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Erick Herbin

Publications and source records attributed to Erick Herbin.

10 recordsLinked to original sources
arXiv2020

Hölder regularity of a Wiener integral in abstract space

In this article, we propose a way to consider processes indexed by a collection $\mathcal{A}$ of subsets of a general set $\mathcal{T}$. A large class of vector spaces, manifolds and continuous $\mathbb{R}$-trees are particular cases. Lattice-theoretic and topological assumptions are considered separately with a view to clarifying the exposition. We then define a Wiener-type integral $Y_A = \int_A f\,\text dX$ for all $A\in\mathcal{A}$ for a deterministic function $f:\mathcal{T} \rightarrow \mathbb{R}$ and a set-indexed Lévy process $X$. It is a particular case of Raput and Rosinski [40], but our setting enables a quicker construction and yields more properties about the sample paths of $Y.$ Finally, bounds for the Hölder regularity of $Y$ are given which indicate how the regularities of $f$ and $X$ contributes to that of $Y$. This follows the works of Jaffard [24] and Balança and Herbin [6].

math.PR↗
arXiv2015

Local Hölder regularity for set-indexed processes

In this paper, we study the Hölder regularity of set-indexed stochastic processes defined in the framework of Ivanoff-Merzbach. The first key result is a Kolmogorov-like Hölder-continuity Theorem, whose novelty is illustrated on an example which could not have been treated with anterior tools. Increments for set-indexed processes are usually not simply written as $X_U-X_V$, hence we considered different notions of Hölder-continuity. Then, the localization of these properties leads to various definitions of Hölder exponents, which we compare to one another. In the case of Gaussian processes, almost sure values are proved for these exponents, uniformly along the sample paths. As an application, the local regularity of the set-indexed fractional Brownian motion is proved to be equal to the Hurst parameter uniformly, with probability one.

math.PR↗
arXiv2012

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 $L^2$-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.

math.PR↗
arXiv2012

From almost sure local regularity to almost sure Hausdorff dimension for Gaussian fields

Fine regularity of stochastic processes is usually measured in a local way by local Hölder exponents and in a global way by fractal dimensions. Following a previous work of Adler, we connect these two concepts for multiparameter Gaussian random fields. More precisely, we prove that almost surely the Hausdorff dimensions of the range and the graph in any ball $B(t_0,ρ)$ are bounded from above using the local Hölder exponent at $t_0$. We define the deterministic local sub-exponent of Gaussian processes, which allows to obtain an almost sure lower bound for these dimensions. Moreover, the Hausdorff dimensions of the sample path on an open interval are controlled almost surely by the minimum of the local exponents. Then, we apply these generic results to the cases of the multiparameter fractional Brownian motion, the multifractional Brownian motion whose regularity function $H$ is irregular and the generalized Weierstrass function, whose Hausdorff dimensions were unknown so far.

math.PR↗
arXiv2012

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 Hölder 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.

math.PR↗
arXiv2012

The set-indexed Lévy process: Stationarity, Markov and sample paths properties

We present a satisfactory definition of the important class of Lévy processes indexed by a general collection of sets. We use a new definition for increment stationarity of set-indexed processes to obtain different characterizations of this class. As an example, the set-indexed compound Poisson process is introduced. The set-indexed Lévy process is characterized by infinitely divisible laws and a Lévy-Khintchine representation. Moreover, the following concepts are discussed: projections on flows, Markov properties, and pointwise continuity. Finally the study of sample paths leads to a Lévy-Itô decomposition. As a corollary, the semimartingale property is proved.

math.PR↗
arXiv2008

Stochastic 2-microlocal analysis

A lot is known about the Hölder regularity of stochastic processes, in particular in the case of Gaussian processes. Recently, a finer analysis of the local regularity of functions, termed 2-microlocal analysis, has been introduced in a deterministic frame: through the computation of the so-called 2-microlocal frontier, it allows in particular to predict the evolution of regularity under the action of (pseudo-) differential operators. In this work, we develop a 2-microlocal analysis for the study of certain stochastic processes. We show that moments of the increments allow, under fairly general conditions, to obtain almost sure lower bounds for the 2-microlocal frontier. In the case of Gaussian processes, more precise results may be obtained: the incremental covariance yields the almost sure value of the 2-microlocal frontier. As an application, we obtain new and refined regularity properties of fractional Brownian motion, multifractional Brownian motion, stochastic generalized Weierstrass functions, Wiener and stable integrals.

math.PR↗
arXiv2008

Stationarity and Self-similarity Characterization of the Set-indexed Fractional Brownian Motion

The set-indexed fractional Brownian motion (sifBm) has been defined by Herbin-Merzbach (2006) for indices that are subsets of a metric measure space. In this paper, the sifBm is proved to statisfy a strenghtened definition of increment stationarity. This new definition for stationarity property allows to get a complete characterization of this process by its fractal properties: The sifBm is the only set-indexed Gaussian process which is self-similar and has stationary increments. Using the fact that the sifBm is the only set-indexed process whose projection on any increasing path is a one-dimensional fractional Brownian motion, the limitation of its definition for a self-similarity parameter 0<H<1/2 is studied, as illustrated by some examples. When the indexing collection is totally ordered, the sifBm can be defined for 0<H<1.

math.PR↗
arXiv2006

The Multiparameter Fractional Brownian Motion

We define and study the multiparameter fractional Brownian motion. This process is a generalization of both the classical fractional Brownian motion and the multiparameter Brownian motion, when the condition of independence is relaxed. Relations with the Lévy fractional Brownian motion and with the fractional Brownian sheet are discussed. Different notions of stationarity of the increments for a multiparameter process are studied and applied to the fractional property. Using self-similarity we present a characterization for such processes. Finally, behavior of the multiparameter fractional Brownian motion along increasing paths is analysed.

math.PR↗