Lifetime and compactness of range for super-Brownian motion with a general branching mechanism

Let X be a super-Brownian motion with a general (time-space) homogeneous branching mechanism. We study a relation between lifetime and compactness of range for X. Under a restricted condition on the branching mechanism, we show that the set X survives is the same as that the range of X is unbounded. (For α-branching super-Brownian motion, 1 < α ⩽ 2, similar results were obtained earlier by Iscoe (1988) and Dynkin (1991).) We also give an interesting example in that case X dies out in finite time, but it has an unbounded range.