  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.
