[Paper Review] On structure of regular Dirichlet subspaces for one-dimensional Brownian motion

Liping Li, Jiangang Ying|arXiv (Cornell University)|Dec 5, 2014

Mathematical Dynamics and Fractals10 references3 citations

TL;DR

This paper investigates the structure of regular Dirichlet subspaces for one-dimensional Brownian motion using the trace method. It shows that the trace of such a subspace on the boundary $F = G^c$ of a measure-dense open set $G$ is a non-local Dirichlet form, while its orthogonal complement corresponds to a time-changed absorbing Brownian motion via darning transform, revealing a decomposition of the trace process into local and non-local components.

ABSTRACT

The main purpose of this paper is to explore the structure of regular subspaces of 1-dim Brownian motion. As outlined in \cite{FMG} every such regular subspace can be characterized by a measure-dense set $G$. When $G$ is open, $F=G^c$ is the boundary of $G$ and, before leaving $G$, the diffusion associated with the regular subspace is nothing but Brownian motion. Their traces on $F$ still inherit the inclusion relation, in other words, the trace Dirichlet form of regular subspace on $F$ is still a regular subspace of trace Dirichlet form of one-dimensional Brownian motion on $F$. Moreover we have proved that the trace of Brownian motion on $F$ may be decomposed into two part, one is the trace of the regular subspace on $F$, which has only the non-local part and the other comes from the orthogonal complement of the regular subspace, which has only the local part. Actually the orthogonal complement of regular subspace corresponds to a time-changed Brownian motion after a darning transform.

Motivation & Objective

• To understand the intrinsic structure of regular Dirichlet subspaces of one-dimensional Brownian motion beyond domain differences.
• To analyze how the trace of a Dirichlet subspace on the boundary $F = G^c$ relates to the trace of the original Brownian motion.
• To decompose the trace Dirichlet form of Brownian motion on $F$ into a non-local part (from the subspace) and a local part (from its orthogonal complement).
• To characterize the orthogonal complement of a regular Dirichlet subspace as a time-changed absorbing Brownian motion via darning transform.

Proposed method

• The trace method is applied to decompose the Dirichlet form of one-dimensional Brownian motion on the boundary $F = G^c$ of a measure-dense open set $G$.
• The trace of the regular Dirichlet subspace on $F$ is shown to inherit the inclusion relation and to contain only the non-local part of the trace form.
• The orthogonal complement of the subspace trace is identified as a strongly local Dirichlet form, corresponding to a time-changed absorbing Brownian motion.
• A darning transform is used to construct a regular representation of the orthogonal complement, showing equivalence to a time-changed Brownian motion on a modified state space $\mathbb{R}^*_0$.
• The Beurling-Deny decomposition is applied to the trace forms to separate local and non-local components.
• The analysis relies on the characterization of regular Dirichlet subspaces via a scaling function $s(x) = \int_0^x \mathbf{1}_G(y)\,dy$ and speed measure being Lebesgue measure.

Experimental results

Research questions

• RQ1 How does the trace of a regular Dirichlet subspace on the boundary $F = G^c$ relate to the trace of the original one-dimensional Brownian motion?
• RQ2 Can the trace Dirichlet form of Brownian motion on $F$ be decomposed into a non-local part and a local part?
• RQ3 What is the probabilistic interpretation of the orthogonal complement of a regular Dirichlet subspace in the trace space?
• RQ4 How does the darning transform relate to the regular representation of the orthogonal complement of the trace subspace?
• RQ5 Does the trace of the regular Dirichlet subspace on $F$ contain only the non-local component of the trace form?

Key findings

• The trace of a regular Dirichlet subspace on $F = G^c$ is a regular Dirichlet subspace of the trace of Brownian motion on $F$, and it contains only the non-local part of the Beurling-Deny decomposition.
• The orthogonal complement of the trace subspace is a strongly local Dirichlet form, and its Beurling-Deny decomposition contains only the diffusion (local) part.
• The orthogonal complement has a regular representation as a time-changed absorbing Brownian motion on $\mathbb{R}^*_0$, with speed measure $\mu^*_0 = m^*_0$.
• The darning transform yields a common regular representation for the D-space of the orthogonal complement, showing equivalence to a time-changed absorbing Brownian motion.
• The trace process of Brownian motion on $F$ decomposes into two orthogonal parts: one from the subspace (non-local) and one from its complement (local), with the latter being equivalent to a time-changed absorbing Brownian motion.
• The decomposition is unique up to a constant if the subspace is recurrent, and the trace forms satisfy $\check{\mathcal{F}}_{\text{e}} = \check{\mathcal{F}}^{(s)}_{\text{e}} \oplus \check{\mathcal{G}}^{(s)}$.