Class of smooth functions in Dirichlet spaces
### Class of smooth functions in Dirichlet spaces

#### P.J. Fitzsimmons, Liping Li

Given a regular Dirichlet form (E,F) on a fixed domain E of R^{d}, we first indicate
that the basic assumption C^{1}_{c}(E) be contained in F is equivalent to the fact that each coordinate function
f_{i}(x) = x_{i} locally belongs to F. Our research starts from these two different
viewpoints. On one hand, we shall explore when C^{1}_{c}(E) is a special standard core of F and
give some useful characterizations. On the other hand, we shall describe Fukushima's
decompositions of (E,F) with respect to the coordinates functions, especially discuss when
their martingale part is a standard Brownian motion and what we can say about their zero
energy part. Finally, when we put these two kinds of discussions together, an interesting class
of stochastic differential equations is raised. They have uncountably many solutions that do not
depend on the initial condition.

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September 26, 2017