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Published April 2009 | Submitted
Journal Article Open

Stochastic variational integrators


This paper presents a continuous and discrete Lagrangian theory for stochastic Hamiltonian systems on manifolds, akin to the Ornstein–Uhlenbeck theory of Brownian motion in a force field. The main result is to derive governing SDEs 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 analogue 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 discrete variational principle. The paper shows that the discrete flow of an SVI is almost surely symplectic and in the presence of symmetry almost surely momentum-map preserving. A first-order mean-squared 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.

Additional Information

© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. Received on 21 October 2007. Revised on 13 February 2008. We wish to thank Andreu Lazaro, Jerry Marsden and Juan-Pablo Ortega for stimulating discussions.

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