On smoothing properties of transition semigroups associated to a class   of SDEs with jumps
  We prove smoothing properties of nonlocal transition semigroups associated to a class of stochastic differential equations (SDE) driven by additive pure-jump L\'evy noise. In particular, we assume that the L\'evy process driving the SDE is the sum of a subordinated Wiener process and of an independent arbitrary L\'evy process, 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 $L_p$ to continuous bounded functions. A key intermediate step is the study of regularizing properties of the transition semigroup associated to the subordinated Wiener process in terms of negative moments of the subordinator. 
Seiichiro Kusuoka, Carlo Marinelli
math.PR math.AP