Symplectic Integration of Stochastic Hamiltonian Systems

Preface
Acknowledgments
Contents
Notation and Symbols
1 Stochastic Hamiltonian Systems
	1.1 Stratonovich Stochastic Differential Equations
	1.2 Stochastic Hamiltonian Systems
		1.2.1 Stochastic Symplectic Structure
		1.2.2 Stochastic Variational Principle
		1.2.3 Stochastic θ-Generating Function
	1.3 Non-canonical Stochastic Hamiltonian Systems
		1.3.1 Stochastic Forced Hamiltonian Systems
		1.3.2 Stochastic Poisson Systems
	1.4 Rough Hamiltonian Systems
	1.5 End Notes
2 Stochastic Structure-Preserving Numerical Methods
	2.1 Stochastic Numerical Methods
		2.1.1 Strong and Weak Convergence
		2.1.2 Numerical Methods via Taylor Expansions
		2.1.3 Stochastic Runge–Kutta Methods
	2.2 Stochastic Symplectic Methods
		2.2.1 Symplectic Methods via Padé Approximation
		2.2.2 Symplectic Methods via Generating Function
		2.2.3 Stochastic Galerkin Variational Integrators
		2.2.4 Stochastic (Rough) Symplectic Runge–Kutta Methods
	2.3 Application of Symplectic Methods to Stochastic Poisson Systems
	2.4 Superiority of Stochastic Symplectic Methods via Large Deviation Principle (LDP)
		2.4.1 LDP for Numerical Methods
		2.4.2 Asymptotical Preservation for LDP
	2.5 Stochastic Pseudo-Symplectic Methods and Stochastic Methods Preserving Invariants
		2.5.1 Stochastic Pseudo-Symplectic Methods
		2.5.2 Stochastic Numerical Methods Preserving Invariants
	2.6 End Notes
3 Stochastic Modified Equations and Applications
	3.1 Stochastic Modified Equations
		3.1.1 Modified Equation via Backward Kolmogorov Equation
		3.1.2 Modified Equation via Stochastic Generating Function
		3.1.3 Modified Equation for Rough Symplectic Methods
	3.2 Stochastic Numerical Methods Based on Modified Equation
		3.2.1 High Order Methods via Backward KolmogorovEquation
		3.2.2 High Order Symplectic Methods via GeneratingFunction
	3.3 Conformal Symplectic and Ergodic Methods for Stochastic Langevin Equation
	3.4 End Notes
4 Infinite-Dimensional Stochastic Hamiltonian Systems
	4.1 Infinite-Dimensional Stochastic Hamiltonian Systems
	4.2 Stochastic Maxwell Equation
		4.2.1 Stochastic Exponential Integrator
		4.2.2 Stochastic Symplectic Runge–Kutta Discretizations
	4.3 Stochastic Schrödinger Equation
		4.3.1 Stochastic Symplectic Runge–Kutta Discretizations
		4.3.2 Approximating Large Deviation Rate Functions via Spectral Galerkin Method
	4.4 Stochastic Wave Equation with Cubic Nonlinearity
		4.4.1 Spectral Galerkin Method and Exponential Integrability
		4.4.2 Exponentially Integrable Full Discretization
	4.5 End Notes
Appendix A
	A.1 Stochastic Stratonovich Integral
	A.2 Large Deviation Principle
	A.3 Exponential Integrability
	A.4 Stochastic Multi-Symplectic Structure
References