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A note on reflected Dirichlet forms

Marcel Schmidt

TL;DR

This work develops an algebraic construction of the active reflected Dirichlet form for a general Dirichlet form by decomposing it into a main part and a killing part and then maximizing the extensions of each part. It proves that the active main part ${\\mathcal{E}}^{(M)}_a$ is the maximal Dirichlet form dominating the original form, and that, in the absence of killing, the active reflected form ${\\mathcal{E}}^{\rm ref}_a$ is the maximal Silverstein extension; when killing is present, a maximal Silverstein extension need not exist, as shown by a concrete counterexample. The paper also establishes density of continuous functions in the domain of the active main part for regular forms, enabling a topological realization of the reflected process on a compactification minus a point, and clarifies the relation between reflected forms and extended/energy form frameworks. Overall, it advances both the analytic construction and the topological interpretation of reflected Dirichlet structures, with implications for boundary behavior of Markov processes and extensions beyond their lifetime.

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.

A note on reflected Dirichlet forms

TL;DR

This work develops an algebraic construction of the active reflected Dirichlet form for a general Dirichlet form by decomposing it into a main part and a killing part and then maximizing the extensions of each part. It proves that the active main part is the maximal Dirichlet form dominating the original form, and that, in the absence of killing, the active reflected form is the maximal Silverstein extension; when killing is present, a maximal Silverstein extension need not exist, as shown by a concrete counterexample. The paper also establishes density of continuous functions in the domain of the active main part for regular forms, enabling a topological realization of the reflected process on a compactification minus a point, and clarifies the relation between reflected forms and extended/energy form frameworks. Overall, it advances both the analytic construction and the topological interpretation of reflected Dirichlet structures, with implications for boundary behavior of Markov processes and extensions beyond their lifetime.

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.

Paper Structure

This paper contains 11 sections, 27 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dirichlet forms and domination
  4. Energy forms and extended Dirichlet spaces
  5. Reflected Dirichlet forms
  6. The main part
  7. The killing part and the reflected Dirichlet form
  8. The (active) reflected Dirichlet form and its maximality
  9. Reflected regular Dirichlet forms
  10. Closed forms on metrizable topological vector spaces
  11. A lemma on monotone forms

Key Result

Lemma 2.1

Let ${\mathcal{E}}$ be a Dirichlet form. For $f,f_1,\ldots,f_n \in L^2(m)$ the inequalities imply In particular, $D({\mathcal{E}}) \cap L^\infty(m)$ is an algebra and $D({\mathcal{E}})$ is a lattice.

Theorems & Definitions (74)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Remark
  • Remark
  • Lemma 2.3
  • proof
  • Remark
  • Remark
  • Lemma 2.4
  • ...and 64 more