A probabilistic proof of non-explosion of a non-linear PDE system

Using a representation in terms of a two-type branching particle system, we prove that positive solutions of the system (u) over dot = Au + uv, (v) ov er dot = Bv + uv remain bounded for suitable bounded initial conditions, pr ovided A and B generate processes with independent increments and one of th e processes is transient with a uniform power decay of its semigroup. For t he case of symmetric stable processes on R-1, this answers a question raise d in [4].