Excursion Laws of Markov Processes in Classical Duality

Abstract

We study excursion laws of two Markov processes $X, \hat{X}$ in duality, out of closed, homogeneous optional sets $M, \hat{M}$ generated by a pair of dual terminal times $R$ and $\hat{R}$. The duality assumptions enable one to compute the laws of $(R, X_R)$ using the pair of exit systems of the two processes. With this, one is able to compute the conditional laws of the excursions, given the boundary process. It turns out that these depend only on the values of the boundary process at the beginning and end of the excursions. We obtain for all $x, y(x \neq y)$ the laws $p^{x,y}$ of the excursions conditioned to start at $x$ and end at $y$. Under these laws, the excursion process is a homogeneous Markov process. It's transition laws are computed. We use the results above to treat excursions that straddle perfect terminal times.