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

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.