## 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 \({\mathbb {R}}\) 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 \(\tfrac{1}{24}\). 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 \({\mathbb {R}}\).

This is a preview of subscription content, access via your institution.

## References

- 1.
Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)

- 2.
Agoritsas, E., Lecomte, V., Giamarchi, T.: Static fluctuations of a thick one-dimensional interface in the 1+1 directed polymer formulation. Phys. Rev. E.

**87**, 042406 (2013) - 3.
Agoritsas, E., Lecomte, V., Giamarchi, T.: Static fluctuations of a thick one-dimensional interface in the 1+1 directed polymer formulation: numerical study. Phys. Rev. E.

**87**, 062405 (2013) - 4.
Bertini, L., Cancrini, N.: The stochastic heat equation: Feynman–Kac formula and intermittence. J. Stat. Phys.

**78**, 1377–1401 (1997) - 5.
Bertini, L., Giacomin, G.: Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys.

**183**, 571–607 (1995) - 6.
Borodin, A., Corwin, I., Ferrari, P.: Free energy fluctuations for directed polymers in random media in 1 + 1 dimension. Commun. Pure Appl. Math.

**67**, 1129–1214 (2014) - 7.
Corwin, I.: The Kardar–Parisi–Zhang equation and universality class. Random Matrices

**1**, 1130001 (2012) - 8.
Da Prato, G., Debussche, A., Tubaro, L.: A modified Kardar–Parisi–Zhang model. Electron. Commun. Probab.

**12**, 442–453 (2007) - 9.
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge University Press, Cambridge (1992). xviii+454 pp

- 10.
Echeverria, P.: A criterion for invariant measures of Markov processes. Z. Wahrsch. Verw. Gebiete

**61**, 1–16 (1982) - 11.
Funaki, T.: Regularity properties for stochastic partial differential equations of parabolic type. Osaka J. Math.

**28**, 495–516 (1991) - 12.
Funaki, T.: A stochastic partial differential equation with values in a manifold. J. Funct. Anal.

**109**, 257–288 (1992) - 13.
Funaki, T.: The scaling limit for a stochastic PDE and the separation of phases. Probab. Theory Relat. Fields

**102**, 221–288 (1995) - 14.
Funaki, T.: Infinitesimal invariance for the coupled KPZ equations, to appear in Memoriam Marc Yor—Séminaire de Probabilités XLVII, Lect. Notes Math. vol. 2137, Springer, Berlin (2015)

- 15.
Funaki, T., Spohn, H.: Motion by mean curvature from the Ginzburg–Landau \(\nabla \phi \) interface model. Commun. Math. Phys.

**185**, 1–36 (1997) - 16.
Hairer, M.: Solving the KPZ equation. Ann. Math.

**178**, 559–664 (2013) - 17.
Hohenberg, P.C., Halperin, B.I.: Theory of dynamic critical phenomena. Rev. Mod. Phys.

**49**, 435–475 (1977) - 18.
Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York (1991)

- 19.
Kipnis, C., Landim, C.: Scaling Limits of Interacting Particle Systems. Springer, New York (1999)

- 20.
Komorowski, T., Landim, C., Olla, S.: Fluctuations in Markov Processes: Time Symmetry and Martingale Approximation. Springer, Heidelberg (2012)

- 21.
Krylov, N.V., Rozovskii, B.L.: Stochastic evolution equations, J. Soviet Math.,

**16**, 1233–1277 (1981), translated from Current Problems in Math.,**14**(Russian), 71–147 (1979) - 22.
Major, P.: Multiple Wiener-Itô Integrals, with Applications to Limit Theorems. Lecture Notes in Mathematics, vol. 849. Springer, Berlin (1981)

- 23.
Mueller, C.: On the support of solutions to the heat equation with noise. Int. J. Probab. Stoch. Process.

**37**, 225–245 (1991) - 24.
Sasamoto, T., Spohn, H.: Superdiffusivity of the 1D lattice Kardar–Parisi–Zhang equation. J. Stat. Phys.

**137**, 917–935 (2009) - 25.
Shiga, T.: Two contrasting properties of solutions for one-dimensional stochastic partial differential equations. Can. J. Math.

**46**, 415–437 (1994)

## Acknowledgments

The authors thank Makiko Sasada for pointing out a simple proof of (3.32). T. Funaki was supported in part by the JSPS Grants (A) 22244007, (B) 26287014 and 26610019. J. Quastel was supported by Natural Sciences and Engineering Research Council of Canada, a Killam Fellowship, and the Institute for Advanced Study.

## Author information

### Affiliations

### Corresponding author

## Rights and permissions

## About this article

### Cite this article

Funaki, T., Quastel, J. KPZ equation, its renormalization and invariant measures.
*Stoch PDE: Anal Comp* **3, **159–220 (2015). https://doi.org/10.1007/s40072-015-0046-x

Received:

Published:

Issue Date:

### Keywords

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

### Mathematics Subject Classification

- 60H15
- 82C28