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On the Hausdorff Dimension of Regular Points of Inviscid Burgers Equation with Stable Initial Data

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Abstract

Consider an inviscid Burgers equation whose initial data is a Lévy α-stable process Z with α>1. We show that when Z has positive jumps, the Hausdorff dimension of the set of Lagrangian regular points associated with the equation is strictly smaller than 1/α, as soon as α is close to 1. This gives a partially negative answer to a Conjecture of Janicki and Woyczynski (J. Stat. Phys. 86(1–2):277–299, 1997). Along the way, we contradict a recent Conjecture of Z. Shi (http://www.proba.jussieu.fr/pageperso/smalldev/pbfile/pb4.pdf) about the lower tails of integrated stable processes.

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References

  1. Bernyk, V., Dalang, R.C., Peskir, G.: The law of the supremum of a stable Levy process with no negative jumps. To appear in the Annals of Probability. Available at http://arxiv.org/pdf/0706.1503 (2007)

  2. Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  3. Bertoin, J.: The inviscid Burgers equation with Brownian initial velocity. Commun. Math. Phys. 193(2), 397–406 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Bingham, N.H.: Fluctuation theory in continuous time. Adv. Appl. Probab. 7(4), 705–766 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  5. Caravenna, F., Deuschel, J.-D.: Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction. To appear in the Annals of Probability. Available at http://arxiv.org/pdf/math/0703434 (2007)

  6. Carraro, L., Duchon, J.: Equation de Burgers avec conditions initiales à accroissements indépendants et homogènes. Ann. Inst. H. Poincaré Anal. Non Linéaire 15(4), 431–458 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chaumont, L.: Conditionings and path decomposition for Lévy processes. Stoch. Process. Appl. 64(1), 39–54 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chaumont, L.: Excursion normalisée, méandre et pont pour les processus de Lévy stables. Bull. Sci. Math. 121(5), 377–403 (1997)

    MathSciNet  MATH  Google Scholar 

  9. Devulder, A., Shi, Z., Simon, T.: The lower tail problem for the area of a symmetric stable process. Talk given at the conference Small Deviations Probabilities and Related Topics II, 2005. Abstract available at http://www.pdmi.ras.ru/EIMI/2005/sd/talk/simon.pdf

  10. Devulder, A., Shi, Z., Simon, T.: The lower tail problem for integrated stable processes. Unpublished manuscript

  11. Goldman, M.: On the first passage of the integrated Wiener process. Ann. Math. Stat. 42(6), 2150–2155 (1971)

    Article  MATH  Google Scholar 

  12. Handa, K.: A remark on shocks in inviscid Burgers’ turbulence. In: Nonlinear Waves and Weak Turbulence with Applications in Oceanography and Condensed Matter Physics, pp. 339–345. Progr. Nonlinear Differential Equations Appl., vol. 11. Birkhäuser, Boston (1993)

    Google Scholar 

  13. Holley, R.: Remarks on the FKG inequalities. Commun. Math. Phys. 36(2), 227–231 (1974)

    Article  ADS  MathSciNet  Google Scholar 

  14. Janicki, A.W., Woyczynski, W.A.: Hausdorff dimension of regular points in stochastic Burgers flows with Lévy α-stable initial data. J. Stat. Phys. 86(1–2), 277–299 (1997)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  15. Marsalle, L.: Hausdorff measures and capacities for increase times of stable processes. Potential Anal. 9(2), 181–200 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. 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)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  17. Novikov, A.A.: The martingale approach in problems on the time of the first crossing of nonlinear boundaries. Tr. Mat. Inst. Steklov 158, 130–152 (1981)

    MATH  Google Scholar 

  18. Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes. Chapman & Hall, New York (1994)

    MATH  Google Scholar 

  19. Shi, Z.: Lower tails of some integrated processes. Abstract available on the website Small Deviations for Stochastic Processes and Related Topics, Problem Panel, Problem 4. http://www.proba.jussieu.fr/pageperso/smalldev/pbfile/pb4.pdf

  20. Simon, T.: The lower tail problem for homogeneous functionals of stable processes with no negative jumps. ALEA Lat. Am. J. Probab. Math. Stat. 3, 165–179 (2007)

    MathSciNet  MATH  Google Scholar 

  21. Sinai, Ya.G.: Statistics of shocks in solutions of inviscid Burgers equation. Commun. Math. Phys. 148(3), 601–621 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Woyczynski, W.A.: Burgers-KPZ Turbulence Göttingen Lectures. Lect. Notes Math., vol. 1700. Springer, Berlin (1998)

    MATH  Google Scholar 

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  1. Equipe d’Analyse et Probabilités, Université d’Evry-Val d’Essonne, Boulevard François Mitterrand, 91025, Evry Cedex, France

    Thomas Simon

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Simon, T. On the Hausdorff Dimension of Regular Points of Inviscid Burgers Equation with Stable Initial Data. J Stat Phys 131, 733–747 (2008). https://doi.org/10.1007/s10955-008-9508-0

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  • DOI: https://doi.org/10.1007/s10955-008-9508-0

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