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). Here we derive and solve, in Laplace space, the recursion relation for the n order path PDF for any arbitrarily inhomogeneous semi-Markovian random walk in a one-dimensional (1D) chain of L states. 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 obtain the Green’s function of the random walk, and 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.

💡 Research Summary

The paper tackles the long‑standing problem of describing random walks on a one‑dimensional chain of L states when the walk is both inhomogeneous (state‑dependent transition probabilities) and semi‑Markovian (state‑specific waiting‑time distributions). Traditional Green‑function representations write the propagator as an infinite sum over all possible paths, but they do not give explicit formulas for the probability density function (PDF) of each path length. The authors fill this gap by deriving a closed‑form recursion relation for the n‑step path PDF in Laplace space and then solving it analytically.

Starting from the semi‑Markov definition, each state i has a waiting‑time density ψ_i(t) with Laplace transform ϕ_i(s). The transition matrix T contains the probabilities of jumping from state i to j. By combining these, the authors define a kernel matrix K(s) with elements K_{ij}(s)=T_{ij} ϕ_j(s). The n‑step path PDF, denoted P_n(s) in Laplace space, satisfies a linear recursion that couples it only to the previous ⌊L/2⌋ shorter PDFs:

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