On conditioned limit structure of the Markov branching process without finite second moment

Consider the continuous-time Markov Branching Process. In the critical case we consider a situation when the generating function of intensity of transformation of particles has the infinite second moment, but its tail regularly varies in sense of Karamata. First we discuss limit properties of transition functions of the process. We prove local limit theorems and investigate ergodic properties of the process. Further we investigate lim- iting probability function conditioned to be never extinct. Hereupon, we obtain a new stochastic population process as a continuous-time Markov chain called the Markov Q-Process. We study main properties of Markov Q-Process