Collision local times, historical stochastic calculus, and competing superprocesses. (English) Zbl 0899.60081

Summary: Branching measure-valued diffusion models are investigated that can be regarded as pairs of historical Brownian motions modified by a competitive interaction mechanism under which individuals from each population have their longevity or fertility adversely affected by collisions with individuals from the other population. For three or fewer spatial dimensions, such processes are constructed using a new fixed-point technique as the unique solution of a strong equation driven by another pair of more explicitly constructible measure-valued diffusions. This existence and uniqueness is used to establish well-posedness of the related martingale problem and hence the strong Markov property for solutions. Previous work of the authors has shown that in four or more dimensions models with the analogous definition do not exist. The definition of the model and its study require a thorough understanding of random measures called collision local times. These gauge the extent to which two measure-valued processes or an \(\mathbb{R}^d\)-valued process and a measure-valued process “collide” as time evolves. This study and the substantial amount of related historical stochastic calculus that is developed are germane to several other problems beyond the scope of the paper. Moreover, new results are obtained on the branching particle systems imbedded in historical processes and on the existence and regularity of superprocesses with immigration, and these are also of independent interest.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J55 Local time and additive functionals
60H99 Stochastic analysis
60G57 Random measures