An elementary proof of the uniqueness of invariant product measures for some infinite dimensional processes (Q1598477)

Let \(T\) denote the unit circle. Consider a finite range elliptic infinite-dimensional diffusion on the torus \(T^{\mathbb{Z}^d}\), \(d\geq 1\), whose coefficients are bounded and satisfy other regularity conditions. Let \(\nu\) be a product measure which is invariant for the process considered. Then (i) if \(\nu\) is translation invariant, then it is the unique invariant, translation invariant measure; (ii) if \(d=1\) or 2, then \(\nu\) is the unique invariant measure; (iii) if \(\nu\) is Lebesgue measure, then it is the unique invariant measure. These results generalize previous results of the author [Probab. Theory Relat. Fields 110, No.~3, 369-395 (1998; Zbl 0929.60081)]. The proofs are elementary. The results obtained are partially extended to some interacting particle systems with state space \([0,1]^{\mathbb{Z}^d}\).