Journal of Statistical Physics

, Volume 167, Issue 6, pp 1546–1554 | Cite as

The Inviscid Burgers Equation with Fractional Brownian Initial Data: The Dimension of Regular Lagrangian Points



Fractional Brownian motion, H-FBM, of index \(0<H<1\) is considered as initial velocity in the inviscid Burgers equation. It is shown that the Hausdorff dimension of regular Lagrangian points at any moment t is equal to H. This fact validates the Sinai-Frisch conjecture known since 1992. We find also that the integrated H-FBM does not exceed a fixed positive level in the interval \((-T, T)\) with probability having the log-asymptotics: \((H-1+o(1))\mathrm{log} T\).


Burgers equation Fractional Brownian motion One-sided exit problem Persistence probability 



I am very grateful to the reviewers for their careful reading of the article and constructive comments. This research had been announced in the proposal that was supported by the Russian Science Foundation through the research project 17-11-01052.


  1. 1.
    Aurzada, F., Dereich, S.: Universality of the asymptotics of the one-sided exit problem for integrated processes. Ann. Inst. Henri Poincare Probab. Stat. 49(1), 236–251 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Aurzada, F., Guillotin-Plantard, N., Pene, F.: Persistence probabilities for stationary increment processes. arXiv:1606.00236, 2016
  3. 3.
    Aurzada, F., Simon, T.: Persistence probabilities and exponents. Levy matters, Vol. 183–221, Lecture Notes in Math., 2149, Springer, Berlin (2015)Google Scholar
  4. 4.
    Bray, A.J., Majumdar, S.N., Schehr, G.: Persistence and first-passage properties in non-equilibrium systems. Adv. Phys. 62(3), 225–361 (2013)ADSCrossRefGoogle Scholar
  5. 5.
    Dembo, A., Ding, J., Gao, F.: Persistence of iterated partial sums. Ann. Inst. Henri Poincare Probab. Stat. 49(3), 873–884 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, New York (1995)MATHGoogle Scholar
  7. 7.
    Fernique, X.: Regularite des trajectories des functions aleatoires gaussiannes. Lecture Notes in Mathematics. 480. Springer, Berlin (1975)Google Scholar
  8. 8.
    Handa, K.: A remark on shocks in inviscid turbulence. In: Fitzmanrice, N., et al. (eds.) Nonlinear Waves and Turbulence, pp. 339–345. Birkhauser, Boston (1993)CrossRefGoogle Scholar
  9. 9.
    Lifshits, M.: Lectures on Gaussian Processes. Springer, Berlin (2012)CrossRefMATHGoogle Scholar
  10. 10.
    Molchan, G.: Maximum of fractional Brownian motion: probabilities of small values. Commun. Math. Phys. 205(1), 97–111 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Molchan, G.: Survival, exponents for some Gaussian processes. Int. J. Stoch. Anal. (2012). doi: 10.1155/2012/137271. Article ID 137271
  12. 12.
    Molchan, G., Khokhlov, A.: Small values of the maximum for the integral of fractional Brownian motion. J. Stat. Phys. 114(3–4), 923–946 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Sinai, Y.G.: Statistics of shocks in solutions of the inviscid Burgers equation. Commun. Math. Phys. 148, 601–621 (1992)ADSMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    She, Z., Aurell, E., Frish, U.: The inviscid Burgers equation with initial data of Brownian type. Commun. Math. Phys. 148, 623–642 (1992)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Vergassola, M., Dubrulle, B., Frisch, U., Noullez, A.: Burgers equation, Devil’s starcases and the mass distribution for large-scale structures. Astron. Astrophys. 289, 325–356 (1994)ADSGoogle Scholar

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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Institute of Earthquake Prediction Theory and Mathematical GeophysicsRussian Academy of ScienceMoscowRussian Federation

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