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

ESAIM: Probability and Statistics

Consider an infinite dimensional diffusion process process on ${T}^{{𝐙}^{d}}$, where $T$ is the circle, defined by the action of its generator $L$ on ${C}^{2}\left({T}^{{𝐙}^{d}}\right)$ local functions as $Lf\left(\eta \right)={\sum }_{i\in {𝐙}^{d}}\left(\frac{1}{2}{a}_{i}\frac{{\partial }^{2}f}{\partial {\eta }_{i}^{2}}+{b}_{i}\frac{\partial f}{\partial {\eta }_{i}}\right)$. 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...

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

ESAIM: Probability and Statistics

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

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