Behaviour of linear multifractional stable motion: membership of a critical Hölder space

Abstract

The study of path behaviour of stochastic processes is a classical topic in probability theory and related areas. In this frame, a natural question one can address is: whether or not sample paths belong to a critical Hölder space? The answer to this question is negative in the case of Brownian motion and many other stochastic processes: it is well-known that despite the fact that Brownian paths satisfy, on each compact interval I, a Hölder condition of any order strictly less than 1 / 2, they fail to belong to the critical Hölder space . In this article, we show that a different phenomenon happens in the case of linear multifractional stable motion (LMSM): for any given compact interval one can find a critical Hölder space to which sample paths belong. Among other things, this result improves an upper estimate, recently derived by Biermé and Lacaux on behaviour of LMSM, by showing that the logarithmic factor in it is not needed.

Keywords:

• Linear multifractional stable motions
• wavelet series representations
• moduli of continuity
• Hölder regularity

Acknowledgements

The authors are very grateful to the editor and to the anonymous associate editor and two referees for their valuable comments and suggestions which have led to great improvements of the article.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work has been partially supported by ANR [grant number ANR-11-BS01-0011] (AMATIS), GDR 3475 (Analyse Multifractale), and [grant number ANR-11-LABX-0007-01] (CEMPI).