  We investigate the anisotropic stable JCIR process which is a
multi-dimensional extension of the stable JCIR process but also a
multi-dimensional analogue of the classical JCIR process. We prove that the
heat kernel of the anisotropic stable JCIR process exists and it satisfies an
a-priori bound in a weighted anisotropic Besov norm. Based on this regularity
result we deduce the strong Feller property and prove, for the subcritical
case, exponential ergodicity in total variation. Also, we show that in the
one-dimensional case the corresponding heat kernel is smooth.
