For the anisotropic stable JCIR process, the heat kernel exists and obeys a weighted anisotropic Besov bound, the strong Feller property holds, and in the subcritical case convergence to the invariant measure is exponential in total variation.
Boundary behavior of multi-type continuous-state branching processes with immigration
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abstract
In this article we provide a sufficient condition for a continuous-state branching process with immigration (CBI process) to not hit its boundary, i.e. for non-extinction. Our result applies to arbitrary dimension $d \geq 1$ and is formulated in terms of an integrability condition for its immigration and branching mechanisms $F$ and $R$. The proof is based on a suitable comparison with one-dimensional CBI processes and an existing result for one-dimensional CBI processes. The same technique is also used to provide a sufficient condition for transience of multi-type CBI processes.
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On the anisotropic stable JCIR process
For the anisotropic stable JCIR process, the heat kernel exists and obeys a weighted anisotropic Besov bound, the strong Feller property holds, and in the subcritical case convergence to the invariant measure is exponential in total variation.