Abstract
We present a sufficient condition for fractional Laplacian with gradient perturbation to generate a sub-Markovian C 0-semigroup on $${L^1(\mathbb{R}^d, dx)}$$ . The condition also yields the ultracontractivity of the semigroup and an upper on-diagonal estimate of the associated transition kernel. Based on the subordination technique, the extension to general pure jump Levy process with gradient perturbation is studied. As a direct application, we obtain sufficient conditions for the strong Feller property of stochastic differential equations driven by additive Levy process.
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