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 ConusLehigh 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 ShiuInstitute 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