241 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
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