Branching diffusion processes in a bounded domain with absorbing boundary

Let $\mu_{xn}(U)$, $U\subset\mathscr X$, be the number of particles of the $n$-th generation in the set $U$ provided initially there was a single particle which was located at the point $x$. It is proved that, for a subcritical branching process, finite-dimensional distributions of the conditional random measure $\mu_{xn}$, $\mu_{xn}(\mathscr X)>0$, converge to finite-dimensional distributions of a fixed random measure $\mu$ independent of the initial distribution. An equation for the generating functional of this measure is found, as well as a sufficient condition for its expectation to be finite. For a critical branching process the limit distribution is given explicitly.