Preprint:
On the chaotic character of the stochastic heat equation, II
Daniel Conus, Mathew Joseph, Davar Khoshnevisan, and Shang-Yuan 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 spatially-correlated 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 \(2\psi-1\) 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 DMS-0747758 (M.J.) and DMS-1006903 (D.K.).
Pre/E-Prints. This paper is available in
| Daniel Conus | Lehigh University, Department
of Mathematics, Christmas--Saucon 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 84112-0090, U.S.A.
(joseph [at sign] math [dot sign] utah [dot sign] edu &
davar[at sign] math [dot sign] utah [dot sign] edu) |
| Shang-Yuan 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 Shang-Yuan Shiu