  We study a catalytic branching process (CBP) with any finite set of
catalysts. This model describes a system of particles where the movement is
governed by a Markov chain with arbitrary finite or countable state space and
the branching may only occur at the points of catalysis. The results obtained
generalize and strengthen those known in cases of CBP with a single catalyst
and of branching random walk on d-dimensional integer lattice with a finite
number of sources of particles generation. We propose to classify CBP with N
catalysts as supercritical, critical or subcritical according to the value of
the Perron root of a specified NxN matrix. Such classification agrees with the
moment analysis performed here for local and total particles numbers. By
introducing the criticality set C we also consider the influence of catalysts
parameters on the process behavior. The proof is based on construction of
auxiliary multi-type Bellman-Harris processes with the help of hitting times
under taboo and on application of multidimensional renewal theorems.
  Keywords and phrases: catalytic branching process, classification, hitting
times under taboo, moment analysis, multi-type Bellman-Harris process.
