Scaling limits of Wick ordered KPZ equation

Terence Chan

Research output: Contribution to journalArticle

Abstract

Consider the KPZ equation u?(t, x) = ?u(t, x) + \?u(t, x)\2 + W(t, x), x ? Rd, where W(t, x) is a space-time white noise. This paper investigates the question of whether, for some exponents ? and z, k-?u(kZt, kx) converges in some sense as k ? 8, and if so, what are the values of these exponents. The non-linear term in the KPZ equation is interpreted as a Wick product and the equation is solved in a suitable space of stochastic distributions. The main tools for establishing the scaling properties of the solution are those of white noise analysis, in particular, the Wiener chaos expansion. A notion of convergence in law in the sense of Wiener chaos is formulated and convergence in this sense of k-?u(kzt, kx) as k ? 8 is established for various values of ? and Z depending on the dimension d.

Original languageEnglish
Pages (from-to)671-690
Number of pages20
JournalCommunications in Mathematical Physics
Volume209
Issue number3
Publication statusPublished - 2000

Fingerprint

KPZ Equation
Scaling Limit
Exponent
Convergence in Law
Wick Product
White Noise Analysis
Space-time White Noise
Chaos Expansion
Chaos
Scaling
Converge
Term

Cite this

Chan, Terence. / Scaling limits of Wick ordered KPZ equation. In: Communications in Mathematical Physics. 2000 ; Vol. 209, No. 3. pp. 671-690.
@article{1f29ef427cec4c6e849168f924014a91,
title = "Scaling limits of Wick ordered KPZ equation",
abstract = "Consider the KPZ equation u?(t, x) = ?u(t, x) + \?u(t, x)\2 + W(t, x), x ? Rd, where W(t, x) is a space-time white noise. This paper investigates the question of whether, for some exponents ? and z, k-?u(kZt, kx) converges in some sense as k ? 8, and if so, what are the values of these exponents. The non-linear term in the KPZ equation is interpreted as a Wick product and the equation is solved in a suitable space of stochastic distributions. The main tools for establishing the scaling properties of the solution are those of white noise analysis, in particular, the Wiener chaos expansion. A notion of convergence in law in the sense of Wiener chaos is formulated and convergence in this sense of k-?u(kzt, kx) as k ? 8 is established for various values of ? and Z depending on the dimension d.",
author = "Terence Chan",
year = "2000",
language = "English",
volume = "209",
pages = "671--690",
journal = "Communications in Mathematical Physics",
issn = "0010-3616",
publisher = "Springer",
number = "3",

}

Scaling limits of Wick ordered KPZ equation. / Chan, Terence.

In: Communications in Mathematical Physics, Vol. 209, No. 3, 2000, p. 671-690.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Scaling limits of Wick ordered KPZ equation

AU - Chan, Terence

PY - 2000

Y1 - 2000

N2 - Consider the KPZ equation u?(t, x) = ?u(t, x) + \?u(t, x)\2 + W(t, x), x ? Rd, where W(t, x) is a space-time white noise. This paper investigates the question of whether, for some exponents ? and z, k-?u(kZt, kx) converges in some sense as k ? 8, and if so, what are the values of these exponents. The non-linear term in the KPZ equation is interpreted as a Wick product and the equation is solved in a suitable space of stochastic distributions. The main tools for establishing the scaling properties of the solution are those of white noise analysis, in particular, the Wiener chaos expansion. A notion of convergence in law in the sense of Wiener chaos is formulated and convergence in this sense of k-?u(kzt, kx) as k ? 8 is established for various values of ? and Z depending on the dimension d.

AB - Consider the KPZ equation u?(t, x) = ?u(t, x) + \?u(t, x)\2 + W(t, x), x ? Rd, where W(t, x) is a space-time white noise. This paper investigates the question of whether, for some exponents ? and z, k-?u(kZt, kx) converges in some sense as k ? 8, and if so, what are the values of these exponents. The non-linear term in the KPZ equation is interpreted as a Wick product and the equation is solved in a suitable space of stochastic distributions. The main tools for establishing the scaling properties of the solution are those of white noise analysis, in particular, the Wiener chaos expansion. A notion of convergence in law in the sense of Wiener chaos is formulated and convergence in this sense of k-?u(kzt, kx) as k ? 8 is established for various values of ? and Z depending on the dimension d.

UR - http://www.scopus.com/inward/record.url?scp=0034348846&partnerID=8YFLogxK

M3 - Article

VL - 209

SP - 671

EP - 690

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -