Fluctuation exponent of the KPZ/stochastic Burgers equation

Authors:
M. Balázs, J. Quastel and T. Seppäläinen

Journal:
J. Amer. Math. Soc. **24** (2011), 683-708

MSC (2010):
Primary 60H15, 82C22; Secondary 35R60, 60K35

DOI:
https://doi.org/10.1090/S0894-0347-2011-00692-9

Published electronically:
January 19, 2011

MathSciNet review:
2784327

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the stochastic heat equation \[ \partial _tZ= \partial _x^2 Z - Z \dot W \] on the real line, where $\dot W$ is space-time white noise. $h(t,x)=-\operatorname {log} Z(t,x)$ is interpreted as a solution of the KPZ equation, and $u(t,x)=\partial _x h(t,x)$ as a solution of the stochastic Burgers equation. We take $Z(0,x)=\exp \{B(x)\}$, where $B(x)$ is a two-sided Brownian motion, corresponding to the stationary solution of the stochastic Burgers equation. We show that there exist $0< c_1\le c_2 <\infty$ such that \[ c_1t^{2/3}\le \operatorname {Var}(\operatorname {log} Z(t,x) )\le c_2 t^{2/3}. \] Analogous results are obtained for some moments of the correlation functions of $u(t,x)$. In particular, it is shown there that the bulk diffusivity satisfies \[ c_1t^{1/3}\le D_\textrm {bulk}(t) \le c_2 t^{1/3}.\] The proof uses approximation by weakly asymmetric simple exclusion processes, for which we obtain the microscopic analogies of the results by coupling.

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Additional Information

**M. Balázs**

Affiliation:
Department of Stochastics, Budapest University of Technology and Economics, 1 Egry Jozsef u, H ep V 7, Budapest, 1111 Hungary

Email:
balazs@math.bme.hu

**J. Quastel**

Affiliation:
Departments of Mathematics and Statistics, University of Toronto, 40 St. George Street, Room 6290, Toronto, ON M5S 1L2 Canada

MR Author ID:
322635

Email:
quastel@math.toronto.edu

**T. Seppäläinen**

Affiliation:
Department of Mathematics, University of Wisconsin–Madison, 480 Lincoln Drive, Madison, Wisconsin 53706-1388

Email:
seppalai@math.wisc.edu

Keywords:
Kardar-Parisi-Zhang equation,
stochastic heat equation,
stochastic Burgers equation,
random growth,
asymmetric exclusion process,
anomalous fluctuations,
directed polymers.

Received by editor(s):
October 16, 2009

Received by editor(s) in revised form:
October 28, 2010

Published electronically:
January 19, 2011

Additional Notes:
The first author is supported by the Hungarian Scientific Research Fund (OTKA) grants K-60708 and F-67729, by the Bolyai Scholarship of the Hungarian Academy of Sciences, and by the Morgan Stanley Mathematical Modeling Center.

The second author is supported by the Natural Sciences and Engineering Research Council of Canada.

The third author is supported by the National Science Foundation grant DMS-0701091 and by the Wisconsin Alumni Research Foundation.

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.