Quasidiffusions are defined as time-changed Brownian motions on certain closed subsets of \(\mathbb {R}\) . They can be represented explicitly using Dirichlet forms, which involve certain speed measures. The Fukushima subspace of a Dirichlet form refers to another regular Dirichlet form on the same state space, but with a smaller Dirichlet space. This paper aims to comprehensively address the problem of Fukushima subspaces for quasidiffusions. The primary finding provides a full characterization of all Fukushima subspaces and their structures. Additionally, criteria are established for the existence of proper Fukushima subspaces and minimal Fukushima subspaces.
