SPDEs with rough noise in space: Hölder continuity of the solution

Raluca M. Balan; Maria Jolis; Lluís Quer-Sardanyons

doi:10.1016/j.spl.2016.09.003

SPDEs with rough noise in space: Hölder continuity of the solution

Raluca M. Balan, Maria Jolis, Lluís Quer-Sardanyons

Department of Mathematics

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

3 Citations (Scopus)

Abstract

© 2016 Elsevier B.V. We consider the stochastic wave and heat equations with affine multiplicative Gaussian noise which is white in time and behaves in space like the fractional Brownian motion with index H∈(14,12). The existence and uniqueness of the solution to these equations has been proved recently by the authors. In the present note we show that these solutions have modifications which are Hölder continuous in space of order smaller than H, and Hölder continuous in time of order smaller than γ, where γ=H for the wave equation and γ=H/2 for the heat equation.

Original language	English
Pages (from-to)	310-316
Journal	Statistics and Probability Letters
Volume	119
DOIs	https://doi.org/10.1016/j.spl.2016.09.003
Publication status	Published - 1 Dec 2016

Keywords

Fractional noise
Hölder continuous paths
Stochastic heat equation
Stochastic wave equation

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title = "SPDEs with rough noise in space: H{\"o}lder continuity of the solution",

abstract = "{\textcopyright} 2016 Elsevier B.V. We consider the stochastic wave and heat equations with affine multiplicative Gaussian noise which is white in time and behaves in space like the fractional Brownian motion with index H∈(14,12). The existence and uniqueness of the solution to these equations has been proved recently by the authors. In the present note we show that these solutions have modifications which are H{\"o}lder continuous in space of order smaller than H, and H{\"o}lder continuous in time of order smaller than γ, where γ=H for the wave equation and γ=H/2 for the heat equation.",

keywords = "Fractional noise, H{\"o}lder continuous paths, Stochastic heat equation, Stochastic wave equation",

author = "Balan, {Raluca M.} and Maria Jolis and Llu{\'i}s Quer-Sardanyons",

year = "2016",

month = dec,

day = "1",

doi = "10.1016/j.spl.2016.09.003",

language = "English",

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journal = "Statistics and Probability Letters",

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TY - JOUR

T1 - SPDEs with rough noise in space: Hölder continuity of the solution

AU - Balan, Raluca M.

AU - Jolis, Maria

AU - Quer-Sardanyons, Lluís

PY - 2016/12/1

Y1 - 2016/12/1

N2 - © 2016 Elsevier B.V. We consider the stochastic wave and heat equations with affine multiplicative Gaussian noise which is white in time and behaves in space like the fractional Brownian motion with index H∈(14,12). The existence and uniqueness of the solution to these equations has been proved recently by the authors. In the present note we show that these solutions have modifications which are Hölder continuous in space of order smaller than H, and Hölder continuous in time of order smaller than γ, where γ=H for the wave equation and γ=H/2 for the heat equation.

AB - © 2016 Elsevier B.V. We consider the stochastic wave and heat equations with affine multiplicative Gaussian noise which is white in time and behaves in space like the fractional Brownian motion with index H∈(14,12). The existence and uniqueness of the solution to these equations has been proved recently by the authors. In the present note we show that these solutions have modifications which are Hölder continuous in space of order smaller than H, and Hölder continuous in time of order smaller than γ, where γ=H for the wave equation and γ=H/2 for the heat equation.

KW - Fractional noise

KW - Hölder continuous paths

KW - Stochastic heat equation

KW - Stochastic wave equation

U2 - 10.1016/j.spl.2016.09.003

DO - 10.1016/j.spl.2016.09.003

M3 - Article

VL - 119

SP - 310

EP - 316

JO - Statistics and Probability Letters

JF - Statistics and Probability Letters

SN - 0167-7152

ER -