We prove a Kolmogorov test for super-Brownian motion started at the Dirac mass at the origin. More precisely, we determine the functions $g$ such that for all $t$ small enough, the support of the process at time $t$ will be contained in the ball of radius $g(t)$ centered at 0. As a consequence, we get a necessary and sufficient condition for the existence in certain space-time domains of a solution of the associated semilinear partial differential equation that blows up at the origin.
Jean-Stéphane Dhersin. Jean-François Le Gall. "Kolmogorov's test for super-Brownian motion." Ann. Probab. 26 (3) 1041 - 1056, July 1998. https://doi.org/10.1214/aop/1022855744