Abstract
We consider the inviscid Burgers equation where the initial datum is given by a stable (Lévy) noise. The asymptotic behavior of the tail distribution of the solution is described; the decay is much faster in the case when the stable noise is completely skewed to the left.
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REFERENCES
M. Avellaneda and W. E, Statistical properties of shocks in Burgers turbulence, Commun. Math. Phys. 172:13–38 (1995).
M. Avellaneda, Statistical properties of shocks in Burgers turbulence, II: Tail probabilities for velocities, shock-strengths and rarefaction intervals, commun. Math. Phys. 169:45–59 (1995).
J. Bertoin, Lévy Processes (Cambridge University Press, Cambridge, 1996).
J. M. Burgers, The Nonlinear Diffusion Equation (Dordrecht, Reidel, 1974).
A. J. Chorin, Lectures on Turbulence Theory (Publish or Perish, Boston, 1975).
J. D. Cole, On a quasi linear parabolic equation occurring in aerodynamies, Quart. Appl. Math. 9:225–236 (1951).
R. Davis and S. Resnick, Limit theory for moving averages of random variables with regularly varying tail probabilities, Ann. Probab. 13:179–195 (1985).
C. Dellacherie and P. A. Meyer, Probabilités et potentiel, Vol. II. Théorie des martingales (Hermann, Paris, 1980).
T. Funaki, D. Surgailis, and W. A. Woyczynski, Gibbs-Cox random fields and Burgers' turbulence, Ann. Appl. Probab. 5:461–492 (1995).
B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Variables (Addison-Wesley, Cambridge, 1954).
E. Hopf, The partial differential equation u 1 + uu x = μu xx , Comm. Pure Appl. Math. 3:201–230 (1950).
I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables (Wolters-Noordhoff, Groningen, 1971).
J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes (Springer, Berlin, 1987).
A. W. Janicki and W. A. Woyczynski, Hausdorff dimension of regular points in stochastic flows with Lévy α-stable initial data, J. Stat. Phys. 86:277–299 (1997).
S. A. Molchanov, D. Surgailis, and W. A. Woyczynski, Hyperbolic asymptotics in Burgers' turbulence and extremal processes, Commun. Math. Phys. 168:209–226 (1995).
R. Ryan, Large-deviation analysis of Burgers turbulence with white-noise initial data, Comm. Pure Appl. Math. 51:47–75 (1998).
G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance (Chapman and Hall, London, 1994).
Z. S. She, E. Aurell, and U. Frisch, The inviscid Burgers equation with initial data of Brownian type, Commun. Math. Phys. 148:623–641 (1992).
W. A. Woyczynski, Göttingen Lectures on Burgers-KPZ Turbulence, Lecture Notes in Maths, Springer (to appear).
V. M. Zolotarev, One-dimensional Stable Distributions (Amer. Math. Soc., Providence, 1986).
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Bertoin, J. Large-Deviations Estimates in Burgers Turbulence with Stable Noise Initial Data. Journal of Statistical Physics 91, 655–667 (1998). https://doi.org/10.1023/A:1023081728243
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DOI: https://doi.org/10.1023/A:1023081728243