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Path-probability density functions for semi-Markovian random walks.

O Flomenbom¹, R J Silbey

¹Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|November 13, 2007

Summary

Researchers derived and solved a recursion relation for path probability density functions (PDFs) in semi-Markovian random walks. This provides explicit expressions for any path PDF, offering a detailed description of these complex random walk systems.

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Area of Science:

Statistical Physics
Stochastic Processes
Mathematical Physics

Background:

The Green's function in random walks is typically an infinite sum of path probability density functions (PDFs).
A recent study provided a closed-form Green's function for inhomogeneous semi-Markovian random walks in 1D using path-PDF calculations.

Purpose of the Study:

To derive and solve the recursion relation for the n-order path PDF in 1D inhomogeneous semi-Markovian random walks.
To obtain explicit expressions for any path PDF in these systems.

Main Methods:

Solving a recursion relation for the n-order path PDF in Laplace space.
Utilizing the z-transform of the recursion relation to find the generating function for path PDFs.

Main Results:

The recursion relation connects n-order path PDFs to shorter path PDFs, featuring n independent coefficients with a universal formula.
The generating function for path PDFs was derived, allowing recovery of the Green's function.
Explicit expressions for any path PDF of the random walk were obtained.

Conclusions:

The derived expressions offer the most detailed description to date for arbitrarily inhomogeneous semi-Markovian random walks in 1D.
This work advances the understanding of complex stochastic processes and their analytical solutions.