On smoothing properties of transition semigroups associated to a class of SDEs with jumps
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 4, p. 1347-1370

We prove smoothing properties of nonlocal transition semigroups associated to a class of stochastic differential equations (SDE) in d driven by additive pure-jump Lévy noise. In particular, we assume that the Lévy process driving the SDE is the sum of a subordinated Wiener process Y (i.e. Y=WT, where T is an increasing pure-jump Lévy process starting at zero and independent of the Wiener process W) and of an arbitrary Lévy process independent of Y, that the drift coefficient is continuous (but not necessarily Lipschitz continuous) and grows not faster than a polynomial, and that the SDE admits a Feller weak solution. By a combination of probabilistic and analytic methods, we provide sufficient conditions for the Markovian semigroup associated to the SDE to be strong Feller and to map L p ( d ) to continuous bounded functions. A key intermediate step is the study of regularizing properties of the transition semigroup associated to Y in terms of negative moments of the subordinator T.

Nous établissons des propriétés de lissage de semi-groupes de transition non locaux associés à une classe d’équations différentielles stochastiques dans d dirigées par un bruit additif de Lévy sans partie continue. En particulier, nous supposons que le processus de Lévy est la somme d’un processus de Wiener subordonné Y (i.e. Y=WT, où T est un processus croissant de Lévy sans partie continue, avec T 0 =0, indépendant du processus de Wiener W) et d’un processus de Lévy arbitraire indépendant de Y; que le coefficient de dérive est continu (mais pas nécessairement lipschitzien) et à croissance polynomiale; et que la EDS admet une solution faible fellerienne. Par une combinaison de méthodes probabilistes et analytiques, nous fournissons des conditions suffisantes pour le semi-groupe markovien associé à l’EDS soit fortement fellérien et envoye L p ( d ) dans les fonctions continues bornées. Une étape intermédiaire essentielle est l’étude de certaines propriétés régularisantes du semi-groupe de transition associé à Y qui dépendent de moments négatifs du subordinateur T.

DOI : https://doi.org/10.1214/13-AIHP559
Classification:  60G30,  60G51,  60H07,  60H10,  60J35
Keywords: Lévy processes, subordination, transition semigroups, non-local operators, Malliavin calculus
@article{AIHPB_2014__50_4_1347_0,
     author = {Kusuoka, Seiichiro and Marinelli, Carlo},
     title = {On smoothing properties of transition semigroups associated to a class of SDEs with jumps},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {4},
     year = {2014},
     pages = {1347-1370},
     doi = {10.1214/13-AIHP559},
     zbl = {06377557},
     mrnumber = {3269997},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_4_1347_0}
}
Kusuoka, Seiichiro; Marinelli, Carlo. On smoothing properties of transition semigroups associated to a class of SDEs with jumps. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 4, pp. 1347-1370. doi : 10.1214/13-AIHP559. http://www.numdam.org/item/AIHPB_2014__50_4_1347_0/

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