One-sided versus two-sided stochastic processes

Stochastic processes can be parametrised by time (such as occurs in Markov chains), in which case conditioning is one-sided (on the past) or by one-dimensional space (which is the case, for example, for one-dimensional Markov fields), where conditioning is two-sided (on the right and on the left). I will discuss some examples, in particular generalising this distinction to g-measures versus Gibbs measures, where, instead of a Markovian dependence, the weaker property of continuity (in the product topology) is considered. In particular I will discuss when the two descriptions (one-sided or two-sided) produce the same objects and when they are different. We show moreover the role one-dimensional entropic repulsion plays in this setting. Based on joint work with R. Bissacot, E. Endo and A. Le Ny, and S. Shlosman