One-Sided Versus Two-Sided Stochastic Descriptions

Abstract

It is well-known that discrete-time finite-state Markov Chains, which are described by onesided conditional probabilities which describe a dependence on the past as only dependent on the present, can also be described as one-dimensional Markov Fields, that is, nearestneighbor Gibbs measures for finite-spin models, which are described by two-sided conditional probabilities. In such Markov Fields the time interpretation of past and future is being replaced by the space interpretation of an interior volume, surrounded by an exterior to the left and to the right. If we relax the Markov requirement to weak dependence, that is, continuous dependence, either on the past (generalising the Markov-Chain description) or on the external configuration (generalising the Markov-Field description), it turns out this equivalence breaks down, and neither class contains the other. In one direction this result has been known for a few years, in the opposite direction a counterexample was found recently. Our counterexample is based on the phenomenon of entropic repulsion in long-range Ising (or "Dyson") models.

Markov and almost Markov properties in one, two or more directions

“…In short, as observed before, in long-range settings, neglecting what lies outside borders is not equivalent to forget history, except for short-sighted approximations (see e.g. [30] for a recent review).…”

“…The system is in a minus-like phase with strictly negative magnetisation left of the interface point, and in a plus-like phase with positive magnetisation to the right of the interface point. From this penomenon it is intuitively plausible, and in fact not that hard to prove [9,30], that the one-dimensional "Dobrushin boundaries" repel the interface point, due to entropic reasons. As such, it is a phenomenon which works at low temperatures, but not at T = 0.…”