We call \( W_{ij} \) for the transition probability and we represent it as a matrix.

• Both \( W \) and \( w \) represent probabilities and they have to be normalized, meaning that at each time step we have

$$ \begin{equation*} \sum_i w_i(t) = 1, \end{equation*} $$ and $$ \begin{equation*} \sum_j W(j\rightarrow i) = 1. \end{equation*} $$ Here we have written the previous matrix \( W_{ij}=W(j\rightarrow i) \).