## Electronic Journal of Probability

### Markov Processes with Identical Bridges

P. Fitzsimmons

#### 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.

#### Article information

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

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

https://projecteuclid.org/euclid.ejp/1454101772

Digital Object Identifier
doi:10.1214/EJP.v3-34

Mathematical Reviews number (MathSciNet)
MR1641066

Zentralblatt MATH identifier
0907.60066

Rights

#### Citation

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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