58 research outputs found

    Noise and dissipation in rigid body motion

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    Using the rigid body as an example, we illustrate some features of stochastic geometric mechanics. These features include: i) a geometric variational motivation for the noise structure involving Lie-Poisson brackets and momentum maps, ii) stochastic coadjoint motion with double bracket dissipation, iii) the Lie-Poisson Fokker-Planck description and its stationary solutions, iv) random dynamical systems, random attractors and SRB measures connected to statistical physics

    Variational Principles on Geometric Rough Paths and the L\'{e}vy Area Correction

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    In this paper, we describe two effects of the L\'evy area correction on the invariant measure of stochastic rigid body dynamics on geometric rough paths. From the viewpoint of dynamics, the L\'evy area correction introduces an additional deterministic torque into the rigid body motion equation on geometric rough paths. When the dynamics is driven by coloured noise, and for rigid body dynamics with double-bracket dissipation, theoretical and numerical results show that this additional deterministic torque shifts the centre of the probability distribution function by shifting the Hamiltonian function in the exponent of the Gibbsian invariant measure.Comment: 39 pages, 9 figures, 3rd version, comments welcom

    Bridge Simulation and Metric Estimation on Landmark Manifolds

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    We present an inference algorithm and connected Monte Carlo based estimation procedures for metric estimation from landmark configurations distributed according to the transition distribution of a Riemannian Brownian motion arising from the Large Deformation Diffeomorphic Metric Mapping (LDDMM) metric. The distribution possesses properties similar to the regular Euclidean normal distribution but its transition density is governed by a high-dimensional PDE with no closed-form solution in the nonlinear case. We show how the density can be numerically approximated by Monte Carlo sampling of conditioned Brownian bridges, and we use this to estimate parameters of the LDDMM kernel and thus the metric structure by maximum likelihood

    Variational principles on geometric rough paths and the Lévy area correction

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    In this paper, we describe two effects of the Lévy area correction on the invariant measure of stochastic rigid body dynamics on geometric rough paths. From the viewpoint of dynamics, the Lévy area correction introduces an additional deterministic torque into the rigid body motion equation on geometric rough paths. When the rigid body dynamics is driven by colored noise, and damped by double-bracket dissipation, our theoretical and numerical results show that the additional deterministic torque due to the the Lévy area correction shifts the center of the probability distribution function by shifting the Hamiltonian function in the exponent of the Gibbsian invariant measure
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