For the Markov process we have a transition probability from a position $x=jl$ to a position $x=il$ given by $$\begin{equation*} W_{ij}(\epsilon)=W(il-jl,\epsilon)=\left\{\begin{array}{cc}\frac{1}{2} & |i-j| = 1\\ 0 & \mathrm{else} \end{array} \right. , \end{equation*}$$ where $W_{ij}$ is normally called the transition probability and we can represent it, see below, as a matrix. Here we have specialized to a case where the transition probability is known.

Our new PDF $w_i(t=\epsilon)$ is now related to the PDF at $t=0$ through the relation $$\begin{equation*} w_i(t=\epsilon) =\sum_{j} W(j\rightarrow i)w_j(t=0). \end{equation*}$$

This equation represents the discretized time-development of an original PDF with equal probability of jumping left or right.