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July 2017 Moment bounds for a class of fractional stochastic heat equations
Mohammud Foondun, Wei Liu, McSylvester Omaba
Ann. Probab. 45(4): 2131-2153 (July 2017). DOI: 10.1214/16-AOP1108

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. 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)].

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Mohammud Foondun. Wei Liu. McSylvester Omaba. "Moment bounds for a class of fractional stochastic heat equations." Ann. Probab. 45 (4) 2131 - 2153, July 2017. https://doi.org/10.1214/16-AOP1108

Information

Received: 1 October 2014; Revised: 1 February 2016; Published: July 2017
First available in Project Euclid: 11 August 2017

zbMATH: 1378.60089
MathSciNet: MR3693959
Digital Object Identifier: 10.1214/16-AOP1108

Subjects:
Primary: 60H15
Secondary: 82B44

Keywords: intermittence , Stochastic partial differential equations

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 4 • July 2017
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