2007/08/31 by Nawaf Bou-Rabee, Houman Owhadi · 2 citations
Mathematics · #math.PR #msc:65Cxx #msc:37Jxx
paper · pdf · doi:10.1093/imanum/drn018
published as IMA Journal of Numerical Analysis, 2008, Vol. 29, 421-443 · 21 pages, 8 figures
arxiv created 2007/10/23 · arxiv updated 2009/12/01
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.