One dimensional stochastic partial differential equations and the branching measure diffusion
An existence theorem in the spirit of \textit{H. J. Keisler} [see: An infinitesimal approach to stochastic analysis. Mem. Am. Math. Soc. 297 (1984; Zbl 0529.60062)], is proved for the simple one-dimensional diffusion equation driven by white noise modulated by a nonlinear function of the solution. This is used to obtain a density and a stochastic partial differential equation in one dimension for the critical branching diffusion studied by \textit{D. A. Dawson} and \textit{K. J. Hochberg} [see Ann. Probab. 7, 693-703 (1979; Zbl 0411.60084), and Carlton Math. Lecture Notes No.22, 4 (1978; Zbl 0381.60044)].
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