Small random perturbation of dynamical systems: recursive multiscale analysis

Martinelli, Fabio; Sbano, Luca; Scoppola, Elisabetta

doi:10.1080/17442509408833923

Cited by 7 publications

(9 citation statements)

References 6 publications

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“…The main result from the paper by Martinelli et al [14] states that with high probability the trajectories starting near stable fixed points eventually contract together exponentially fast.…”

Section: Proof Of Stepmentioning

confidence: 98%

“…It should be noted that (H1)-(H4) are, for our purposes, quite restrictive. They are necessary in order to apply the result due to Martinelli et al [14] which provides the contraction result required to show the attractor is a point, see Sect. 5.2.…”

Section: Single Point Random Attractormentioning

confidence: 99%

“…Here, W t is a two-sided Brownian motion of dimension d and in the proof ε > 0 has to be taken sufficiently small, although observations from numerical simulations suggest that random global attractors generated by (2) should in fact be single points for any ε > 0. A result of Martinelli et al [14] provides the contraction that is central to the main result in Sect. 5.…”

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confidence: 92%

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Collapse of attractors for ODEs under small random perturbations

Tearne

2007

Probab. Theory Relat. Fields

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This paper concerns comparisons between attractors for random dynamical systems and their corresponding noiseless systems. It is shown that if a random dynamical system has negative time trajectories that are transient or explode with probability one, then the random attractor cannot contain any open set. The result applies to any Polish space and when applied to autonomous stochastic differential equations with additive noise requires only a mild dissipation of the drift. Additionally, following observations from numerical simulations in a previous paper, analytical results are presented proving that the random global attractors for a class of gradient-like stochastic differential equations consist of a single random point. Comparison with the noiseless system reveals that arbitrarily small non-degenerate additive white noise causes the deterministic global attractor, which may have non-zero dimension, to 'collapse'. Unlike existing results of this type, no order preserving property is necessary.

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Section: Proof Of Stepmentioning

confidence: 98%

Section: Single Point Random Attractormentioning

confidence: 99%

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confidence: 92%

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Collapse of attractors for ODEs under small random perturbations

Tearne

2007

Probab. Theory Relat. Fields

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“…Another approach, based on large deviation techniques, has been introduced in [34,35,46]. Besides several technical assumptions, assuming for (1.1) that b has only finitely many fixed points and that σ is small enough, these works prove synchronization by noise.…”

Section: Comments On the Existing Literaturementioning

confidence: 99%

Synchronization by noise

Flandoli

Gess

Scheutzow

2016

Probab. Theory Relat. Fields

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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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“…For the related effect of synchronisation in master-slave systems we refer to [10] and the references therein. Approaches to synchronisation by noise based on local stability have been introduced in [4], with extensions in [14,11], and large deviation techniques have been employed in [24,23,30]. For synchronisation for discrete time random dynamical systems (RDS) see [20,21,22,27] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Synchronisation by noise for the stochastic quantisation equation in dimensions $2$ and $3$

Gess,

Tsatsoulis

2019

Preprint

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We prove uniform synchronisation by noise with rates for the stochastic quantisation equation in dimensions two and three. The proof relies on a combination of coming down from infinity estimates and the framework of order-preserving Markov semigroups derived in [Butkovsky, Scheutzow;2019]. In particular, it is shown that this framework can be applied to the case of state spaces given in terms of Hölder spaces of negative exponent.

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Small random perturbation of dynamical systems: recursive multiscale analysis

Cited by 7 publications

References 6 publications

Collapse of attractors for ODEs under small random perturbations

Collapse of attractors for ODEs under small random perturbations

Synchronization by noise

Synchronisation by noise for the stochastic quantisation equation in dimensions $2$ and $3$

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