### Abstract

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 ○ 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 Lp (ℝ^{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.

Original language | English |
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Pages (from-to) | 1347-1370 |

Number of pages | 24 |

Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |

Volume | 50 |

Issue number | 4 |

DOIs | |

Publication status | Published - Nov 1 2014 |

Externally published | Yes |

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### Keywords

- Lévy processes
- Malliavin calculus
- Non-local operators
- Subordination
- Transition semigroups

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annales de l'institut Henri Poincare (B) Probability and Statistics*,

*50*(4), 1347-1370. https://doi.org/10.1214/13-AIHP559