Abstract
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) description and its stationary solutions , (iv) random dynamical systems , random attractors and SRB measures connected to statistical physics .
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Acknowledgements
We are grateful to many people for fruitful and encouraging discussions, including S. Albeverio, J.-M. Bismut, N. Bou-Rabee, M.D. Chekroun, G. Chirikjian, D.O. Crisan, A.-B. Cruzeiro, J. Eldering, M. Engel, N. Grandchamps, P. Lynch, J.-P. Ortega, G. Pavliotis, V. Putkaradze, T. Ratiu and C. Tronci. The simulations were run with the Imperial College High Performance Computing Service. We also acknowledge the Bernoulli Centre for Advanced Studies at EPFL where parts of this work were elaborated. AA acknowledges partial support from an Imperial College London Roth Award and AC from a CAPES Research Award BEX 11784-13-0. All the authors are also supported by the European Research Council Advanced Grant 267382 FCCA held by DH. Last, but not least, we want to mention the inspiration of the lectures of H. Dumpty on broken symmetry, and stochastic processes on coset spaces.
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Arnaudon, A., De Castro, A.L., Holm, D.D. (2017). Noise and Dissipation in Rigid Body Motion. In: Albeverio, S., Cruzeiro, A., Holm, D. (eds) Stochastic Geometric Mechanics . CIB-SGM 2015. Springer Proceedings in Mathematics & Statistics, vol 202. Springer, Cham. https://doi.org/10.1007/978-3-319-63453-1_1
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