A note on reflected Dirichlet forms

TL;DR

This paper provides an algebraic construction of reflected Dirichlet forms, explores their maximal extensions, and offers an alternative construction for regular forms on compactified spaces.

Contribution

It introduces an algebraic method for constructing reflected Dirichlet forms and analyzes their maximal Silverstein extensions, especially in the absence of killing parts.

Findings

Reflected Dirichlet forms are the maximal Silverstein extension without killing.

Dirichlet forms may lack a maximal Silverstein extension if killing is present.

An alternative construction for regular forms on compactified spaces is provided.

Abstract

In this paper we give an algebraic construction of the (active) reflected Dirich- let form. We prove that it is the maximal Silverstein extension whenever the given form does not possess a killing part and we prove that Dirichlet forms need not have a maximal Silverstein extension if a killing is present. For regular Dirichlet forms we provide an alternative construction of the reflected process on a compactification (minus one point) of the underlying space.