On symmetric linear diffusions

Abstract:

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 $(\mathcal {E},\mathcal {F})$ be a regular and local Dirichlet form on $L^2(I,m)$, where $I$ is an interval and $m$ is a fully supported Radon measure on $I$. We shall first present a complete representation for $(\mathcal {E},\mathcal {F})$, which shows that $(\mathcal {E},\mathcal {F})$ 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 $C_c^\infty (I)$ being a special standard core of $(\mathcal {E},\mathcal {F})$ and shall identify the closure of $C_c^\infty (I)$ in $(\mathcal {E},\mathcal {F})$ when $C_c^\infty (I)$ is contained but not necessarily dense in $\mathcal {F}$ relative to the $\mathcal {E}_1^{1/2}$-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.