Asymptotic properties of some space-time fractional stochastic equations

Mohammud Foondun, Erkan Nane

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

Consider non-linear time-fractional stochastic heat type equations of the following type,
∂βtut(x)=−ν(−Δ)α/2ut(x)+I1−βt[λσ(u)F⋅(t,x)]
in (d+1) dimensions, where v > 0, β ε (0,1), α ε (0,2]. The operator ∂βt is the Caputo fractional derivative while -(-Δ)α/2 is the generator of an isotropic stable process and It1-β is the fractional integral operator. The forcing noise denoted by F(t,x) is a Gaussian noise. And the multiplicative non-linearity σ:R→R is assumed to be globally Lipschitz continuous. Mijena and Nane (Stochastic Process Appl 125(9):3301–3326, 2015) have introduced these time fractional SPDEs. These types of time fractional stochastic heat type equations can be used to model phenomenon with random effects with thermal memory. Under suitable conditions on the initial function, we study the asymptotic behaviour of the solution with respect to time and the parameter λ. In particular, our results are significant extensions of those in Ann Probab (to appear), Foondun and Khoshnevisan (Electron J Probab 14(21): 548–568, 2009), Mijena and Nane (2015) and Mijena and Nane (Potential Anal 44:295–312, 2016). Along the way, we prove a number of interesting properties about the deterministic counterpart of the equation.
LanguageEnglish
Number of pages27
JournalMathematische Zeitschrift
Early online date3 Jan 2017
DOIs
Publication statusE-pub ahead of print - 3 Jan 2017

Fingerprint

Asymptotic Properties
Stochastic Equations
Fractional
Space-time
Heat
Fractional Integral Operator
Caputo Fractional Derivative
Stable Process
Gaussian Noise
Random Effects
Forcing
Lipschitz
Stochastic Processes
Multiplicative
Asymptotic Behavior
Nonlinearity
Generator
Electron
Operator
Model

Keywords

  • space-time-fractional stochastic partial differential equations
  • fractional Duhamel’s principle
  • Caputo derivatives
  • noise excitability

Cite this

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Asymptotic properties of some space-time fractional stochastic equations. / Foondun, Mohammud; Nane, Erkan.

In: Mathematische Zeitschrift, 03.01.2017.

Research output: Contribution to journalArticle

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AB - Consider non-linear time-fractional stochastic heat type equations of the following type,∂βtut(x)=−ν(−Δ)α/2ut(x)+I1−βt[λσ(u)F⋅(t,x)]in (d+1) dimensions, where v > 0, β ε (0,1), α ε (0,2]. The operator ∂βt is the Caputo fractional derivative while -(-Δ)α/2 is the generator of an isotropic stable process and It1-β is the fractional integral operator. The forcing noise denoted by F(t,x) is a Gaussian noise. And the multiplicative non-linearity σ:R→R is assumed to be globally Lipschitz continuous. Mijena and Nane (Stochastic Process Appl 125(9):3301–3326, 2015) have introduced these time fractional SPDEs. These types of time fractional stochastic heat type equations can be used to model phenomenon with random effects with thermal memory. Under suitable conditions on the initial function, we study the asymptotic behaviour of the solution with respect to time and the parameter λ. In particular, our results are significant extensions of those in Ann Probab (to appear), Foondun and Khoshnevisan (Electron J Probab 14(21): 548–568, 2009), Mijena and Nane (2015) and Mijena and Nane (Potential Anal 44:295–312, 2016). Along the way, we prove a number of interesting properties about the deterministic counterpart of the equation.

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KW - noise excitability

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