Régularité du temps local du processus symétrique stable en norme besov

Abstract

Let be a real-valued symmetric stable process of index β,1 < β ≤ 2, i.e., a Lévy process with characteristic function Let be the local time of the process X. The existence of an a.s continuous (as a function of two variables) version of L(t, x) has been proved in [1,6]The mean zero Gaussian process with covariance is said to be associated with Markov process X if: where is the symmetric transition density of X Marcus and Rosen used in [11] the Dynkin isomorphism theorem (cf. Lemma 3.2), which relates sample path properties of the local time L(t, x) to those of the associated Gaussian process G of X, to study the almost sure variation in the spatial variable of the local time of X. In the special case the process X is a Brownian motion and it is known (cf. [4,9]) that for each fixed t > 0 and p < ∞, the random function satisfies a.s a Hölder condition in the Lp norm, with the modulus of smoothness In this paper the Holder condition for the local time of the symmetric stable process of index 1 < β ≤ 2 in the Lp norm is studied. The main result – Theorem 3.1 – states that for each t > 0 and 1 ≤ p < ∞-the modulus of smoothness of the local time in the Lp norm-satisfies a. s Our theorem can be regarded as an extension of the result of [4]. The main tool of the proof is the Dynkin isomorphism theorem and some regularity results for Gaussian processes presented in [8]