Building a stationary stochastic process from a finite-dimensional marginal (Q2715675)

Let \(\mathbb{U}\) be a finite alphabet, and let \(\mathbb{U}^\mathbb{Z}\) be the space of bi-infinite sequences on \(\mathbb{U}\). A stationary stochastic process is a probability measure \(\mu\) on \(\mathbb{U}^\mathbb{Z}\) so that, for any \(W\in\mathbb{N}\), \(b_0,b_1, \dots,b_W \in\mathbb{U}\), and any \(k\in\mathbb{Z}\) NEWLINE\[NEWLINE\mu\{a\in \mathbb{U}^\mathbb{Z}\mid a_0=b_0, \dots,a_W= b_W\}=\mu \{a\in\mathbb{U}^\mathbb{Z}\mid a_k=b_0, \dots,a_{k+W}= b_W \}NEWLINE\]NEWLINE The author adresses the following questions: If \(\mathbb{U}\) is finite, and \(U \subset \mathbb{Z}^d\), and \(\mu_U\) is a probability measure on \(\mathbb{U}^U\) that ``looks like'' the marginal projection of a stationary stochastic process on \(\mathbb{U}^{\mathbb{Z}^d}\), then can we ``extend'' \(\mu_U\) to such a process? The author shows that an extendible, locally stationary measure with full support can be ``embedded'' in any ergodic \(\mathbb{Z}^d\)-dynamical system, in the sense that it is a marginal projection of a stationary \(\mathbb{Z}^d\)-process generated by a partition on that system. The author also shows that ``almost all'' extendible measures have extensions which are ergodic, weakly mixing or quasiperiodic.