Electronic Journal of Probability

Markov Processes with Identical Bridges

P. Fitzsimmons

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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.

Article information

Electron. J. Probab., Volume 3 (1998), paper no. 12, 12 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]

Bridge law eigenfunction transition density

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Fitzsimmons, P. Markov Processes with Identical Bridges. Electron. J. Probab. 3 (1998), paper no. 12, 12 pp. doi:10.1214/EJP.v3-34. https://projecteuclid.org/euclid.ejp/1454101772

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