In this paper we consider the unique nonnegative solution to the following generalized version of the stochastic differential equation for a continuous-state branching process.

A continuous-state polynomial branching process is constructed as the pathwise unique solution of a stochastic integral equation with absorbing boundary condition. The process can also be obtained from a spectrally positive Lévy process through Lamperti type transformations. The extinction and explosion probabilities and the mean extinction and explosion times are computed explicitly. Some of those are also new for the classical linear branching process. We present necessary and sufficient conditions for the process to extinguish or explode in finite times. In the critical or subcritical case, we give a construction of the process coming down from infinity. Finally, it is shown that the continuous-state polynomial branching process arises naturally as the rescaled limit of a sequence of discrete-state processes.MSC (2010): primary 60J80; secondary 60H30, 92D15, 92D25.

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ABSTRACT:The thermal degradation of natural rubber (NR) in air at a constant heating rate was studied by using of the thermogravimetry (TG) and thermogravimetry and differential thermal analysis (TG-DTA) simultaneous techniques. It indicates that the temperature of thermal degradation of gel and sol of NR rises linearly along with the increment of the heating rate, whereas the heating rate has little effect on the degree of thermal degradation. Accompanying other side reactions, the thermal degradation of NR is not a simple random chain scission process, and it is an exothermic reaction. The dynamic variation of molecular structure of NR during the thermal degradation was studied using Fourier transform infrared spectroscopy (FTIR). It shows that the products of the thermal degradation of both sol and gel of NR are hydroperoxide, carbonyl, and hydroxyl compounds. The formation of gel makes the temperature of the thermal degradation of NR decrease and the rate of the thermal degradation increase; thus, the thermal stability of NR is reduced.