Smoothness of continuous state branching with immigration semigroups



Chazal, M, Loeffen, R and Patie, P ORCID: 0000-0003-4221-0439
(2018) Smoothness of continuous state branching with immigration semigroups. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 459 (2). pp. 619-660.

[img] Text
Main_CBI_ArXiV.pdf - Author Accepted Manuscript

Download (627kB) | Preview

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.

Item Type: Article
Uncontrolled Keywords: Continuous state branching processes with immigration, Bernstein functions, Non-self-adjoint integro-differential operators, Laguerre polynomials, Markov semigroups, Spectral theory
Depositing User: Symplectic Admin
Date Deposited: 24 Apr 2020 13:01
Last Modified: 18 Jan 2023 23:53
DOI: 10.1016/j.jmaa.2017.10.071
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/3084569