On the Spatial Asymptotic Behavior of Stochastic Flows in Euclidean Space

Imkeller, Peter; Scheutzow, Michael

doi:10.1214/aop/1022677255

Cited by 15 publications

(23 citation statements)

References 10 publications

(13 reference statements)

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“…For a short introduction to stochastic flows we will follow [8,Section 2]. Let {F (x, t)} t≥0 be a family of R d -valued continuous semimartingales indexed by x ∈ R d on a filtered probability space (Ω,F , (F t ) t≥0 , P).…”

Section: Definition Of Stochastic Flowsmentioning

confidence: 99%

Pesin’s Formula for Random Dynamical Systems on $$\mathbf {R}^d$$ R d

Biskamp

2014

J Dyn Diff Equat

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Pesin's formula relates the entropy of a dynamical system with its positive Lyapunov exponents. It is well known, that this formula holds true for random dynamical systems on a compact Riemannian manifold with invariant probability measure which is absolutely continuous with respect to the Lebesgue measure. We will show that this formula remains true for random dynamical systems on R d which have an invariant probability measure absolutely continuous to the Lebesgue measure on R d . Finally we will show that a broad class of stochastic flows on R d of a Kunita type satisfies Pesin's formula.

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Section: Definition Of Stochastic Flowsmentioning

confidence: 99%

Pesin’s Formula for Random Dynamical Systems on $$\mathbf {R}^d$$ R d

Biskamp

2014

J Dyn Diff Equat

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“…In the case D = R d + Corollary 3.5 implies the following assertion. Corollary 3.6 Let (G) and (M) be in force with M i of the form (12). Assume that…”

Section: Deterministic Invariant Regionsmentioning

confidence: 99%

Invariance and monotonicity for stochastic delay differential equations

Čhuešhov¹,

Scheutzow²

2013

Discrete &Amp; Continuous Dynamical Systems - B

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We study invariance and monotonicity properties of Kunita-type stochastic differential equations in R d with delay. Our first result provides sufficient conditions for the invariance of closed subsets of R d . Then we present a comparison principle and show that under appropriate conditions the stochastic delay system considered generates a monotone (orderpreserving) random dynamical system. Several applications are considered.

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“…This paper is devoted to estimate the asymptotics of those jump processes in the [7] framework. It is therefore an alternative approach to the theory of flows of diffeomorphisms, theory well established in [1][2][3][4] and [8][9][10][11][12] among others; we deal here with the special class of volume preserving diffeomorphisms; the qualitative behaviour for this special class is the same as for the general class as presently established in the scientific literature. The methodologies developed here could be also used to construct the lifting of diffemorphism flows to the phase space, using the transfer energy matrix computed in [6]: we shall not touch here this last point.…”

Section: Introductionmentioning

confidence: 99%