Skip to main content
SpringerLink
Log in

Zero Temperature Limit for Directed Polymers and Inviscid Limit for Stationary Solutions of Stochastic Burgers Equation

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We consider a space-continuous and time-discrete polymer model for positive temperature and the associated zero temperature model of last passage percolation type. In our previous work, we constructed and studied infinite-volume polymer measures and one-sided infinite minimizers for the associated variational principle, and used these objects for the study of global stationary solutions of the Burgers equation with positive or zero viscosity and random kick forcing, on the entire real line. In this paper, we prove that in the zero temperature limit, the infinite-volume polymer measures concentrate on the one-sided minimizers and that the associated global solutions of the viscous Burgers equation with random kick forcing converge to the global solutions of the inviscid equation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. den Hollander, F.: Random Polymers. Lecture Notes in Mathematics. Lectures from the 37th Probability Summer School held in Saint-Flour, vol. 1974. Springer, Berlin (2007)

    MATH  Google Scholar 

  2. Giacomin, G.: Random Polymer Models. Imperial College Press, London (2007)

    Book  Google Scholar 

  3. Comets, F.: Directed Polymers in Random Environments. Lecture Notes in Mathematics. Lecture Notes from the 46th Probability Summer School held in Saint-Flour, vol. 2175. Springer, Cham (2017)

    MATH  Google Scholar 

  4. Howard, C.D., Newman, C.M.: Euclidean models of first-passage percolation. Probab. Theory Relat. Fields 108, 153–170 (1997). https://doi.org/10.1007/s004400050105

    Article  MathSciNet  MATH  Google Scholar 

  5. Howard, C.D., Newman, C.M.: From greedy lattice animals to Euclidean first-passage percolation. In: Perplexing problems in probability, vol. 44 of Progr. Probab., pp. 107–119. Birkhäuser Boston, Boston (1999)

  6. Howard, C.D., Newman, C.M.: Geodesics and spanning trees for Euclidean first-passage percolation. Ann. Probab. 29(2), 577–623 (2001)

    MathSciNet  MATH  Google Scholar 

  7. Wüthrich, M.V.: Asymptotic behaviour of semi-infinite geodesics for maximal increasing subsequences in the plane. In: In and Out of equilibrium (Mambucaba, 2000), vol. 51 of Progr. Probab., pp. 205–226. Birkhäuser Boston, Boston (2002)

  8. Cator, E., Pimentel, L.P.R.: A shape theorem and semi-infinite geodesics for the Hammersley model with random weights. ALEA 8, 163–175 (2011)

    MathSciNet  MATH  Google Scholar 

  9. Cator, E., Pimentel, L.P.R.: Busemann functions and equilibrium measures in last passage percolation models. Probab. Theory Relat. Fields 154(1–2), 89–125 (2012)

    Article  MathSciNet  Google Scholar 

  10. Damron, M., Hanson, J.: Busemann functions and infinite geodesics in two-dimensional first-passage percolation. Commun. Math. Phys. 325(3), 917–963 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  11. Bakhtin, Y.: Inviscid Burgers equation with random kick forcing in noncompact setting. Electron. J. Probab. 21, 50 (2016)

    Article  MathSciNet  Google Scholar 

  12. Georgiou, N., Rassoul-Agha, F., Seppäläinen, T.: Variational formulas and cocycle solutions for directed polymer and percolation models. Commun. Math. Phys. 346(2), 741–779 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  13. Rassoul-Agha, F., Seppäläinen, T., Yilmaz, A.: Variational formulas and disorder regimes of random walks in random potentials. To appear in Bernoulli (2016)

  14. Georgiou, N., Rassoul-Agha, F., Seppäläinen, T.: Stationary cocycles and Busemann functions for the corner growth model. ArXiv e-prints, October (2015)

  15. Auffinger, A., Hanson, J., Damron, M.: 50 years of first passage percolation. ArXiv e-prints, November (2015)

  16. Bakhtin, Y., Khanin, K.: Localization and Perron–Frobenius theory for directed polymers. Mosc. Math. J. 10(4), 667–686 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Georgiou, N., Rassoul-Agha, F., Seppäläinen, T., Yilmaz, A.: Ratios of partition functions for the log-gamma polymer. Ann. Probab. 43(5), 2282–2331 (2015)

    Article  MathSciNet  Google Scholar 

  18. Bakhtin, Y., Li, L.: Thermodynamic limit for directed polymers and stationary solutions of the Burgers equation. Accepted at Commun. Pure Appl. Math. (2016)

  19. Freidlin, M.I., Wentzell, A.D.: Random perturbations of dynamical systems, volume 260 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg, 3rd edn. Translated from the 1979 Russian original by Joseph Szücs (2012)

  20. Sinaĭ, Y.G.: Two results concerning asymptotic behavior of solutions of the Burgers equation with force. J. Stat. Phys. 64(1–2), 1–12 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  21. Weinan, E., Khanin, K., Mazel, A., Sinai, Y.: Invariant measures for Burgers equation with stochastic forcing. Ann. of Math. 151(3), 877–960 (2000)

