Spectral properties for a class of continuous state branching processes with immigration. (Q1405092)

The paper is a part of general program which consists in finding nontrivial classes of measure-valued processes with a good \(L^{2}\) [see also the author, Ann. Probab. 28, No. 2, 667-684 (2000; Zbl 1044.60037), Probab. Theory Relat. Fields 122, No. 3, 431-469 (2002; Zbl 1001.60079) and Ann. Probab. 31, No. 3, 1377--1412 (2003; Zbl 1043.60076)]. In the present paper the author constructs non-trivial invariant measures (including Gamma measure or path space measures of Lévy subordinators) for a class of continuous state branching processes with immigration. Long time behaviour and hypercontractivity of the transition semigroups are studied.