Moment bounds for a class of fractional stochastic heat equations

• Mohammud Foondun ^[1] ; Wei Liu ^[2] ; McSylvester Omaba ^[3]
  1. [1] University of Strathclyde

     University of Strathclyde

     Reino Unido

     
  2. [2] Shanghai Normal University

     Shanghai Normal University

     China

     
  3. [3] Loughborough University

     Loughborough University

     Charnwood District, Reino Unido

     

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 4, 2017, págs. 2131-2153
• Idioma: inglés
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• Resumen

  • We consider fractional stochastic heat equations of the form ∂ut(x)∂t=−(−Δ)α/2ut(x)+λσ(ut(x))F˙(t,x)∂ut(x)∂t=−(−Δ)α/2ut(x)+λσ(ut(x))F˙(t,x). Here, F˙F˙ denotes the noise term. Under suitable assumptions, we show that the second moment of the solution grows exponentially with time. Since we do not assume that the initial condition is bounded below, this solves an open problem stated in [Probab. Theory Related Fields 152 (2012) 681–701]. Along the way, we prove a number of other interesting results about continuity properties and noise excitation indices. These extend and complement results in [Stochastic Process. Appl. 124 (2014) 3429–3440], [Khoshnevisan and Kim (2013)] and [Khoshnevisan and Kim (2014)].