On general boundary conditions for one-dimensional diffusions with symmetry (Q2445048)

Let \(X^0\) be a minimal diffusion on a one-dimensional open interval \(I\). The author studies the \(m\)-symmetry of part processes of \(X^0\) on each relatively compact subinterval of \(I\) and calculates corresponding symmetric forms via integration by parts. Then, all possible symmetric diffusion extensions of \(X^0\) in terms of boundary conditions are characterized both in the \(L^2\)-setting and the \(C_b\)-setting, where \(C_b\) denotes the space of all bounded (finely) continuous functions. Finally, the author studies symmetric diffusion extensions of \(X^0\), which exhaust all possible symmetric diffusion-extensions of \(X^0\) obtained by adding regular boundary points or their identification to \(I\).