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 & Continuous Dynamical Systems - B, 2016, 21 (9) : 3003-3014. doi: 10.3934/dcdsb.2016084
References:
[1]

L. Arnold, Random Dynamical Systems,, Springer-Verlag, (1998). doi: 10.1007/978-3-662-12878-7. Google Scholar

[2]

P. Baxendale and T. Harris, Isotropic stochastic flows,, Ann. Probab., 14 (1986), 1155. doi: 10.1214/aop/1176992360. Google Scholar

[3]

A. Carverhill, Survey: Lyapunov exponents for stochastic flows on manifolds,, in Lyapunov Exponents, (1986), 292. doi: 10.1007/BFb0076849. Google Scholar

[4]

G. Dimitroff, Some Properties of Isotropic Brownian and Ornstein-Uhlenbeck Flows,, Ph.D thesis, (2006). Google Scholar

[5]

G. Dimitroff and M. Scheutzow, Attractors and expansion for Brownian flows,, Electron. J. Probab., 16 (2011), 1193. doi: 10.1214/EJP.v16-894. Google Scholar

[6]

F. Flandoli, B. Gess and M. Scheutzow, Synchronization by noise,, Probab. Theory Rel. Fields, (2016). doi: 10.1007/s00440-016-0716-2. Google Scholar

[7]

F. Flandoli, B. Gess and M. Scheutzow, Synchronization by noise for order-preserving random dynamical systems,, Ann. Probab., (2016). doi: 10.1214/16-AOP1088. Google Scholar

[8]

O. Kallenberg, Foundations of Modern Probability,, $2^{nd}$ edition, (2002). doi: 10.1007/978-1-4757-4015-8. Google Scholar

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I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus,, $2^{nd}$ edition, (1991). doi: 10.1007/978-1-4612-0949-2. Google Scholar

[10]

H. Kunita, Stochastic Flows and Stochastic Differential Equations,, Cambridge University Press, (1990). Google Scholar

[11]

Y. Le Jan, Équilibre statistique pour les produits de difféomorphismes aléatoires indépendants,, Ann. Inst. H. Poincaré Probab. Statist., 23 (1987), 111. Google Scholar

[12]

Y. Le Jan and S. Watanabe, Stochastic flows of diffeomorphisms,, in Stochastic analysis (Katata/Kyoto, (1984), 307. doi: 10.1016/S0924-6509(08)70398-2. Google Scholar

[13]

O. Raimond, Flots browniens isotropes sur la sphère,, Ann. Inst. H. Poincaré Probab. Statist., 35 (1999), 313. doi: 10.1016/S0246-0203(99)80014-4. Google Scholar

[14]

M. Scheutzow, Comparison of various concepts of a random attractor: A case study,, Arch. Math. (Basel), 78 (2002), 233. doi: 10.1007/s00013-002-8241-1. Google Scholar

show all references

References:
[1]

L. Arnold, Random Dynamical Systems,, Springer-Verlag, (1998). doi: 10.1007/978-3-662-12878-7. Google Scholar

[2]

P. Baxendale and T. Harris, Isotropic stochastic flows,, Ann. Probab., 14 (1986), 1155. doi: 10.1214/aop/1176992360. Google Scholar

[3]

A. Carverhill, Survey: Lyapunov exponents for stochastic flows on manifolds,, in Lyapunov Exponents, (1986), 292. doi: 10.1007/BFb0076849. Google Scholar

[4]

G. Dimitroff, Some Properties of Isotropic Brownian and Ornstein-Uhlenbeck Flows,, Ph.D thesis, (2006). Google Scholar

[5]

G. Dimitroff and M. Scheutzow, Attractors and expansion for Brownian flows,, Electron. J. Probab., 16 (2011), 1193. doi: 10.1214/EJP.v16-894. Google Scholar

[6]

F. Flandoli, B. Gess and M. Scheutzow, Synchronization by noise,, Probab. Theory Rel. Fields, (2016). doi: 10.1007/s00440-016-0716-2. Google Scholar

[7]

F. Flandoli, B. Gess and M. Scheutzow, Synchronization by noise for order-preserving random dynamical systems,, Ann. Probab., (2016). doi: 10.1214/16-AOP1088. Google Scholar

[8]

O. Kallenberg, Foundations of Modern Probability,, $2^{nd}$ edition, (2002). doi: 10.1007/978-1-4757-4015-8. Google Scholar

[9]

I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus,, $2^{nd}$ edition, (1991). doi: 10.1007/978-1-4612-0949-2. Google Scholar

[10]

H. Kunita, Stochastic Flows and Stochastic Differential Equations,, Cambridge University Press, (1990). Google Scholar

[11]

Y. Le Jan, Équilibre statistique pour les produits de difféomorphismes aléatoires indépendants,, Ann. Inst. H. Poincaré Probab. Statist., 23 (1987), 111. Google Scholar

[12]

Y. Le Jan and S. Watanabe, Stochastic flows of diffeomorphisms,, in Stochastic analysis (Katata/Kyoto, (1984), 307. doi: 10.1016/S0924-6509(08)70398-2. Google Scholar

[13]

O. Raimond, Flots browniens isotropes sur la sphère,, Ann. Inst. H. Poincaré Probab. Statist., 35 (1999), 313. doi: 10.1016/S0246-0203(99)80014-4. Google Scholar

[14]

M. Scheutzow, Comparison of various concepts of a random attractor: A case study,, Arch. Math. (Basel), 78 (2002), 233. doi: 10.1007/s00013-002-8241-1. Google Scholar

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