On the tangent flow of a stochastic differential equation with fast drift
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- by Richard B. Sowers PDF
- Trans. Amer. Math. Soc. 353 (2001), 1321-1334 Request permission
Abstract:
We investigate the behavior of the tangent flow of a stochastic differential equation with a fast drift. The state space of the stochastic differential equation is the two-dimensional cylinder. The fast drift has closed orbits, and we assume that the orbit times vary nontrivially with the axial coordinate. Under a nondegeneracy assumption, we find the rate of growth of the tangent flow. The calculations involve a transformation introduced by Pinsky and Wihstutz.References
- L. Arnold, E. Oeljeklaus, and É. Pardoux, Almost sure and moment stability for linear Itô equations, Lyapunov exponents (Bremen, 1984) Lecture Notes in Math., vol. 1186, Springer, Berlin, 1986, pp. 129–159. MR 850074, DOI 10.1007/BFb0076837
- Peter Baxendale, Brownian motions in the diffeomorphism group. I, Compositio Math. 53 (1984), no. 1, 19–50. MR 762306
- Peter H. Baxendale, Stability along trajectories at a stochastic bifurcation point, Stochastic dynamics (Bremen, 1997) Springer, New York, 1999, pp. 1–25. MR 1678447, DOI 10.1007/0-387-22655-9_{1}
- Denis R. Bell, The Malliavin calculus, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 34, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987. MR 902583
- K. D. Elworthy, Stochastic differential equations on manifolds, London Mathematical Society Lecture Note Series, vol. 70, Cambridge University Press, Cambridge-New York, 1982. MR 675100
- Mark I. Freidlin and Alexander D. Wentzell, Random perturbations of Hamiltonian systems, Mem. Amer. Math. Soc. 109 (1994), no. 523, viii+82. MR 1201269, DOI 10.1090/memo/0523
- R. Z. Has′minskiĭ, Diffusion processes with a small parameter, Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963), 1281–1300 (Russian). MR 0169278
- R. Z. Has′minskiĭ, A limit theorem for solutions of differential equations with a random right hand part, Teor. Verojatnost. i Primenen 11 (1966), 444–462 (Russian, with English summary). MR 0203789
- R. Z. Has′minskiĭ, On the principle of averaging the Itô’s stochastic differential equations, Kybernetika (Prague) 4 (1968), 260–279 (Russian, with Czech summary). MR 260052
- Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
- Hiroshi Kunita, Stochastic flows and stochastic differential equations, Cambridge Studies in Advanced Mathematics, vol. 24, Cambridge University Press, Cambridge, 1990. MR 1070361
- David Nualart, Analysis on Wiener space and anticipating stochastic calculus, Lectures on probability theory and statistics (Saint-Flour, 1995) Lecture Notes in Math., vol. 1690, Springer, Berlin, 1998, pp. 123–227. MR 1668111, DOI 10.1007/BFb0092538
- M. A. Pinsky and V. Wihstutz, Lyapunov exponents of nilpotent Itô systems, Stochastics 25 (1988), no. 1, 43–57. MR 1008234, DOI 10.1080/17442508808833531
Additional Information
- Richard B. Sowers
- Affiliation: Department of Mathematics, University of Illinois at Urbana–Champaign, Urbana, Illinois 61801
- Email: r-sowers@math.uiuc.edu
- Received by editor(s): September 21, 1999
- Received by editor(s) in revised form: July 20, 2000
- Published electronically: December 18, 2000
- Additional Notes: This work was supported by NSF DMS 9615877. The author would like to thank Professor Sri Namachchivaya of the Department of Aeronautical and Astronautical Engineering at the University of Illinois at Urbana-Champaign for pointing out the paper by Pinsky and Wihstutz. The author would also like to thank the anonymous referee who insisted upon notational clarity.
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 353 (2001), 1321-1334
- MSC (1991): Primary 60H10
- DOI: https://doi.org/10.1090/S0002-9947-00-02773-2
- MathSciNet review: 1806739