Preprint:
On the chaotic character of the stochastic heat equation, II
Daniel Conus, Mathew Joseph, Davar Khoshnevisan, and ShangYuan Shiu
Abstract.
Consider the stochastic heat equation
\(\partial_t u = (\kappa/2)\Delta
u+\sigma(u)\dot{F}\), where the solution \(u:=u_t(x)\) is indexed by
\((t\,,x)\in (0\,,\infty)\times{\bf R}^d\),
and \(\dot{F}\) is a centered Gaussian noise that is white in time
and has spatiallycorrelated coordinates. We analyze the
large\(\x\\) fixed\(t\) behavior of the solution \(u\) in different
regimes, thereby study the effect of noise on the solution in various
cases. Among other things, we show that if the spatial correlation
function \(f\) of the noise is of Riesz type, that is \(f(x)\propto \x\^{\alpha}\),
then the "fluctuation exponents"
of the solution are \(\psi\) for the spatial variable and \(2\psi1\) for the time
variable, where \(\psi:=2/(4\alpha)\). Moreover, these exponent relations
hold as long as \(\alpha\in(0\,,d\wedge 2)\); that is precisely when
Dalang's theory (1999) implies the existence
of a solution to our stochastic PDE. These findings bolster
earlier physical predictions [Kardar, Parisi, and Zhang, 1985; Kardar and Zhang, 1987].
Keywords.
The stochastic heat equation, chaos,
intermittency, the parabolic Anderson model, the KPZ equation, critical exponents
AMS Classification (2000) Primary: 60H15; Secondary: 35R60.
Support. Research supported in part by the
NSF grants DMS0747758 (M.J.) and DMS1006903 (D.K.).
Pre/EPrints. This paper is available in
Daniel Conus  Lehigh University, Department
of Mathematics, ChristmasSaucon Hall, 14 East Packer Avenue,
Bethlehem, PA 18015 (daniel [dot sign] conus
[at sign] lehigh [dot sign] edu>)

Mathew Joseph & Davar Khoshnevisan  Department of Mathematics
University of Utah, 155 S, 1400 E JWB 233, Salt Lake City, UT 841120090, U.S.A.
(joseph [at sign] math [dot sign] utah [dot sign] edu &
davar[at sign] math [dot sign] utah [dot sign] edu) 
ShangYuan Shiu  Institute of Mathematics,
Academia Sinica, Taipei 10617 (shiu[dot sign] math
[at sign] sinica [dot sign] edu [dot sign] tw>)

Last Update: November 21, 2011
© 2011  Daniel Conus, Mathew Joseph, Davar Khoshnevisan, and ShangYuan Shiu