Invariant measures of discrete interacting particle systems: Algebraic aspects

Consider a continuous time particle system $\eta^t=(\eta^t(k),k\in \mathbb{L})$, indexed by a lattice $\mathbb{L}$ which will be either $\mathbb{Z}$, $\mathbb{Z}/n\mathbb{Z}$, a segment $\{1,\cdots, n\}$, or $\mathbb{Z}^d$, and taking its values in the set $E_{\kappa}^{\mathbb{L}}$ where $E_{\kappa}=\{0,\cdots,\kappa-1\}$ for some fixed $\kappa\in\{\infty, 2,3,\cdots\}$. Assume that the Markovian evolution of the particle system (PS) is driven by some translation invariant local dynamics with bounded range, encoded by a jump rate matrix $T$. These are standard settings, satisfied by the TASEP, the voter models, the contact processes... The aim of this paper is to provide some sufficient and/or necessary conditions on the matrix $T$ so that this Markov process admits some simple invariant distribution, as a product measure (if $\mathbb{L}$ is any of the spaces mentioned above), as the law of a Markov process indexed by $\mathbb{Z}$ or $[0,n]\cap \mathbb{Z}$ (if $\mathbb{L}=\mathbb{Z}$ or $\{1,\cdots,n\}$), or a Gibbs measure if $\mathbb{L}=\mathbb{Z}/n\mathbb{Z}$. Multiple applications follow: efficient ways to find invariant Markov laws for a given jump rate matrix or to prove that none exists. The voter models and the contact processes are shown not to possess any Markov laws as invariant distribution (for any memory $m$) Footnote: As usual, a random process $X$ indexed by $\mathbb{Z}$ or $\mathbb{N}$ is said to be a Markov chain with memory $m\in\{0,1,2,\cdots\}$ if for $\mathbb{P}(X_k \in A ~| X_{k-i}, i \geq 1)= \mathbb{P}(X_k \in A ~| X_{k-i}, 1\leq i\leq m)$, for any $k$.}. We also prove that some models close to these models do. We exhibit PS admitting hidden Markov chains as invariant distribution and design many PS on $\mathbb{Z}^2$, with jump rates indexed by $2\times 2$ squares, admitting product invariant measures.