Strong uniqueness of finite dimensional Dirichlet operators with singular drifts

TL;DR

This paper proves the strong uniqueness and essential self-adjointness of finite-dimensional Dirichlet operators with singular drifts in -dimensional space, broadening the class of drifts for which these properties hold.

Contribution

It establishes the L^r-uniqueness and self-adjointness of Dirichlet operators with less regular drifts, using elliptic regularity and Dirichlet form techniques.

Findings

Proves L^r-uniqueness for r in (1,2]

Shows essential self-adjointness under singular drift conditions

Allows  ext{nabla} ho in L^d_{loc} or L^{2+ ext{epsilon}}_{loc}

Abstract

We show the $L^r(\mathbb{R}^d, \mu)$-uniqueness for any $r \in (1, 2]$ and the essential self-adjointness of a Dirichlet operator $Lf = \Delta f +\langle \frac{1}{\rho}\nabla \rho , \nabla f \rangle$, $f \in C_0^{\infty}(\mathbb{R}^d)$ with $d \geq 3$ and $\mu=\rho dx$. In particular, $\nabla \rho$ is allowed to be in $L^d_{loc}(\mathbb{R}^d, \mathbb{R}^d)$ or in $L^{2+\varepsilon}_{loc}(\mathbb{R}^d, \mathbb{R}^d)$ for some $\varepsilon>0$, while $\rho$ is required to be locally bounded below and above by strictly positive constants. The main tools in this paper are elliptic regularity results for divergence and non-divergence type operators and basic properties of Dirichlet forms and their resolvents.