The $\alpha$-orthogonal complements of regular subspaces of 1-dim Brownian motion

TL;DR

This paper investigates the orthogonal complement of regular subspaces of 1-dimensional Brownian motion, revealing its connection to the $\alpha$-harmonic equation with Neumann boundary conditions.

Contribution

It characterizes the orthogonal complement of regular subspaces of 1D Brownian motion and links it to solutions of the $\alpha$-harmonic equation with Neumann boundary conditions.

Findings

Orthogonal complement closely related to $\alpha$-harmonic equation.

Provides a characterization of the orthogonal complement in terms of boundary conditions.

Establishes a connection between Dirichlet form subspaces and boundary value problems.

Abstract

Roughly speaking, a regular subspace of a Dirichlet form is a subspace, which is also a regular Dirichlet form, on the same state space. In particular, the domain of regular subspace is a closed subspace of the Hilbert space induced by the domain and $\alpha$-inner product of original Dirichlet form. We shall investigate the orthogonal complement of regular subspace of 1-dimensional Brownian motion in this paper. Our main results indicate that this orthogonal complement has a very close connection with the $\alpha$-harmonic equation under Neumann boundary condition.