## The Annals of Probability

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

L. C. G. Rogers

#### Abstract

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

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

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176994362

Digital Object Identifier
doi:10.1214/aop/1176994362

Mathematical Reviews number (MathSciNet)
MR624683

Zentralblatt MATH identifier
0465.60063

JSTOR
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. https://projecteuclid.org/euclid.aop/1176994362