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Random perturbations of invariant Lagrangian tori of Hamiltonian vector fields. (English. Russian original) Zbl 0952.37013
Math. Notes 64, No. 5, 674-679 (1998); translation from Mat. Zametki 64, No. 5, 783-787 (1998).
The authors consider diffusion type random perturbations of Hamiltonian systems (possibly nonintegrable) having invariant Lagrangian tori (i.e. the form \(dp\wedge dq\) vanishes there) with quasiperiodic motion on them. They consider the corresponding small parameter parabolic problem for distributions with the initial condition \(\delta_{\Lambda,d\mu}\) where \((\delta_{\Lambda,d\mu}\psi(x))=\int_\Lambda\psi d\mu\) and \(\Lambda\) is the corresponding torus. Applying Maslov’s theory of complex germs the authors obtain the leading term of the asymptotics of the solution of the above problem which is completely determined by the torus \(\Lambda.\)
37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
58J37 Perturbations of PDEs on manifolds; asymptotics
Full Text: DOI
[1] V. I. Atnol’d,Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1974).
[2] V. P. Maslov,Operational Methods, Mir Publ., Moscow (1976).
[3] F. Treves,Introduction to Pseudodifferential and Fourier Integral Operators, Vol. 2, Plenum, New York (1982).
[4] V. F. Lasutkin,KAM-Theory and Semiclassical Approximations to Eigenfunctions, Springer, Berlin (1993).
[5] I. M. Gel’fand and G. E. Shilov,Generalized Functions [in Russian], Vols. 1, 3, Fizmatlit, Moscow (1958). · Zbl 0091.11103
[6] A. D. Wentzell and M. I. Freidlin,Fluctuations in Dynamical Systems Under the Action of Small Random Perturbations [in Russian], Nauka, Moscow (1979).
[7] M. I. Freidlin and A. D. Wentzell,Random Perturbations of Hamiltonian Systems, Vol. 109, Mem. Amer. Math. Soc. No. 523, Amer. Math. Soc., Providence (R.I.) (1994). · Zbl 0804.60070
[8] S. Albeverio, A. Hilbert, and V. N. Kolokoltsov,Sur le comportetement asymptotique du noyau associé à une diffusion dégénérée, Preprint No. 320, Inst. Math. Ruhr Universitat Bochum, Bochum (1996).
[9] V. N. Kolokoltsov,Semiclassical Asymptotics for Diffusion. I, Research Report No. 1/97, Nottingham Trent University, Nottingham (1997).
[10] C. Yu. Dobrokhotov, V. N. Kolokol’tsov, and V. M. Olive,Mat. Zametki [Math. Notes],58, No. 2, 301–306 (1995).
[11] V. G. Danilov and V. P. Maslov, ”Quasi-invertibility of functions of ordered operators in the theory of pseudodifferential equations,” in:Contemporary Problems in Mathematics. Fundamental Directions [in Russian], Vol. 6, Itogi Nauki i Tekhniki, VINITI, Moscow (1976), pp. 5–132. · Zbl 0402.35094
[12] V. P. Maslov and I. A. Shishmarev, ”OnT-products of hypoelliptic operators,” in:Contemporary Problems in Mathematics. Fundamental Directions [in Russian], Vol. 8, Itogi Nauki i Tekhniki, VINITI, Moscow (1979), pp. 7–20.
[13] O. A. Oleinik and E. V. Radkevich, ”Second-order equations with nonnegative characteristic form,” in:Mathematical Analysis 1969 [in Russian], Itogi Nauki i Tekhniki, VINITI, Moscow (1971), pp. 7–20.
[14] V. P. Maslov,The Complex WKB Method in Nonlinear Equations [in Russian], Nauka, Moscow (1977). · Zbl 0449.58001
[15] C. Yu. Dobrokhotov and V. M. Olive,Mat. Zametki [Math. Notes],54, No. 4, 45–68 (1993).
[16] V. A. Yakubovich and V. M. Starzhinskii,Linear Differential Equations With Periodic Coefficients and Their Applications [in Russian], Nauka, Moscow (1972).
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