@article{EJP34,
author = {P. Fitzsimmons},
title = {Markov Processes with Identical Bridges},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {3},
year = {1998},
keywords = {Bridge law, eigenfunction, transition density.},
abstract = {Let $X$ and $Y$ be time-homogeneous Markov processes with common state space $E$, and assume that the transition kernels of $X$ and $Y$ admit densities with respect to suitable reference measures. We show that if there is a time $t>0$ such that, for each $x\in E$, the conditional distribution of $(X_s)_{0\le s\le t}$, given $X_0=x=X_t$, coincides with the conditional distribution of $(Y_s)_{0\le s\le t}$, given $Y_0=x=Y_t$, then the infinitesimal generators of $X$ and $Y$ are related by $L^Yf=\psi^{-1}L^X(\psi f)-\lambda f$, where $\psi$ is an eigenfunction of $L^X$ with eigenvalue $\lambda\in{\bf R}$. Under an additional continuity hypothesis, the same conclusion obtains assuming merely that $X$ and $Y$ share a "bridge" law for one triple $(x,t,y)$. Our work extends and clarifies a recent result of I. Benjamini and S. Lee.},
pages = {no. 12, 1-12},
issn = {1083-6489},
doi = {10.1214/EJP.v3-34},
url = {http://ejp.ejpecp.org/article/view/34}}