An asymptotic theory for randomly forced discrete nonlinear heat equations

Mohammud Foondun; Davar Khoshnevisan

doi:10.3150/11-BEJ357

An asymptotic theory for randomly forced discrete nonlinear heat equations

Mohammud Foondun, Davar Khoshnevisan

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

We study discrete nonlinear parabolic stochastic heat equations of the form, u_n+1(x)-u_n(x) = (Lu_n)(x)+ σ(u _n(x))ζ_n(x), for n ∈ Z₊ and x ∈ Z^d, where ζ := {ζ_n(x)} _n≥0,x∈Zd denotes random forcing and L the generator of a random walk on 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.

Original language	English
Pages (from-to)	1042-1060
Number of pages	19
Journal	Bernoulli
Volume	18
Issue number	3
Early online date	28 Jun 2012
DOIs	https://doi.org/10.3150/11-BEJ357
Publication status	Published - 31 Aug 2012
Externally published	Yes

Keywords

intermittency
stochastic heat equations

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N2 - We study discrete nonlinear parabolic stochastic heat equations of the form, un+1(x)-un(x) = (Lun)(x)+ σ(u n(x))ζn(x), for n ∈ Z+ and x ∈ Zd, where ζ := {ζn(x)} n≥0,x∈Zd denotes random forcing and L the generator of a random walk on Zd. 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.

AB - We study discrete nonlinear parabolic stochastic heat equations of the form, un+1(x)-un(x) = (Lun)(x)+ σ(u n(x))ζn(x), for n ∈ Z+ and x ∈ Zd, where ζ := {ζn(x)} n≥0,x∈Zd denotes random forcing and L the generator of a random walk on Zd. 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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An asymptotic theory for randomly forced discrete nonlinear heat equations

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