  A general affine Markov semigroup is formulated as the convolution of a
homogeneous one with a skew convolution semigroup. We provide some sufficient
conditions for the regularities of the homogeneous affine semigroup and the
skew convolution semigroup. The corresponding affine Markov process is
constructed as the strong solution of a system of stochastic equations with
non-Lipschitz coefficients and Poisson-type integrals over some random sets.
Based on this characterization, it is proved that the affine process arises
naturally in a limit theorem for the difference of a pair of reactant processes
in a catalytic branching system with immigration.
