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August, 1981 Characterizing all Diffusions with the $2M - X$ Property
L. C. G. Rogers
Ann. Probab. 9(4): 561-572 (August, 1981). DOI: 10.1214/aop/1176994362


If $(X_t)_{t \geq 0}$ is a Brownian motion on the real line, started at zero, if $M_t \equiv \max\{X_s; s \leq t\}$ and if $Y_t \equiv 2M_t - X_t$ for $t \geq 0$, then $(Y_t)_{t \geq 0}$ is a homogeneous strong Markov process equal in law to the radial part of Brownian motion in three dimensions. This result was discovered by Pitman, and recently Rogers and Pitman have found other one-dimensional diffusions $X$ for which $2M - X$ is again a diffusion. This paper gives a complete characterisation of all such diffusions $X$.


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L. C. G. Rogers. "Characterizing all Diffusions with the $2M - X$ Property." Ann. Probab. 9 (4) 561 - 572, August, 1981.


Published: August, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0465.60063
MathSciNet: MR624683
Digital Object Identifier: 10.1214/aop/1176994362

Primary: 60J25
Secondary: 60J35 , 60J60 , 60J65

Keywords: 2M-X property , Bessel process , Brownian motion , Markov kernel , one dimensional diffusion , Path decomposition , scale function , Speed measure

Rights: Copyright © 1981 Institute of Mathematical Statistics

Vol.9 • No. 4 • August, 1981
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