On regular Dirichlet subspaces of \(H^1(I)\) and associated linear diffusions
Let \(I\) denote either a bounded open interval or \(\mathbb R\). Starting with the classical (regular, recurrent, local) Dirichlet space \[ H^ {1} (I) := \{ u\in L ^ {2} (I) \mid u \text{ absolutely continuous},\;u' \in L ^{2} (I) \}, \] \[ \mathbf D (u, v) := \int _ {I} u' v' d x \] the authors describe completely the regular Dirichlet subspaces of this space. Also, a description of the recurrent regular Dirichlet subspaces, for the case \(I = \mathbb{R}\) is given. The subspaces are indexed by the set of functions \(s\) on \(I\), which are: absolutely continuous, strictly increasing, \(s'(x) = 0\) or \(1\) a.e. on \(I\), positive on \(I\), resp. on \((0, +\infty )\) if \( I = \mathbb{R}\). In the case \(I = \mathbb{R}\), the recurrent regular subspaces correspond with those \(s\) satisfying moreover \(s(\pm \infty ) = \pm \infty \). The associated process is constructed from the reflected Brownian motion, by a time change and a state space transformation.
scientific article; zbMATH DE number 2165811