P (n)P n is \much worse" than the Martin boundary for P in a very simple situation). Finally, from a probabilistic point of view it would be more natural to ask rst about suitable measure-theoretical objects, and only then about topological ones. A function f being bounded harmonic is equivalent to the sequence f(y n) being a bounded backward martingale with respect to the decreasing sequence of coordinate-algebras A 1 n = fy k : k ng determined by the behaviour of the chain at times k n. Thus, for any bounded harmonic function f there exists the limit b f(y) = lim f(y n). The function b f is measurable with respect to the stationary-algebra in the path space consisting of the sets which are invariant under the time shift, and, conversely, every such bounded b f determines a harmonic function. If the path space is a Lebesgue …