Weak synchronization for isotropic flows

Michael  Cranston; Benjamin  Gess; Michael  Scheutzow

doi:10.3934/dcdsb.2016084

Article Contents

2016, Volume 21, Issue 9: 3003-3014. Doi: 10.3934/dcdsb.2016084

This issue Previous Article The transport equation and zero quadratic variation processes Next Article An integral inequality for the invariant measure of some finite dimensional stochastic differential equation

Weak synchronization for isotropic flows

1.
Department of Mathematics, University of California, Irvine
2.
Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig
3.
Institut für Mathematik, MA 7-5, Technische Universität Berlin, 10623 Berlin

Received: October 2015

Revised: January 2016

Published: November 2016

Abstract / Introduction Related Papers Cited by

Abstract

We study Brownian flows on manifolds for which the associated Markov process is strongly mixing with respect to an invariant probability measure and for which the distance process for each pair of trajectories is a diffusion $r$. We provide a sufficient condition on the boundary behavior of $r$ at $0$ which guarantees that the statistical equilibrium of the flow is almost surely a singleton and its support is a weak point attractor. The condition is fulfilled in the case of negative top Lyapunov exponent, but it is also fulfilled in some cases when the top Lyapunov exponent is zero. Particular examples are isotropic Brownian flows on $S^{d-1}$ as well as isotropic Ornstein-Uhlenbeck flows on $\mathbb{R}^d$.

Keywords:

Mathematics Subject Classification: Primary: 37B25; Secondary: 37G35, 37H15.

Citation:

Related Papers

Cited by

References

[1]	L. Arnold, Random Dynamical Systems, Springer-Verlag, New York, 1998.doi: 10.1007/978-3-662-12878-7.
[2]	P. Baxendale and T. Harris, Isotropic stochastic flows, Ann. Probab., 14 (1986), 1155-1179.doi: 10.1214/aop/1176992360.
[3]	A. Carverhill, Survey: Lyapunov exponents for stochastic flows on manifolds, in Lyapunov Exponents, (Bremen 1984), volume 1186 of Lecture Notes in Math., Springer, (1986), 292-307.doi: 10.1007/BFb0076849.
[4]	G. Dimitroff, Some Properties of Isotropic Brownian and Ornstein-Uhlenbeck Flows, Ph.D thesis, Technische Universität Berlin, https://depositonce.tu-berlin.de/bitstream/11303/1627/1/Dokument_37.pdf, 2006.
[5]	G. Dimitroff and M. Scheutzow, Attractors and expansion for Brownian flows, Electron. J. Probab., 16 (2011), 1193-1213.doi: 10.1214/EJP.v16-894.
[6]	F. Flandoli, B. Gess and M. Scheutzow, Synchronization by noise, Probab. Theory Rel. Fields, 2016, arXiv:1411.1340.doi: 10.1007/s00440-016-0716-2.
[7]	F. Flandoli, B. Gess and M. Scheutzow, Synchronization by noise for order-preserving random dynamical systems, Ann. Probab., 2016, arXiv:1503.08737.doi: 10.1214/16-AOP1088.
[8]	O. Kallenberg, Foundations of Modern Probability, $2^{nd}$ edition, Springer-Verlag, New York, 2002.doi: 10.1007/978-1-4757-4015-8.
[9]	I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, $2^{nd}$ edition, Springer-Verlag, New York, 1991.doi: 10.1007/978-1-4612-0949-2.
[10]	H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge, 1990.
[11]	Y. Le Jan, Équilibre statistique pour les produits de difféomorphismes aléatoires indépendants, Ann. Inst. H. Poincaré Probab. Statist., 23 (1987), 111-120.
[12]	Y. Le Jan and S. Watanabe, Stochastic flows of diffeomorphisms, in Stochastic analysis (Katata/Kyoto, 1982), volume 32 of North-Holland Math. Library, North-Holland, Amsterdam, (1984), 307-332.doi: 10.1016/S0924-6509(08)70398-2.
[13]	O. Raimond, Flots browniens isotropes sur la sphère, Ann. Inst. H. Poincaré Probab. Statist., 35 (1999), 313-354.doi: 10.1016/S0246-0203(99)80014-4.
[14]	M. Scheutzow, Comparison of various concepts of a random attractor: A case study, Arch. Math. (Basel), 78 (2002), 233-240.doi: 10.1007/s00013-002-8241-1.