Sturm-Liouville Theory and Decay Parameter for Quadratic Markov Branching Processes

TL;DR

This paper establishes a connection between the decay parameter of quadratic Markov branching processes and the first eigenvalue of a Sturm-Liouville operator, providing explicit bounds and illustrating with examples.

Contribution

It introduces a novel link between decay parameters of QMBPs and Sturm-Liouville eigenvalues, with explicit bounds derived via Hardy inequalities.

Findings

Decay parameter equals the first eigenvalue of a Sturm-Liouville operator.

Explicit upper and lower bounds for the decay parameter are provided.

The Hardy index is deeply analyzed and estimated in relation to the decay parameter.

Abstract

For a quadratic Markov branching process (QMBP), we show that the decay parameter is equal to the first eigenvalue of a Sturm-Liouville operator associated with the PDE that the generating function of the transition probability satisfies. The proof is based on the spectral properties of the Sturm-Liouville operator. Both the upper and lower bounds of the decay parameter are given explicitly by means of a version of Hardy inequality. Two examples are provided to illustrate our results. The important quantity, the Hardy index, which is closely linked with the decay parameter of QMBP, is deeply investigated and estimated.