### Abstract

Consider the KPZ equation u?(t, x) = ?u(t, x) + \?u(t, x)\2 + W(t, x), x ? R^{d}, where W(t, x) is a space-time white noise. This paper investigates the question of whether, for some exponents ? and z, k^{-?}u(k^{Z}t, 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(k^{z}t, kx) as k ? 8 is established for various values of ? and Z depending on the dimension d.

Original language | English |
---|---|

Pages (from-to) | 671-690 |

Number of pages | 20 |

Journal | Communications in Mathematical Physics |

Volume | 209 |

Issue number | 3 |

Publication status | Published - 2000 |

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### Cite this

*Communications in Mathematical Physics*,

*209*(3), 671-690.

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*Communications in Mathematical Physics*, vol. 209, no. 3, pp. 671-690.

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

Research output: Contribution to journal › Article

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 -