November  2016, 21(9): 3003-3014. doi: 10.3934/dcdsb.2016084

Weak synchronization for isotropic flows

1. 

Department of Mathematics, University of California, Irvine, United States

2. 

Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany

3. 

Institut für Mathematik, MA 7-5, Technische Universität Berlin, 10623 Berlin, Germany

Received  October 2015 Revised  January 2016 Published  October 2016

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 $\mathbb{R}^d$.
Citation: Michael Cranston, Benjamin Gess, Michael Scheutzow. Weak synchronization for isotropic flows. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3003-3014. doi: 10.3934/dcdsb.2016084
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.

show all references

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.

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