Since both $\boldsymbol{W}$ and $\boldsymbol{w}$ 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.