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

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