Conditional Log-Laplace Functionals of Immigration Superprocesses with Dependent Spatial Motion

Abstract

Abstract. A non-critical branching immigration superprocess with dependent spatial motion is constructed and characterized as the solution of a stochastic equation driven by a time-space white noise and an orthogonal martingale measure. A representation of its conditional logLaplace functionals is established, which gives the uniqueness of the solution and hence its Markov property. Some properties of the superprocess including an ergodic theorem are also obtained. (2000): Primary 60J80, 60G57; Secondary 60J35 Mathematics Subject Classification

Conditional Entrance Laws for Superprocesses With Dependent Spatial Motion

“…Some similar results were obtained earlier in [21,22] for the model of Skoulakis and Adler [14]. However, the conditional log-Laplace functional in [12] does not give automatically a decomposition of the SDSM. The main difficulty is that under the conditional probability given {W (ds, dy)} we only have the a.s. Markov property of the finite dimensional distributions of the SDSM, which does not imply immediately the full conditional Markov property.…”

“…gives a generalized drift in the underlying migration. This observation was confirmed to some extend in [12] by characterizing the conditional log-Laplace functional of {X t : t ≥ 0} given {W (ds, dy)}. Some similar results were obtained earlier in [21,22] for the model of Skoulakis and Adler [14].…”

A stochastic log-Laplace equation

“…An easier approach in deriving (1.8) is available. We refer the reader to [16] for the treatment of a related model which adds immigration structure to a branching interacting system studied in [4] and [19]. In this paper, we use the Wong-Zakai approximation because this is part of the conjecture in [18] and the main purpose of the current paper is to solve that conjecture.…”

Ergodic theory for a superprocess over a stochastic flow

“…A similar characterization of the conditional log-Laplace functional of the model of [4,14] was given in [11]. The next theorem characterizes the conditional log-Laplace functional of the weighted occupation time of {X t }.…”