Table of Contents
Preface vii
1 Introduction to Superprocesses 1
1.1 Branching particle system 1
1.2 The log-Laplace equation 7
1.3 The moment duality 8
1.4 The SPDE for the density 13
1.5 The SPDE for the distribution 21
1.6 Historical remarks 22
2 Superprocesses in Random Environments 25
2.1 Introduction and main result 25
2.2 The moment duality 28
2.3 Conditional martingale problem 30
2.4 Historical remarks 33
3 Linear SPDE 35
3.1 An equation on measure space 35
3.2 A duality representation 44
3.3 Two estimates 53
3.4 Historical remarks 64
4 Particle Representations for a Class of Nonlinear SPDEs 65
4.1 Introduction 65
4.2 Solution for the system 67
4.3 A nonlinear SPDE 79
4.4 Historical remarks 80
5 Stochastic Log-Laplace Equation 83
5.1 Introduction 83
5.2 Approximation and two estimates 85
5.3 Existence and uniqueness 93
5.4 Conditional log-Laplace transform 96
5.5 Historical remarks 104
6 SPDEs for Density Fields of the Superprocesses in Random Environment 105
6.1 Introduction 105
6.2 Derivation of SPDE 108
6.3 A convolution representation 111
6.4 An estimate in spatial increment 114
6.5 Estimates in time increment 116
6.6 Historical remarks 124
7 Backward Doubly Stochastic Differential Equations 125
7.1 Introduction and basic definitions 125
7.2 Itô-Pardoux-Peng formula 126
7.3 Uniqueness of solution 128
7.4 Historical remarks 130
8 From SPDE to BSDE 131
8.1 The SPDE for the distribution 131
8.2 Existence of solution to SPDE 135
8.3 From BSDE to SPDE 141
8.4 Uniqueness for SPDE 143
8.5 Historical remarks 147
Appendix: Some Auxiliary Results 149
A.1 Martingale representation theorems 149
A.2 Weak convergence 154
A.3 Relation among strong existence, weak existence and path-wise uniqueness 155
Bibliography 157
Index 163
