A limit shape theorem for periodic stochastic dispersion

Dolgopyat, Dmitry; Калошин, В. А.; Koralov, Leonid

doi:10.1002/cpa.20032

Cited by 12 publications

(14 citation statements)

References 15 publications

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“…This is done using the invariance properties with respect to time reversal of IBFs. These properties are not shared by the model of [8] and hence are a novelty in the present subject. The paper is divided into several sections.…”

Section: Introductionmentioning

confidence: 82%

“…Nevertheless these bounds turn out to be far from each other in some examples and there is little hope to match these bounds with the methods from [6] or [14]. We will follow a different approach which first appeared in [8], wherein a class of periodic stochastic flows on R 2 (or stochastic flows on the torus) is considered.…”

Section: Introductionmentioning

confidence: 99%

“…[8] develop a similar limit theorem (even with a stronger assertion) using the fact that their model essentially lives on a compact manifold. Although we will sometimes follow the lines of thought of [8] in the first part, we will see that to get the assertion we will have to replace the methods relying on the assumption of periodicity (which means perfect dependence of particles which are far from each other) on R 2 by different ones. This is done using the invariance properties with respect to time reversal of IBFs.…”

Section: Introductionmentioning

confidence: 99%

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A Weak Limit Shape Theorem for Planar Isotropic Brownian Flows

Bargen

2011

Stoch. Dyn.

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It has been shown by various authors under different assumptions that the diameter of a bounded non-trivial set γ under the action of a stochastic flow grows linearly in time. We show that the asymptotic linear expansion speed if properly defined is deterministic i.e. we show for a 2-dimensional isotropic Brownian flow Φ with a positive Lyapunov exponent that there exists a non-random set B such that we have for > 0, arbitrary connected γ ⊂⊂ R 2 consisting of at least two different points and arbitrarily large times T thatDefinition 1.2 A function b : R d → R d×d is an isotropic covariance tensor if x → b(x) is C ∞ and all derivatives of any order are bounded, b(0) = E d (the d-dimensional unit matrix), x → b(x) is not constant and b(x) = O * b(Ox)O holds for any x ∈ R d and any orthogonal matrix O.

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Section: Introductionmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Weak Limit Shape Theorem for Planar Isotropic Brownian Flows

Bargen

2011

Stoch. Dyn.

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“…• Shape of poisoned area Theorem 5. [10] Let the original set Ω ⊂ R 2 be bounded and contain a continuous curve with positive diameter and let assumptions (A)-(E) below be satisfied. Then there is a compact convex non random set B, independent of Ω, such that for any ε > 0 almost surely…”

Section: Theorem 4 [9] Let Assumptions (A)-(d) Be Satisfied For Almmentioning

confidence: 99%

Long time behaviour of periodic stochastic flows

Калошин¹,

Dolgopyat²,

Koralov³

2006

XIVth International Congress on Mathematical Physics

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We consider the evolution of a set carried by a space periodic incompressible stochastic flow in a Euclidean space. We report on three main results obtained in [8, 9, 10] concerning long time behaviour for a typical realization of the stochastic flow. First, at time t most of the particles are at a distance of order √ t away from the origin. Moreover, we prove a Central Limit Theorem for the evolution of a measure carried by the flow, which holds for almost every realization of the flow. Second, we show the existence of a zero measure full Hausdorff dimension set of points, which escape to infinity at a linear rate. Third, in the 2-dimensional case, we study the set of points visited by the original set by time t. Such a set, when scaled down by the factor of t, has a limiting non random shape.

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“…Nevertheless, upper and lower bounds for the linear growth turn out to be far from each other in some examples. In the case of planar periodic stochastic flows (stochastic flows on the torus) Dolgopyat, Kaloshin and Koralov [7] used a new approach based on the so-called stable norm, to identify the precise deterministic linear growth rate of such flows. By this approach, van Bargen [15] identified the precise deterministic growth rate for planar IBFs, which have a strictly positive top-Lyapunov exponent.…”

mentioning

confidence: 99%

Asymptotic support theorem for planar isotropic Brownian flows

Biskamp¹

2013

Ann. Probab.

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It has been shown by various authors that the diameter of a given nontrivial bounded connected set $\mathcal{X}$ grows linearly in time under the action of an isotropic Brownian flow (IBF), which has a nonnegative top-Lyapunov exponent. In case of a planar IBF with a positive top-Lyapunov exponent, the precise deterministic linear growth rate K of the diameter is known to exist. In this paper we will extend this result to an asymptotic support theorem for the time-scaled trajectories of a planar IBF $\varphi$, which has a positive top-Lyapunov exponent, starting in a nontrivial compact connected set $\mathcal{X}\subseteq \mathbf{R}^2$; that is, we will show convergence in probability of the set of time-scaled trajectories in the Hausdorff distance to the set of Lipschitz continuous functions on [0,1] starting in 0 with Lipschitz constant K.Comment: Published in at http://dx.doi.org/10.1214/11-AOP701 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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A limit shape theorem for periodic stochastic dispersion

Cited by 12 publications

References 15 publications

A Weak Limit Shape Theorem for Planar Isotropic Brownian Flows

A Weak Limit Shape Theorem for Planar Isotropic Brownian Flows

Long time behaviour of periodic stochastic flows

Asymptotic support theorem for planar isotropic Brownian flows

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