The McKean martingale for the homogeneous \(R\)-\(D\)-systems (Q1343455)

This article is part of the author's work on the probabilistic description of the solution of nonlinear partial differential equations (PDE) [see also ibid. 43, No. 2, 137-142 (1991) resp. ibid. 43, No. 2, 162-167 (1991; Zbl 0729.60081) and Math. USSR, Sb. 62, No. 2, 525-540 (1989), translation from Mat. Sb., Nov. Ser. 134(176), No. 4(12), 530-545 (1987; Zbl 0663.60066)]. The author chooses a nonlinearity which generalizes the case \(f(u) = u^2 - u\) treated by H. P. McKean and gives an explicit probabilistic description of the solution of the PDE in this case. It is in terms of a martingale which contains the Jirina branching process \(M\), and the author shows that \(M\) is spatially homogeneous iff the nonlinear part in the PDE has this property.