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.
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- 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: firstname.lastname@example.org
- Eric Cator
- Affiliation: Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
- Email: email@example.com
- Konstantin Khanin
- Affiliation: Department of Mathematics, University of Toronto, 40 St George Street, Toronto, Ontario, M5S 2E4, Canada
- MR Author ID: 207695
- Email: firstname.lastname@example.org
- 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