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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 \(T\).

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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