ON SYMMETRIC LINEAR DIFFUSIONS.

The main purpose of this paper is to explore the structure of local and regular Dirichlet forms associated with symmetric one-dimensional diffusions, which are also called symmetric linear diffusions. Let (E,F) be a regular and local Dirichlet form on L²(I,m), where I is an interval and m is a fully supported Radon measure on I. We shall first present a complete representation for (E,F), which shows that (E,F) lives on at most countable disjoint "effective" intervals with an "adapted" scale function on each interval, and any point outside these intervals is a trap of the one-dimensional diffusion. Furthermore, we shall give a necessary and sufficient condition for C8 c (I) being a special standard core of (E,F) and shall identify the closure of C8 c (I) in (E,F) when C8 c (I) is contained but not necessarily dense in F relative to the E1/2 1 -norm. This paper is partly motivated by a result of Hamza's that was stated in a theorem of Fukushima, Oshima, and Takeda's and that provides a different point of view to this theorem. To illustrate our results, many examples are provided. [ABSTRACT FROM AUTHOR]