Electronic Journal of Probability

Inviscid Burgers equation with random kick forcing in noncompact setting

Yuri Bakhtin

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We develop ergodic theory of the inviscid Burgers equation with random kick forcing in noncompact setting. The results are parallel to those in our recent work on the Burgers equation with Poissonian forcing. However, the analysis based on the study of one-sided minimizers of the relevant action is different. In contrast with previous work, finite time coalescence of the minimizers does not hold, and hyperbolicity (exponential convergence of minimizers in reverse time) is not known. In order to establish a One Force — One Solution principle on each ergodic component, we use an extremely soft method to prove a weakened hyperbolicity property and to construct Busemann functions along appropriate subsequences.

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Electron. J. Probab., Volume 21 (2016), paper no. 37, 50 pp.

Received: 7 July 2015
Accepted: 12 May 2016
First available in Project Euclid: 19 May 2016

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Primary: 37L40: Invariant measures 37L55: Infinite-dimensional random dynamical systems; stochastic equations [See also 35R60, 60H10, 60H15] 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 37H99: None of the above, but in this section 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G55: Point processes

SPDE Burgers equation last passage percolation invariant distributions Busemann functions One Force – One Solution Principle

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Bakhtin, Yuri. Inviscid Burgers equation with random kick forcing in noncompact setting. Electron. J. Probab. 21 (2016), paper no. 37, 50 pp. doi:10.1214/16-EJP4413. https://projecteuclid.org/euclid.ejp/1463683782

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  • [1] Ludwig Arnold, Random dynamical systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.
  • [2] Yuri Bakhtin, Burgers equation with random boundary conditions, Proc. Amer. Math. Soc. 135 (2007), no. 7, 2257–2262 (electronic).
  • [3] Yuri Bakhtin, The Burgers equation with Poisson random forcing, Ann. Probab. 41 (2013), no. 4, 2961–2989.
  • [4] Yuri Bakhtin, Eric Cator, and Konstantin Khanin, Space-time stationary solutions for the Burgers equation, J. Amer. Math. Soc. 27 (2014), no. 1, 193–238.
  • [5] Jérémie Bec and Konstantin Khanin, Burgers turbulence, Phys. Rep. 447 (2007), no. 1-2, 1–66.
  • [6] Alexandre Boritchev and Konstantin Khanin, On the hyperbolicity of minimizers for 1D random Lagrangian systems, Nonlinearity 26 (2013), no. 1, 65–80.
  • [7] 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.
  • [8] Eric Cator and Leandro P.R. Pimentel, A shape theorem and semi-infinite geodesics for the Hammersley model with random weights, ALEA 8 (2011), 163–175.
  • [9] Michael Damron and Jack Hanson, Busemann functions and infinite geodesics in two-dimensional first-passage percolation, Comm. Math. Phys. 325 (2014), no. 3, 917–963.
  • [10] 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.
  • [11] Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998.
  • [12] N. Georgiou, F. Rassoul-Agha, and T. Seppäläinen, Variational formulas and cocycle solutions for directed polymer and percolation models, ArXiv e-prints, arXiv:1311.3016 (2013).
  • [13] N. Georgiou, F. Rassoul-Agha, and T. Seppäläinen, Stationary cocycles and Busemann functions for the corner growth model, ArXiv e-prints, arXiv:1510.00859 (2015).
  • [14] 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.
  • [15] Viet Ha Hoang and Konstantin Khanin, Random Burgers equation and Lagrangian systems in non-compact domains, Nonlinearity 16 (2003), no. 3, 819–842.
  • [16] Wassily Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (1963), 13–30.
  • [17] C. Douglas Howard and Charles M. Newman, Euclidean models of first-passage percolation, Probability Theory and Related Fields 108 (1997), 153–170, 10.1007/s004400050105.
  • [18] 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.
  • [19] C. Douglas Howard and Charles M. Newman, Geodesics and spanning trees for Euclidean first-passage percolation, Ann. Probab. 29 (2001), no. 2, 577–623.
  • [20] R. Iturriaga and K. Khanin, Burgers turbulence and random Lagrangian systems, Comm. Math. Phys. 232 (2003), no. 3, 377–428.
  • [21] Harry Kesten, On the speed of convergence in first-passage percolation, Ann. Appl. Probab. 3 (1993), no. 2, 296–338.
  • [22] Sergei Kuksin, Andrey Piatnitski, and Armen Shirikyan, A coupling approach to randomly forced nonlinear PDEs. II, Comm. Math. Phys. 230 (2002), no. 1, 81–85.
  • [23] Sergei Kuksin and Armen Shirikyan, A coupling approach to randomly forced nonlinear PDE’s. I, Comm. Math. Phys. 221 (2001), no. 2, 351–366.
  • [24] Sergei B. Kuksin, Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2006.
  • [25] Charles M. Newman, A surface view of first-passage percolation, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) (Basel), Birkhäuser, 1995, pp. 1017–1023.
  • [26] F. Rassoul-Agha, T. Seppäläinen, and A. Yilmaz, Variational formulas and disorder regimes of random walks in random potentials, ArXiv e-prints, arXiv:1410.4474 (2014).
  • [27] Toufic M. Suidan, Stationary measures for a randomly forced Burgers equation, Comm. Pure Appl. Math. 58 (2005), no. 5, 620–638.
  • [28] Cédric Villani, Optimal transport. old and new, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009.
  • [29] 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.