    Article  MathSciNet  Google Scholar 

  22. Iturriaga, R., Khanin, K.: Burgers turbulence and random Lagrangian systems. Commun. Math. Phys. 232(3), 377–428 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  23. Gomes, D., Iturriaga, R., Khanin, K., Padilla, P.: Viscosity limit of stationary distributions for the random forced Burgers equation. Mosc. Math. J. 5(3), 613–631 (2005)

    MathSciNet  MATH  Google Scholar 

  24. Dirr, N., Souganidis, P.E.: Large-time behavior for viscous and nonviscous Hamilton–Jacobi equations forced by additive noise. SIAM J. Math. Anal. 37(3), 777–796 (2005)

    Article  MathSciNet  Google Scholar 

  25. Bakhtin, Y.: Burgers equation with random boundary conditions. Proc. Am. Math. Soc. 135(7), 2257–2262 (2007)

    Article  MathSciNet  Google Scholar 

  26. Debussche, A., Vovelle, J.: Invariant measure of scalar first-order conservation laws with stochastic forcing. Probab. Theory Relat. Fields 163(3–4), 575–611 (2015)

    Article  MathSciNet  Google Scholar 

  27. Hoang, V.H., Khanin, K.: Random Burgers equation and Lagrangian systems in non-compact domains. Nonlinearity 16(3), 819–842 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  28. Suidan, T.M.: Stationary measures for a randomly forced Burgers equation. Commun. Pure Appl. Math. 58(5), 620–638 (2005)

    Article  MathSciNet  Google Scholar 

  29. Bakhtin, Y.: The Burgers equation with Poisson random forcing. Ann. Probab. 41(4), 2961–2989 (2013)

    Article  MathSciNet  Google Scholar 

  30. Bakhtin, Y., Cator, E., Khanin, K.: Space-time stationary solutions for the Burgers equation. J. Am. Math. Soc. 27(1), 193–238 (2014)

    Article  MathSciNet  Google Scholar 

  31. Kesten, H.: On the speed of convergence in first-passage percolation. Ann. Appl. Probab. 3(2), 296–338 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  32. Weinan, E., Mattingly, J.C., Sinai, Y.: Gibbsian dynamics and ergodicity for the stochastically forced Navier–Stokes equation. Commun. Math. Phys. 224(1), 83–106 (2001). (Dedicated to Joel L. Lebowitz)

    Article  ADS  MathSciNet  Google Scholar 

  33. Bricmont, J., Kupiainen, A., Lefevere, R.: Ergodicity of the 2D Navier–Stokes equations with random forcing. Commun. Math. Phys. 224(1), 65–81 (2001). (Dedicated to Joel L. Lebowitz)

    Article  ADS  MathSciNet  Google Scholar 

  34. Kuksin, S., Shirikyan, A.: Stochastic dissipative PDEs and Gibbs measures. Commun. Math. Phys. 213(2), 291–330 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  35. Hairer, M., Mattingly, J.C.: Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing. Ann. of Math. 164(3), 993–1032 (2006)

    Article  MathSciNet  Google Scholar 

  36. Hairer, M., Mattingly, J.C.: Spectral gaps in Wasserstein distances and the 2D stochastic Navier–Stokes equations. Ann. Probab. 36(6), 2050–2091 (2008)

    Article  MathSciNet  Google Scholar 

  37. Hairer, M., Mattingly, J.C.: A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs. Electron. J. Probab. 16(23), 658–738 (2011)

    Article  MathSciNet  Google Scholar 

  38. Kuksin, S.B.: The Eulerian limit for 2D statistical hydrodynamics. J. Stat. Phys. 115(1–2), 469–492 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  39. Kuksin, S.B.: Eulerian limit for 2D Navier–Stokes equation and damped/driven KdV equation as its model. Tr. Mat. Inst. Steklova 259(Anal. i Osob. Ch. 2), 134–142 (2007)

    MathSciNet  MATH  Google Scholar 

  40. Kuksin, S.B.: On distribution of energy and vorticity for solutions of 2D Navier–Stokes equation with small viscosity. Commun. Math. Phys. 284(2), 407–424 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  41. Bakhtin, Y.: Existence and uniqueness of stationary solutions for 3D Navier–Stokes system with small random forcing via stochastic cascades. J. Stat. Phys. 122(2), 351–360 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  42. Newman, C.M.: A surface view of first-passage percolation. In: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pp. 1017–1023. Birkhäuser, Basel (1995)

  43. Young, L.-S.: Stochastic stability of hyperbolic attractors. Ergod. Theory Dyn. Syst. 6(2), 311–319 (1986)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

YB gratefully acknowledges partial support from NSF through Grant DMS-1460595.

Author information

Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, 251 Mercer St, New York, NY, 10012, USA

    Yuri Bakhtin & Liying Li

Authors
  1. Yuri Bakhtin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Liying Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuri Bakhtin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bakhtin, Y., Li, L. Zero Temperature Limit for Directed Polymers and Inviscid Limit for Stationary Solutions of Stochastic Burgers Equation. J Stat Phys 172, 1358–1397 (2018). https://doi.org/10.1007/s10955-018-2104-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-018-2104-z

Keywords

Search

Navigation