Propagation of support for super-Brownian motion with general branching mechanism

We study the spatial propagation of super-Brownian motion on ${\mathbb{R}}^d$ with general critical or subcritical (spatially dependent) branching mechanisms. Under local spatial lower bounds satisfying a Keller-Osserman type integrability condition, we establish a quantitative upper bound for the short-time probability that the support exits a prescribed neighborhood of its initial support. The estimate has a Gaussian-tail form and is obtained through weighted occupation times, Feynman-Kac representations, singular elliptic boundary blow-up estimates, and mild comparison arguments for log-Laplace equations. As an application, we derive the compact support property directly for spatially dependent branching mechanisms satisfying suitable local lower bounds. This yields a sufficient compact-support criterion expressed in terms of the inverse Keller integral. In particular, for spatially dependent super-Brownian motions with stable branching, we generalize the compact support results of Engländer-Pinsky and Ren.