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1998 Markov Processes with Identical Bridges
P. Fitzsimmons
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Electron. J. Probab. 3: 1-12 (1998). DOI: 10.1214/EJP.v3-34


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.


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P. Fitzsimmons. "Markov Processes with Identical Bridges." Electron. J. Probab. 3 1 - 12, 1998.


Accepted: 5 July 1998; Published: 1998
First available in Project Euclid: 29 January 2016

zbMATH: 0907.60066
MathSciNet: MR1641066
Digital Object Identifier: 10.1214/EJP.v3-34

Primary: 60J25
Secondary: 60J35

Keywords: Bridge law , eigenfunction , Transition density

Vol.3 • 1998
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