Has the following conjecture been prooved, or has any step in the direction of its proof been done?


"ANY Probabilistic Cellular Automata converge fast on the stationary probability distribution iff the infinite system is ergodic and converge slowly iff the infinite system is non-ergodic".


"Fast" means that the distance between the asymptotic probability distribution and the probability distribution at time $t$ decreases exponentially with time. "Slow" if the speed of convergence is slower than exponential.



Ergodicity and convergence time in Probabilistic Cellular Automata
yes, a result in this direction was stated for a family of reversible Markov stochastic dynamics. See <https://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8215259> and the article "Ergodicity of PCA: equivalence between spatial and temporal mixing conditions" downloadable there 
<http://users.math.uni-potsdam.de/~louis/>
Best.


