Stochastic Variational Integrators

Details Stochastic Variational Integrators Stochastic Variational Integrators

2007-08-16

arxiv.org/abs/0708.2187v2

IMA Journal of Numerical Analysis, 2008, Vol. 29, 421-443

Mathematics

Probability

21 pages, 8 figures

Scientific paper

10.1093/imanum/drn018

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.

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