Limit Theorems for Superprocesses : Rescaled Processes, Immigration Superprocesses and Homogeneous Superprocesses

This monograph treats recent results on limit theorems for superprocesses (SPs). Chapter 1 is devoted to a class of SPs with deterministic immigration, and deals with a convergence problem for rescaled processes. When such an SP with dependent spatial motion (SDSM) is given, under a proper scaling the rescaled SPs converge to an SP with coalescing spatial motion (SCSM) in the sense of probability distribution on the space of measure-valued continuous paths (pdmc). In Chapter 2 we discuss a distinct convergence problem for a class of SDSMs with deterministic immigration rate. When the immigration rate converges to a non-vanishing deterministic one, then we can prove that under a suitable scaling, the rescaled SPs associated with SDSM converge to a class of immigration SPs associated with SCSM in the sense of pdmc. This rescaled limit does not provide only with a new class of SPs but gives also a new type of limit theorem. A class of homogeneous SPs of diffusion type with spatially dependent parameters and its asymptotic behaviors are discussed in Chapter 3. If the underlying diffusion is recurrent, we prove a limit theorem on the moment of such SPs as time t goes to infinity.