Spread rate of catalytic branching symmetric stable processes

We study the growth order of the maximal displacement of branching symmetric $\alpha$-stable processes. We assume the branching rate measure $\mu$ is in the Kato class and $\mu$ has a compact support on ${\mathbb R}^d$. We show that the maximal displacement exponentially grows and its order is determined by the index $\alpha$ and the spectral bottom of the corresponding Schr\"odinger-type operator.