TY - JOUR

T1 - On the chaotic character of the stochastic heat equation, II

AU - Conus, Daniel

AU - Joseph, Mathew

AU - Khoshnevisan, Davar

AU - Shiu, Shang Yuan

N1 - Funding Information:
This research was supported in part by the NSFs grant DMS-0747758 (M.J.) and DMS-1006903 (D.K.).

PY - 2013/8

Y1 - 2013/8

N2 - Consider the stochastic heat equation ∂t u = (x/2)Δ u+σ(u)Ḟ, where the solution u:= u t(x) is indexed by (t, x) ∈ (0, ∞) × ℝd, and Ḟ is a centered Gaussian noise that is white in time and has spatially-correlated coordinates. We analyze the large-{double pipe}x{double pipe} 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)∝ {double pipe}x{double pipe}-α, then the "fluctuation exponents" of the solution are ψ for the spatial variable and 2ψ-1 for the time variable, where ψ:=2/(4-α). Moreover, these exponent relations hold as long as α ∈ (0, d {n-ary logical and} 2); that is precisely when Dalang's theory [Dalang, Electron J Probab 4:(Paper no. 6):29, 1999] implies the existence of a solution to our stochastic PDE. These findings bolster earlier physical predictions [Kardar et al., Phys Rev Lett 58(20):889-892, 1985; Kardar and Zhang, Phys Rev Lett 58(20):2087-2090, 1987].

AB - Consider the stochastic heat equation ∂t u = (x/2)Δ u+σ(u)Ḟ, where the solution u:= u t(x) is indexed by (t, x) ∈ (0, ∞) × ℝd, and Ḟ is a centered Gaussian noise that is white in time and has spatially-correlated coordinates. We analyze the large-{double pipe}x{double pipe} 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)∝ {double pipe}x{double pipe}-α, then the "fluctuation exponents" of the solution are ψ for the spatial variable and 2ψ-1 for the time variable, where ψ:=2/(4-α). Moreover, these exponent relations hold as long as α ∈ (0, d {n-ary logical and} 2); that is precisely when Dalang's theory [Dalang, Electron J Probab 4:(Paper no. 6):29, 1999] implies the existence of a solution to our stochastic PDE. These findings bolster earlier physical predictions [Kardar et al., Phys Rev Lett 58(20):889-892, 1985; Kardar and Zhang, Phys Rev Lett 58(20):2087-2090, 1987].

KW - Chaos

KW - Critical exponents

KW - Intermittency

KW - The KPZ equation

KW - The parabolic Anderson model

KW - The stochastic heat equation

UR - http://www.scopus.com/inward/record.url?scp=84880641638&partnerID=8YFLogxK

U2 - 10.1007/s00440-012-0434-3

DO - 10.1007/s00440-012-0434-3

M3 - 期刊論文

AN - SCOPUS:84880641638

VL - 156

SP - 483

EP - 533

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 3-4

ER -