Markov snakes and superprocesses (Q1902864)

\textit{J.-F. Le Gall} introduced a class of path-valued Markov processes [ibid. 95, No. 1, 25-46 (1993; Zbl 0794.60076)] and obtained a representation of superdiffusions in terms of them. He used these to investigate path properties of superdiffusions and to give a probabilistic approach to the study of a class of nonlinear partial differential equations. The authors present an isomorphism theorem which establishes a one-to-one relation between a class of these path processes, which they call Markov snakes, and a class of superprocesses. This provides two different sets of tools which can be used to study these which are complementary. They also establish the weak convergence of a class of discrete Markov snakes, called simple snakes, which arise from a simple random walk on \(\beta Z_+\) with reflection at zero as \(\beta\to0\). They also consider the class of simple L-snakes (corresponding to the random walk killed on the first return to 0). The corresponding Brownian L-snake is used to describe the limit distributions for additive functionals of the simple L-snake.