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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the stochastic heat equation with spatially-colored random forcing

Authors: Mohammud Foondun and Davar Khoshnevisan
Journal: Trans. Amer. Math. Soc. 365 (2013), 409-458
MSC (2010): Primary 60H15; Secondary 35R60
Published electronically: August 8, 2012
Erratum: Tran. Amer. Math. Soc. 366 (2014), 561-562
MathSciNet review: 2984063
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Abstract: We consider the stochastic heat equation of the following form:

$\displaystyle \frac {\partial }{\partial t}u_t(x) = (\mathcal {L} u_t)(x) +b(u_t(x)) + \sigma (u_t(x))\dot {F}_t(x)$$\displaystyle \quad \text {for }t>0,\ x\in \mathbf {R}^d,$    

where $ \mathcal {L}$ is the generator of a Lévy process and $ \dot {F}$ is a spatially-colored, temporally white, Gaussian noise. We will be concerned mainly with the long-term behavior of the mild solution to this stochastic PDE.

For the most part, we work under the assumptions that the initial data $ u_0$ is a bounded and measurable function and $ \sigma $ is nonconstant and Lipschitz continuous. In this case, we find conditions under which the preceding stochastic PDE admits a unique solution which is also weakly intermittent. In addition, we study the same equation in the case where $ \mathcal {L}u$ is replaced by its massive/dispersive analogue $ \mathcal {L}u-\lambda u$, where $ \lambda \in \mathbf {R}$. We also accurately describe the effect of the parameter $ \lambda $ on the intermittence of the solution in the case where $ \sigma (u)$ is proportional to $ u$ [the ``parabolic Anderson model''].

We also look at the linearized version of our stochastic PDE, that is, the case where $ \sigma $ is identically equal to one [any other constant also works]. In this case, we study not only the existence and uniqueness of a solution, but also the regularity of the solution when it exists and is unique.

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Additional Information

Mohammud Foondun
Affiliation: School of Mathematics, Loughborough University, Leicestershire, LE11 3TU United Kingdom

Davar Khoshnevisan
Affiliation: Department of Mathematics, University of Utah, 155 South 1400 East JWB 233, Salt Lake City, Utah 84112–0090

Keywords: The stochastic heat equation, spatially-colored homogeneous noise, Lévy processes.
Received by editor(s): April 18, 2011
Published electronically: August 8, 2012
Additional Notes: This research was supported in part by grants from the National Science Foundation.
Article copyright: © Copyright 2012 American Mathematical Society