Concave Majorant of Stochastic Processes and Burgers Turbulence

Raphaël Lachièze-Rey ()
Additional contact information
Raphaël Lachièze-Rey: Univ. Lille 1

Journal of Theoretical Probability, 2012, vol. 25, issue 2, 313-332

Abstract: 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.

Keywords: Lévy processes; Burgers equation; Convex hull; 60G17; 60D05; 35Q35; 35R60; 60J99 (search for similar items in EconPapers)
Date: 2012
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10959-011-0354-7 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:25:y:2012:i:2:d:10.1007_s10959-011-0354-7

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1007/s10959-011-0354-7

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().