## Space-time stationary solutions for the Burgers equation

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- by Yuri Bakhtin, Eric Cator and Konstantin Khanin
- J. Amer. Math. Soc.
**27**(2014), 193-238 - DOI: https://doi.org/10.1090/S0894-0347-2013-00773-0
- Published electronically: May 14, 2013
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## Abstract:

We construct space-time stationary solutions of the $1$D Burgers equation with random forcing in the absence of periodicity or any other compactness assumptions. More precisely, for the forcing given by a homogeneous Poisson point field in space-time we prove that there is a unique global solution with any prescribed average velocity. These global solutions serve as one-point random attractors for the infinite-dimensional dynamical system associated with solutions to the Cauchy problem. The probability distribution of the global solutions defines a stationary distribution for the corresponding Markov process. We describe a broad class of initial Cauchy data for which the distribution of the Markov process converges to the above stationary distribution.

Our construction of the global solutions is based on a study of the field of action-minimizing curves. We prove that for an arbitrary value of the average velocity, with probability 1 there exists a unique field of action-minimizing curves initiated at all of the Poisson points. Moreover, action-minimizing curves corresponding to different starting points merge with each other in finite time.

## References

- D. Aldous and P. Diaconis,
*Hammersley’s interacting particle process and longest increasing subsequences*, Probab. Theory Related Fields**103**(1995), no. 2, 199–213. MR**1355056**, DOI 10.1007/BF01204214 - Ludwig Arnold,
*Random dynamical systems*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR**1723992**, DOI 10.1007/978-3-662-12878-7 - Yuri Bakhtin,
*Burgers equation with random boundary conditions*, Proc. Amer. Math. Soc.**135**(2007), no. 7, 2257–2262. MR**2299503**, DOI 10.1090/S0002-9939-07-08736-9 - Yuri Bakhtin,
*Burgers equation with Poisson random forcing*, Accepted at Ann. of Probab., available at http://arxiv.org/abs/1109.5668 (2012). - Eric Cator and Leandro P. R. Pimentel,
*Busemann functions and equilibrium measures in last passage percolation models*, Probab. Theory Related Fields**154**(2012), no. 1-2, 89–125. MR**2981418**, DOI 10.1007/s00440-011-0363-6 - Eric Cator and Leandro P. R. Pimentel,
*A shape theorem and semi-infinite geodesics for the Hammersley model with random weights*, ALEA Lat. Am. J. Probab. Math. Stat.**8**(2011), 163–175. MR**2783936** - J. Theodore Cox, Alberto Gandolfi, Philip S. Griffin, and Harry Kesten,
*Greedy lattice animals. I. Upper bounds*, Ann. Appl. Probab.**3**(1993), no. 4, 1151–1169. MR**1241039** - D. J. Daley and D. Vere-Jones,
*An introduction to the theory of point processes. Vol. I*, 2nd ed., Probability and its Applications (New York), Springer-Verlag, New York, 2003. Elementary theory and methods. MR**1950431** - Weinan E,
*Aubry-Mather theory and periodic solutions of the forced Burgers equation*, Comm. Pure Appl. Math.**52**(1999), no. 7, 811–828. MR**1682812**, DOI 10.1002/(SICI)1097-0312(199907)52:7<811::AID-CPA2>3.0.CO;2-D - Weinan E, K. Khanin, A. Mazel, and Ya. Sinai,
*Invariant measures for Burgers equation with stochastic forcing*, Ann. of Math. (2)**151**(2000), no. 3, 877–960. MR**1779561**, DOI 10.2307/121126 - Albert Fathi,
*The weak KAM theorem in lagrangian dynamics*, Cambridge Studies in Advanced Mathematics, Cambridge University Press, in press, 2012. - Alberto Gandolfi and Harry Kesten,
*Greedy lattice animals. II. Linear growth*, Ann. Appl. Probab.**4**(1994), no. 1, 76–107. MR**1258174** - Diogo Gomes, Renato Iturriaga, Konstantin Khanin, and Pablo Padilla,
*Viscosity limit of stationary distributions for the random forced Burgers equation*, Mosc. Math. J.**5**(2005), no. 3, 613–631, 743 (English, with English and Russian summaries). MR**2241814**, DOI 10.17323/1609-4514-2005-5-3-613-631 - Viet Ha Hoang and Konstantin Khanin,
*Random Burgers equation and Lagrangian systems in non-compact domains*, Nonlinearity**16**(2003), no. 3, 819–842. MR**1975784**, DOI 10.1088/0951-7715/16/3/303 - C. Douglas Howard and Charles M. Newman,
*Euclidean models of first-passage percolation*, Probab. Theory Related Fields**108**(1997), no. 2, 153–170. MR**1452554**, DOI 10.1007/s004400050105 - C. Douglas Howard and Charles M. Newman,
*From greedy lattice animals to Euclidean first-passage percolation*, Perplexing problems in probability, Progr. Probab., vol. 44, Birkhäuser Boston, Boston, MA, 1999, pp. 107–119. MR**1703127** - C. Douglas Howard and Charles M. Newman,
*Geodesics and spanning trees for Euclidean first-passage percolation*, Ann. Probab.**29**(2001), no. 2, 577–623. MR**1849171**, DOI 10.1214/aop/1008956685 - R. Iturriaga and K. Khanin,
*Burgers turbulence and random Lagrangian systems*, Comm. Math. Phys.**232**(2003), no. 3, 377–428. MR**1952472**, DOI 10.1007/s00220-002-0748-6 - Kurt Johansson,
*Transversal fluctuations for increasing subsequences on the plane*, Probab. Theory Related Fields**116**(2000), no. 4, 445–456. MR**1757595**, DOI 10.1007/s004400050258 - Harry Kesten,
*On the speed of convergence in first-passage percolation*, Ann. Appl. Probab.**3**(1993), no. 2, 296–338. MR**1221154** - Ulrich Krengel,
*Ergodic theorems*, De Gruyter Studies in Mathematics, vol. 6, Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel. MR**797411**, DOI 10.1515/9783110844641 - Pierre-Louis Lions,
*Generalized solutions of Hamilton-Jacobi equations*, Research Notes in Mathematics, vol. 69, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR**667669** - Charles M. Newman,
*A surface view of first-passage percolation*, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 1017–1023. MR**1404001** - Cédric Villani,
*Optimal transport*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009. Old and new. MR**2459454**, DOI 10.1007/978-3-540-71050-9 - Mario V. Wüthrich,
*Asymptotic behaviour of semi-infinite geodesics for maximal increasing subsequences in the plane*, In and out of equilibrium (Mambucaba, 2000) Progr. Probab., vol. 51, Birkhäuser Boston, Boston, MA, 2002, pp. 205–226. MR**1901954**

