Generalized dimensions of images of measures under Gaussian processes

Kenneth Falconer; Yimin Xiao

doi:10.1016/j.aim.2013.11.002

Generalized dimensions of images of measures under Gaussian processes

Kenneth Falconer, Yimin Xiao

Pure Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We show that for certain Gaussian random processes and fields X:R^N→R^d,
D_q(μx) = min {d, 1/α Dq (μ)} a.s., for an index α which depends on Hölder properties and strong local nondeterminism of X, where q>1, where D_q denotes generalized q-dimension μ_Xis the image of the measure μ under X. In particular this holds for index-α fractional Brownian motion, for fractional Riesz–Bessel motions and for certain infinity scale fractional Brownian motions.

Original language	English
Pages (from-to)	492-517
Number of pages	26
Journal	Advances in Mathematics
Volume	252
DOIs	https://doi.org/10.1016/j.aim.2013.11.002
Publication status	Published - 15 Feb 2014

Keywords

Gaussian process
Local nondeterminism
Generalised dimension
Fractional Brownian

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10.1016/j.aim.2013.11.002

FalconerXiaoAIMRevised
© 2013 Elsevier Inc. All rights reserved. NOTICE: this is the author’s version of a work that was accepted for publication in Advances in Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Advances in Mathematics, Vol 252 (2014). DOI: 10.1016/j.aim.2013.11.002
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http://arxiv.org/pdf/1212.2383.pdf

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title = "Generalized dimensions of images of measures under Gaussian processes",

abstract = "We show that for certain Gaussian random processes and fields X:RN→Rd, Dq(μx) = min \{d, 1/α Dq (μ)\} a.s., for an index α which depends on H{\"o}lder properties and strong local nondeterminism of X, where q>1, where Dq denotes generalized q-dimension μX is the image of the measure μ under X. In particular this holds for index-α fractional Brownian motion, for fractional Riesz–Bessel motions and for certain infinity scale fractional Brownian motions.",

keywords = "Gaussian process, Local nondeterminism, Generalised dimension, Fractional Brownian",

author = "Kenneth Falconer and Yimin Xiao",

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AU - Xiao, Yimin

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PY - 2014/2/15

Y1 - 2014/2/15

N2 - We show that for certain Gaussian random processes and fields X:RN→Rd, Dq(μx) = min {d, 1/α Dq (μ)} a.s., for an index α which depends on Hölder properties and strong local nondeterminism of X, where q>1, where Dq denotes generalized q-dimension μX is the image of the measure μ under X. In particular this holds for index-α fractional Brownian motion, for fractional Riesz–Bessel motions and for certain infinity scale fractional Brownian motions.

AB - We show that for certain Gaussian random processes and fields X:RN→Rd, Dq(μx) = min {d, 1/α Dq (μ)} a.s., for an index α which depends on Hölder properties and strong local nondeterminism of X, where q>1, where Dq denotes generalized q-dimension μX is the image of the measure μ under X. In particular this holds for index-α fractional Brownian motion, for fractional Riesz–Bessel motions and for certain infinity scale fractional Brownian motions.

KW - Gaussian process

KW - Local nondeterminism

KW - Generalised dimension

KW - Fractional Brownian

UR - https://www.scopus.com/pages/publications/84889603022

U2 - 10.1016/j.aim.2013.11.002

DO - 10.1016/j.aim.2013.11.002

M3 - Article

SN - 0001-8708

VL - 252

SP - 492

EP - 517

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -

Generalized dimensions of images of measures under Gaussian processes

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Kenneth Falconer

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Generalized dimensions of images of measures under Gaussian processes

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