Three classes of nonlinear stochastic partial differential equations / Jie Xiong.

The study of measure-valued processes in random environments has seen some intensive research activities in recent years whereby interesting nonlinear stochastic partial differential equations (SPDEs) were derived. Due to the nonlinearity and the non-Lipschitz continuity of their coefficients, new techniques and concepts have recently been developed for the study of such SPDEs. These include the conditional Laplace transform technique, the conditional mild solution, and the bridge between SPDEs and some kind of backward stochastic differential equations. This volume provides an introduction to these topics with the aim of attracting more researchers into this exciting and young area of research. It can be considered as the first book of its kind. The tools introduced and developed for the study of measure-valued processes in random environments can be used in a much broader area of nonlinear SPDEs.

Includes bibliographical references (pages 157-162) and index.

1. Introduction to superprocesses. 1.1. Branching particle system. 1.2. The log-Laplace equation. 1.3. The moment duality. 1.4. The SPDE for the density. 1.5. The SPDE for the distribution. 1.6. Historical remarks -- 2. Superprocesses in random environments. 2.1. Introduction and main result. 2.2. The moment duality. 2.3. Conditional martingale problem. 2.4. Historical remarks -- 3. Linear SPDE. 3.1. An equation on measure space. 3.2. A duality representation. 3.3. Two estimates. 3.4. Historical remarks -- 4. Particle representations for a class of nonlinear SPDEs. 4.1. Introduction. 4.2. Solution for the system. 4.3. A nonlinear SPDE. 4.4. Historical remarks -- 5. Stochastic log-Laplace equation. 5.1. Introduction. 5.2. Approximation and two estimates. 5.3. Existence and uniqueness. 5.4. Conditional log-Laplace transform. 5.5. Historical remarks -- 6. SPDEs for density fields of the superprocesses in random environment. 6.1. Introduction. 6.2. Derivation of SPDE. 6.3. A convolution representation. 6.4. An estimate in spatial increment. 6.5. Estimates in time increment. 6.6. Historical remarks -- 7. Backward doubly stochastic differential equations. 7.1. Introduction and basic definitions. 7.2. Itô-Pardoux-Peng formula. 7.3. Uniqueness of solution. 7.4. Historical remarks -- 8. From SPDE to BSDE. 8.1. The SPDE for the distribution. 8.2. Existence of solution to SPDE. 8.3. From BSDE to SPDE. 8.4. Uniqueness for SPDE. 8.5. Historical remarks.

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