Abstract
We provide sufficient conditions for synchronization by noise, i.e. under these conditions we prove that weak random attractors for random dynamical systems consist of single random points. In the case of SDE with additive noise, these conditions are also essentially necessary. In addition, we provide sufficient conditions for the existence of a minimal weak point random attractor consisting of a single random point. As a result, synchronization by noise is proven for a large class of SDE with additive noise. In particular, we prove that the random attractor for an SDE with drift given by a (multidimensional) double-well potential and additive noise consists of a single random point. All examples treated in Tearne (Probab Theory Relat Fields 141(1–2):1–18, 2008) are also included.
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Notes
For notation and background on RDS see Sect. 1.2 below.
For example, every metric space (E, d) that is both locally compact and \(\sigma \)-compact allows an equivalent metric \(d'\) such that \((E,d')\) is Heine-Borel [49].
Note that we have \(\varphi _t(\omega ,\cdot )\in C^{1,\delta }_{loc}\) and not only \(\varphi _t(\omega ,\cdot )\in C^{1,\beta }_{loc}\) for every \(\beta <\delta \) due to the additive noise in (3.6). This may be seen by considering the transformation \(\tilde{\varphi }= \varphi - \sigma W\) and studying the regularity of the corresponding pathwise ODE.
In fact, [48, Theorem, p. 243] assumes b to be smooth. However, it is an easy exercise to see that only \(b \in C^\delta \) for some \(\delta >0\) is required for the proof (cf. also [29, Theorem 10.4.1] for an according regularity result for linear, non-degenerate second order PDE with Hölder coefficients).
Note that this property is implied by (3.25).
The existence of (weak) random attractors under a weak coercivity condition has been obtained in [21].
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Acknowledgments
We thank the anonymous referees for their careful reading or our manuscript and their many insightful comments and suggestions that helped to improve the presentation of the manuscript and simplify several proofs.
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BG has been partially supported by the Research Project “Random dynamical systems and regularization by noise for stochastic partial differential equations” funded by the German Research Foundation.
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Flandoli, F., Gess, B. & Scheutzow, M. Synchronization by noise. Probab. Theory Relat. Fields 168, 511–556 (2017). https://doi.org/10.1007/s00440-016-0716-2
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DOI: https://doi.org/10.1007/s00440-016-0716-2
Keywords
- Synchronization
- Random dynamical system
- Random attractor
- Lyapunov exponent
- Stochastic differential equation
- Statistical equilibrium
Mathematics Subject Classification
- 37B25
- 37G35
- 37H15