Exponential growth of the numbers of particles for branching symmetric α -stable processes

Abstract

We study the exponential growth of the numbers of particles for a branching symmetric $\alpha$-stable process in terms of the principal eigenvalue of an associated Schrödinger operator. Here the branching rate and the branching mechanism can be state-dependent. In particular, the branching rate can be a measure belonging to a certain Kato class and is allowed to be singular with respect to the Lebesgue measure. We calculate the principal eigenvalues and give some examples.