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August 2012 An asymptotic theory for randomly forced discrete nonlinear heat equations
Mohammud Foondun, Davar Khoshnevisan
Bernoulli 18(3): 1042-1060 (August 2012). DOI: 10.3150/11-BEJ357


We study discrete nonlinear parabolic stochastic heat equations of the form, $u_{n+1}(x) - u_n(x) = (\mathcal{L} u_n)(x) + \sigma(u_n(x))\xi_n(x)$, for $n\in {\mathbf Z}_+$ and $x\in {\mathbf Z}^d$, where ${\boldsymbol\xi}:=\{\xi_n(x)\}_{n\ge 0,x\in{\mathbf Z}^d}$ denotes random forcing and $\mathcal{L}$ the generator of a random walk on ${\mathbf Z}^d$. Under mild conditions, we prove that the preceding stochastic PDE has a unique solution that grows at most exponentially in time. And that, under natural conditions, it is “weakly intermittent.” Along the way, we establish a comparison principle as well as a finite support property.


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Mohammud Foondun. Davar Khoshnevisan. "An asymptotic theory for randomly forced discrete nonlinear heat equations." Bernoulli 18 (3) 1042 - 1060, August 2012.


Published: August 2012
First available in Project Euclid: 28 June 2012

zbMATH: 1260.60119
MathSciNet: MR2948912
Digital Object Identifier: 10.3150/11-BEJ357

Rights: Copyright © 2012 Bernoulli Society for Mathematical Statistics and Probability


Vol.18 • No. 3 • August 2012
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