21 research outputs found
Uniform multifractal structure of stable trees
In this work, we investigate the spectrum of singularities of random stable
trees with parameter . We consider for that purpose the scaling
exponents derived from two natural measures on stable trees: the local time
and the mass measure , providing as well a purely
geometrical interpretation of the latter exponent. We first characterise the
uniform component of the multifractal spectrum which exists at every level
of stable trees and corresponds to large masses with scaling index
for the mass measure
(or equivalently for the local
time). In addition, we investigate the distribution of vertices appearing at
random levels with exceptionally large masses of index
. Finally, we discuss more precisely the
order of the largest mass existing on any subset of a stable
tree, characterising the former with the packing dimension of the set .Comment: 50 pages. Major overhaul of the paper, correcting Theorem 4 and
adding the study of the mass measure spectru
Image sets of fractional Brownian sheets
Let be an -fractional Brownian
sheet with Hurst index . The main objective of
the present paper is to study the Hausdorff dimension of the image sets
, and , in the dimension
case . Following the seminal work of
Kaufman (1989), we establish uniform dimensional properties on , answering
questions raised by Khoshnevisan et al (2006) and Wu and Xiao (2009).
For the purpose of this work, we introduce a refinement of the sectorial
local-nondeterminism property which can be of independent interest to the study
of other fine properties of fractional Brownian sheets.Comment: 14 pages, 1 figur
Some sample path properties of multifractional Brownian motion
The geometry of the multifractional Brownian motion (mBm) is known to present
a complex and surprising form when the Hurst function is greatly irregular.
Nevertheless, most of the literature devoted to the subject considers
sufficiently smooth cases which lead to sample paths locally similar to a
fractional Brownian motion (fBm). The main goal of this paper is therefore to
extend these results to a more general frame and consider any type of
continuous Hurst function. More specifically, we mainly focus on obtaining a
complete characterization of the pointwise H\"older regularity of the sample
paths, and the Box and Hausdorff dimensions of the graph. These results, which
are somehow unusual for a Gaussian process, are illustrated by several
examples, presenting in this way different aspects of the geometry of the mBm
with irregular Hurst functionsComment: 33 pages, 2 figure
A set-indexed Ornstein-Uhlenbeck process
The purpose of this article is a set-indexed extension of the well-known
Ornstein-Uhlenbeck process. The first part is devoted to a stationary
definition of the random field and ends up with the proof of a complete
characterization by its -continuity, stationarity and set-indexed Markov
properties. This specific Markov transition system allows to define a general
\emph{set-indexed Ornstein-Uhlenbeck (SIOU) process} with any initial
probability measure. Finally, in the multiparameter case, the SIOU process is
proved to admit a natural integral representation.Comment: 13 page
2-microlocal analysis of martingales and stochastic integrals
Recently, a new approach in the fine analysis of stochastic processes sample
paths has been developed to predict the evolution of the local regularity under
(pseudo-)differential operators. In this paper, we study the sample paths of
continuous martingales and stochastic integrals. We proved that the almost sure
2-microlocal frontier of a martingale can be obtained through the local
regularity of its quadratic variation. It allows to link the H\"older
regularity of a stochastic integral to the regularity of the integrand and
integrator processes. These results provide a methodology to predict the local
regularity of diffusions from the fine analysis of its coefficients. We
illustrate our work with examples of martingales with unusual complex
regularity behavior and square of Bessel processes.Comment: 40 pages, 3 figure
Fine regularity of stochastic processes and 2-microlocal analysis
Les travaux présentés dans cette thèse s'intéressent à la géométrie fractale de processus stochastiques à travers le prisme d'un outil appelé l'analyse 2-microlocale. Ce dernier est issu d'une autre branche des mathématiques, l'analyse fonctionnelle et l'étude des équations aux dérivées partielles, et s'est avéré être pertinent pour décrire la géométrie fine de fonctions déterministes ou de processus aléatoires, généralisant notamment les exposants de Hölder classiques. Nous envisageons ainsi dans ce manuscrit différentes classes de processus, traitant en premier lieu le cas des martingales continues et de l'intégrale stochastique d'Ito. La régularité 2-microlocale de ces derniers fait notamment apparaître un autre concept, la pseudo frontière 2-microlocale, étroitement lié à son aîné. Nous appliquons également ce formalisme d'étude à une classe de processus gaussiens : le mouvement brownien multifractionnaire. Nous caractérisons ainsi sa régularité 2-microlocale et hölderienne, et