### Uniqueness of invariant product measures for elliptic infinite dimensional diffusions and particle spin systems

Consider an infinite dimensional diffusion process process on ${T}^{{\mathbf{Z}}^{d}}$, where $T$ is the circle, defined by the action of its generator $L$ on ${C}^{2}\left({T}^{{\mathbf{Z}}^{d}}\right)$ local functions as $Lf\left(\eta \right)={\sum}_{i\in {\mathbf{Z}}^{d}}\left(\frac{1}{2}{a}_{i}\frac{{\partial}^{2}f}{\partial {\eta}_{i}^{2}}+{b}_{i}\frac{\partial f}{\partial {\eta}_{i}}\right)$. Assume that the coefficients, ${a}_{i}$ and ${b}_{i}$ are smooth, bounded, finite range with uniformly bounded second order partial derivatives, that ${a}_{i}$ is only a function of ${\eta}_{i}$ and that ${inf}_{i,\eta}{a}_{i}\left(\eta \right)\>0$. Suppose $\nu $ is an invariant product measure. Then, if $\nu $ is the Lebesgue measure or if $d=1,2$, it is the unique invariant measure. Furthermore, if $\nu $ is translation invariant, then...