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On smoothing properties of transition semigroups associated to a class of SDEs with jumps. (English. French summary) Zbl 1319.60127
Summary: We prove smoothing properties of nonlocal transition semigroups associated to a class of stochastic differential equations (SDE) in $$\mathbb{R} ^{d}$$ 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 $$L_{p}(\mathbb{R} ^{d})$$ 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 $$T$$.

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J35 Transition functions, generators and resolvents 60J75 Jump processes (MSC2010) 60G51 Processes with independent increments; Lévy processes 47D07 Markov semigroups and applications to diffusion processes 60H07 Stochastic calculus of variations and the Malliavin calculus 60G30 Continuity and singularity of induced measures
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