The Annals of Probability

Characterizing all Diffusions with the $2M - X$ Property

L. C. G. Rogers

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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 $X$.

Article information

Ann. Probab., Volume 9, Number 4 (1981), 561-572.

First available in Project Euclid: 19 April 2007

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Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J65: Brownian motion [See also 58J65] 60J60: Diffusion processes [See also 58J65]

Brownian motion Bessel process one dimensional diffusion scale function speed measure 2M-X property path decomposition Markov kernel


Rogers, L. C. G. Characterizing all Diffusions with the $2M - X$ Property. Ann. Probab. 9 (1981), no. 4, 561--572. doi:10.1214/aop/1176994362.

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