Abstract
We study the nonlinear fractional stochastic heat equation in the spatial domain \({\mathbb {R}}\) driven by space-time white noise. The initial condition is taken to be a measure on \({\mathbb {R}}\), such as the Dirac delta function, but this measure may also have non-compact support. Existence and uniqueness, as well as upper and lower bounds on all pth moments \((p\ge 2)\), are obtained. These bounds are uniform in the spatial variable, which answers an open problem mentioned in Conus and Khoshnevisan (Probab Theory Relat Fields 152(3–4):681–701, 2012). We improve the weak intermittency statement by Foondun and Khoshnevisan (Electron J Probab 14(21):548–568, 2009), and we show that the growth indices (of linear type) introduced in Conus and Khoshnevisan (Probab Theory Relat Fields 152(3–4):681–701, 2012) are infinite. We introduce the notion of “growth indices of exponential type” in order to characterize the manner in which high peaks propagate away from the origin, and we show that the presence of a fractional differential operator leads to significantly different behavior compared with the standard stochastic heat equation.
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Acknowledgments
The authors thank Davar Khoshnevisan for interesting discussions on this problem. A preliminary version of this work was presented at a conference at Michigan State University, East Lansing, USA, August 2013. The authors also thank two anonymous referees for a careful reading of this paper and many useful suggestions, which led to significant improvements.
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Le Chen and Robert C. Dalang were supported in part by the Swiss National Foundation for Scientific Research.
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Chen, L., Dalang, R.C. Moments, intermittency and growth indices for the nonlinear fractional stochastic heat equation. Stoch PDE: Anal Comp 3, 360–397 (2015). https://doi.org/10.1007/s40072-015-0054-x
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DOI: https://doi.org/10.1007/s40072-015-0054-x
Keywords
- Nonlinear fractional stochastic heat equation
- Parabolic Anderson model
- Rough initial data
- Intermittency
- Growth indices
- Stable processes