Reproduction of initial distributions from the first hitting time distribution for birth-and-death processes

TL;DR

This paper demonstrates that for birth-and-death processes, the initial distribution can be reconstructed from the first hitting time distribution using a differential operator derived from the process's spectral properties.

Contribution

It introduces a method to reproduce initial distributions from hitting time data via spectral analysis and differential operators for birth-and-death processes.

Findings

Reproduction of initial distributions from hitting times is possible.

Spectral theory provides the framework for this reproduction.

Application to asymmetric random walks extends the method.

Abstract

For birth-and-death processes, we show that every initial distribution is reproduced from the first hitting time distribution. The reproduction is done by applying to the distribution function a differential operator defined through the eigenfunction of the generator. Using the spectral theory for generalized second-order differential operators, we study asymmetric random walks.