## Electronic Communications in Probability

### Linear Expansion of Isotropic Brownian Flows

#### Abstract

We consider an isotropic Brownian flow on $R^d$ for $d\geq 2$ with a positive Lyapunov exponent, and show that any nontrivial connected set almost surely contains points whose distance from the origin under the flow grows linearly with time. The speed is bounded below by a fixed constant, which may be computed from the covariance tensor of the flow. This complements earlier work, which showed that stochastic flows with bounded local characteristics and zero drift cannot grow at a linear rate faster than linear.

#### Article information

Source
Electron. Commun. Probab., Volume 4 (1999), paper no. 12, 91-101.

Dates
Accepted: 27 August 1999
First available in Project Euclid: 2 March 2016

https://projecteuclid.org/euclid.ecp/1456938434

Digital Object Identifier
doi:10.1214/ECP.v4-1010

Mathematical Reviews number (MathSciNet)
MR1741738

Zentralblatt MATH identifier
0938.60048

Subjects
Primary: 60H20: Stochastic integral equations

Rights

#### Citation

Cranston, Michael; Scheutzow, Michael; Steinsaltz, David. Linear Expansion of Isotropic Brownian Flows. Electron. Commun. Probab. 4 (1999), paper no. 12, 91--101. doi:10.1214/ECP.v4-1010. https://projecteuclid.org/euclid.ecp/1456938434

#### References

• Peter Baxendale The Lyapunov spectrum of a stochastic flow of diffeomorphisms. In Lyapunov Exponents: 322–337. (Ludwig Arnold and Volker Wihstutz, editors.) 1186 of Lecture Notes in Mathematics, Springer Verlag (1983).
• Peter Baxendale and Theodore E. Harris, Isotropic stochastic flows, The Annals of Probability 14 (1986), 1155–1179. John Wiley & Sons, New York, (1968).
• Rene Carmona and Frederic Cerou, Transport by incompressible random velocity fields: Simulations & mathematical conjectures. In Stochastic Partial Differential Equations: Six Perspectives, 153–181. American Mathematical Society (1999).
• Michael Cranston and Yves Le Jan, Asymptotic curvature for stochastic dynamical systems. In Stochastic Dynamics (Bremen: 1997), 327–338. (Hans Crauel, Matthias Gundlach editors.) Springer Verlag (1999).
• Michael Cranston and Yves Le Jan, Geometric evolution under isotropic stochastic flow. Electronic Journal of Probability, 3(4):1–36 (1998).
• Michael Cranston, Michael Scheutzow, and David Steinsaltz, Linear and near-linear bounds for stochastic dispersion. To appear, Annals of Probability.
• Gedeon Dagan, Transport in heterogeneous porous formations:spatial moments, ergodicity, and effective dispersion. Water Resources Research, 26:1281–1290 (1990).
• Russ E. Davis, Lagrangian ocean studies. Annual Review of Fluid Mechanics, 23:43–64 (1991).
• Krzysztof Gawedzki and Antti Kupiainen, Universality in turbulence: an exactly soluble model. Lecture notes (1995). Reprint
• Theodore E. Harris, Brownian motions on the homeomorphisms of the plane. The Annals of Probability, 9:232–254 (1981).
• Peter Hall and Christopher C. Heyde, Martingale Limit Theory and its Application. Academic Press, New York, London (1980).
• Kiyoshi Itô, Isotropic random current. In Proceedings of the Third Berkeley Symposium in Mathematical Statistics and Probability, v. 2. (Jerzy Neyman, editor.) University of California Press, Berkeley (1956).
• Ioannis Karatzas and Steven Shreve, Brownian Motion and Stochastic Calculus. Springer-Verlag, New York (1988).
• Yves Le Jan, On isotropic Brownian motions. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 70:609–620 (1985).
• Stanislav Molchanov, Topics in statistical oceanography. In Stochastic modelling in physical oceanography, (R. Adler, P. Müller, and B. L. Rozovskii editors.) Birkhaeuser, Boston: 343–380 (1996).
• Stanislav Molchanov and Alexander Ruzmaikin, Lyapunov exponents and distributions of magnetic fields in dynamo models. In The Dynkin Festschrift: Markov Processes and their Applications. (Mark Freidlin, editor.) Birkhäuser, Boston: 287–306 (1994).
• Daniel W. Stroock and S.R. Srinivasa Varadhan, Multidimensional Diffusion Processes. Springer-Verlag, New York (1979).
• A. M. Yaglom, Some classes of random fields in n-dimensional space, related to stationary random processes. Theory of Probability and its Applications, 2:273–320, (1957).