Hostname: page-component-76c49bb84f-7l7zb Total loading time: 0 Render date: 2025-07-11T14:40:05.566Z Has data issue: false hasContentIssue false
  • English
  • Français

A CLASS OF GROWTH MODELS RESCALING TO KPZ

Part of: Miscellaneous topics - Partial differential equations Stochastic analysis Time-dependent statistical mechanics (dynamic and nonequilibrium)

Published online by Cambridge University Press:  19 November 2018

MARTIN HAIRER  and
JEREMY QUASTEL
MARTIN HAIRER
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK; m.hairer@imperial.ac.uk
JEREMY QUASTEL
Affiliation:
Department of Mathematics, University of Toronto, M5S 1L2, Canada; quastel@math.toronto.edu

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a large class of $1+1$-dimensional continuous interface growth models and we show that, in both the weakly asymmetric and the intermediate disorder regimes, these models converge to Hopf–Cole solutions to the KPZ equation.

MSC classification

Primary: 60H15: Stochastic partial differential equations
Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations 82C28: Dynamic renormalization group methods

Information

Type
Research Article
Information
Forum of Mathematics, Pi , Volume 6 , 2018 , e3
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2018

References

Amir, G., Corwin, I. and Quastel, J., ‘Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions’, Comm. Pure Appl. Math. 64(4) (2011), 466–537.Google Scholar
Alberts, T., Khanin, K. and Quastel, J., ‘Intermediate disorder regime for directed polymers in dimension 1+1’, Phys. Rev. Lett. 105(9) (2010), 090603.Google Scholar
Ben-Artzi, M., ‘Lectures on viscous Hamilton-Jacobi equations’, unpublished lecture notes (2007). URL: http://www.ma.huji.ac.il/∼mbartzi/.Google Scholar
Borodin, A. and Corwin, I., ‘Macdonald processes’, Probab. Theory Related Fields 158(1–2) (2014), 225–400.Google Scholar
Bertini, L. and Giacomin, G., ‘Stochastic Burgers and KPZ equations from particle systems’, Comm. Math. Phys. 183(3) (1997), 571–607.Google Scholar
Bogoliubow, N. N. and Parasiuk, O. S., ‘Über die Multiplikation der Kausalfunktionen in der Quantentheorie der Felder’, Acta Math. 97(1957) 227–266.Google Scholar
Bertini, L., Presutti, E., Rüdiger, B. and Saada, E., ‘Dynamical fluctuations at the critical point: convergence to a nonlinear stochastic PDE’, Teor. Veroyatnost. i Primenen. 38(4) (1993), 689–741.Google Scholar
Balázs, M., Quastel, J. and Seppäläinen, T., ‘Fluctuation exponent of the KPZ / stochastic Burgers equation’, J. Amer. Math. Soc. 24(3) (2011), 683–708.Google Scholar
Corwin, I. and Tsai, L.-C., ‘KPZ equation limit of higher-spin exclusion processes’, Ann. Probab. 45(3) (2017), 1771–1798.Google Scholar
Dembo, A. and Tsai, L.-C., ‘Weakly asymmetric non-simple exclusion process and the Kardar-Parisi-Zhang equation’, Comm. Math. Phys. 341(1) (2016), 219–261.Google Scholar
Furlan, M. and Gubinelli, M., ‘Weak universality for a class of 3d stochastic reaction-diffusion models’, Preprint, 2017, arXiv:1708.03118.Google Scholar
Friz, P. and Hairer, M., A Course on Rough Paths, Universitext (Springer, New York, 2014), xiv+251. With an introduction to Regularity Structures. doi:10.1007/978-3-319-08332-2.Google Scholar
Funaki, T. and Quastel, J., ‘KPZ equation, its renormalization and invariant measures’, Stoch. Partial Differ. Equ. Anal. Comput. 3(2) (2015), 159–220.Google Scholar
Gonçalves, P. and Jara, M., ‘Nonlinear fluctuations of weakly asymmetric interacting particle systems’, Arch. Ration. Mech. Anal. 212(2) (2014), 597–644.Google Scholar
Gubinelli, M. and Perkowski, N., ‘KPZ reloaded’, Comm. Math. Phys. 349(1) (2017), 165–269.Google Scholar
