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

Ramírez, Alejandro F.

ESAIM: Probability and Statistics, Tome 6 (2002), pp. 147-155.

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 it is the unique invariant, translation invariant measure. Now, consider an infinite particle spin system, with state space ${\{0,1\}}^{{𝐙}^d}$, defined by the action of its generator on local functions $f$ by $Lf\left(\eta \right)={\sum }_{x\in {𝐙}^d}c(x,\eta )(f\left({\eta }^x\right)-f\left(\eta \right))$, where ${\eta }^x$ is the configuration obtained from $\eta$ altering only the coordinate at site $x$. Assume that $c(x,\eta )$ are of finite range, bounded and that ${inf}_{x,\eta }c(x,\eta )>0$. Then, if $\nu$ is an invariant product measure for this process, $\nu$ is unique when $d=1,2$. Furthermore, if $\nu$ is translation invariant, it is the unique invariant, translation invariant measure. The proofs of these results show how elementary methods can give interesting information for general processes.

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