This paper presents a continuous and discrete Lagrangian theory for stochastic Hamiltonian systems on manifolds. The main result is to derive stochastic governing equations for such systems from a critical point of a stochastic action. Using this result the paper derives Langevin-type equations for constrained mechanical systems and implements a stochastic analog of Lagrangian reduction. These are easy consequences of the fact that the stochastic action is intrinsically defined. Stochastic variational integrators (SVIs) are developed using a discretized stochastic variational principle. The paper shows that the discrete flow of an SVI is a.s. symplectic and in the presence of symmetry a.s. momentum-map preserving. A first-order mean-square convergent SVI for mechanical systems on Lie groups is introduced. As an application of the theory, SVIs are exhibited for multiple, randomly forced and torqued rigid-bodies interacting via a potential.
We study stochastic Poisson integrators for a class of stochastic Poisson systems driven by Stratonovich noise. Such geometric integrators preserve Casimir functions and the Poisson map property. For this purpose, we propose explicit stochastic Poisson integrators based on a splitting strategy, and analyse their qualitative and quantitative properties: preservation of Casimir functions, existence of almost sure or moment bounds, asymptotic preserving property, and strong and weak rates of convergence. The construction of the schemes and the theoretical results are illustrated through extensive…
We construct an efficient integrator for stochastic differential systems driven by Levy processes. An efficient integrator is a strong approximation that is more accurate than the corresponding stochastic Taylor approximation, to all orders and independent of the governing vector fields. This holds provided the driving processes possess moments of all orders and the vector fields are sufficiently smooth. Moreover the efficient integrator in question is optimal within a broad class of perturbations for half-integer global root mean-square orders of convergence. We obtain these results using the…
Explicit stabilized integrators are an efficient alternative to implicit or semi-implicit methods to avoid the severe timestep restriction faced by standard explicit integrators applied to stiff diffusion problems. In this paper, we provide a fully discrete strong convergence analysis of a family of explicit stabilized methods coupled with finite element methods for a class of parabolic semilinear deterministic and stochastic partial differential equations. Numerical experiments including the semilinear stochastic heat equation with space-time white noise confirm the theoretical findings.
In this paper, we consider the strong convergence order of the exponential integrator for the stochastic heat equation driven by an additive fractional Brownian motion with Hurst parameter $H\in(\frac12,1)$. By showing the strong order one of accuracy of the exponential integrator under appropriote assumptions, we present the first super-convergence result in temporal direction on full discretizations for stochastic partial differential equations driven by infinite dimensional fractional Brownian motions with Hurst parameter $H\in(\frac12,1)$. The proof is a combination of Malliavin calculus, …
Analysis of an exponential integrator for stochastic PDEs driven by Riesz noise
Universite de Pau et des Pays de l’Adour, E2S UPPA（波城大学）；Chalmers University of Technology and University of Gothenburg（查尔姆斯理工大学和哥德堡大学）；Universitat Autònoma de Barcelona（巴塞罗那自治大学）；Swedish Defence Research Agency (FOI)（瑞典国防研究局）
Algebraic structure of stochastic expansions and efficient simulation
Norwegian University of Science and Technology（挪威科技大学）；Maxwell Institute for Mathematical Sciences, and School of Mathematical and Computer Sciences, Heriot-Watt University（赫瑞瓦特大学数学与计算机科学学院及麦克斯韦数学科学研究所）；Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas（西班牙高等科学研究理事会数学科学研究所）；University of Bergen（卑尔根大学）
The rates of strong convergence for various approximation schemes are investigated for a class of stochastic differential equations (SDEs) which involve a random time change given by an inverse subordinator. SDEs to be considered are unique in two different aspects: i) they contain two drift terms, one driven by the random time change and the other driven by a regular, non-random time variable; ii) the standard Lipschitz assumption is replaced by that with a time-varying Lipschitz bound. The difficulty imposed by the first aspect is overcome via an approach that is significantly different from…
