The Annals of Probability

Asymptotic support theorem for planar isotropic Brownian flows

Moritz Biskamp

Full-text: Open access


It has been shown by various authors that the diameter of a given nontrivial bounded connected set $\mathcal{X}$ grows linearly in time under the action of an isotropic Brownian flow (IBF), which has a nonnegative top-Lyapunov exponent. In case of a planar IBF with a positive top-Lyapunov exponent, the precise deterministic linear growth rate $K$ of the diameter is known to exist. In this paper we will extend this result to an asymptotic support theorem for the time-scaled trajectories of a planar IBF $\varphi$, which has a positive top-Lyapunov exponent, starting in a nontrivial compact connected set $\mathcal{X}\subseteq\mathbf{R}^{2}$; that is, we will show convergence in probability of the set of time-scaled trajectories in the Hausdorff distance to the set of Lipschitz continuous functions on $[0,1]$ starting in $0$ with Lipschitz constant $K$.

Article information

Ann. Probab., Volume 41, Number 2 (2013), 699-721.

First available in Project Euclid: 8 March 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G17: Sample path properties 37C10: Vector fields, flows, ordinary differential equations
Secondary: 60G15: Gaussian processes 37H10: Generation, random and stochastic difference and differential equations [See also 34F05, 34K50, 60H10, 60H15]

Stochastic flows isotropic Brownian flows asymptotic expansion asymptotic support theorem


Biskamp, Moritz. Asymptotic support theorem for planar isotropic Brownian flows. Ann. Probab. 41 (2013), no. 2, 699--721. doi:10.1214/11-AOP701.

Export citation


  • [1] Arnold, L. (1998). Random Dynamical Systems. Springer, Berlin.
  • [2] Baxendale, P. and Dimitroff, G. (2009). Uniform shrinking and expansion under isotropic Brownian flows. J. Theoret. Probab. 22 620–639.
  • [3] Baxendale, P. and Harris, T. E. (1986). Isotropic stochastic flows. Ann. Probab. 14 1155–1179.
  • [4] Carmona, R. A. and Cerou, F. (1999). Transport by incompressible random velocity fields: Simulations & mathematical conjectures. In Stochastic Partial Differential Equations: Six Perspectives. Math. Surveys Monogr. 64 153–181. Amer. Math. Soc., Providence, RI.
  • [5] Cranston, M., Scheutzow, M. and Steinsaltz, D. (1999). Linear expansion of isotropic Brownian flows. Electron. Commun. Probab. 4 91–101.
  • [6] Cranston, M., Scheutzow, M. and Steinsaltz, D. (2000). Linear bounds for stochastic dispersion. Ann. Probab. 28 1852–1869.
  • [7] Dolgopyat, D., Kaloshin, V. and Koralov, L. (2004). A limit shape theorem for periodic stochastic dispersion. Comm. Pure Appl. Math. 57 1127–1158.
  • [8] Itô, K. (1956). Isotropic random current. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 19541955, Vol. II 125–132. Univ. California Press, Berkeley.
  • [9] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [10] Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics 24. Cambridge Univ. Press, Cambridge.
  • [11] Le Jan, Y. (1985). On isotropic Brownian motions. Z. Wahrsch. Verw. Gebiete 70 609–620.
  • [12] Lisei, H. and Scheutzow, M. (2001). Linear bounds and Gaussian tails in a stochastic dispersion model. Stoch. Dyn. 1 389–403.
  • [13] Scheutzow, M. (2009). Chaining techniques and their application to stochastic flows. In Trends in Stochastic Analysis. London Mathematical Society Lecture Note Series 353 35–63. Cambridge Univ. Press, Cambridge.
  • [14] Scheutzow, M. and Steinsaltz, D. (2002). Chasing balls through martingale fields. Ann. Probab. 30 2046–2080.
  • [15] van Bargen, H. (2011). A weak limit shape theorem for planar isotropic Brownian flows. Stoch. Dyn. 11 593–626.
  • [16] Yaglom, A. M. (1957). Some classes of random fields in $n$-dimensional space, related to stationary random processes. Theory Probab. Appl. 2 273–320.