New perspectives on Ray's theorem for the local times of diffusions (Q1394532)

Let \(X\) be a transient regular diffusion on some interval \(I\subset\mathbb{R}\), starting from a point \(x\in I\), conditioned to die at a fixed point \(y\in I\), and let \(\{L^r_\infty;\,r\in I\}\) be its total cumulated local time. The authors prove a relation between the law of \(L^\cdot_\infty\), and some related Gaussian process. This relation provides a single global description of \(L^\cdot_\infty\) on the hole interval \(I\). It follows an explicit description of the local time in terms of \(0\)th-order squared Bessel processes. Several results on local times of diffusions follow easily, as the Markov property of \(L^\cdot_\infty\), and Dynkin's isomorphism theorem.