Symplectic Integration of Stochastic Hamiltonian Systems: (2314 Lecture Notes in Mathematics)

This book provides an accessible overview concerning the stochastic numerical methods inheriting long-time dynamical behaviours of finite and infinite-dimensional stochastic Hamiltonian systems. The long-time dynamical behaviours under study involve symplectic structure, invariants, ergodicity and invariant measure. The emphasis is placed on the systematic construction and the probabilistic superiority of stochastic symplectic methods, which preserve the geometric structure of the stochastic flow of stochastic Hamiltonian systems. The problems considered in this book are related to several fascinating research hotspots: numerical analysis, stochastic analysis, ergodic theory, stochastic ordinary and partial differential equations, and rough path theory. This book will appeal to researchers who are interested in these topics.

Table of Contents:
Chapter 1 Deterministic Hamiltonian System.- Chapter 2 Stochastic Hamiltonian System.- Chapter 3 Stochastic Structure Preserving Numerical Integrators.- Chapter 4 Stochastic Modified Equation and Its Applications.- Chapter 5 Stochastic Hamiltonian Partial Differential Equation.

About the Author :
Jialin Hong is a professor at the Chinese Academy of Sciences. He obtained his Ph.D. in 1994 at Jilin University. He works in various directions including structure-preserving algorithms for dynamical systems involving symplectic and multi-symplectic methods for Hamiltonian ODEs and PDEs, Lie group methods and applications, numerical dynamics including chaos, bifurcations for discrete systems, numerical methods for stochastic ordinary differential systems, stochastic partial differential equations and backward stochastic differential equations, almost periodic dynamical systems, and ergodic theory. Liying Sun is a postdoctoral researcher in the Chinese Academy of Sciences. She works in stochastic differential equations and their numerical methods. She has been investigating regularity properties and strong convergence of numerical approximations for stochastic partial differential equations, weak convergence and numerical longtime behaviors of numerical approximations for stochastic partial differential equations, structure-preserving numerical methods including symplectic integrators and energy-preserving integrators for stochastic Hamiltonian system.

Review :
"Each chapter concludes with a brief summary and some open questions for further research. Finally, the book contains a brief appendix and a list of 274 references, including very recent ones from the year of publication of the book itself ... . Some proofs are presented in the book itself, some proofs are sketched, and some results are mentioned with references to the literature. The analysis is complemented by a few numerical results and almost 20 figures." (Hendrik Ranocha, zbMATH 1550.65004, 2025)