Flows of stochastic dynamical systems: ergodic theory

Carverhill, Andrew

doi:10.1080/17442508508833343

Cited by 124 publications

(50 citation statements)

References 10 publications

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“…For measure-preserving stochastic flows with conditions (C) Lyapunov exponents λ 1 , λ 2 exist by multiplicative ergodic theorem for stochastic flows of diffeomorphisms (see [3], thm. 2.1).…”

Section: Nondegeneracy Assumptionsmentioning

confidence: 99%

A limit shape theorem for periodic stochastic dispersion

Dolgopyat

Калошин

Koralov

2004

Comm Pure Appl Math

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Abstract. We consider the evolution of a connected set on the plane carried by a periodic incompressible stochastic flow. While for almost every realization of the random flow at time t most of the particles are at a distance of order √ t away from the origin, there is a measure zero set of points, which escape to infinity at the linear rate. We study the set of points visited by the original set by time t, and show that such a set, when scaled down by the factor of t, has a limiting non random shape.

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Section: Nondegeneracy Assumptionsmentioning

confidence: 99%

A limit shape theorem for periodic stochastic dispersion

Dolgopyat

Калошин

Koralov

2004

Comm Pure Appl Math

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“…In fact the Canonical SDS on the flat torus is an example which is not ergodic: this is clear from the fact that parallel translation of a frame cannot alter its angle of inclination. Thus, we cannot deduce from [3] that the spectrum of Theorem 1 above is independent of u E OM. Our aim in this section is to show that this is nevertheless true.…”

Section: Introductionmentioning

confidence: 94%

Lyapunov Exponents for a Stochastic Analogue of the Geodesic Flow

Carverhill¹,

Elworthy²

1986

Transactions of the American Mathematical Society

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ABSTRACT.New invariants for a Riemannian manifold are defined as Lyapunov exponents of a stochastic analogue of the geodesic flow. A lower bound is given reminiscent of corresponding results for the geodesic flow, and an upper bound is given for surfaces of positive curvature.For surfaces of constant negative curvature a direct method via the Doob rt-transform is used to determine the full Lyapunov structure relating the stable manifolds to the horocycles.

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“…Then, c(x 1 , x 2 ) can take only two values: either 0 or 1. In fact, it is possible to show (see [17,18,19,20]) that the following picture holds. There exists a random periodic sequence of points x(k) = x * (ω) + k, k ∈ Z, x * ∈ [0, 1) such that c(x 1 , x 2 ) = 0 for all x 1 , x 2 ∈ (x * (ω) + k, x * (ω) + k + 1) and c(x 1 , x 2 ) = 1 for x 1 ∈ (x * (ω) + k − 1, x * (ω) + k) and x 2 ∈ (x * (ω) + k, x * (ω) + k + 1).…”

Section: Two Related Statistical Modelsmentioning

confidence: 99%

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Bec

Khanin

2003

Journal of Statistical Physics

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The inviscid Burgers equation with random and spatially smooth forcing is considered in the limit when the size of the system tends to infinity. For the one-dimensional problem, it is shown both theoretically and numerically that many of the features of the space-periodic case carry over to infinite domains as intermediate time asymptotics. In particular, for large time T we introduce the concept of T -global shocks replacing the notion of main shock which was considered earlier in the periodic case (1997, E et al., Phys. Rev. Lett. 78, 1904. In the case of spatially extended systems these objects are no anymore global. They can be defined only for a given time scale and their spatial density behaves as ρ(T ) ∼ T −2/3 for large T . The probability density function p(A) of the age A of shocks behaves asymptotically as A −5/3 . We also suggest a simple statistical model for the dynamics and interaction of shocks and discuss an analogy with the problem of distribution of instability islands for a simple first-order stochastic differential equation.

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Flows of stochastic dynamical systems: ergodic theory

Cited by 124 publications

References 10 publications

A limit shape theorem for periodic stochastic dispersion

A limit shape theorem for periodic stochastic dispersion

Lyapunov Exponents for a Stochastic Analogue of the Geodesic Flow

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