LEADER 03722oam 2200469 450 001 9910809486703321 005 20190911112729.0 010 $a981-4452-36-X 035 $a(OCoLC)844311148 035 $a(MiFhGG)GVRL8RAZ 035 $a(EXLCZ)992670000000372491 100 $a20140422h20132013 uy 0 101 0 $aeng 135 $aurun|---uuuua 181 $ctxt 182 $cc 183 $acr 200 10$aThree classes of nonlinear stochastic partial differential equations /$fJie Xiong, University of Macau, China & The University of Tennessee, Knoxville, USA 210 $aSingapore $cWorld Scientific Pub. Co.$d2013 210 1$aNew Jersey :$cWorld Scientific,$d[2013] 210 4$d?2013 215 $a1 online resource (xi, 164 pages) $cillustrations 225 0 $aGale eBooks 300 $aDescription based upon print version of record. 311 $a981-4452-35-1 320 $aIncludes bibliographical references and index. 327 $aPreface; Contents; 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 327 $a4.2 Solution for the system4.3 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 327 $a7.2 Ito-Pardoux-Peng formula7.3 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; Appendix Some Auxiliary Results; A.1 Martingale representation theorems; A.2 Weak convergence; A.3 Relation among strong existence, weak existence and pathwise uniqueness; Bibliography; Index 330 $aThe 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 606 $aStochastic partial differential equations 606 $aDifferential equations, Nonlinear 615 0$aStochastic partial differential equations. 615 0$aDifferential equations, Nonlinear. 676 $a515.353 700 $aXiong$b Jie$0736517 801 0$bMiFhGG 801 1$bMiFhGG 906 $aBOOK 912 $a9910809486703321 996 $aThree classes of nonlinear stochastic partial differential equations$93940099 997 $aUNINA