We discuss various properties of Probabilistic Cellular Automata, such as the structure of the set of stationary measures and multiplicity of stationary measures (or phase transition) for reversible models.

We discuss various properties of Probabilistic Cellular Automata, such as the structure of the set of stationary measures and multiplicity of stationary measures (or phase transition) for reversible models.

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...

Consider an infinite dimensional diffusion process process on TZd, where T is the circle, defined by the action of its generator L on C2(TZd) 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, ai and bi are smooth, bounded, finite range with uniformly bounded second order partial derivatives, that ai is only a function of ${\eta }_i$ and that ${inf}_{i,\eta }{a}_i\left(\eta \right)>0$. Suppose ν is an invariant product measure. Then, if ν is the Lebesgue measure or if d=1,2, it is the unique invariant measure. Furthermore, if ν is translation...