Measure-Valued Branching Markov Processes, 2nd

This book provides a compact introduction to the theory of measure-valued branching processes. Using an approach based on skew convolution semigroups, it focuses on the systematic treatment of immigration superprocesses and generalized Ornstein–Uhlenbeck processes, two classes which are connected by fluctuation theorems. Measure-valued branching processes arise as high density limits of branching particle systems. The first part of the book gives an analytic construction of a special class of such processes, the Dawson–Watanabe superprocesses, which includes the finite-dimensional continuous-state branching process as an example. By constructing superprocesses with Borel right underlying motions and general branching mechanisms, the existence of their Borel right realizations is shown. Transformations are then used to derive the existence and regularity of several different forms of the superprocesses. This technique simplifies the constructions and gives useful new perspectives. Martingale problems of superprocesses are then discussed under Feller type assumptions. The second part investigates the immigration structures associated with measure-valued branching processes, starting with some characterizations of their entrance laws. A theory of stochastic equations for one-dimensional continuous-state branching processes with or without immigration is developed, which plays a key role in the construction of measure flows of those processes. The volume is aimed at researchers in measure-valued processes, branching processes, stochastic analysis, biological and genetic models, and graduate students in probability theory and stochastic processes.