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In this work we develop an original and thorough analysis of the(non)-smoothness properties of the semigroups, and their heat kernels,associated to a large class of continuous state branching processes withimmigration. Our approach is based on an in-depth analysis of the regularity ofthe absolutely continuous part of the invariant measure combined with asubstantial refinement of Ogura's spectral expansion of the transition kernels.In particular, we find new representations for the eigenfunctions andeigenmeasures that allow us to derive delicate uniform bounds that are usefulfor establishing the uniform convergence of the spectral representation of thesemigroup acting on linear spaces that we identify. We detail several exampleswhich illustrate the variety of smoothness properties that CBI transitionkernels may enjoy and also reveal that our results are sharp. Finally, ourtechnique enables us to provide the (eventually) strong Feller property as wellas the rate of convergence to equilibrium in the total variation norm.

Smoothness of continuous state branching with immigration semigroups