Jialin Hong – författare

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.

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.

This book provides some recent advance in the study of stochastic nonlinear Schrödinger equations and their numerical approximations, including the well-posedness, ergodicity, symplecticity and multi-symplecticity. It gives an accessible overview of the existence and uniqueness of invariant measures for stochastic differential equations, introduces geometric structures including symplecticity and (conformal) multi-symplecticity for nonlinear Schrödinger equations and their numerical approximations, and studies the properties and convergence errors of numerical methods for stochastic nonlinear Schrödinger equations.This book will appeal to researchers who are interested in numerical analysis, stochastic analysis, ergodic theory, partial differential equation theory, etc.

This book provides some recent advance in the study of stochastic nonlinear Schrödinger equations and their numerical approximations, including the well-posedness, ergodicity, symplecticity and multi-symplecticity. It gives an accessible overview of the existence and uniqueness of invariant measures for stochastic differential equations, introduces geometric structures including symplecticity and (conformal) multi-symplecticity for nonlinear Schrödinger equations and their numerical approximations, and studies the properties and convergence errors of numerical methods for stochastic nonlinear Schrödinger equations.This book will appeal to researchers who are interested in numerical analysis, stochastic analysis, ergodic theory, partial differential equation theory, etc.

This book presents the latest developments and progress in the numerical study of the stochastic functional differential equation, with a particular emphasis on the longtime asymptotics and probabilistic characteristics of numerical methods used to solve such equation. The longtime asymptotics under investigation include the time-independent convergence analysis in both the strong and weak senses, the numerical invariant measure, and the ergodicity of numerical methods. Additionally, the probabilistic characteristics of numerical solutions explored in this book encompass the density function, limit theorems, and the Freidlin–Wentzell type large deviation principle. The topics presented here lie at the intersection of several fascinating areas: numerical analysis, stochastic analysis, ergodicity theory, Malliavin calculus, large deviation theory, and probability theory, providing a rich framework to deepen our understanding of stochastic functional differential equations from both theoretical and numerical perspectives. This book will appeal to researchers interested in these topics.

This book presents the latest developments and progress in the numerical study of the stochastic functional differential equation, with a particular emphasis on the longtime asymptotics and probabilistic characteristics of numerical methods used to solve such equation. The longtime asymptotics under investigation include the time-independent convergence analysis in both the strong and weak senses, the numerical invariant measure, and the ergodicity of numerical methods. Additionally, the probabilistic characteristics of numerical solutions explored in this book encompass the density function, limit theorems, and the Freidlin-Wentzell type large deviation principle. The topics presented here lie at the intersection of several fascinating areas: numerical analysis, stochastic analysis, ergodicity theory, Malliavin calculus, large deviation theory, and probability theory, providing a rich framework to deepen our understanding of stochastic functional differential equations from both theoretical and numerical perspectives. This book will appeal to researchers interested in these topics.

This book provides an overview of structure-preserving discrete approximations for the probabilistic characteristics of stochastic differential equations, which are essential for understanding stochastic systems in fields such as finance, physics, and engineering. It highlights recent advances in the study of key probabilistic features of discretized systems. In particular, this book presents methods for density approximation and examines the impact of numerical discretizations on hitting probabilities for stochastic ordinary and partial differential equations. The preservation of important asymptotic properties, such as large deviation principles and weak intermittency for parabolic stochastic partial differential equations, is also investigated. A distinctive feature of this book is its demonstration of Malliavin calculus and its adaptation to the analysis of probabilistic properties in discrete settings. This book is intended for graduate students and researchers with backgrounds in probability theory, stochastic analysis, and numerical analysis who are interested in the analysis and numerical approximation of stochastic differential equations.

The stochastic Maxwell equations play an essential role in many fields, including fluctuational electrodynamics, statistical radiophysics, integrated circuits, and stochastic inverse problems. This book provides some recent advances in the investigation of numerical approximations of the stochastic Maxwell equations via structure-preserving algorithms. It presents an accessible overview of the construction and analysis of structure-preserving algorithms with an emphasis on the preservation of geometric structures, physical properties, and asymptotic behaviors of the stochastic Maxwell equations. A friendly introduction to the simulation of the stochastic Maxwell equations with some structure-preserving algorithms is provided using MATLAB for the reader’s convenience.The objects considered in this book are related to several fascinating mathematical fields: numerical analysis, stochastic analysis, (multi-)symplectic geometry, large deviations principle, ergodic theory, partial differential equation, probability theory, etc. This book will appeal to researchers who are interested in these topics.