Let us recapitulate some of our results about Markov chains and random walks.

• The time development of our PDF $w(t)$, after

one time-step from $t=0$ is given by $$\begin{equation*} w_i(t=\epsilon) = W(j\rightarrow i)w_j(t=0). \end{equation*}$$

This equation represents the discretized time-development of an original PDF. We can rewrite this as a $$\begin{equation*} w_i(t=\epsilon) = W_{ij}w_j(t=0). \end{equation*}$$ with the transition matrix $W$ for a random walk given by $$\begin{equation*} W_{ij}(\epsilon)=W(il-jl,\epsilon)=\left\{\begin{array}{cc}\frac{1}{2} & |i-j| = 1\\ 0 & \mathrm{else} \end{array} \right. \end{equation*}$$