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 $$ \begin{equation*} W_{ij}(\epsilon)=W(il-jl,\epsilon)=\left\{\begin{array}{cc}\frac{1}{2} & |i-j| = 1\\ 0 & \mathrm{else} \end{array} \right. \end{equation*} $$