Bernoulli

  • Bernoulli
  • Volume 18, Number 3 (2012), 1042-1060.

An asymptotic theory for randomly forced discrete nonlinear heat equations

Mohammud Foondun and Davar Khoshnevisan

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Abstract

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.

Article information

Source
Bernoulli, Volume 18, Number 3 (2012), 1042-1060.

Dates
First available in Project Euclid: 28 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.bj/1340887013

Digital Object Identifier
doi:10.3150/11-BEJ357

Mathematical Reviews number (MathSciNet)
MR2948912

Zentralblatt MATH identifier
1260.60119

Keywords
intermittency stochastic heat equations

Citation

Foondun, Mohammud; Khoshnevisan, Davar. An asymptotic theory for randomly forced discrete nonlinear heat equations. Bernoulli 18 (2012), no. 3, 1042--1060. doi:10.3150/11-BEJ357. https://projecteuclid.org/euclid.bj/1340887013


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