Roughly speaking, the regular subspace of a Dirichlet form is also a regular Dirichlet form on the same state space. It inherits the same form of the original Dirichlet form but possesses a smaller domain. What we are concerned in this paper are the regular subspaces of associated Dirichlet forms of skew product diffusions. A skew product diffusion X is a symmetric Markov process on the product state space and expressed as
where is a symmetric diffusion on for , and A is a positive continuous additive functional of . One of our main results indicates that any skew product type regular subspace of X, say
can be characterized as follows: the associated smooth measure of is equal to that of A, and corresponds to a regular subspace of for . Furthermore, we shall make some discussions on rotationally invariant diffusions on , which are special skew product diffusions on . Our main purpose is to extend a regular subspace of rotationally invariant diffusion on to a new regular Dirichlet form on . More precisely, fix a regular Dirichlet form on the state space . Its part Dirichlet form on is denoted by . Let be a regular subspace of . We want to find a regular subspace of such that the part Dirichlet form of on is exactly . If exists, we call it a regular extension of . We shall prove that, under a mild assumption, any rotationally invariant type regular subspace of has a unique regular extension.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11271240
Funding statement: Research supported in part by NSFC grant 11271240.
This work was initiated when the first author visited the University of California, San Diego. He would like to thank Professor Patrick J. Fitzsimmons for his hospitality and many helpful discussions. We also want to thank the anonymous reviewers for pointing out the article  that we missed before.
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