On the tangent flow of a stochastic differential equation with fast drift

Author:
Richard B. Sowers

Journal:
Trans. Amer. Math. Soc. **353** (2001), 1321-1334

MSC (1991):
Primary 60H10

DOI:
https://doi.org/10.1090/S0002-9947-00-02773-2

Published electronically:
December 18, 2000

MathSciNet review:
1806739

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**1.**L. Arnold, E. Oeljeklaus, and É. Pardoux,*Almost sure and moment stability for linear Ito equations*, Lyapunov exponents (Bremen, 1984), Springer, Berlin, 1986, pp. 129-159. MR**87m:60123****2.**Peter Baxendale,*Brownian motions in the diffeomorphism group. I*, Compositio Math.**53**(1984), no. 1, 19-50. MR**86e:58086****3.**-,*Stability along trajectories at a stochastic bifurcation point*, Stochastic dynamics (Bremen, 1997), Springer, New York, 1999, pp. 1-25. MR**2000a:60112****4.**Denis R. Bell,*The Malliavin calculus*, Longman Scientific & Technical, Harlow, 1987. MR**88m:60155****5.**K. D. Elworthy,*Stochastic differential equations on manifolds*, Cambridge University Press, Cambridge, 1982 MR**84d:58080****6.**Mark I. Freidlin and Alexander D. Wentzell,*Random perturbations of Hamiltonian systems*, Mem. Amer. Math. Soc.**109**(1994), no. 523, viii+82. MR**94j:35064****7.**R. Z. Has'minski,*Diffusion processes with a small parameter*, Izv. Akad. Nauk SSSR Ser. Mat.**27**(1963), 1281-1300. MR**29:6530****8.**-,*A limit theorem for solutions of differential equations with a random right hand part*, Teor. Verojatnost. i Primenen**11**(1966), 444-462. MR**34:3637****9.**-,*On the principle of averaging the Ito's stochastic differential equations*, Kybernetika (Prague)**4**(1968), 260-279. MR**41:4681****10.**Tosio Kato,*Perturbation theory for linear operators*, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition. MR**96a:47025****11.**Hiroshi Kunita,*Stochastic flows and stochastic differential equations*, Cambridge University Press, Cambridge, 1990. MR**91m:60107****12.**David Nualart,*Analysis on Wiener space and anticipating stochastic calculus*, Lectures on probability theory and statistics (Saint-Flour, 1995), Springer, Berlin, 1998, pp. 123-227. MR**99k:60144****13.**M. A. Pinsky and V. Wihstutz,*Lyapunov exponents of nilpotent Ito systems*, Stochastics**25**(1988), no. 1, 43-57. MR**91a:60156**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
60H10

Retrieve articles in all journals with MSC (1991): 60H10

Additional Information

**Richard B. Sowers**

Affiliation:
Department of Mathematics, University of Illinois at Urbana–Champaign, Urbana, Illinois 61801

Email:
r-sowers@math.uiuc.edu

DOI:
https://doi.org/10.1090/S0002-9947-00-02773-2

Keywords:
Floquet,
Lyapunov exponent,
Pinsky-Wihstutz,
stochastic averaging

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.

Article copyright:
© Copyright 2000
American Mathematical Society