Path-probability density functions for semi-Markovian random walks

In random walks, the path representation of the Green's function is an infinite sum over the length of path probability density functions (PDFs). Recently, a closed-form expression for the Green's function of an arbitrarily inhomogeneous semi-Markovian random walk in a one-dimensional (1D) chain of L states was obtained by utilizing path-PDFs calculations. Here we derive and solve, in Laplace space, the recursion relation for the n order path PDF for the same system. The recursion relation relates the n order path PDF to L/2 (round towards zero for an odd L) shorter path PDFs and has n independent coefficients that obey a universal formula. The z transform of the recursion relation straightforwardly gives the generating function for path PDFs, from which we recover the Green's function of the random walk, but, moreover, derive an explicit expression for any path PDF of the random walk. These expressions give the most detailed description of arbitrarily inhomogeneous semi-Markovian random walks in 1D.