that are needed in our construction of discrete time models. The reader is invited to refer to [@CRV; @VanicatThesis] for details.

#### Preliminaries.

We are interested in describing a system of interacting particles on a one dimensional lattice with $L$ sites. Each site of the lattice can carry at most $s$ particles. Note that when $s=1$, there is at most one particle per site and this class of model is known as exclusion processes. Paradigmatic examples of exclusion processes are the ASEP and the SSEP [@DerrReview]. For $s>1$ the exclusion constraint is partially relaxed and the corresponding class of models is called generalized exclusion process. For each site $i$ of the lattice, we define a local state variable $\tau_i \in \{0,1,\dots,s \}$ denoting the number of particles lying on site $i$. A configuration of particles on the lattice is thus efficiently encapsulated in a $L$-uplet $\bm{\tau}=(\tau_1,\tau_2,\dots,\tau_L)$. We will specify later on a stochastic dynamics on this configuration space, which motivates the introduction of the quantity $\cP_t(\bm{\tau})$, which stands for the probability to observe the system in the configuration $\bm{\tau}$ at time $t$.

The time evolution of $\cP_t(\bm{\tau})$ obeys a master equation. It will be particularly efficient to formulate this master 