[Paper Review] Regular Dirichlet extensions of one-dimensional Brownian motion

Liping Li, Jiangang Ying|arXiv (Cornell University)|Jun 2, 2016

Mathematical Dynamics and Fractals11 references3 citations

TL;DR

This paper characterizes all regular Dirichlet extensions of one-dimensional Brownian motion by showing they decompose into countably many invariant intervals and a polar set, with each interval governed by a scale function in a specific class. The key contribution is proving that a pure jump Dirichlet form can have proper regular Dirichlet subspaces, resolving a long-standing open problem.

ABSTRACT

The regular Dirichlet extension is the dual concept of regular Dirichlet subspace. The main purpose of this paper is to characterize all the regular Dirichlet extensions of one-dimensional Brownian motion and to explore their structures. It is shown that every regular Dirichlet extension of one-dimensional Brownian motion may essentially decomposed into at most countable disjoint invariant intervals and an $\mathcal{E}$-polar set relative to this regular Dirichlet extension. On each invariant interval the regular Dirichlet extension is characterized uniquely by a scale function in a given class. To explore the structure of regular Dirichlet extension we apply the idea introduced in [17], we formulate the trace Dirichlet forms and attain the darning process associated with the restriction to each invariant interval of the orthogonal complement of $H^1_\mathrm{e}(\mathbb{R})$ in the extended Dirichlet space of the regular Dirichlet extension. As a result, we find an answer to a long-standing problem whether a pure jump Dirichlet form has proper regular Dirichlet subspaces.

Motivation & Objective

• To characterize all regular Dirichlet extensions of one-dimensional Brownian motion, defined as regular Dirichlet forms containing the standard Dirichlet form of Brownian motion.
• To resolve the long-standing open question of whether a pure jump Dirichlet form can admit a proper regular Dirichlet subspace.
• To explore the structural properties of such extensions via trace Dirichlet forms and darning processes on invariant intervals.
• To clarify the role of the 'small jump' component in generating regular Dirichlet subspaces, particularly in non-local settings.
• To establish a duality between regular Dirichlet extensions and subspaces by analyzing orthogonal complements in the extended Dirichlet space.

Proposed method

• Decomposes the state space into at most countably many disjoint invariant intervals and an $\mathcal{E}$-polar set relative to the extension.
• Uses the theory of trace Dirichlet forms to analyze the restriction of the extended Dirichlet space to each invariant interval.
• Applies the darning process construction to the orthogonal complement of $H^1_{\mathrm{e}}(\mathbb{R})$ in the extended Dirichlet space of the extension.
• Characterizes each extension on an invariant interval via a scale function $\mathtt{s}$ satisfying $\mathtt{s}' = 0$ or $1$ a.e., ensuring the associated diffusion is a time-changed Brownian motion.
• Employs the Beurling-Deny decomposition and Lévy system analysis to compare jump components and identify structural differences between extensions.
• Uses the Revuz correspondence to show equivalence of Lévy systems for two symmetric pure jump Hunt processes, despite different Dirichlet forms.

Experimental results

Research questions

• RQ1 Can a pure jump Dirichlet form admit a proper regular Dirichlet subspace?
• RQ2 How can all regular Dirichlet extensions of one-dimensional Brownian motion be fully characterized?
• RQ3 What is the role of the 'small jump' component in the structure of regular Dirichlet extensions?
• RQ4 How do darning processes and trace Dirichlet forms help describe the structure of extensions on invariant intervals?
• RQ5 What is the relationship between the Dirichlet form and the associated Hunt process in terms of irreducibility and hitting probabilities?

Key findings

• Every regular Dirichlet extension of one-dimensional Brownian motion decomposes into at most countably many disjoint invariant intervals and an $\mathcal{E}$-polar set.
• On each invariant interval, the extension is uniquely characterized by a scale function $\mathtt{s} \in \mathbf{S}(\mathbb{R})$, i.e., strictly increasing and absolutely continuous with $\mathtt{s}' = 0$ or $1$ a.e.
• The trace Dirichlet form on each invariant interval corresponds to a darning process obtained by collapsing the interval into a single point, reflecting the local structure of the extension.
• A pure jump Dirichlet form can admit a proper regular Dirichlet subspace, as demonstrated by the example on the Cantor set $K$ in Example 3.20.
• The associated Hunt process $\check{X}$ of the extended form $\check{\mathcal{E}}$ is not irreducible; it only jumps between endpoints $a_n$ and $b_n$ of intervals where $a_n > -\infty$, $b_n < \infty$, and $K \setminus \{a_n, b_n\}$ is $\check{\mathcal{E}}$-polar.
• The trace of Brownian motion $\check{B}$ on $K$ is irreducible and hits any point in $K$ with positive probability, even though its energy form $\frac{1}{2}\check{\mathbf{D}}$ does not reflect this behavior directly.