Moment bounds for a class of fractional stochastic heat equations

Mohammud Foondun, Wei Liu, McSylvester Omaba

Research output: Contribution to journalArticle

Abstract

We consider fractional stochastic heat equations of the form $\frac{\partial u_t(x)}{\partial t} = -(-\Delta)^{\alpha/2} u_t(x)+\lambda \sigma (u_t(x)) \dot F(t,\, x)$. Here $\dot F$ denotes the noise term. Under suitable assumptions, we show that the second moment of the solution grows exponentially with time. In particular, this answers an open problem in \cite{CoKh}. Along the way, we prove a number of other interesting properties which extend and complement results in \cite{foonjose}, \cite{Khoshnevisan:2013aa} and \cite{Khoshnevisan:2013ab}.
LanguageEnglish
Pages1-24
Number of pages24
JournalAnnals of Probability
Publication statusAccepted/In press - 19 Sep 2014

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Stochastic Heat Equation
Open Problems
Fractional
Complement
Denote
Moment
Partial
Term
Form
Class

Keywords

  • intermittence
  • stochastic partial differential equations

Cite this

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title = "Moment bounds for a class of fractional stochastic heat equations",
abstract = "We consider fractional stochastic heat equations of the form $\frac{\partial u_t(x)}{\partial t} = -(-\Delta)^{\alpha/2} u_t(x)+\lambda \sigma (u_t(x)) \dot F(t,\, x)$. Here $\dot F$ denotes the noise term. Under suitable assumptions, we show that the second moment of the solution grows exponentially with time. In particular, this answers an open problem in \cite{CoKh}. Along the way, we prove a number of other interesting properties which extend and complement results in \cite{foonjose}, \cite{Khoshnevisan:2013aa} and \cite{Khoshnevisan:2013ab}.",
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language = "English",
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Moment bounds for a class of fractional stochastic heat equations. / Foondun, Mohammud; Liu, Wei; Omaba, McSylvester.

In: Annals of Probability, 19.09.2014, p. 1-24.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Moment bounds for a class of fractional stochastic heat equations

AU - Foondun, Mohammud

AU - Liu, Wei

AU - Omaba, McSylvester

PY - 2014/9/19

Y1 - 2014/9/19

N2 - We consider fractional stochastic heat equations of the form $\frac{\partial u_t(x)}{\partial t} = -(-\Delta)^{\alpha/2} u_t(x)+\lambda \sigma (u_t(x)) \dot F(t,\, x)$. Here $\dot F$ denotes the noise term. Under suitable assumptions, we show that the second moment of the solution grows exponentially with time. In particular, this answers an open problem in \cite{CoKh}. Along the way, we prove a number of other interesting properties which extend and complement results in \cite{foonjose}, \cite{Khoshnevisan:2013aa} and \cite{Khoshnevisan:2013ab}.

AB - We consider fractional stochastic heat equations of the form $\frac{\partial u_t(x)}{\partial t} = -(-\Delta)^{\alpha/2} u_t(x)+\lambda \sigma (u_t(x)) \dot F(t,\, x)$. Here $\dot F$ denotes the noise term. Under suitable assumptions, we show that the second moment of the solution grows exponentially with time. In particular, this answers an open problem in \cite{CoKh}. Along the way, we prove a number of other interesting properties which extend and complement results in \cite{foonjose}, \cite{Khoshnevisan:2013aa} and \cite{Khoshnevisan:2013ab}.

KW - intermittence

KW - stochastic partial differential equations

M3 - Article

SP - 1

EP - 24

JO - Annals of Probability

T2 - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

ER -