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
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
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.
 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.
Jinho
Baik, Percy
Deift, and Kurt
Johansson, On the distribution of the length of
the longest increasing subsequence of random permutations, J. Amer. Math. Soc. 12 (1999), no. 4, 1119–1178. MR 1682248
(2000e:05006), 10.1090/S0894034799003070
 3.
Márton
Balázs and Timo
Seppäläinen, Exact connections between current
fluctuations and the second class particle in a class of deposition
models, J. Stat. Phys. 127 (2007), no. 2,
431–455. MR 2314355
(2008e:82056), 10.1007/s1095500792913
 4.
Márton
Balázs and Timo
Seppäläinen, Fluctuation bounds for the asymmetric simple
exclusion process, ALEA Lat. Am. J. Probab. Math. Stat.
6 (2009), 1–24. MR 2485877
(2010c:60280)
 5.
AlbertLászló
Barabási and H.
Eugene 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 (1997), no. 3,
571–607. MR 1462228
(99e:60212), 10.1007/s002200050044
 7.
Sérgio
Bezerra, Samy
Tindel, and Frederi
Viens, Superdiffusivity for a Brownian polymer in a continuous
Gaussian environment, Ann. Probab. 36 (2008),
no. 5, 1642–1675. MR 2440919
(2010a:60352), 10.1214/07AOP363
 8.
Patrick
Billingsley, Convergence of probability measures, John Wiley
& Sons, Inc., New YorkLondonSydney, 1968. MR 0233396
(38 #1718)
 9.
Terence
Chan, Scaling limits of Wick ordered KPZ equation, Comm. Math.
Phys. 209 (2000), no. 3, 671–690. MR 1743612
(2001f:60072), 10.1007/PL00020963
 10.
Patrik
L. Ferrari and Herbert
Spohn, Scaling limit for the spacetime covariance of the
stationary totally asymmetric simple exclusion process, Comm. Math.
Phys. 265 (2006), no. 1, 1–44. MR 2217295
(2007g:82038a), 10.1007/s0022000615490
 11.
Dieter
Forster, David
R. Nelson, and Michael
J. Stephen, Largedistance and longtime properties of a randomly
stirred fluid, Phys. Rev. A (3) 16 (1977),
no. 2, 732–749. MR 0459274
(56 #17468)
 12.
Helge
Holden, Bernt
Øksendal, Jan
Ubøe, and Tusheng
Zhang, Stochastic partial differential equations, Probability
and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1996. A
modeling, white noise functional approach. MR 1408433
(98f:60124)
 13.
Kurt
Johansson, Transversal fluctuations for increasing subsequences on
the plane, Probab. Theory Related Fields 116 (2000),
no. 4, 445–456. MR 1757595
(2001e:60210), 10.1007/s004400050258
 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.
M. Newman, and M.
S. T. Piza, Superdiffusivity in firstpassage percolation,
Probab. Theory Related Fields 106 (1996), no. 4,
559–591. MR 1421992
(98a:60151), 10.1007/s004400050075
 18.
Olivier
Mejane, Upper bound of a volume exponent for directed polymers in a
random environment, Ann. Inst. H. Poincaré Probab. Statist.
40 (2004), no. 3, 299–308 (English, with
English and French summaries). MR 2060455
(2005e:60239), 10.1016/S02460203(03)000724
 19.
Carl
Mueller, On the support of solutions to the heat equation with
noise, Stochastics Stochastics Rep. 37 (1991),
no. 4, 225–245. MR 1149348
(93e:60122)
 20.
M. Petermann.
Superdiffusivity of directed polymers in random environment. Ph.D. thesis, University of Zürich, 2000.
 21.
M.
S. T. Piza, Directed polymers in a random environment: some results
on fluctuations, J. Statist. Phys. 89 (1997),
no. 34, 581–603. MR 1484057
(99d:82036), 10.1007/BF02765537
 22.
Michael
Prähofer and Herbert
Spohn, Current fluctuations for the totally asymmetric simple
exclusion process, In and out of equilibrium (Mambucaba, 2000)
Progr. Probab., vol. 51, Birkhäuser Boston, Boston, MA, 2002,
pp. 185–204. MR 1901953
(2003e:60224)
 23.
Giuseppe
Da Prato and Jerzy
Zabczyk, Stochastic equations in infinite dimensions,
Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge
University Press, Cambridge, 1992. MR 1207136
(95g:60073)
 24.
Jeremy
Quastel and Benedek
Valko, 𝑡^{1/3} Superdiffusivity of finiterange asymmetric
exclusion processes on ℤ, Comm. Math. Phys.
273 (2007), no. 2, 379–394. MR 2318311
(2008h:60414), 10.1007/s0022000702422
 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.
John
B. Walsh, An introduction to stochastic partial differential
equations, École d’été de
probabilités de SaintFlour, XIV—1984, Lecture Notes in
Math., vol. 1180, Springer, Berlin, 1986, pp. 265–439. MR 876085
(88a:60114), 10.1007/BFb0074920
 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)
Similar Articles
Retrieve articles in Journal of the American Mathematical Society
with MSC (2010):
60H15,
82C22,
35R60,
60K35
Retrieve articles in all journals
with MSC (2010):
60H15,
82C22,
35R60,
60K35
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
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.
