### Abstract

The Kardar–Parisi–Zhang (KPZ) equation is a stochastic partial differential equation which is ill-posed because of the inconsistency between the nonlinearity and the roughness of the forcing noise. However, its Cole–Hopf solution, defined as the logarithm of the solution of the linear stochastic heat equation (SHE) with a multiplicative noise, is a mathematically well-defined object. In fact, Hairer (Ann Math 178:559–694, 2013) has recently proved that the solution of SHE can actually be derived through the Cole–Hopf transform of the solution of the KPZ equation with a suitable renormalization under periodic boundary conditions. This transformation is unfortunately not well adapted to studying the invariant measures of these Markov processes. The present paper introduces a different type of regularization for the KPZ equation on the whole line ℝ or under periodic boundary conditions, which is appropriate from the viewpoint of studying the invariant measures. The Cole–Hopf transform applied to this equation leads to an SHE with a smeared noise having an extra complicated nonlinear term. Under time average and in the stationary regime, it is shown that this term can be replaced by a simple linear term, so that the limit equation is the linear SHE with an extra linear term with coefficient
_{24}
^{1}
. The methods are essentially stochastic analytic: The Wiener–Itô expansion and a similar method for establishing the Boltzmann–Gibbs principle are used. As a result, it is shown that the distribution of a two-sided geometric Brownian motion with a height shift given by Lebesgue measure is invariant under the evolution determined by the SHE on ℝ.

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

Pages (from-to) | 159-220 |

Number of pages | 62 |

Journal | Stochastics and Partial Differential Equations: Analysis and Computations |

Volume | 3 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2013 Jan 1 |

Externally published | Yes |

### Fingerprint

### Keywords

- Cole
- Hopf transform
- Invariant measure
- KPZ equation
- Stochastic partial differential equation

### ASJC Scopus subject areas

- Statistics and Probability
- Modelling and Simulation
- Applied Mathematics

### Cite this

**Kpz equation, its renormalization and invariant measures.** / Funaki, Tadahisa; Quastel, Jeremy.

Research output: Contribution to journal › Article

*Stochastics and Partial Differential Equations: Analysis and Computations*, vol. 3, no. 2, pp. 159-220. https://doi.org/10.1007/s40072-015-0046-x

}

TY - JOUR

T1 - Kpz equation, its renormalization and invariant measures

AU - Funaki, Tadahisa

AU - Quastel, Jeremy

PY - 2013/1/1

Y1 - 2013/1/1

N2 - The Kardar–Parisi–Zhang (KPZ) equation is a stochastic partial differential equation which is ill-posed because of the inconsistency between the nonlinearity and the roughness of the forcing noise. However, its Cole–Hopf solution, defined as the logarithm of the solution of the linear stochastic heat equation (SHE) with a multiplicative noise, is a mathematically well-defined object. In fact, Hairer (Ann Math 178:559–694, 2013) has recently proved that the solution of SHE can actually be derived through the Cole–Hopf transform of the solution of the KPZ equation with a suitable renormalization under periodic boundary conditions. This transformation is unfortunately not well adapted to studying the invariant measures of these Markov processes. The present paper introduces a different type of regularization for the KPZ equation on the whole line ℝ or under periodic boundary conditions, which is appropriate from the viewpoint of studying the invariant measures. The Cole–Hopf transform applied to this equation leads to an SHE with a smeared noise having an extra complicated nonlinear term. Under time average and in the stationary regime, it is shown that this term can be replaced by a simple linear term, so that the limit equation is the linear SHE with an extra linear term with coefficient 24 1 . The methods are essentially stochastic analytic: The Wiener–Itô expansion and a similar method for establishing the Boltzmann–Gibbs principle are used. As a result, it is shown that the distribution of a two-sided geometric Brownian motion with a height shift given by Lebesgue measure is invariant under the evolution determined by the SHE on ℝ.

AB - The Kardar–Parisi–Zhang (KPZ) equation is a stochastic partial differential equation which is ill-posed because of the inconsistency between the nonlinearity and the roughness of the forcing noise. However, its Cole–Hopf solution, defined as the logarithm of the solution of the linear stochastic heat equation (SHE) with a multiplicative noise, is a mathematically well-defined object. In fact, Hairer (Ann Math 178:559–694, 2013) has recently proved that the solution of SHE can actually be derived through the Cole–Hopf transform of the solution of the KPZ equation with a suitable renormalization under periodic boundary conditions. This transformation is unfortunately not well adapted to studying the invariant measures of these Markov processes. The present paper introduces a different type of regularization for the KPZ equation on the whole line ℝ or under periodic boundary conditions, which is appropriate from the viewpoint of studying the invariant measures. The Cole–Hopf transform applied to this equation leads to an SHE with a smeared noise having an extra complicated nonlinear term. Under time average and in the stationary regime, it is shown that this term can be replaced by a simple linear term, so that the limit equation is the linear SHE with an extra linear term with coefficient 24 1 . The methods are essentially stochastic analytic: The Wiener–Itô expansion and a similar method for establishing the Boltzmann–Gibbs principle are used. As a result, it is shown that the distribution of a two-sided geometric Brownian motion with a height shift given by Lebesgue measure is invariant under the evolution determined by the SHE on ℝ.

KW - Cole

KW - Hopf transform

KW - Invariant measure

KW - KPZ equation

KW - Stochastic partial differential equation

UR - http://www.scopus.com/inward/record.url?scp=84941209125&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84941209125&partnerID=8YFLogxK

U2 - 10.1007/s40072-015-0046-x

DO - 10.1007/s40072-015-0046-x

M3 - Article

AN - SCOPUS:84941209125

VL - 3

SP - 159

EP - 220

JO - Stochastics and Partial Differential Equations: Analysis and Computations

JF - Stochastics and Partial Differential Equations: Analysis and Computations

SN - 2194-0401

IS - 2

ER -