Abstract
The asymptotic solution of the inviscid Burgers equations with initial potential ψ is closely related to the convex hull of the graph of ψ.
In this paper, we study this convex hull, and more precisely its extremal points, if ψ is a stochastic process. The times where those extremal points are reached, called extremal times, form a negligible set for Lévy processes, their integrated processes, and Itô processes. We examine more closely the case of a Lévy process with bounded variation. Its extremal points are almost surely countable, with accumulation only around the extremal values. These results are derived from the general study of the extremal times of ψ+f, where ψ is a Lévy process and f a smooth deterministic drift.
These results allow us to show that, for an inviscid Burgers turbulence with a compactly supported initial potential ψ, the only point capable of being Lagrangian regular is the time T where ψ reaches its maximum, and that is indeed a regular point if and only if 0 is regular for both half-lines. As a consequence, if the turbulence occurs on a non-compact interval, there are almost surely no Lagrangian regular points.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Avellaneda, M., E, W.: Statistical properties of shocks in Burgers turbulence. Commun. Math. Phys. 172, 13–38 (1995)
Avellaneda, M.: Statistical properties of shocks in Burgers turbulence, II: Tail probabilities for velocities, shock-strengths and rarefaction intervals. Commun. Math. Phys. 169, 45–59 (1995)
Bertoin, J.: Lévy Processes. Cambridge Tracts in Mathematics (1996)
Cole, J.D.: On a quasi linear parabolic equation occurring in aerodynamics. Q. Appl. Math. 9, 225–236 (1951)
Bertoin, J.: Regularity of the half-line for Lévy processes. Bull. Sci. Math. 121(5), 345–354 (1997)
Bertoin, J.: Large deviation estimates in Burgers turbulence with stable noise initial data. J. Stat. Phys. 91(3/4), 655–667 (1998)
Bertoin, J.: Structure of shocks in Burgers turbulence with stable noise initial data. Commun. Math. Phys. 203, 397–406 (1998)
Bertoin, J.: The convex minorant of the Cauchy process. Electron. Commun. Probab. 5, 51–55 (2000)
Davydov, Yu.: Enveloppes convexes des processus gaussiens. Ann. Inst. Henri Poincaré Probab. Stat. 38(6), 847–861 (2002)
Groenboom, P.: The concave majorant of Brownian motion. Ann. Probab. 11(4), 1016–1027 (1983)
Hopf, E.: The partial differential equation u t +uu x =μu xx . Commun. Pure Appl. Math. 3, 201–230 (1950)
Pitman, J.W.: Remarks on the convex minorant of Brownian motion. In: Seminar on Stochastic Processes Evanston (1982)
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, Berlin (1991)
She, Z.S., Aurell, E., Frisch, U.: The inviscid Burgers equation with initial data of Brownian type. Commun. Math. Phys. 148, 623–641 (1992)
Sinai, Y.G.: Statistics of shocks in solutions of inviscid Burgers equation. Commun. Math. Phys. 148, 601–621 (1992)
Skorohod, A.V.: Random Processes with Independent Increments. Kluwer Academic, Dordrecht (1991)
Winkel, M.: Limit clusters in the inviscid Burgers turbulence with certain random initial velocities. J. Stat. Phys. 107, 893–917 (2002)
Winkel, M.: Burgers turbulence initialized by a regenerative impulse. Stoch. Process. Appl. 93, 241–268 (2001)
Woyczinski, W.A.: Burgers KPZ-Turbulence. Springer, Berlin (1999)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lachièze-Rey, R. Concave Majorant of Stochastic Processes and Burgers Turbulence. J Theor Probab 25, 313–332 (2012). https://doi.org/10.1007/s10959-011-0354-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10959-011-0354-7