An alternative approach to super-Brownian motion with a locally infinite branching mass

Fleischmann and Mueller (Probab. Theory Related Fields 107 (1997) 325) constructed a super-Brownian motion in R1 with a locally infinite branching rate function, and they showed that this super-Brownian motion has a strong killing property in the critical case. In this paper, we first construct, via a limiting procedure, a super-Brownian motion which is equivalent to the super-Brownian motion with a locally infinite branching. From this construction, one can easily see the connection between the superprocess and a killed Brownian motion. Next, by taking advantage of the new construction, we give a new proof of the strong killing property of the process.