## Bibliographic Information

**Yuri Bakhtin**- Affiliation: School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, Georgia 30332-0160
- MR Author ID: 648835
- ORCID: 0000-0003-1125-4543
- Email: bakhtin@math.gatech.edu
**Eric Cator**- Affiliation: Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
- Email: e.cator@math.ru.nl
**Konstantin Khanin**- Affiliation: Department of Mathematics, University of Toronto, 40 St George Street, Toronto, Ontario, M5S 2E4, Canada
- MR Author ID: 207695
- Email: khanin@math.toronto.edu
- Received by editor(s): May 30, 2012
- Received by editor(s) in revised form: March 21, 2013, March 24, 2013, and March 25, 2013
- Published electronically: May 14, 2013
- Additional Notes: The first author was supported by NSF CAREER Award DMS-0742424 and grant 040.11.264 from the Netherlands Organisation for Scientific Research (NWO). He is grateful for the hospitality of the Fields Institute in Toronto, Delft Technical University, and CRM in Barcelona where parts of this work have been written.

The second author is grateful for the hospitality of the Fields Institute in Toronto.

The third author was supported by NSERC Discovery Grant RGPIN 328565 - © Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**27**(2014), 193-238 - MSC (2010): Primary 37L40; Secondary 37L55, 35R60, 37H99, 60K35, 60G55
- DOI: https://doi.org/10.1090/S0894-0347-2013-00773-0
- MathSciNet review: 3110798