# 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>) 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) 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