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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

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
Published electronically: January 19, 2011
MathSciNet review: 2784327
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Abstract: We consider the stochastic heat equation

$\displaystyle \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

$\displaystyle 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

$\displaystyle c_1t^{1/3}\le D_{\rm bulk}(t) \le c_2 t^{1/3}.$

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
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

DOI: http://dx.doi.org/10.1090/S0894-0347-2011-00692-9
PII: S 0894-0347(2011)00692-9
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.