Markovian Extensions of Symmetric Second Order Elliptic Differential Operators

TL;DR

This paper classifies Markovian self-adjoint extensions of symmetric elliptic operators on bounded domains, linking them to Dirichlet forms and boundary conditions, and analyzes their properties like heat kernel bounds.

Contribution

It provides a complete classification of Markovian extensions via Dirichlet forms and boundary conditions, with explicit correspondence and multiple equivalent characterizations.

Findings

Explicit classification of Markovian extensions

Characterization of domains via Wentzell boundary conditions

Analysis of properties like heat kernel bounds and recurrence

Abstract

Let $\Omega\subset{\mathbb R}^n$ be bounded with a smooth boundary $\Gamma$ and let $S$ be the symmetric operator in $L^2(\Omega)$ given by the minimal realization of a second order elliptic differential operator. We give a complete classification of the Markovian self-adjoint extensions of $S$ by providing an explicit one-to-one correspondence between such extensions and the class of Dirichlet forms in $L^2(\Gamma)$ which are additively decomposable by the bilinear form of the Dirichlet-to-Neumann operator plus a Markovian form. By such a result two further equivalent classifications are provided: the first one is expressed in terms of an additive decomposition of the bilinear forms associated to the extensions, the second one uses the additive decomposition of the resolvents provided by Krein's formula. The Markovian part of the decomposition allows to characterize the operator domain of the corresponding extension in terms of Wentzell-type boundary conditions. Some properties of the extensions, and of the corresponding Dirichlet forms, semigroups and heat kernels, like locality, regularity, irreducibility, recurrence, transience, ultracontractivity and Gaussian bounds are also discussed.