## The Annals of Probability

### A central limit theorem for the KPZ equation

#### Abstract

We consider the KPZ equation in one space dimension driven by a stationary centred space–time random field, which is sufficiently integrable and mixing, but not necessarily Gaussian. We show that, in the weakly asymmetric regime, the solution to this equation considered at a suitable large scale and in a suitable reference frame converges to the Hopf–Cole solution to the KPZ equation driven by space–time Gaussian white noise. While the limiting process depends only on the integrated variance of the driving field, the diverging constants appearing in the definition of the reference frame also depend on higher order moments.

#### Article information

Source
Ann. Probab., Volume 45, Number 6B (2017), 4167-4221.

Dates
Revised: August 2016
First available in Project Euclid: 12 December 2017

https://projecteuclid.org/euclid.aop/1513069258

Digital Object Identifier
doi:10.1214/16-AOP1162

Mathematical Reviews number (MathSciNet)
MR3737909

Zentralblatt MATH identifier
06838118

#### Citation

Hairer, Martin; Shen, Hao. A central limit theorem for the KPZ equation. Ann. Probab. 45 (2017), no. 6B, 4167--4221. doi:10.1214/16-AOP1162. https://projecteuclid.org/euclid.aop/1513069258

#### References

• [1] Akhiezer, N. I. (1965). The Classical Moment Problem and Some Related Questions in Analysis. Hafner Publishing, New York.
• [2] Alberts, T., Khanin, K. and Quastel, J. (2010). Intermediate disorder regime for directed polymers in dimension $1+1$. Phys. Rev. Lett. 105 090603. Available at arXiv:1003.1885.
• [3] Amir, G., Corwin, I. and Quastel, J. (2011). Probability distribution of the free energy of the continuum directed random polymer in $1+1$ dimensions. Comm. Pure Appl. Math. 64 466–537.
• [4] Avram, F. and Taqqu, M. S. (1987). Noncentral limit theorems and Appell polynomials. Ann. Probab. 15 767–775.
• [5] Avram, F. and Taqqu, M. S. (2006). On a Szegö type limit theorem and the asymptotic theory of random sums, integrals and quadratic forms. In Dependence in Probability and Statistics. Lect. Notes Stat. 187 259–286. Springer, New York.
• [6] Baik, J., Deift, P. and Johansson, K. (1999). On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 1119–1178.
• [7] Balázs, M., Quastel, J. and Seppäläinen, T. (2011). Fluctuation exponent of the KPZ/stochastic Burgers equation. J. Amer. Math. Soc. 24 683–708.
• [8] Bertini, L. and Giacomin, G. (1997). Stochastic Burgers and KPZ equations from particle systems. Comm. Math. Phys. 183 571–607.
• [9] Bogoliubow, N. N. and Parasiuk, O. S. (1957). Über die Multiplikation der Kausalfunktionen in der Quantentheorie der Felder. Acta Math. 97 227–266.
• [10] Borodin, A. and Corwin, I. (2014). Macdonald processes. Probab. Theory Related Fields 158 225–400.
• [11] Catellier, R. and Chouk, K. (2013). Paracontrolled distributions and the 3-dimensional stochastic quantization equation. Preprint. Available at arXiv:1310.6869.
• [12] Corwin, I. and Tsai, L.-C. (2015). KPZ equation limit of higher-spin exclusion processes. Preprint. Available at arXiv:1505.04158.
• [13] Cramér, H. and Wold, H. (1936). Some Theorems on Distribution Functions. J. Lond. Math. Soc. S 1-11 290.
• [14] Da Prato, G. and Zabczyk, J. (2014). Stochastic Equations in Infinite Dimensions, 2nd ed. Encyclopedia of Mathematics and Its Applications 152. Cambridge Univ. Press, Cambridge.
• [15] Dembo, A. and Tsai, L.-C. (2016). Weakly asymmetric non-simple exclusion process and the Kardar–Parisi–Zhang equation. Comm. Math. Phys. 341 219–261.
• [16] Edwards, S. F. and Wilkinson, D. R. (1982). The surface statistics of a granular aggregate. Proc. Roy. Soc. London Ser. A 381 17–31.
• [17] Friz, P. and Victoir, N. (2006). A note on the notion of geometric rough paths. Probab. Theory Related Fields 136 395–416.
• [18] Friz, P. K. and Hairer, M. (2014). A Course on Rough Paths. Universitext. Springer, Cham.
• [19] Giraitis, L. and Surgailis, D. (1986). Multivariate Appell polynomials and the central limit theorem. In Dependence in Probability and Statistics. Progr. Probab. Statist. 11 21–71. Birkhäuser Boston, Boston, MA.
• [20] Glimm, J. and Jaffe, A. (1987). Quantum Physics: A Functional Integral Point of View, 2nd ed. Springer, New York.
• [21] Gonçalves, P. and Jara, M. (2014). Nonlinear fluctuations of weakly asymmetric interacting particle systems. Arch. Ration. Mech. Anal. 212 597–644.
• [22] Hairer, M. (2013). Solving the KPZ equation. Ann. of Math. (2) 178 559–664.
• [23] Hairer, M. (2014). A theory of regularity structures. Invent. Math. 198 269–504.
• [24] Hairer, M. (2015). Introduction to regularity structures. Braz. J. Probab. Stat. 29 175–210.
• [25] Hairer, M. and Pardoux, É. (2015). A Wong–Zakai theorem for stochastic PDEs. J. Math. Soc. Japan 67 1551–1604. Available at arXiv:1409.3138.
• [26] Hairer, M. and Quastel, J. (2015). A class of growth models rescaling to KPZ. Preprint. Available at arXiv:1512.07845.
• [27] Hepp, K. (1969). On the equivalence of additive and analytic renormalization. Comm. Math. Phys. 14 67–69.
• [28] Kardar, M., Parisi, G. and Zhang, Y.-C. (1986). Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56 889–892.
• [29] Kupiainen, A. (2016). Renormalization group and stochastic PDEs. Annales Henri Poincaré 17 497–535.
• [30] Lukkarinen, J. and Marcozzi, M. (2015). Wick polynomials and time-evolution of cumulants. Preprint. Available at arXiv:1503.05851.
• [31] Mourrat, J.-C. and Weber, H. (2014). Convergence of the two-dimensional dynamic Ising–Kac model to $\phi^{4}_{2}$. Preprint. Available at arXiv:1410.1179.
• [32] O’Connell, N. and Yor, M. (2002). A representation for non-colliding random walks. Electron. Commun. Probab. 7 1–12. (electronic).
• [33] Peccati, G. and Taqqu, M. S. (2011). Wiener Chaos: Moments, Cumulants and Diagrams. Bocconi & Springer Series 1. Springe, Milan.
• [34] Sasamoto, T. and Spohn, H. (2010). Exact height distributions for the KPZ equation with narrow wedge initial condition. Nuclear Phys. B 834 523–542.
• [35] Surgailis, D. (1983). On Poisson multiple stochastic integrals and associated equilibrium Markov processes. In Theory and Application of Random Fields. Lecture Notes in Control and Information Sciences 49 233–248. Springer, Berlin.
• [36] Villani, C. (2009). Optimal Transport Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer, Berlin.
• [37] Zimmermann, W. (1969). Convergence of Bogoliubov’s method of renormalization in momentum space. Comm. Math. Phys. 15 208–234.