REGULARITY - 2012 - Annual activity report

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REGULARITY - 2012

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Section: New Results

Fine regularity of Lévy processes

Participant : Paul Balança.

This ongoing work focuses on the fine regularity of one-parameter Lévy processes. The main idea of this study is to use the framework of stochastic 2-microlocal analysis (introduced and developed in [16] ,[33] ) to refine sample paths results obtained in [65] .

The latter describes entirely the multifractal spectrum of Lévy processes, i.e. the Hausdorff geometry of level sets ${(E_{h})}_{h \in 𝐑_{+}}$ of the pointwise exponent. These are usually called the iso-Hölder sets of $X$ and are given by

E_{h} = \{t \in 𝐑 : α_{X, t} = h\} for every h \in 𝐑_{+} \cup {+ \infty} .

The multifractal spectrum is itself defined as the localized Hausdorff dimension of the previous sets, i.e.

d_{X} (h, V) = {dim}_{ℋ} (E_{h} \cap V) for every h \in 𝐑_{+} \cup {+ \infty} and V \in 𝒪 (open sets in R).

(11)

[65] states that under a mild assumption on the Lévy measure $π$ , a Lévy process $X$ with no Brownian component almost surely satisfies

\forall V \in 𝒪; d_{X} (h, V) = \{\begin{matrix} β h & if h \in [0, 1 / β]; \\ - \infty & if h \in (1 / β, + \infty], \end{matrix}

(12)

where the Blumenthal-Getoor exponent $β$ is given by

β = inf \{δ \geq 0 : \int_{𝐑^{d}} (1 \land {∥ x ∥}^{δ}) π (d x) < \infty\} .

(13)

Since classic multifractal analysis focuses on the pointwise exponent, it is natural from our point of view to integrate the 2-microlocal frontier into this description. More precisely, we focus on the dichotomy usual/unusual regularity, corresponding to the sets ${({\tilde{E}}_{h})}_{h \in 𝐑_{+}}$ and ${({\hat{E}}_{h})}_{h \in 𝐑_{+}}$ :

{\tilde{E}}_{h} = \{t \in E_{h} : \forall s^{'} \in 𝐑; σ_{X, t} (s^{'}) = (h + s^{'}) \land 0\} and {\hat{E}}_{h} = E_{h} ∖ {\tilde{E}}_{h} .

The collection ${({\tilde{E}}_{h})}_{h \in 𝐑_{+}}$ represents times at which the 2-microlocal behaviour is rather common (i.e. the slope is equal to one), whereas at points which belong ${({\hat{E}}_{h})}_{h \in 𝐑_{+}}$ , the 2-microlocal frontier has an unusual form.

Then, our main result states that sample paths of a Lévy process $X$ with no Brownian component almost surely satisfy

\forall V \in 𝒪; {dim}_{ℋ} ({\tilde{E}}_{h} \cap V) = \{\begin{matrix} β h & if h \in [0, 1 / β]; \\ - \infty & if h \in (1 / β, + \infty] . \end{matrix}

(14)

Furthermore, the collection of sets ${({\hat{E}}_{h})}_{h \in 𝐑_{+}}$ enjoys almost surely

\forall V \in 𝒪; {dim}_{ℋ} ({\hat{E}}_{h} \cap V) \leq \{\begin{matrix} 2 β h - 1 & if h \in (1 / 2 β, 1 / β); \\ - \infty & if h \in [0, 1 / 2 β] \cup [1 / β, + \infty] . \end{matrix}

(15)

These results clearly extend those obtained in [65] since we know that the pointwise exponent is completely characterize by the 2-microlocal frontier. Moreover, it also proves that from a Hausdorff dimension point of view, the common regularity is a 2-microlocal frontier with a slope equal to one.

Nevertheless, equation (15 ) also exhibits some unusual behaviours, corresponding to times ${({\hat{E}}_{h})}_{h \in 𝐑_{+}}$ , that are not captured by the classic multifractal spectrum. The existence of such particular times highly depends on the structure of the Lévy measure, and not only the value of the Blumenthal-Getoor exponent which is therefore not sufficient to characterize entirely the fine regularity. This last aspect of the study illustrates the fact that 2-microlocal analysis is an interesting tool for the study of stochastic processes' regularity since some sample paths' properties can not be captured by common tools such as Hölder exponents.

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