Since both $W$ and $w$ represent probabilities, they have to be normalized, i.e., we require that at each time step we have $$\begin{equation*} \sum_i w_i(t) = 1, \end{equation*}$$ and $$\begin{equation*} \sum_j W(j\rightarrow i) = 1, \end{equation*}$$ which applies for all $j$-values. The further constraints are $0 \le W_{ij} \le 1$ and $0 \le w_{j} \le 1$. Note that the probability for remaining at the same place is in general not necessarily equal zero.