Central limit theorems for supercritical branching nonsymmetric Markov processes (Q516129)

Let \(\xi = \{ {\xi _t},{\Pi _x}\} \) be a Hunt process on a locally compact separable metric space \(E\). The paper considers a branching system characterized by the following properties: i) each individual has a random birth and death time; ii) given that an individual is born at \(x \in E\), the conditional distribution of its path is determined by \({\Pi _x}\); iii) given the path \(\xi \) of an individual up to time \(t\) and given that the particle is alive at time \(t\), its probability of dying in the interval \([t,t + dt)\) is \(\beta ({\xi _t})dt + o(dt)\); iv) when an individual dies at \(x \in E\), it splits into \(n\) individuals all positioned at \(x\), with probability \({p_n}(x)\); v) when an individual reaches a cemetery point \(\partial \), it disappears from the system; vi) all the individuals, once born, evolve independently. The branching mechanism is supposed to satisfy the condition \(\mathop {\sup }\limits_{x \in E} \sum\limits_{n = 0}^\infty {{n^2}{p_n}(x)} < \infty \). The authors establish a spatial central limit theorem for supercritical branching processes, which generalizes and unifies their results in [J. Funct. Anal. 266, No. 3, 1716--1756 (2014; Zbl 1292.60034)] for symmetric processes. To this end, they develop the spectral theory of nonsymmetric strongly continuous semigroups.