From one dimensional diffusions to symmetric Markov processes
Let \(X^0\) be an absorbing diffusion on a one-dimensional regular interval \(I\), with scale function \(s\), speed measure \(m\), and no killing inside. The first aim of the paper is to identify the Dirichlet form of \(X^0\) on \(L^2(I,m)\) and its extended Dirichlet space in terms of the scale function \(s\). Once the Dirichlet form has been identified, one can consider all possible symmetric extension of \(X^0\). The maximal extension corresponds to the concept of the reflected Dirichlet space. The author also discusses the possibility of symmetric extensions of a time changed transient reflecting Brownian motion on a closed domain in \(\mathbb R^d\), \(d\geq 3\), possessing two branches of infinite cones.
scientific article; zbMATH DE number 5710796
scientific article