Gu, Y., Ryzhik, L. and Zeitouni, O., ‘The Edwards-Wilkinson limit of the random heat equation in dimensions three and higher’, Preprint, 2017, arXiv:1710.00344.Google Scholar
Hairer, M., ‘Solving the KPZ equation’, Ann. of Math. (2) 178(2) (2013), 559–664.Google Scholar
Hairer, M., ‘A theory of regularity structures’, Invent. Math. 198(2) (2014), 269–504.Google Scholar
Hairer, M., ‘Introduction to regularity structures’, Braz. J. Probab. Stat. 29(2) (2015), 175–210.Google Scholar
Hairer, M., ‘Regularity structures and the dynamical 𝛷3 4 model’, inCurrent Developments in Mathematics 2014 (Int. Press, Somerville, MA, 2016), 1–49.Google Scholar
Hepp, K., ‘On the equivalence of additive and analytic renormalization’, Comm. Math. Phys. 14(1969) 67–69.Google Scholar
Halpin-Healy, T. and Zhang, Y.-C., ‘Kinetic roughening phenomena, stochastic growth, directed polymers and all that’, Phys. Rep. 254(4–6) (1995), 215–414.Google Scholar
Hairer, M. and Labbé, C., ‘Multiplicative stochastic heat equations on the whole space’, J. Eur. Math. Soc. (JEMS) 20(4) (2018), 1005–1054.Google Scholar
Hairer, M. and Pardoux, É., ‘A Wong-Zakai theorem for stochastic PDEs’, J. Math. Soc. Japan 67(4) (2015), 1551–1604.Google Scholar
Hairer, M., Pardoux, É. and Piatnitski, A., ‘Random homogenization of a highly oscillatory singular potential’, SPDEs: Anal. Comput. 1(4) (2013), 571–605.Google Scholar
Hairer, M. and Shen, H., ‘A central limit theorem for the KPZ equation’, Ann. Probab. 45(6B) (2017), 4167–4221.Google Scholar
Hairer, M. and Xu, W., ‘Large-scale limit of interface fluctuation models’, Preprint, 2018, arXiv:1802.08192.Google Scholar
Kardar, M., Parisi, G. and Zhang, Y.-C., ‘Dynamic scaling of growing interfaces’, Phys. Rev. Lett. 56(9) (1986), 889–892.Google Scholar
Kruskal, J. B. Jr., ‘On the shortest spanning subtree of a graph and the traveling salesman problem’, Proc. Amer. Math. Soc. 7(1956) 48–50.Google Scholar
Krug, J. and Spohn, H., ‘Kinetic roughening of growing surfaces’, inSolids Far from Equilibrium (Cambridge University Press, Cambridge, 1991), 479–583.Google Scholar
Meyer, Y., ‘Wavelets and operators’, inCambridge Studies in Advanced Mathematics vol. 37 (Cambridge University Press, Cambridge, 1992), xvi+224. Translated from the 1990 French original by D. H. Salinger. doi:10.1017/CBO9780511623820.Google Scholar
Moreno-Flores, G., Quastel, J. and Remenik, D., In preparation.Google Scholar
Magnen, J. and Unterberger, J., ‘Diffusive limit for 3-dimensional KPZ equation: the Cole-Hopf case’, Preprint, 2017, arXiv:1702.03122.Google Scholar
Mueller, C., ‘On the support of solutions to the heat equation with noise’, Stochastics and Stochastic Reports 37(4) (1991), 225–245.Google Scholar
Nualart, D., ‘The Malliavin calculus and related topics’, inProbability and its Applications (New York), second edn, (Springer, Berlin, 2006), xiv+382.Google Scholar
Pardoux, É. and Piatnitski, A., ‘Homogenization of a singular random one-dimensional PDE with time-varying coefficients’, Ann. Probab. 40(3) (2012), 1316–1356.Google Scholar
Quastel, J. and Valkó, B., ‘Diffusivity of lattice gases’, Arch. Ration. Mech. Anal. 210(1) (2013), 269–320.Google Scholar
Spohn, H., Large Scale Dynamics of Interacting Particles, Texts and Monographs in Physics (Springer, Berlin, 1991).Google Scholar
Sweedler, M. E., ‘Hopf algebras’, inMathematics Lecture Note Series (W. A. Benjamin, Inc., New York, 1969), vii+336.Google Scholar
Walsh, J. B., ‘An introduction to stochastic partial differential equations’, inÉcole d’été de probabilités de Saint-Flour, XIV—1984, Lecture Notes in Mathematics, 1180 (Springer, Berlin, 1986), 265–439.Google Scholar
Zimmermann, W., ‘Convergence of Bogoliubov’s method of renormalization in momentum space’, Comm. Math. Phys. 15(1969) 208–234.Google Scholar