Analysis of the gradient of the solution to a stochastic heat equation via fractional Brownian motion

Mohammud Foondun, Davar Khoshnevisan, Pejman Mahboubi

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

Consider the stochastic partial differential equation \(\partial _{t} u = Lu+\sigma (u)\xi \), where \(\xi \) denotes space–time white noise and \(L:=-(-\Delta )^{\alpha /2}\) denotes the fractional Laplace operator of index \(\alpha /2\in (1/2\,,1]\). We study the detailed behavior of the approximate spatial gradient \(u_{t}(x)-u_{t}(x-\varepsilon )\) at fixed times \(t>0\), as \(\varepsilon \downarrow 0\). We discuss a few applications of this work to the study of the sample functions of the solution to the KPZ equation as well.
LanguageEnglish
Pages133–158
Number of pages26
JournalStochastic and Partial Differential Equations: Analysis and Computations
Volume3
Issue number2
DOIs
Publication statusPublished - 1 Jun 2015

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Stochastic Heat Equation
Fractional Brownian Motion
KPZ Equation
Space-time White Noise
Gradient
Denote
Stochastic Partial Differential Equations
Laplace Operator
Fractional
Partial

Keywords

  • stochastic heat equation
  • fractional Brownian motion
  • gradient estimates
  • KPZ equation

Cite this

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Analysis of the gradient of the solution to a stochastic heat equation via fractional Brownian motion. / Foondun, Mohammud; Khoshnevisan, Davar; Mahboubi, Pejman.

In: Stochastic and Partial Differential Equations: Analysis and Computations, Vol. 3, No. 2, 01.06.2015, p. 133–158.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Analysis of the gradient of the solution to a stochastic heat equation via fractional Brownian motion

AU - Foondun, Mohammud

AU - Khoshnevisan, Davar

AU - Mahboubi, Pejman

PY - 2015/6/1

Y1 - 2015/6/1

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AB - Consider the stochastic partial differential equation \(\partial _{t} u = Lu+\sigma (u)\xi \), where \(\xi \) denotes space–time white noise and \(L:=-(-\Delta )^{\alpha /2}\) denotes the fractional Laplace operator of index \(\alpha /2\in (1/2\,,1]\). We study the detailed behavior of the approximate spatial gradient \(u_{t}(x)-u_{t}(x-\varepsilon )\) at fixed times \(t>0\), as \(\varepsilon \downarrow 0\). We discuss a few applications of this work to the study of the sample functions of the solution to the KPZ equation as well.

KW - stochastic heat equation

KW - fractional Brownian motion

KW - gradient estimates

KW - KPZ equation

U2 - 10.1007/s40072-015-0045-y

DO - 10.1007/s40072-015-0045-y

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SP - 133

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JO - Stochastic and Partial Differential Equations: Analysis and Computations

T2 - Stochastic and Partial Differential Equations: Analysis and Computations

JF - Stochastic and Partial Differential Equations: Analysis and Computations

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