On some properties of a class of fractional stochastic heat equations

Wei Liu; Kuanhou Tian; Mohammud Foondun

doi:10.1007/s10959-016-0684-6

On some properties of a class of fractional stochastic heat equations

Wei Liu, Kuanhou Tian, Mohammud Foondun^*

^*Corresponding author for this work

Mathematics And Statistics

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20 Citations (Scopus)

34 Downloads (Pure)

Abstract

We consider nonlinear parabolic stochastic equations of the form ∂tu=Lu+λσ(u)ξ˙∂tu=Lu+λσ(u)ξ˙ on the ball B(0,R)B(0,R) , where ξ˙ξ˙ denotes some Gaussian noise and σσ is Lipschitz continuous. Here LL corresponds to a symmetric αα -stable process killed upon exiting B(0, R). We will consider two types of noises: space-time white noise and spatially correlated noise. Under a linear growth condition on σσ , we study growth properties of the second moment of the solutions. Our results are significant extensions of those in Foondun and Joseph (Stoch Process Appl, 2014) and complement those of Khoshnevisan and Kim (Proc AMS, 2013, Ann Probab, 2014).

Original language	English
Number of pages	24
Journal	Journal of Theoretical Probability
DOIs	https://doi.org/10.1007/s10959-016-0684-6
Publication status	Published - 19 May 2016

Keywords

fractional laplacian
heat kernel
stochastic heat equation
stochastic partial differential equations

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Liu-et-al-JTP-2016-On-some-properties-of-a-classFinal published version, 515 KBLicence: CC BY 4.0

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title = "On some properties of a class of fractional stochastic heat equations",

abstract = "We consider nonlinear parabolic stochastic equations of the form ∂tu=Lu+λσ(u)ξ˙∂tu=Lu+λσ(u)ξ˙ on the ball B(0,R)B(0,R) , where ξ˙ξ˙ denotes some Gaussian noise and σσ is Lipschitz continuous. Here LL corresponds to a symmetric αα -stable process killed upon exiting B(0, R). We will consider two types of noises: space-time white noise and spatially correlated noise. Under a linear growth condition on σσ , we study growth properties of the second moment of the solutions. Our results are significant extensions of those in Foondun and Joseph (Stoch Process Appl, 2014) and complement those of Khoshnevisan and Kim (Proc AMS, 2013, Ann Probab, 2014).",

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TY - JOUR

T1 - On some properties of a class of fractional stochastic heat equations

AU - Liu, Wei

AU - Tian, Kuanhou

AU - Foondun, Mohammud

PY - 2016/5/19

Y1 - 2016/5/19

N2 - We consider nonlinear parabolic stochastic equations of the form ∂tu=Lu+λσ(u)ξ˙∂tu=Lu+λσ(u)ξ˙ on the ball B(0,R)B(0,R) , where ξ˙ξ˙ denotes some Gaussian noise and σσ is Lipschitz continuous. Here LL corresponds to a symmetric αα -stable process killed upon exiting B(0, R). We will consider two types of noises: space-time white noise and spatially correlated noise. Under a linear growth condition on σσ , we study growth properties of the second moment of the solutions. Our results are significant extensions of those in Foondun and Joseph (Stoch Process Appl, 2014) and complement those of Khoshnevisan and Kim (Proc AMS, 2013, Ann Probab, 2014).

AB - We consider nonlinear parabolic stochastic equations of the form ∂tu=Lu+λσ(u)ξ˙∂tu=Lu+λσ(u)ξ˙ on the ball B(0,R)B(0,R) , where ξ˙ξ˙ denotes some Gaussian noise and σσ is Lipschitz continuous. Here LL corresponds to a symmetric αα -stable process killed upon exiting B(0, R). We will consider two types of noises: space-time white noise and spatially correlated noise. Under a linear growth condition on σσ , we study growth properties of the second moment of the solutions. Our results are significant extensions of those in Foondun and Joseph (Stoch Process Appl, 2014) and complement those of Khoshnevisan and Kim (Proc AMS, 2013, Ann Probab, 2014).

KW - fractional laplacian

KW - heat kernel

KW - stochastic heat equation

KW - stochastic partial differential equations

UR - https://www.scopus.com/pages/publications/84969776674

UR - http://link.springer.com/journal/10959

U2 - 10.1007/s10959-016-0684-6

DO - 10.1007/s10959-016-0684-6

M3 - Article

AN - SCOPUS:84969776674

SN - 0894-9840

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

ER -

On some properties of a class of fractional stochastic heat equations

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Keywords

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