Supersymmetric time-continuous discrete random walks

TL;DR

This paper extends supersymmetric techniques to analyze one-dimensional discrete random walks, providing exact solutions for homogeneous cases and generalizations to multi-axis scenarios, with potential applications in bistable processes.

Contribution

It introduces a supersymmetric approach to discrete random walks, including a formal second-order master derivative and solutions for homogeneous and multi-axis cases.

Findings

Exact matrix solutions for homogeneous random walks.

Extension of supersymmetric methods to multi-axis cases.

Potential applications in bistable and multistable processes.

Abstract

We apply the supersymmetric procedure to one-step random walks in one dimension at the level of the usual master equation, extending a study initiated by H.R. Jauslin [Phys. Rev. A {\bf 41}, 3407 (1990)]. A discussion of the supersymmetric technique for this discrete case is presented by introducing a formal second-order discrete master derivative and its ``square root", and we solve completely, and in matrix form, the cases of homogeneous random walks (constant jumping rates). A simple generalization of Jauslin's results to two uncorrelated axes is also provided. There may be many applications, especially to bistable and multistable one-step processes.