This paper is divided into 7 chapters．The central theme of these lectures is the construction and study of a new class of superpro-cesses named as“superprocesses arising from interacting stochastic flows”(abbreviated to SAISF).

Preface

Chapter 1 From particle systems to measure-valued processes

  1.1 Measure-valued feller processes

  1.2 Independent particle systems:dynamical law of large numbers

  1.3 Exchangeable particle systems

  1.4 Random probability measures moment measures and exchangeable sequences

  1.5 Weak convergence and the martingale problem

  1.6 Branching particle systems

Chapter 2 Random measures and canonical measure-valued processes

  2.1 State spaces for measure-valued processes

  2.2 Random measures and laplace functions

  2.3 Poisson cluster random measures and the canonical representation of infinitely divisible random measures

  2.4 The structure of random measures

  2.5 Markov transition kernels and Laplace functionals

  2.6 Weak convergence of processes

Chapter 3 Construction of superprocesses arising from interacting stochastic flows

  3.1  Introduction

  3.2 Approximation of diffusion processes generated by Gm

  3.3 Function-valued dual processes of SAISF

  3.4 Branching particle systems generated by interacting stochastic flows

  3.5 SAISF with positive continuous branching density

  3.6 SAISF with Borel branching density

  3.7 Examples of SAISF

  3.8 Construction of SAISF with immigration

  3.9 1-dimensional SAISF with measure-valued catalysts

Chapter 4 Probabilistic properties of superprocesses arising from stochastic flows

  4.1 SAISF's without active underlying motions

  4.2 Absolutely continuity and stochastic partial differential equations for the 1-dimensional SAISF with active underlying motions

  4.3 Rescaled limits of the SAISF

Chapter 5 Superprocesses with branching mechanism depending on population size

  5.1 Construction of limit dual process for SAISF depending on population size

  5.2 Existence of SAISF depending on population size

  5.3 Superprocesses depending on population size and the nonlinear functional of a class of 1-dimensional diffusion process

  5.4 Some probabilistic properties of superprocesses depending on population size

Chapter 6 Stochastic flows of mappings

  6.1 Notations and symbols

  6.2 Construction of a family of probability measures on(E, e)

  6.3 Construction of stochastic flows of mappings and its kernels

Chapter 7 State decomposition of superprocess of stochastic flows

  7.1  Introduction

  7.2 Martingale description

  7.3 State decomposition of superprocess of stochastic flows

Bibliography