Sturm-Liouville systems for the survival probability in first-passage time problems

Abstract

We derive a Sturm-Liouville system of equations for the exact calculation of the survival probability in first-passage time problems. This system is the one associated with the Wiener-Hopf integral equation obtained from the theory of random walks. The derived approach is an alternative to the existing literature and we tested it against direct calculations from both discrete- and continuous-time random walks in a manageable, but meaningful, example. Within this framework, the Sparre Andersen theorem results to be a boundary condition for the system.

10 pages

References in corpus (2)

• Unified Solution of the Expected Maximum of a Random Walk and the Discrete Flux to a Spherical Trap
• On the Gap and Time Interval between the First Two Maxima of Long Random Walks