Le drap linéaire fractionnaire stable (LFSS) est un champ aléatoire qui généralise le drap brownien fractionnaire en remplaçant la mesure gaussienne dans sa représentation linéaire fractionnaire par une mesure α-stable, . Dans cette Note nous étendons certaines propriétés du temps local montrées pour le cas gaussien au cas symétrique α-stable. Le -LFSS, , est défini sur et prend ses valeurs dans , le cas correspondant au mouvement linéaire fractionnaire stable (LFSM). Ce champ est principalement paramétré par . Nous considérons un champ aléatoire à valeurs dans , le -LFSS, , défini en prenant d copies indépendantes d'un -LFSS. Nous montrons que, si , alors le -LFSS de paramètre H admet un temps local. De plus, dans le cas où ses trajectoires sont continues, i.e., pour , quand les paramètres vérifient , nous établissons la bicontinuité de ce temps local.
Linear fractional stable sheets (LFSS) are a class of random fields containing the class of fractional Brownian sheets (FBS) by allowing, in the linear fractional representation of the FBS, the random measure to be α-stable with . In this Note, we extend some properties of the local time shown in the Gaussian case to the symmetric α-stable case. For any , an -LFSS is a real valued random field defined on . When , the process is called linear fractional stable motion (LFSM). For , an -LFSS is mainly parameterized by a multidimensional index . Let be fixed, we consider a random field defined on and taking its values in , an -LFSS, whose components are d independent copies of the same -LFSS. We show that, if , then the -LFSS with index H has a local time. Moreover, when the sample path of the LFSS is continuous, that is, for , when , we show that the local time is jointly continuous.
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Antoine Ayache 1 ; François Roueff 2 ; Yimin Xiao 3
@article{CRMATH_2007__344_10_635_0, author = {Antoine Ayache and Fran\c{c}ois Roueff and Yimin Xiao}, title = {Joint continuity of the local times of linear fractional stable sheets}, journal = {Comptes Rendus. Math\'ematique}, pages = {635--640}, publisher = {Elsevier}, volume = {344}, number = {10}, year = {2007}, doi = {10.1016/j.crma.2007.03.028}, language = {en}, }
TY - JOUR AU - Antoine Ayache AU - François Roueff AU - Yimin Xiao TI - Joint continuity of the local times of linear fractional stable sheets JO - Comptes Rendus. Mathématique PY - 2007 SP - 635 EP - 640 VL - 344 IS - 10 PB - Elsevier DO - 10.1016/j.crma.2007.03.028 LA - en ID - CRMATH_2007__344_10_635_0 ER -
Antoine Ayache; François Roueff; Yimin Xiao. Joint continuity of the local times of linear fractional stable sheets. Comptes Rendus. Mathématique, Volume 344 (2007) no. 10, pp. 635-640. doi : 10.1016/j.crma.2007.03.028. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.03.028/
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