Abstract
We review the geometry of diffusion processes of differential forms on smooth compact manifolds, as a basis for the random representations of the kinematic dynamo equations on these manifolds. We realize these representations in terms of sequences of ordinary (for almost all times) differential equations. We construct the random symplectic geometry and the random Hamiltonian structure for these equations, and derive a new class of Poincaré-Cartan invariants of magnetohydrodynamics. We obtain a random Liouville invariant. We work out in detail the case of R 3.
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References
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland/ Kodansha, 1981.
P. Malliavin, Géométrie Différentielle Stochastique, Les Presses Univ., Montreal, 1978.
D.L. Rapoport, Random diffeomorphisms and integration of the classical Navier-Stokes equations, Rep. Math. Phys. 49, no. 1, 1–27 (2002).
D.L. Rapoport, Random representations for viscous fluids and the passive magnetic fields transported by them, in Proceedings, Third International Conference on Dynamical Systems and Differential Equations, Kennesaw, May 2000, Discrete & Cont. Dyn. Sys. special issue, S. Hu (ed.), 2000.
D.L. Rapoport, On the geometry of the random representations for viscous fluids and a remarkable pure noise representation, Rep. Math. Phys. 50, no. 2, 211–250 (2002).
D.L. Rapoport, Martingale problem approach to the representations of the Navier-Stokes equations on smooth boundary manifolds and semispace, Rand. Oper. Stoch. Eqts. 11, no. 2, 109–136 (2003).
D.L. Rapoport, Random symplectic geometry and the realization of the random representations of the Navier-Stokes equations by ordinary differential equations, Rand. Oper. Stoch. Eqts. vol. 11, no. 2, 359–380 (2003).
K. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Univ. Press, Cambridge, 1994.
P. Baxendale and K.D. Elworthy, Flows of Stochastic Systems, Z. Wahrschein. verw. Gebiete 65, 245–267 (1983).
V.I. Arnold and B.A. Khesin, Topological Methods in Hydrodynamics, Springer, New York, 1999.
J.M. Bismut, Mécanique Analytique, Springer LNM 866, 1982.
K.D. Elworthy, Stochastic Flows on Riemannian Manifolds, in Diffusion Processes and Related Problems in Analysis, M. Pinsky et al. (eds.), vol. II, Birkhäuser, 1992.
Elworthy, K.D. and Li, X.M., Formula for the derivative of heat semigroups, J. Func. Anal. 125, 252–286 (1994).
V.I. Arnold, Mathematical Methods in Classical Mechanics, Springer Verlag, 1982.
M. Taylor, Partial Differential Equations, vol. I, Springer Verlag, 1995.
M. Ghill & S. Childress, Stretch, Twist and Fold, Springer Verlag, 1995
D. Ebin & J. Marsden, Groups of diffeomorphisms and the motion of incompressible fluids, Ann. Math. 92, 102–163 (1971).
H.K. Moffatt & R.L. Ricca, Helicity and Calugareanu invariants, Procds. Roy. London A 439 (1992), 411–429.
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© 2005 Birkhäuser Verlag Basel/Switzerland
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Rapoport, D.L. (2005). Stochastic Geometry Approach to the Kinematic Dynamo Equation of Magnetohydrodynamics. In: Rodrigues, J.F., Seregin, G., Urbano, J.M. (eds) Trends in Partial Differential Equations of Mathematical Physics. Progress in Nonlinear Differential Equations and Their Applications, vol 61. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7317-2_17
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DOI: https://doi.org/10.1007/3-7643-7317-2_17
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