One-sided versus two-sided points of view

Abstract:
Finite-state, discrete-time Markov chains coincide with Markov fields on Z, (which are nearest-neighbour Gibbs measures in one dimension). That is, the one-sided Markov property and the two-sided Markov property are equivalent. We discuss to what extent this remains true if we try to weaken the Markov property to the almost Markov property, which is a form of continuity of conditional probabilities. The generalisation of the one-sided Markov measures leads to the so-called "g-measures" (aka "chains with complete connection", "uniform martingales",..), whereas the two-sided genralisation leads to the class of Gibbs or DLR measures, as studied in statistical mechanics. It was known before that there exist g-measures which are not Gibbs measures. It is shown here that neither class includes the other. We consider this issue in particular on the example of long-range, Dyson model, Gibbs measures. This is a joint work with R.Bissacot, E.Endo and A. Le Ny.