Stochastic processes can be parametrised by time (such as occurs in Markov chains), in which case conditioning is one-sided (on the past), as naturally occurs in Dynamical Systems, or by one-dimensional space (which is the case, for example, for one-dimensional Markov fields), as is natural in Statistical Mechanics, where the 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.