The solution for the Green's function G i j ( t ; L ) for a semi-Markovian random walk in an arbitrarily heterogeneous environment in 1D was recently given using the path representation. (The function G i j ( t ; L ) is the PDF for occupying state i at time t given that the process started at state j exactly at time 0.) A semi-Markovian random walk in 1D is defined as follows: a random walk whose dynamics are described by the (possibly) state- and direction-dependent JT-PDFs, ψ ψ i j ( t ) , for transitions between states i and i ± 1, that generates stochastic trajectories of uncorrelated waiting times that are not-exponential distributed. ψ ψ i j ( t ) obeys the normalization conditions (see fig. 1)