  Continuous state branching processes with immigration are studied. We are
particularly concerned with the associated (non-symmetric) Dirichlet form.
After observing that gamma distributions are only reversible distributions for
this class of models, we prove that every generalized gamma convolution is a
stationary distribution of the process with suitably chosen branching mechanism
and with continuous immigration. For such non-reversible processes, the strong
sector condition is discussed in terms of a characteristic called the Thorin
measure. In addition, some connections with notion from noncommutative
probability theory will be pointed out through calculations involving the
Stieltjes transform.
