Théorèmes limites pour certaines fonctionnelles associées aux processus stables sur l’espace de Hölder

Abstract

In this paper we study the Hölder regularity property of the local time of a symmetric stable process of index 1 < α ≤ 2 and of its fractional derivative as a doubly indexed process with respect to the space and the time variables. As an application we establish some limit theorems for occupation times of one-dimensional symmetric stable processes in the space of Hölder continuous functions. Our results generalize those obtained by Fitzsimmons and Getoor for stable processes in the space on continuous functions. The limiting processes are fractional derivatives and Hilbert transforms of local times. 2000 Mathematics Subject Classification. 60F25, 60G52. Mots-clés. Processus stables, temps local, transformé de Hilbert, dérivée fractionnaire, fonctionnels additives, norme de Hölder. Cette note aété complétée au moment où le second auteurétait en visite au Centre de Recerca Matemàtica (CRM) Barcelona, Spain.

Convergence in law for certain additive functionals of symmetric stable processes under strong topology

“…We will always denote by (L x t , (t,x) ∈ R + × R) its local time. The following lemma due to Ait Ouahra and Eddahbi [2] presents the mixed Hölder regularity of local time in x and in t. where · p = [E| · | p ] 1/p and C is a constant depending only of α and p.…”

“…Let {X t , t 0} be a symmetric stable process with an index α ∈ (1,2]. This means that X is a real-valued process with stationary independent increment such that,…”

On fractional derivatives of the local time of a symmetric stable process as a doubly indexed process

“…(ii) A similar result for mixed regularity of local time can be found in [10] for Brownian motion case, also in [2] for the case of local time of symmetric stable Lévy process, in [23] for local time of fractional Brownian motion and for the local time of the multifractional Brownian motion in [9].…”

“…Fractional derivatives of the Brownian local time and related functionals have arisen naturally in some new limit theorems for occupation times of a linear Brownian motion (see, e.g., Yamada [25]). Similar limit theorems of stable processes have been obtained by Fitzsimmons and Getoor [15], Ait Ouahra and Eddahbi [2] and recently by Ait Ouahra and Ouali [3] for fractional Brownian motion. Fractional derivatives of local time have been discussed for physical purposes in [14].…”

“…Let ( , ≥ 0) be a symmetric stable process with index in (1,2]. It is well known that this process is a real valued process with stationary independent increments such that = − | | , ∈ ℝ.…”