Fluctuation exponent of the KPZ/stochastic Burgers equation
Authors:
M. Balázs, J. Quastel and T. Seppäläinen
Journal:
J. Amer. Math. Soc. 24 (2011), 683708
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 on the real line, where is spacetime white noise. is interpreted as a solution of the KPZ equation, and as a solution of the stochastic Burgers equation. We take , where is a twosided Brownian motion, corresponding to the stationary solution of the stochastic Burgers equation. We show that there exist such that Analogous results are obtained for some moments of the correlation functions of . In particular, it is shown there that the bulk diffusivity satisfies 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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 1.
 T. Alberts, K. Khanin, and J. Quastel.
The intermediate disorder regime for directed polymers in dimension 1 + 1. Phys. Rev. Lett., 105, 2010.
 2.
 J. Baik, P. Deift, and K. Johansson.
On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc., 12:1119 1178, 1999. MR 1682248 (2000e:05006)
 3.
 M. Balázs and T. Seppäläinen.
Exact connections between current fluctuations and the second class particle in a class of deposition models. J. Stat. Phys., 127(2), 2007. MR 2314355 (2008e:82056)
 4.
 M. Balázs and T. Seppäläinen.
Fluctuation bounds for the asymmetric simple exclusion process. ALEA Lat. Am. J. Probab. Math. Stat., 6:124, 2009. MR 2485877 (2010c:60280)
 5.
 A.L. Barabasi and H. E. Stanley.
Fractal concepts in surface growth. Cambridge University Press, Cambridge, 1995. MR 1600794 (99b:82072)
 6.
 Lorenzo Bertini and Giambattista Giacomin.
Stochastic Burgers and KPZ equations from particle systems. Comm. Math. Phys., 183(3):571607, 1997. MR 1462228 (99e:60212)
 7.
 S. Bezerra, S. Tindel, and F. Viens.
Superdiffusivity for a Brownian polymer in a continuous Gaussian environment. Ann. Probab., 36(5):16421675, 2008. MR 2440919 (2010a:60352)
 8.
 P. Billingsley.
Convergence of probability measures. Wiley, 1968. MR 0233396 (38:1718)
 9.
 Terence Chan.
Scaling limits of Wick ordered KPZ equation. Comm. Math. Phys., 209(3):671690, 2000. MR 1743612 (2001f:60072)
 10.
 P. L. Ferrari and H. Spohn.
Scaling limit for the spacetime covariance of the stationary totally asymmetric simple exclusion process. Comm. Math. Phys., 265(1):144, 2006. MR 2217295 (2007g:82038a)
 11.
 Dieter Forster, David R. Nelson, and Michael J. Stephen.
Largedistance and longtime properties of a randomly stirred fluid. Phys. Rev. A (3), 16(2):732749, 1977. MR 0459274 (56:17468)
 12.
 H. Holden, B. Øksendal, J. Ubøe, and T. Zhang.
Stochastic partial differential equations. A modeling, white noise functional approach. Birkhäuser Boston, Boston, 1996. MR 1408433 (98f:60124)
 13.
 K. Johansson.
Transversal fluctuations for increasing subsequences on the plane. Probab. Theory Related Fields, 116:445456, 2000. MR 1757595 (2001e:60210)
 14.
 K. Kardar, G. Parisi, and Y.Z. Zhang.
Dynamic scaling of growing interfaces. Phys. Rev. Lett., 56:889892, 1986.
 15.
 T. Kriecherbauer and J. Krug.
A pedestrian's view on interacting particle systems, KPZ universality, and random matrices. J. Phys. A: Math. Theor., 43, 2001.
 16.
 H. Krug and H. Spohn.
Kinetic roughening of growing surfaces, pages 412525. Cambridge Univ. Press., 1991.
 17.
 C. Licea, C. Newman, and M. Piza.
Superdiffusivity in firstpassage percolation. Prob. Th. Rel. Fields, 106:559591, 1996. MR 1421992 (98a:60151)
 18.
 O. Mejane.
Upper bound of a volume exponent for directed polymers in a random environment. Ann. Inst. H. Poincaré Probab. Statist., 40:299308, 2004. MR 2060455 (2005e:60239)
 19.
 C. Mueller.
On the support of solutions to the heat equation with noise. Stochastics Stochastics Rep., 37(4):225245, 1991. MR 1149348 (93e:60122)
 20.
 M. Petermann.
Superdiffusivity of directed polymers in random environment. Ph.D. thesis, University of Zürich, 2000.
 21.
 M. Piza.
Directed polymers in a random environment: Some results on fluctuations. J. Statist. Phys., 89:581603, 1997. MR 1484057 (99d:82036)
 22.
 M. Prähofer and H. Spohn.
Current fluctuations for the totally asymmetric simple exclusion process. Progress in Probability. Birkhäuser, 2002. MR 1901953 (2003e:60224)
 23.
 G. Da Prato and J. Zabczyk.
Stochastic equations in infinite dimensions. Cambridge University Press, Cambridge, 1992. MR 1207136 (95g:60073)
 24.
 J. Quastel and B. Valkó.
superdiffusivity of finiterange asymmetric exclusion processes on . Comm. Math. Phys., 273(2):379394, 2007. MR 2318311 (2008h:60414)
 25.
 T. Seppäläinen.
Scaling for a onedimensional directed polymer with boundary conditions. To appear in Ann. Probab., arXiv:0911.2446, 2009.
 26.
 T. Seppäläinen and B. Valkó.
Bounds for scaling exponents for a 1+1 dimensional directed polymer in a Brownian environment. To appear in Alea, arXiv:1006.4864, 2010.
 27.
 J. Walsh.
An introduction to stochastic partial differential equations, volume 1180 of Lecture Notes in Mathematics, pages 265439. SpringerVerlag, 1986. MR 876085 (88a:60114)
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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 537061388
Email:
seppalai@math.wisc.edu
DOI:
http://dx.doi.org/10.1090/S089403472011006929
PII:
S 08940347(2011)006929
Keywords:
KardarParisiZhang 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 K60708 and F67729, 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 DMS0701091 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.
