A purely atomic superprocess with dependent spatial motion is characterized as the pathwise unique solution of a stochastic partial differential equation, which is driven by a time-space white noise defining the spatial motion and a sequence of independent Brownian motions defining the branching mechanism.

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doi.org/10.1142/S0219025703001377
Infinite Dimensional Analysis, Quantum Probability and Related Topics
School of Mathematics and Statistics

Dawson, D.A, Li, Z. (Zenghu), & Wang, H. (Hao). (2003). A degenerate stochastic partial differential equation for the purely atomic superprocess with dependent spatial motion. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 6(4), 597–607. doi:10.1142/S0219025703001377