Stratonovich calculus with spatial parameters and anticipative problems in multiplicative ergodic theory

Arnold, Ludwig; Imkeller, Peter

doi:10.1016/0304-4149(95)00081-x

Cited by 28 publications

(28 citation statements)

References 13 publications

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“…The third method, which is based on Dudley's entropy theorem and the Gaussian isoperimetric inequality, provides a stronger result in the sense that the upper bound is a constant instead of a random variable [cf. Arnold and Imkeller (1996). A similar result can also be found in Dalang, Khoshnevisan and Nualart (2007a).…”

Section: Modulus Of Continuitysupporting

confidence: 82%

“…We summarize three methods for deriving a sharp modulus of continuity for all anisotropic Gaussian random fields satisfying Condition (C1). The first method is to use an extension of the powerful Garsia-Rodemich-Rumsey continuity lemma [see Arnold and Imkeller (1996), Funaki, Kikuchi and Potthoff (2006), Dalang, Khoshnevisan and Nualart (2007a)]; the second is the "minorizing metric" method of Kwapień and Risiński (2004); and the third is based on the Gaussian isoperimetric inequality. While the first two methods have wider applicability, the third method produces more precise results for Gaussian random fields.…”

Section: Introductionmentioning

confidence: 99%

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Sample Path Properties of Anisotropic Gaussian Random Fields

Xiao¹

Lecture Notes in Mathematics

139

227

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Anisotropic Gaussian random fields arise in probability theory and in various applications. Typical examples are fractional Brownian sheets, operator-scaling Gaussian fields with stationary increments, and the solution to the stochastic heat equation.This paper is concerned with sample path properties of anisotropic Gaussian random fields in general. Let X = {X(t), t ∈ R N } be a Gaussian random field with values in R d and with parameters H 1 , . . . , H N . Our goal is to characterize the anisotropic nature of X in terms of its parameters explicitly.Under some general conditions, we establish results on the modulus of continuity, small ball probabilities, fractal dimensions, hitting probabilities and local times of anisotropic Gaussian random fields. An important tool for our study is the various forms of strong local nondeterminism.

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Section: Modulus Of Continuitysupporting

confidence: 82%

Section: Introductionmentioning

confidence: 99%

Sample Path Properties of Anisotropic Gaussian Random Fields

Xiao¹

Lecture Notes in Mathematics

139

227

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“…(3.2) is a well-known implication of the fundamental continuity lemma of Garsia, Rodemich and Rumsey, see for example Barlow and Yor [11, formula (3.b),] or Arnold and Imkeller [5]. h…”

Section: 1mentioning

confidence: 99%

“…Then by [5,Theorem 9], the linear SDE (2.1) can be diagonalized by means of a random (anticipative) coordinate transformation w t h t xv t , so that the linear cocycle…”

Section: Resonant Casementioning

confidence: 99%

Normal forms for stochastic differential equations

Arnold¹,

Imkeller²

1998

Probab Theory Relat Fields

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We address the following problem from the intersection of dynamical systems and stochastic analysis: Two SDE dx t m j0 f j x t d j t and dx t m j0 g j x t d j t in R d with smooth coecients satisfying f j 0 g j 0 0 are said to be smoothly equivalent if there is a smooth random dieomorphism (coordinate transformation) hx with hxY 0 0 and hhxY 0 id which conjugates the corresponding local¯ows,where h t xs xt s À xt is the (ergodic) shift on the canonical Wiener space. The normal form problem for SDE consists in ®nding the``simplest possible'' member in the equivalence class of a given SDE, in particular in giving conditions under which it can be linearized (g j x hf j 0x).We develop a mathematically rigorous normal form theory for SDE which justi®es the engineering and physics literature on that problem. It is based on the multiplicative ergodic theorem and uses a uniform (with respect to a spatial parameter) Stratonovich calculus which allows the handling of non-adapted initial values and coecients in the stochastic version of the cohomological equation. Our Probab. Theory Relat. Fields 110, 559±588 (1998) Key words and phrases: Stochastic dierential equation ± Stochastic normal form ± Stochastic cohomological equation ± Anticipative calculus ± Random dynamical system ± Multiplicative ergodic theory ± Ane stochastic dierential equation ± Noisy Dung-van der Pol oscillator. main result (Theorem 3.2) is that an SDE is (formally) equivalent to its linearization if the latter is nonresonant.As a by-product, we prove a general theorem on the existence of a stationary solution of an anticipative ane SDE.The study of the Dung-van der Pol oscillator with small noise concludes the paper.

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“…This is a simple generalization of the GRR inequality byArnold and Imkeller (1996).18 The factor in front of d(t, s) is 4. We do not assume the convexity of T. For example, when T is a subset of R 2 , it is possible that the open disc centered at (t + s)/2 and of diameter d(t, s) is not included in T.…”

mentioning

confidence: 98%