Smoothness of continuous state branching with immigration semigroups

TL;DR

This paper thoroughly analyzes the smoothness and regularity properties of semigroups and heat kernels associated with continuous state branching processes with immigration, providing new spectral representations and convergence results.

Contribution

It introduces novel spectral expansion techniques and bounds, enhancing understanding of the smoothness and convergence properties of CBI semigroups.

Findings

Derived new eigenfunction and eigenmeasure representations.

Established uniform bounds for spectral convergence.

Proved the strong Feller property and convergence rates.

Abstract

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 with immigration. Our approach is based on an in-depth analysis of the regularity of the absolutely continuous part of the invariant measure combined with a substantial refinement of Ogura's spectral expansion of the transition kernels. In particular, we find new representations for the eigenfunctions and eigenmeasures that allow us to derive delicate uniform bounds that are useful for establishing the uniform convergence of the spectral representation of the semigroup acting on linear spaces that we identify. We detail several examples which illustrate the variety of smoothness properties that CBI transition kernels may enjoy and also reveal that our results are sharp. Finally, our technique enables us to provide the (eventually) strong Feller property as well as the rate of convergence to equilibrium in the total variation norm.