Abstract
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.
| 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 | |
| Publication status | Published - 31 Aug 2012 |
| Externally published | Yes |
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Keywords
- intermittency
- stochastic heat equations
Cite this
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An asymptotic theory for randomly forced discrete nonlinear heat equations. / Foondun, Mohammud; Khoshnevisan, Davar.
In: Bernoulli, Vol. 18, No. 3, 31.08.2012, p. 1042-1060.Research output: Contribution to journal › Article
TY - JOUR
T1 - An asymptotic theory for randomly forced discrete nonlinear heat equations
AU - Foondun, Mohammud
AU - Khoshnevisan, Davar
PY - 2012/8/31
Y1 - 2012/8/31
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.
KW - intermittency
KW - stochastic heat equations
UR - http://www.scopus.com/inward/record.url?scp=84873403136&partnerID=8YFLogxK
UR - http://projecteuclid.org/current/euclid.bj
U2 - 10.3150/11-BEJ357
DO - 10.3150/11-BEJ357
M3 - Article
VL - 18
SP - 1042
EP - 1060
JO - Bernoulli Society for Mathematical Statistics and Probability
JF - Bernoulli Society for Mathematical Statistics and Probability
SN - 1350-7265
IS - 3
ER -