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
AN - SCOPUS:84873403136
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 -