  Quasidiffusions are, by definition, time-changed Brownian motions on certain
closed subset of $\mathbb{R}$. They admit an explicit representation of
Dirichlet forms in terms of so-called speed measures. The Fukushima subspace of
a Dirichlet form means another regular Dichichlet form on the same state space
but having a smaller Dirichlet space. In this paper we aim to solve the problem
of Fukushima subspaces for quasidiffusions. The main result obtains all
Fukushima subspaces and characterizes their structures. In addition, we will
also give criteria for the uniqueness of Fukushima subspaces and the existence
of minimal Fukushima subspace.