déterminons dans un deuxième temps la forme générale de la dimension fractale de ses trajectoires. Dans notre étude portant sur les processus de Lévy, nous combinons le formalisme 2-microlocale à l'analyse multifractale, permettant alors de mettre en évidence des comportements géométriques n'étant pas captés par les outils usuels. Nous obtenons également en corollaire le spectre multifractal des processus fractionnaires de Lévy. Enfin, dans une dernière partie, nous nous intéressons à la définition et aux propriétés de certains processus de Markov multiparamètres, pouvant être plus généralement indicés par des ensembles.The work presented in this thesis concerns the study of the fractal geometry of stochastic processes using the formalism of 2-microlocal analysis. The latter has been introduced in another branch of mathematics -functional analysis- but has also proved to be relevant to describe the geometry of deterministic functions or random processes, extending in particular the classic Hölder exponents. Several classes of processes are investigated in this manuscript, beginning with continuous martingales and Ito integrals. In particular, the characterisation of the 2-microlocal regularity of the latter leads to the introduction of a closely related concept: the pseudo 2-microlocal frontier. We also investigate using this formalism a class of Gaussian processes called multifractional Brownian motion and obtain a fine description of its Hölder and 2-microlocal behaviours. In addition, we characterize entirely the Hausdorff and Box dimensions of its graph. In our study of Lévy processes, we combine the 2-microlocal formalism and multifractal analysis to describe their regularity, exhibiting in particular some subtle geometrical behaviours which are not captured by classic tools. Furthermore, as a corollary of this result, we also determine the multifractal spectrum of another family of processes: the fractional Lévy processes. Lastly, we also define a class of multiparameter and set-indexed Markov processes and study its properties
Régularité fine de processus stochastiques et analyse 2-microlocale
The work presented in this thesis concerns the study of the fractal geometry of stochastic processes using the formalism of 2-microlocal analysis. The latter has been introduced in another branch of mathematics -functional analysis- but has also proved to be relevant to describe the geometry of deterministic functions or random processes, extending in particular the classic Hölder exponents. Several classes of processes are investigated in this manuscript, beginning with continuous martingales and Ito integrals. In particular, the characterisation of the 2-microlocal regularity of the latter leads to the introduction of a closely related concept: the pseudo 2-microlocal frontier. We also investigate using this formalism a class of Gaussian processes called multifractional Brownian motion and obtain a fine description of its Hölder and 2-microlocal behaviours. In addition, we characterize entirely the Hausdorff and Box dimensions of its graph. In our study of Lévy processes, we combine the 2-microlocal formalism and multifractal analysis to describe their regularity, exhibiting in particular some subtle geometrical behaviours which are not captured by classic tools. Furthermore, as a corollary of this result, we also determine the multifractal spectrum of another family of processes: the fractional Lévy processes. Lastly, we also define a class of multiparameter and set-indexed Markov processes and study its properties.Les travaux présentés dans cette thèse s'intéressent à la géométrie fractale de processus stochastiques à travers le prisme d'un outil appelé l'analyse 2-microlocale. Ce dernier est issu d'une autre branche des mathématiques, l'analyse fonctionnelle et l'étude des équations aux dérivées partielles, et s'est avéré être pertinent pour décrire la géométrie fine de fonctions déterministes ou de processus aléatoires, généralisant notamment les exposants de Hölder classiques. Nous envisageons ainsi dans ce manuscrit différentes classes de processus, traitant en premier lieu le cas des martingales continues et de l'intégrale stochastique d'Ito. La régularité 2-microlocale de ces derniers fait notamment apparaître un autre concept, la pseudo frontière 2-microlocale, étroitement lié à son aîné. Nous appliquons également ce formalisme d'étude à une classe de processus gaussiens : le mouvement brownien multifractionnaire. Nous caractérisons ainsi sa régularité 2-microlocale et hölderienne, et déterminons dans un deuxième temps la forme générale de la dimension fractale de ses trajectoires. Dans notre étude portant sur les processus de Lévy, nous combinons le formalisme 2-microlocale à l'analyse multifractale, permettant alors de mettre en évidence des comportements géométriques n'étant pas captés par les outils usuels. Nous obtenons également en corollaire le spectre multifractal des processus fractionnaires de Lévy. Enfin, dans une dernière partie, nous nous intéressons à la définition et aux propriétés de certains processus de Markov multiparamètres, pouvant être plus généralement indicés par des ensembles