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

2012/08/14 by Kusuoka, Seiichiro, Marinelli, Carlo
#Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1208.2860

Abstract

We prove smoothing properties of nonlocal transition semigroups associated to a class of stochastic differential equations (SDE) 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 and of an independent arbitrary Lévy 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.

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