Algorithmical Unsolvability of the Ergodicity Problem for Locally Interacting Processes with Continuous Time

We prove algorithmical unsolvability of the ergodicity problem for a class of one-dimensional translation-invariant random processes with local interaction with continuous time, also known as interacting particle systems. The set of states of every component is finite, the interaction occurs only between nearest neighbors, only one particle can change its state at a time and all rates are 0 or 1.