Dynamic uniqueness for stochastic chains with unbounded memory

We say that a probability kernel exhibits dynamic uniqueness (DU) if all the stochastic chains starting from a fixed past coincide on the future tail $\sigma$-algebra. Our first theorem is a set of properties that are pairwise equivalent to DU which allow us to understand how it compares to other more classical concepts. In particular, we prove that DU is equivalent to a weak-${?}^2$ summability condition on the kernel. As a corollary to this theorem, we prove that the Bramson–Kalikow and the long-range Ising models both exhibit DU if and only if their kernels are ${?}^2$ summable. Finally, if we weaken the condition for DU, asking for coincidence on the future $\sigma$-algebra for almost every pair of pasts, we obtain a condition that is equivalent to $\beta$-mixing (weak-Bernoullicity) of the compatible stationary chain. As a consequence, we show that a modification of the weak-${?}^2$ summability condition on the kernel is equivalent to the $\beta$-mixing of the compatible stationary chain.