A new approach to the $L^p$-theory of $-\Delta + b\cdot\nabla$, and its applications to Feller processes with general drifts

Abstract

We develop a detailed regularity theory of $-\Delta +b\cdot\nabla$ in $L^p(\mathbb R^d)$, for a wide class of vector fields. The $L^p$-theory allows us to construct associated strong Feller process in $C_\infty(\mathbb R^d)$. Our starting object is an operator-valued function, which, we prove, coincides with the resolvent of an operator realization of $-\Delta + b\cdot \nabla$, the generator of a holomorphic $C_0$-semigroup on $L^p(\mathbb R^d)$. Then the very form of the operator-valued function yields crucial information about smoothness of the domain of the generator.