On smoothing properties of transition semigroups associated to a class of SDEs with jumps

Kusuoka, Seiichiro; Marinelli, Carlo

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 4, pp. 1347-1370.

Nous établissons des propriétés de lissage de semi-groupes de transition non locaux associés à une classe d’équations différentielles stochastiques dans ${ℝ}^d$ 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. $Y=W\circ T$, 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 $L_p\left({ℝ}^d\right)$ 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 $T$.

We prove smoothing properties of nonlocal transition semigroups associated to a class of stochastic differential equations (SDE) in ${ℝ}^d$ 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. $Y=W\circ T$, 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 $L_p\left({ℝ}^d\right)$ 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 $T$.

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