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

TL;DR

This paper develops a comprehensive $L^p$-theory for the operator $- abla^2 + b abla$, enabling the construction of associated Feller processes with general drifts and providing insights into the smoothness of the generator's domain.

Contribution

It introduces a new $L^p$-theory for $- abla^2 + b abla$, linking operator resolvents to semigroup generators and Feller processes with broad drift conditions.

Findings

Established a resolvent operator coinciding with the generator of a holomorphic $C_0$-semigroup.

Proved the operator-valued function provides key regularity information.

Constructed strong Feller processes for a wide class of vector fields.

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.