Annals of Probability

Spatial estimates for stochastic flows in Euclidean space

Salah-Eldin A. Mohammed and Michael K. R. Scheutzow

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We study the behavior for large $|x|$ of Kunita-type stochastic flows $\phi(t, \omega x)$ on $R^d$, driven by continuous spatial semimartingales. For this class of flows we prove new spatial estimates for large $|x|$, under very mild regularity conditions on the driving semimartingale random field. It is expected that the results would be of interest for the theory of stochastic flows on noncompact manifolds as well as in the study of nonlinear filtering, stochastic functional and partial differential equations. Some examples and counterexamples are given.

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Ann. Probab., Volume 26, Number 1 (1998), 56-77.

First available in Project Euclid: 31 May 2002

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Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H20: Stochastic integral equations
Secondary: 60H25: Random operators and equations [See also 47B80]

Stochastic flow spacial semimartingale local characteristics quadratic variation stochastic differential equation (s.d.e.)


Mohammed, Salah-Eldin A.; Scheutzow, Michael K. R. Spatial estimates for stochastic flows in Euclidean space. Ann. Probab. 26 (1998), no. 1, 56--77. doi:10.1214/aop/1022855411.

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  • 1 ARNOLD, L. 1995. Random Dynamical Systems. Preliminary version, Inst. Dynamische Systeme, Universitat Bremen. ¨
  • 2 ARNOLD, L., KLIEMANN, W. and OELJEKLAUS, E. 1989. Lyapunov exponents of linear stochastic systems. Lyapunov Exponents. Lecture Notes in Math. 1186 85 125. Springer, Berlin.
  • 3 ARNOLD, L., OELJEKLAUS, E. and PARDOUX, E. 1986. Almost sure and moment stability for linear Ito equations. Lyapunov Exponents. Lecture Notes in Math. 1186 129 159. Springer, Berlin.
  • 4 BAXENDALE, P. H. 1987. Moment stability and large deviations for linear stochastic differential equations. In Proceedings of the Taniguchi Symposium on ProbabilisticZ. Methods in Mathematical Physics, Katata and Kyoto 1985 N. Ikeda, ed. 31 54. Kinokuniya, Tokyo.
  • 5 BAXENDALE, P. H. Private communication.
  • 6 DOSS, H. 1977. Liens entre equations differentielles stochastiques et ordinaires. Ann. Inst. ´ ´ H. Poincare Probab. Statist. 13 99 125. ´
  • 7 IKEDA, N. and WATANABE, S. 1989. Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland, Amsterdam.
  • 8 JACOD, J. and SHIRYAEV, A. N. 1987. Limit Theorems for Stochastic Processes. Springer, New York.
  • 9 KUNITA, H. 1990. Stochastic Flows and Stochastic Differential Equations. Cambridge Univ. Press.
  • 10 MOHAMMED, S.-E. A. 1984. Stochastic Functional Differential Equations. Research Notes in Mathematics 99. Pitman, Boston.
  • 11 MOHAMMED, S.-E. A. 1990. The Lyapunov spectrum and stable manifolds for stochastic linear delay equations. Stochastics Stochastics Rep. 29 89 131.
  • 12 MOHAMMED, S.-E. A. 1992. Lyapunov exponents and stochastic flows of linear and affine hereditary systems. In Diffusion Processes and Related Problems in Analysis 2M. Pinsky and V. Wihstutz, eds. 141 169. Birkhauser, Boston. ¨
  • 13 MOHAMMED, S.-E. A. and SCHEUTZOW, M. 1996. Lyapunov exponents of linear stochastic functional differential equations drive by semimartingales, I: the multiplicative ergodic theory. Ann. Inst. H. Poincare Probab. Statist. 32 69 105. ´
  • 14 OCONE, D. and PARDOUX, E. 1989. A generalized Ito Ventzell formula. Application to a class of anticipating stochastic differential equations. Ann. Inst. H. Poincare Probab. ´ Statist. 25 39 71.
  • 15 RUELLE, D. 1982. Characteristic exponents and invariant manifolds in Hilbert space. Ann. of Math. 115 243 290.
  • 16 SUSSMAN, H. J. 1978. On the gap between deterministic and stochastic ordinary differential equations. Ann. Probab. 6 19 41.